Research Article On Positive Solutions and Mann Iterative …downloads.hindawi.com/journals/aaa/2014/470181.pdf · e existence of uncountably many positive solutions and converg ence

Research ArticleOn Positive Solutions and Mann Iterative Schemes of a ThirdOrder Difference Equation

Zeqing Liu1 Heng Wu1 Shin Min Kang2 and Young Chel Kwun3

1 Department of Mathematics Liaoning Normal University Dalian Liaoning 116029 China2Department of Mathematics and RINS Gyeongsang National University Jinju 660-701 Republic of Korea3 Department of Mathematics Dong-A University Pusan 614-714 Republic of Korea

Correspondence should be addressed to Young Chel Kwun yckwundauackr

Received 14 October 2013 Accepted 16 December 2013 Published 28 January 2014

Academic Editor Zhi-Bo Huang

Copyright copy 2014 Zeqing Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The existence of uncountably many positive solutions and convergence of the Mann iterative schemes for a third order nonlinearneutral delay difference equation are proved Six examples are given to illustrate the results presented in this paper

1 Introduction and Preliminaries

Recently many researchers studied the oscillation nonoscil-lation and existence of solutions for linear and nonlinearsecond and third order difference equations and systemssee for example [1ndash23] and the references cited therein Bymeans of the Reccati transformation techniques Saker [18]discussed the third order difference equation

Δ3119909119899+ 119901119899119909119899+1

= 0 forall119899 ge 1198990 (1)

and presented some sufficient conditions which ensure thatall solutions are to be oscillatory or tend to zero Utilizing theSchauder fixed point theorem Yan and Liu [22] proved theexistence of a bounded nonoscillatory solution for the thirdorder difference equation

Δ3119909119899+ 119891 (119899 119909

119899 119909119899minus120591) = 0 forall119899 ge 119899

0 (2)

Agarwal [2] established the oscillatory and asymptotic prop-erties for the third order nonlinear difference equation

Δ3119909119899+ 119902119899119891 (119909119899+1) = 0 forall119899 ge 1 (3)

Andruch-Sobiło andMigda [4] studied the third order lineardifference equation of neutral type

Δ3(119909119899minus 119901119899119909120590119899) plusmn 119902119899119909120591119899= 0 forall119899 ge 119899

0 (4)

and obtained sufficient conditions which ensure that allsolutions of the equation are oscillatory Grace andHamedani[6] discussed the difference equation

Δ3(119909119899minus 119909119899minus120591) plusmn 119902119899

1003816100381610038161003816119909119899minus1205901003816100381610038161003816

3 sgn119909119899minus120590

= 0 forall119899 ge 0 (5)

and gave some new criteria for the oscillation of all solutionsand all bounded solutions

Our goal is to discuss solvability and convergence ofthe Mann iterative schemes for the following third ordernonlinear neutral delay difference equation

Δ3(119909119899+ 119887119899119909119899minus120591) + Δℎ (119899 119909

ℎ1119899

119909ℎ2119899

119909ℎ119896119899

)

+119891 (119899 1199091198911119899

1199091198912119899

119909119891119896119899

) = 119888119899 forall119899 ge 119899

0

(6)

where 120591 119896 1198990isin N 119887

119899119899isinN1198990

119888119899119899isinN1198990

sub R ℎ 119891 isin 119862(N1198990

times

R119896R) ℎ119897119899119899isinN1198990

119891119897119899119899isinN1198990

sube N and

lim119899rarrinfin

ℎ119897119899= lim119899rarrinfin

119891119897119899= +infin 119897 isin 1 2 119896 (7)

By employing the Banach fixed point theorem and somenew techniques we establish the existence of uncountablymany positive solutions of (6) conceive a few Mann iter-ative schemes for approximating these positive solutionsand prove their convergence and the error estimates Sixnontrivial examples are included

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 470181 16 pageshttpdxdoiorg1011552014470181

2 Abstract and Applied Analysis

Throughout this paper we assume that Δ is the forwarddifference operator defined by Δ119909

119899= 119909119899+1

minus 119909119899 R =

(minusinfin +infin) R+ = [0 +infin) N0and N denote the sets of

nonnegative integers and positive integers respectively

N119905= 119899 119899 isin N with 119899 ge 119905 forall119905 isin N

120573 = min 1198990minus 120591 inf ℎ

119897119899 119891119897119899 1 le 119897 le 119896 119899 isin N

1198990

isin N

119867119899= max ℎ2

119897119899 119897 isin 1 2 119896 forall119899 isin N

1198990

119865119899= max 1198912


1198990

(8)

and 119897infin

120573represents the Banach space of all real sequences

on N120573with norm

119909 = sup119899isinN120573

1003816100381610038161003816100381610038161003816

119909119899

1198992

1003816100381610038161003816100381610038161003816lt +infin for each 119909 = 119909

119899119899isinN120573

isin 119897infin

120573

119860 (119873119872) = 119909 = 119909119899119899isinN120573

isin 119897infin

120573 119873 le

119909119899

1198992le 119872 119899 isin N

120573

for any 119872 gt 119873 gt 0

(9)

It is easy to see that 119860(119873119872) is a closed and convex subsetof 119897infin120573 By a solution of (6) wemean a sequence 119909

119899119899isinN120573

witha positive integer 119879 ge 119899

0+120591+120573 such that (6) holds for all 119899 ge

119879

Lemma 1 Let 119901119905119905isinN be a nonnegative sequence and 120591 isin N

(i) If lim119899rarrinfin

(11198992) suminfin

119905=119899+1205911199052119901119905

= 0

then lim119899rarrinfin

(11198992) suminfin

119894=1suminfin

119904=119899+119894120591suminfin

119905=119904119901119905= 0

(ii) If lim119899rarrinfin

(11198992) suminfin

119905=119899+1205911199053119901119905

= 0


(11198992) suminfin

119894=1suminfin

119906=119899+119894120591suminfin

119904=119906suminfin

119905=119904119901119905= 0

Proof Note that

0 le1

1198992

infin

sum

119894=1

infin

sum

119904=119899+119894120591

infin

sum

119905=119904

119901119905

=1

1198992

infin

sum

119894=1

(

infin

sum

119905=119899+119894120591

119901119905+

infin

sum

119905=119899+1+119894120591

119901119905+

infin

sum

119905=119899+2+119894120591

119901119905+ sdot sdot sdot )

=1

1198992

infin

sum

119894=1

infin

sum

119905=119899+119894120591

(1 + 119905 minus 119899 minus 119894120591) 119901119905le1

1198992

infin

sum

119894=1

infin

sum

119905=119899+119894120591

119905119901119905

=1

1198992(

infin

sum

119905=119899+120591

119905119901119905+

infin

sum

119905=119899+2120591

119905119901119905+

infin

sum

119905=119899+3120591

119905119901119905+ sdot sdot sdot )

le1

1198992

infin

sum

119905=119899+120591

(1 +119905 minus 119899 minus 120591

120591) 119905119901119905=1

1198992

infin

sum

119905=119899+120591

119905 minus 119899

120591119905119901119905

le1

1198992120591

infin

sum

119905=119899+120591

1199052119901119905997888rarr 0 as 119899 997888rarr infin

(10)

that is

lim119899rarrinfin

1

1198992

infin

sum

119894=1

infin

sum

119904=119899+119894120591

infin

sum

119905=119904

119901119905= 0 (11)

As in the proof of (10) we infer that

0 le1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

infin

sum

119905=119904

119901119905

=1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119905=119906

(1 + 119905 minus 119906) 119901119905

le1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119905=119906

119905119901119905le

1

1198992120591

infin

sum

119905=119899+120591


(12)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

infin

sum

119905=119904

119901119905= 0 (13)

This completes the proof

2 Uncountably Many Positive Solutions andMann Iterative Schemes

In this section using the Banach fixed point theoremand Mann iterative schemes we establish the existence ofuncountably many positive solutions of (6) prove conver-gence of the Mann iterative schemes relative to these positivesolutions and compute the error estimates between theManniterative schemes and the positive solutions

Theorem 2 Assume that there exist twoconstants 119872 and 119873 with 119872 gt 119873 gt 0 and four nonnegativesequences 119875

119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying1003816100381610038161003816119891 (119899 1199061 1199062 119906119896) minus 119891 (119899 1199061 1199062 119906119896)

1003816100381610038161003816

le 119875119899max 1003816100381610038161003816119906119897 minus 119906119897

1003816100381610038161003816 1 le 119897 le 119896

1003816100381610038161003816ℎ (119899 1199061 1199062 119906119896) minus ℎ (119899 1199061 1199062 119906119896)1003816100381610038161003816

le 119877119899max 1003816100381610038161003816119906119897 minus 119906119897

1003816100381610038161003816 1 le 119897 le 119896

forall (119899 119906119897 119906119897) isin N1198990

times (R+ 0)2

1 le 119897 le 119896

(14)

1003816100381610038161003816119891 (119899 1199061 1199062 119906119896)1003816100381610038161003816 le 119876119899

1003816100381610038161003816ℎ (119899 1199061 1199062 119906119896)1003816100381610038161003816 le 119882119899

forall (119899 119906119897) isin N1198990

times (R+ 0) 1 le 119897 le 119896

(15)

lim119899rarrinfin

1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (16)

lim119899rarrinfin

1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (17)

119887119899= minus1 eventually (18)

Abstract and Applied Analysis 3

Then one has the following(a) For any 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge

1198990+ 120591 + 120573 such that for each 119909

0= 119909

0119899119899isinN120573

isin

119860(119873119872) the Mann iterative sequence 119909119898119898isinN0

=

119909119898119899119899isinN120573

119898isinN0

generated by the scheme

119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+1205721198981198992119871

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871

+

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(19)

converges to a positive solution 119911 = 119911119899119899isinN120573

isin 119860(119873119872) of (6)with lim

119899rarrinfin119911119899= +infin and has the following error estimate

1003817100381710038171003817119909119898+1 minus 1199111003817100381710038171003817 le 119890minus(1minus120579)sum

119898

119894=01205721198941003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817 forall119898 isin N0 (20)

where 120572119898119898isinN0

is an arbitrary sequence in [0 1] such thatinfin

sum

119898=0

120572119898= +infin (21)

(b) Equation (6) possesses uncountablymany positive solu-tions in 119860(119873119872)

Proof Firstly we show that (a) holds Put 119871 isin (119873119872) Itfollows from (16)sim(18) that there exist 120579 isin (0 1) and 119879 ge

1198990+ 120591 + 120573 satisfying

120579 =1

1198792

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (22)

1

1198792

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt min 119872 minus 119871 119871 minus 119873

(23)

119887119899= minus1 forall119899 ge 119879 (24)

Define a mapping 119878119871 119860(119873119872) rarr 119897

infin

120573by

119878119871119909119899

=

1198992119871

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879 119878119871119909119879 120573 le 119899 lt 119879

(25)

for each 119909 = 119909119899119899isinN120573

isin 119860(119873119872) In light of (14) (15) (22)(23) and (25) we obtain that for each 119909 = 119909

119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872)

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[119877119904max ℎ2

119897119904 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 1198912

119897119905 1 le 119897 le 119896]

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198792

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus 119871

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816


+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816 ]

le1

1198792

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


(26)which yield that

119878119871 (119860 (119873119872)) sube 119860 (119873119872)

1003817100381710038171003817119878119871119909 minus 1198781198711199101003817100381710038171003817 le 120579

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119860 (119873119872)

(27)

which implies that 119878119871is a contraction in 119860(119873119872) The

Banach fixed point theorem and (27) ensure that 119878119871has a

unique fixed point 119911 = 119911119899119899isinN120573

isin 119860(119873119872) that is

119911119899= 1198992119871

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905]

forall119899 ge 119879

119911119899minus120591

= (119899 minus 120591)2119871

+

infin

sum

119894=1

infin

sum

119906=119899+(119894minus1)120591

infin

sum

119904=119906

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

)

minus 119888119905] forall119899 ge 119879 + 120591

(28)which mean that119911119899minus 119911119899minus120591

= (2119899120591 minus 1205912) 119871

minus

infin

sum

119906=119899

infin

sum

119904=119906

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905]

forall119899 ge 119879 + 120591

(29)which yields thatΔ (119911119899minus 119911119899minus120591)

= 2120591119871 +

infin

sum

119904=119899

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905]

forall119899 ge 119879 + 120591

Δ2(119911119899minus 119911119899minus120591)

= minusℎ (119899 119911ℎ1119899

119911ℎ2119899

119911ℎ119896119899

)

+

infin

sum

119905=119899

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905] forall119899 ge 119879 + 120591

(30)

which gives that

Δ3(119911119899minus 119911119899minus120591)

= minusΔℎ (119899 119911ℎ1119899

119911ℎ2119899

119911ℎ119896119899

)

minus 119891 (119905 1199111198911119899

1199111198912119899

119911119891119896119899

) + 119888119899 forall119899 ge 119879 + 120591

(31)

which together with (24) implies that 119911 = 119911119899119899isinN120573

is apositive solution of (6) in 119860(119873119872) Note that

119873 le119911119899

1198992le 119872 forall119899 isin N

120573 (32)

which guarantees that lim119899rarrinfin

119911119899= +infin It follows from (19)

(22) (24) (25) and (27) that for any 119898 isin N0and 119899 ge 119879

1003816100381610038161003816100381610038161003816

119909119898+1119899

1198992minus119911119899

1198992

1003816100381610038161003816100381610038161003816

=1

1198992

1003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572119898) 119909119898119899

+ 1205721198981198992119871

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

) minus 119888119905)] minus 119911

119899

1003816100381610038161003816100381610038161003816100381610038161003816

le (1 minus 120572119898)

1003816100381610038161003816119909119898119899 minus 1199111198991003816100381610038161003816

1198992+ 120572119898

1003816100381610038161003816119878119871119909119898119899 minus 1198781198711199111198991003816100381610038161003816

1198992

le (1 minus 120572119898)1003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817 + 1205791205721198981003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817

le [1 minus (1 minus 120579) 120572119898]1003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817 forall119898 isin N0 119899 ge 119879

(33)

which implies that


119898

119894=01205721198941003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817 forall119898 isin N0 (34)

That is (20) holds Thus Lemma 1 (20) and (21) guaranteethat lim

119898rarrinfin119909119898= 119911

Next we show that (b) holds Let 1198711 1198712

isin

(119873119872) and 1198711= 1198712 As in the proof of (a) we deduce

similarly that for each 119888 isin 1 2 there exist constants 120579119888isin

(0 1) and 119879119888ge 1198990+ 120591 + 120573 and a mapping 119878

119871119888

satisfying


(22)sim(27) where 120579 119871 and 119879 are replaced by 120579119888 119871119888 and 119879

119888

respectively and the mapping 119878119871119888

has a fixed point 119911119888 =

119911119888

119899119899isinN120573

isin 119860(119873119872) which is a positive solution of (6) in119860(119873119872) with lim

119899rarrinfin119911119888

119899= +infin It follows that

119911119888

119899= 1198992119871119888

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119911119888

ℎ1119904

119911119888

ℎ2119904

119911119888

ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 119911119888

1198911119905

119911119888

1198912119905

119911119888

119891119896119905

) minus 119888119905]

forall119899 ge 119879119888

(35)

which together with (14) and (20) means that for 119899 ge

max1198791 1198792

100381610038161003816100381610038161003816100381610038161003816

1199111

119899

1198992minus1199112

119899

1198992

100381610038161003816100381610038161003816100381610038161003816

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119911

1

ℎ1119904

1199111

ℎ2119904

1199111

ℎ119896119904

)

minus ℎ (119904 1199112

ℎ1119904

1199112

ℎ2119904

1199112

ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119911

1

1198911119905

1199111

1198912119905

1199111

119891119896119905

) minus 119891 (119905 1199112

1198911119905

1199112

1198912119905

1199112

119891119896119905

)10038161003816100381610038161003816

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119911

1

ℎ119897119904

minus 1199112

ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119911

1

119891119897119905

minus 1199112

119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

max 11987921 1198792

2

infin

sum

119894=1

infin

sum

119906=max1198791 1198792+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816 minusmax 1205791 1205792100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

(36)

which yields that

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817ge

10038161003816100381610038161198711 minus 11987121003816100381610038161003816

1 +max 1205791 1205792gt 0 (37)

that is 1199111 = 1199112 This completes the proof

Theorem 3 Assume that there exist two constants119872 and 119873with 119872 gt 119873 gt 0 and four nonnegative sequences 119875

119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14) (15) and

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (38)

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (39)

119887119899= 1 eventually (40)

Then one has the following

(a) For any 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge


0= 119909

0119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+1205721198981198992119871

minus

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

) minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871

minus

infin

sum

119894=1

119879+2119894120591minus1

sum

119906=119879+(2119894minus1)120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

) minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(41)


isin

119860(119873119872) of (6) with lim119899rarrinfin

119911119899= +infin and has the

error estimate (20) where 120572119898119898isinN0

is an arbitrarysequence in [0 1] satisfying (21)



Proof Let 119871 isin (119873119872) It follows from (38)sim(40) that thereexist 120579 isin (0 1) and 119879 ge 119899

0+ 120591 + 120573 satisfying

120579 =1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (42)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)) lt min 119872 minus 119871 119871 minus 119873

(43)

119887119899= 1 forall119899 ge 119879 (44)


infin

120573by

119878119871119909119899

=

1198992119871

minus

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus119888119905] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(45)

for each 119909 = 119909119899119899isinN120573

isin 119860(119873119872) Using (14) (15) (42) (43)and (45) we get that for each 119909 = 119909

119899119899isinN120573

119910 = 119910119899119899isinN120573

isin

119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) = 120579

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus 119871

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)])


(46)

which imply (27) The rest of the proof is similar to the proofof Theorem 2 and is omitted This completes the proof

Theorem 4 Assume that there exist three constants 119887 119872and 119873 with (1 minus 119887)119872 gt 119873 gt 0 and four nonnega-tive sequences 119875

119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14) (15) (38) (39) and0 le 119887119899le 119887 lt 1 eventually (47)

Then one has the following(a) For any 119871 isin (119887119872 + 119873119872) there exist 120579 isin

(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any

1199090

= 1199090119899119899isinN120573

isin 119860(119873119872) the Mann iterativesequence 119909

119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0

generated bythe scheme

119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+1205721198981198992119871 minus 119887119899119909119898119899minus120591

minus

infin

sum

119906=119899

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871 minus 119887119879119909119898119879minus120591

minus

infin

sum

119906=119879

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(48)


isin







Proof Put 119871 isin (119887119872 + 119873119872) It follows from (38) (39) and(47) that there exist 120579 isin (0 1) and 119879 ge 119899

0+ 120591 + 120573 satisfying

120579 = 119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt min 119872 minus 119871 119871 minus 119887119872 minus119873

0 le 119887119899le 119887 lt 1 forall119899 ge 119879

(49)


infin

120573by

119878119871119909119899

=

1198992119871 minus 119887119899119909119899minus120591

minus

infin

sum

119906=119899

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(50)for each 119909 = 119909

119899119899isinN120573

isin 119860(119873119872) In view of (14) (15) and(49) and (50) we obtain that for each 119909 = 119909

119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 11987910038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le 119887119899

100381610038161003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

(119899 minus 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 minus 120591)2

1198992

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le 1198871003817100381710038171003817119909 minus 119910

1003817100381710038171003817

+

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max ℎ2

119897119904 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 1198912

119897119905 1 le 119897 le 119896]

le [119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 120579

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt 119871 +min 119872 minus 119871 119871 minus 119887119872 minus119873 le 119872

119878119871119909119899

1198992

ge 119871 minus 119887119872

minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus 119887119872 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]

gt 119871 minus 119887119872 minusmin 119872 minus 119871 119871 minus 119887119872 minus119873 ge 119873

(51)

which imply (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof

Theorem 5 Assume that there exist constants 119887 119872 and 119873with (1 + 119887)119872 gt 119873 gt 0 and four nonnegative sequences119875119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14)(15) (38) (39) and

minus1 lt 119887 le 119887119899le 0 eventually (52)


(a) For any 119871 isin (119873 (1 + 119887)119872) there exist 120579 isin

(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for

any 1199090= 1199090119899119899isinN120573


119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0

generated by(48) converges to a positive solution 119911 = 119911

119899119899isinN120573

isin







Proof Put 119871 isin (119873 (1 + 119887)119872) It follows from (38) (39) and(52) that there exist 120579 isin (0 1) and 119879 ge 119899

0+ 120591 + 120573 satisfying

120579 = minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (53)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt min (1 + 119887)119872 minus 119871 119871 minus 119873

(54)

minus1 lt 119887 le 119887119899le 0 forall119899 ge 119879 (55)


infin

120573by (50) By virtue of

(15) (50) (53) and (55) we infer that for all 119909 = 119909119899119899isinN120573

119910 = 119910

119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le [minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 minus 119887119872

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 minus 119887119872 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt 119871 minus 119887119872 +min (1 + 119887)119872 minus 119871 119871 minus 119873 le 119872

119878119871119909119899

1198992

ge 119871 minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]

gt 119871 minusmin (1 + 119887)119872 minus 119871 119871 minus 119873 ge 119873

(56)

That is (27) holds The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof

Theorem 6 Assume that there exist constants 119887119872 and 119873 with (1 minus 1119887)119872 gt 119873 gt 0 and fournonnegative sequences 119875

119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882

119899119899isinN1198990

satisfying (14) (15) (38) (39) and

119887119899ge 119887 gt 1 eventually (57)

Then one has the following(a) For any 119871 isin ((1119887)119872 + 119873119872) there exist 120579 isin

(0 1) and 119879 ge 1198990+ 120591 + 120573 such that for any 119909

0=

1199090119899119899isinN120573

isin 119860(119873119872) the Mann iterative sequence119909119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 1205721198981198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(58)



isin






Proof Put 119871 isin ((1119887)119872 + 119873119872) It follows from (38) (39)and (57) that there exist 120579 isin (0 1) and 119879 ge 119899

0+ 120591 +

120573 satisfying

120579 =1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (59)

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt min 119872 minus 119871 119871 minus1

119887119872 minus119873

(60)

119887119899ge 119887 gt 1 forall119899 ge 119879 (61)


infin

120573by

119878119871119909119899

=

1198992119871 minus

119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[(119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus 119888119905)] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(62)

for each 119909 = 119909119899119899isinN120573

isin 119860(119873119872) In view of (14) (15)and (59)∽(62) we obtain that for each 119909 = 119909

119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

+1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

+1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le 119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt 119871 +min 119872 minus 119871 119871 minus1

119887119872 minus119873 le 119872

119878119871119909119899

1198992

ge 119871 minus1

119887119872

minus1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

119887119872 minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

gt 119871 minus1

119887119872 minusmin 119872 minus 119871 119871 minus

1

119887119872 minus119873 ge 119873

(63)


Theorem 7 Assume that there exist constants 119887 119872and 119873 with (1 + 1119887)119872 gt 119873 gt 0 and four nonnegativesequences 119875

119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14) (15) (38) (39) and

119887119899le 119887 lt minus1 eventually (64)



(a) For any 119871 isin (minus(1 + 1119887)119872 minus119873) there exist 120579 isin (0 1)and 119879 ge 119899

0+ 120591 + 120573 such that for any 119909

0= 1199090119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 120572119898 minus 1198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+120572119898 minus 119879

2119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(65)


isin






Proof Put 119871 isin (minus(1 + 1119887)119872 minus119873) It follows from (38)(39) and (64) that there exist 120579 isin (0 1) and 119879 ge 119899

0+ 120591 +

120573 satisfying

120579 = minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (66)

minus1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt min (1 + 1

119887)119872 minus 119871 119871 minus

1

119887119872 minus119873

(67)

119887119899ge 119887 gt 1 forall119899 ge 119879 (68)


infin

120573by

119878119871119909119899

=

minus1198992119871 minus

119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(69)

for each 119909 = 119909119899119899isinN120573

isin 119860(119873119872) Making use of (15) (66)(68) and (69) we conclude that10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le minus1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

minus1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le minus1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

minus1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le minus119871 minus119872

119887minus

1

1198871198992

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]



119887minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt minus119871 minus119872

119887+min (1 + 1

119887)119872 + 119871 minus119871 minus 119873 le 119872

119878119871119909119899

1198992

ge minus119871

+1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge minus119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]

gt minus119871 minusmin (1 + 1

119887)119872 + 119871 minus119871 minus 119873 ge 119873

(70)

which yield (27) The rest of the proof is similar to that ofTheorem 2 and is omitted This completes the proof

3 Examples

In this section we suggest six examples to explain the resultspresented in Section 2

Example 1 Consider the third order nonlinear neutral delaydifference equation

Δ3(119909119899minus 119909119899minus120591) + Δ(

sin2119909119899minus3

1198997) +

1

(1198999 + 21198995 + 1) (1 + 1199092

1198992)

=1198992minus 2119899

1198998 + 1198993 + 1 forall119899 ge 4

(71)

where 120591 isin N is fixed Let 1198990= 4 119896 = 1 and 120573 = min4minus120591 1

and let119872 and 119873 be two positive constants with 119872 gt 119873 and

119887119899= minus1 119888

119899=

1198992minus 2119899

1198998 + 1198993 + 1

119891 (119899 119906) =1

(1198999 + 21198995 + 1) (1 + 1199062)

ℎ (119899 119906) =sin21199061198997

1198911119899= 1198992 119865

119899= 1198994

ℎ1119899= 119899 minus 3 119867

119899= (119899 minus 3)

2 119875

119899=2119872

1198999

119876119899=1

1198999 119877

119899=2

1198997 119882

119899=1

1198997

forall (119899 119906) isin N1198990

timesR

(72)

It is easy to see that (14) (15) and (18) are satisfied Note that

1

1198992

infin

sum

119905=119899+120591

1199052max 119877

119905119867119905119882119905

=1

1198992

infin

sum

119905=119899+120591

1199052max2(119905 minus 3)

2

11990571

1199057

=

infin

sum

119905=119899+120591

2(119905 minus 3)2+ 1

1199055

le2

1198992

infin

sum

119905=119899+120591

1


1

1198992

infin

sum

119905=119899+120591

1199053max 119875

119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119905=119899+120591

1199053max2119872

11990551

1199059

100381610038161003816100381610038161199052minus 2119905

10038161003816100381610038161003816

1199058 + 1199053 + 1

lemax 1 2119872

1198992

infin

sum

119905=119899+120591

1


(73)

which together with Lemma 1 yield that (16) and (17) holdIt follows from Theorem 2 that (71) possesses uncountablymany positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 119899

0+120591+120573 such

that for each 1199090= 1199090119899119899isinN120573


119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0

generated by (19)converges to a positive solution 119911 = 119911

119899119899isinN120573

isin 119860(119873119872) of(71) with lim

119899rarrinfin119911119899= +infin and has the error estimate (20)

where 120572119898119898isinN0

is an arbitrary sequence in [0 1] satisfying(21)


Δ3(119909119899+ 119909119899minus120591) + Δ(

sin211990931198993+1

1198993 (1198996 + 2) (1 + 1199094

21198992minus3)

)

+(minus1)1198991198993(1199091198992minus119899minus1

+ 119909(119899+1)(119899+2)

)

(11989913 + 1198995 + 1) (1 + 1199092

1198992minus119899minus1

+ 1199092

(119899+1)(119899+2))

=1198992minus ln 119899

1198996 + 1198995 + 1 forall119899 ge 5

(74)

where 120591 isin N is fixed Let 1198990= 5 119896 = 2 and 120573 = 5 minus 120591 and let

119872 and 119873 be two positive constants with 119872 gt 119873 and

119887119899= 1 119888

119899=

1198992minus ln 119899

1198996 + 1198995 + 1

119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)

(11989913 + 1198995 + 1) (1 + 1199062 + V2)


ℎ (119899 119906 V) =sin2V

1198993 (1198996 + 2) (1 + 1199064) 119891

1119899= 1198992minus 119899 minus 1

1198912119899= (119899 + 1) (119899 + 2)

119865119899= (119899 + 1)

2(119899 + 2)

2 ℎ

1119899= 21198992minus 3

ℎ2119899= 31198993+ 1

119867119899= (3119899

3+ 1)2

119875119899= 119876119899=

4

11989910 119877

119899= 119882119899=10

1198999

forall (119899 119906 V) isin N1198990

timesR2

(75)

It is clear that (14) (15) and (40) are fulfilled Note that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

10(31199043+ 1)2

119904910

1199049

le160

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le160

1198992

infin

sum

119904=119899

1


(76)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (77)

Observe that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max4(119905 + 1)2(119905 + 2)

2

119905104

119905101199052minus ln 119905

1199056 + 1199055 + 1

le196

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054=196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

119905 minus 119906 + 1

1199054

le196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le196

1198992

infin

sum

119905=119899

1


(78)

which yields that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)

Thus Theorem 3 guarantees that (74) possesses uncount-ably positive solutions in 119860(119873119872) On the other hand forany 119871 isin (119873119872) there exist 120579 isin (0 1) and 119879 ge 120591 +

1198990+ 120573 such that the Mann iterative sequence 119909

119898119898isinN0

=

119909119898119899119899isinN120573

119898isinN0

generated by (41) converges to a positivesolution 119911 = 119911

119899119899isinN120573

isin 119860(119873119872) of (74) with lim119899rarrinfin

119911119899=

+infin and has the error estimate (20) where 120572119898119898isinN0

is anarbitrary sequence in [0 1] satisfying (21)


Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899

119909119899minus120591)

+ Δ(

(minus1)119899 sin (119890minus119899

2|11990951198992minus3|)

11989915 minus radic119899 + 3

+1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|11990921198993+1|)

+(minus1)119899

1198996 (1 + 1199092

119899minus3)minus

1

(1198997 + 21198994 minus 1) (1 + 1199092

119899+4)

=3(minus1)1198991198992

911989910ln3119899 forall119899 ge 7

(80)

where 120591 isin N is fixed Let 1198990= 7 119896 = 2 119887 = 34 and

120573 = min7 minus 120591 4 and let119872 and119873 be two positive constantswith 119872 gt 4119873 and

119887119899=1 + 3 ln 1198992 + 4 ln 119899

119888119899=3(minus1)1198991198992

911989910ln3119899

119891 (119899 119906 V) =(minus1)119899

1198996 (1 + 1199062)minus

1

(1198997 + 21198994 minus 1) (1 + V2)

ℎ (119899 119906 V) =(minus1)119899 sin (119890minus119899

2|119906|)

11989915 minus radic119899 + 3+

1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|V|

1198911119899= 119899 minus 3 119891

2119899= 119899 + 4

119865119899= (119899 + 4)

2 ℎ

1119899= 51198992minus 3

ℎ2119899= 21198993+ 1 119867

119899= (2119899

3+ 1)2

119875119899= 119876119899=3

1198996 119877

119899= 119882119899=

2

11989910

forall (119899 119906 V) isin N1198990

timesR2

(81)

It is not difficult to verify that (14) (15) and (47) are fulfilledNote that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

2(21199043+ 1)2

119904102

11990410

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le18

1198992

infin

sum

119904=119899

1


(82)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (83)


Observe that1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max3(119905 + 4)2

11990563

1199056

100381610038161003816100381610038161003816100381610038161003816

3(minus1)1199051199052

911990510ln3119905

100381610038161003816100381610038161003816100381610038161003816

le12

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054

le12

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le12

1198992

infin

sum

119905=119899

1


(84)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)

That is (38) and (39) hold Consequently Theorem 4 impliesthat (80) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin ((34)119872 +

119873119872) there exist 120579 isin (0 1) and 119879 ge 1198990+ 120591 +

120573 such that the Mann iterative sequence 119909119898119898isinN0

=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 minus 5119899

3

2 + 61198993119909119899minus120591) + Δ(

21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199092

3119899minus7))

+

sin (119899211990931198992minus2)

(radic119899 + 14)22

=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9

(86)

where 120591 isin N is fixed Let 1198990= 9 119896 = 1 119887 = minus56 and

120573 = 9 minus 120591 and let 119872 and 119873 be two positive constants with119872 gt 6119873 and

119887119899=1 minus 5119899

3

2 + 61198993 119888

119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7

119891 (119899 119906) =

sin (1198992119906)

(radic119899 + 14)22

ℎ (119899 119906) =21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199062)

1198911119899= 31198992minus 2 119865

119899= (3119899

2minus 2)2

ℎ1119899= 3119899 minus 7

119867119899= (3119899 minus 7)

2 119875

119899= 119876119899=3

1198999 119877119899= 119882119899=1

1198995

forall (119899 119906) isin N1198990

timesR

(87)

Obviously (14) (15) and (52) are satisfied Note that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max(3119904 minus 7)2

11990451

1199045

le9

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le9

1198992

infin

sum

119904=119899

1


(88)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (89)

Notice that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max

3(31199052minus 2)2

11990593

1199059

100381610038161003816100381610038161003816100381610038161003816

(minus1)1199051199053+ 51199052+ 4119905 minus 2

1199059 + 1199058 + 21199055 + 1199053 + 7

100381610038161003816100381610038161003816100381610038161003816

le27

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le27

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le27

1198992

infin

sum

119905=119899

1


(90)

which gives that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)

That is (38) and (39) hold Thus Theorem 5 shows that(86) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (119873 (16)119872)there exist 120579 isin (0 1) and 119879 ge 119899

0+ 120591 + 120573 such that the Mann

iterative sequence 119909119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0

generatedby (48) converges to a positive solution 119911 = 119911

119899119899isinN120573

isin


119911119899= +infin and has the error

estimate (20) where 120572119898119898isinN0

is an arbitrary sequencein [0 1] satisfying (21)



Δ3(119909119899+ (

120587

2+ 119899 sin 1

119899) 119909119899minus120591)

+ Δ((minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119909

2119899+1)))

+119899 sin (119899119909

119899minus2)

2 + (119899 + 5)16

=

(minus1)119899minus1cos3 (1198992 + 1)11989916 + ln 119899

forall119899 ge 3

(92)


120573 = min3 minus 120591 1 and let119872 and119873 be two positive constantswith (1 minus 2120587)119872 gt 119873 and

119887119899=120587

2+ 119899 sin 1

119899 119888

119899=


119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)

16

ℎ (119899 119906) =(minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119906))

1198911119899= 119899 minus 2 119865

119899= (119899 minus 2)

2

ℎ1119899= 2119899 + 1 119867

119899= (2119899 + 1)

2

119875119899= 119876119899=

1

11989914 119877119899= 119882119899=2

1198999 forall (119899 119906) isin N

1198990

timesR

(93)

Clearly (14) (15) and (61) are satisfied Note that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max2(2119904 + 1)2

11990492

1199049

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199047le18

1198992

infin

sum

119904=119899

1


(94)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max(119905 minus 2)2

119905141

11990514

10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

11990514le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

11990513

le1

1198992

infin

sum

119905=119899

1


(95)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899minus21198995+ 91198992minus 1

1198995 + 31198992 + 2119909119899minus120591) + Δ(

cos ((minus1)119899119890119899)

(119899 + 7)6radic1 +

1003816100381610038161003816119909119899minus21003816100381610038161003816

)

+

sin (1198992119909119899minus1)

1198999 + 31198995 + 21198992 + 1

=(minus1)119899minus11198994+ 41198992+ 119899 minus 1

11989911 + 61198993 + 7119899 + 2 forall119899 ge 6

(97)

where 120591 isin N is fixed Let 1198990= 6 119896 = 1 119887 = minus2 and 120573 =

min6 minus 120591 3 and let 119872 and 119873 be two positive constantswith (12)119872 gt 119873 and

119887119899= minus

21198995+ 91198992minus 1

1198995 + 31198992 + 2

119888119899=(minus1)119899minus11198994+ 41198992+ 119899 minus 1

11989911 + 61198993 + 7119899 + 2

119891 (119899 119906) =

sin (1198992119906)1198999 + 31198995 + 21198992 + 1


ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816

(minus1)119905minus11199054+ 41199052+ 119905 minus 1

11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)

That is (38) and (39) hold Consequently Theorem 7 impliesthat (97) possesses uncountably many positive solutionsin 119860(119873119872) On the other hand for any 119871 isin (minus1198722 minus119873)there exist 120579 isin (0 1) and 119879 ge 119899

0+ 120591 + 120573 such that

the Mann iterative sequence 119909119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0

generated by (65) converges to a positive solution 119911 =

119911119899119899isinN120573


119911119899= +infin and

has the error estimate (20) where 120572119898119898isinN0

is an arbitrarysequence in[0 1] satisfying (21)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by the Science Research Foun-dation of Educational Department of Liaoning Province(L2012380)

References

[1] M H Abu-Risha ldquoOscillation of second-order linear differenceequationsrdquo Applied Mathematics Letters vol 13 no 1 pp 129ndash135 2000

[2] R P Agarwal Difference Equations and Inequalities MarcelDekker New York NY USA 2nd edition 2000

[3] R P Agarwal and J Henderson ldquoPositive solutions and nonlin-ear eigenvalue problems for third-order difference equationsrdquoComputers amp Mathematics with Applications vol 36 no 10-12pp 347ndash355 1998

[4] A Andruch-Sobiło and M Migda ldquoOn the oscillation ofsolutions of third order linear difference equations of neutraltyperdquoMathematica Bohemica vol 130 no 1 pp 19ndash33 2005

[5] Z Dosla and A Kobza ldquoGlobal asymptotic properties of third-order difference equationsrdquo Computers amp Mathematics withApplications vol 48 no 1-2 pp 191ndash200 2004

[6] S R Grace and G G Hamedani ldquoOn the oscillation of certainneutral difference equationsrdquo Mathematica Bohemica vol 125no 3 pp 307ndash321 2000

[7] J Cheng ldquoExistence of a nonoscillatory solution of a second-order linear neutral difference equationrdquo Applied MathematicsLetters vol 20 no 8 pp 892ndash899 2007

[8] LKongQKong andB Zhang ldquoPositive solutions of boundaryvalue problems for third-order functional difference equationsrdquoComputersampMathematics withApplications vol 44 no 3-4 pp481ndash489 2002

[9] I Y Karaca ldquoDiscrete third-order three-point boundary valueproblemrdquo Journal of Computational and Applied Mathematicsvol 205 no 1 pp 458ndash468 2007

[10] W-T Li and J P Sun ldquoExistence of positive solutions of BVPsfor third-order discrete nonlinear difference systemsrdquo AppliedMathematics and Computation vol 157 no 1 pp 53ndash64 2004

[11] W-T Li and J-P Sun ldquoMultiple positive solutions of BVPs forthird-order discrete difference systemsrdquo Applied Mathematicsand Computation vol 149 no 2 pp 389ndash398 2004

[12] Z Liu M Jia S M Kang and Y C Kwun ldquoBounded positivesolutions for a third order discrete equationrdquo Abstract andApplied Analysis vol 2012 Article ID 237036 12 pages 2012

[13] Z Liu S M Kang and J S Ume ldquoExistence of uncountablymany bounded nonoscillatory solutions and their iterativeapproximations for second order nonlinear neutral delay dif-ference equationsrdquo Applied Mathematics and Computation vol213 no 2 pp 554ndash576 2009

[14] Z Liu Y Xu and S M Kang ldquoBounded oscillation criteriafor certain third order nonlinear difference equations withseveral delays and advancesrdquo Computers amp Mathematics withApplications vol 61 no 4 pp 1145ndash1161 2011

[15] M Migda and J Migda ldquoAsymptotic properties of solutions ofsecond-order neutral difference equationsrdquoNonlinear Analysis


Theory Methods and Applications vol 63 no 5ndash7 pp e789ndashe799 2005

[16] N Parhi ldquoNon-oscillation of solutions of difference equationsof third orderrdquoComputersampMathematics withApplications vol62 no 10 pp 3812ndash3820 2011

[17] N Parhi and A Panda ldquoNonoscillation and oscillation ofsolutions of a class of third order difference equationsrdquo Journalof Mathematical Analysis and Applications vol 336 no 1 pp213ndash223 2007

[18] S H Saker ldquoNew oscillation criteria for second-order nonlinearneutral delay difference equationsrdquo Applied Mathematics andComputation vol 142 no 1 pp 99ndash111 2003

[19] S H Saker ldquoOscillation of third-order difference equationsrdquoPortugaliae Mathematica vol 61 no 3 pp 249ndash257 2004

[20] S Stevic ldquoOn a third-order system of difference equationsrdquoApplied Mathematics and Computation vol 218 no 14 pp7649ndash7654 2012

[21] X H Tang ldquoBounded oscillation of second-order delay dif-ference equations of unstable typerdquo Computers amp Mathematicswith Applications vol 44 no 8-9 pp 1147ndash1156 2002

[22] J Yan and B Liu ldquoAsymptotic behavior of a nonlinear delaydifference equationrdquo Applied Mathematics Letters vol 8 no 6pp 1ndash5 1995

[23] Z G Zhang and Q L Li ldquoOscillation theorems for second-order advanced functional difference equationsrdquo Computers ampMathematics with Applications vol 36 no 6 pp 11ndash18 1998

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Throughout this paper we assume that Δ is the forwarddifference operator defined by Δ119909

119899= 119909119899+1

minus 119909119899 R =

(minusinfin +infin) R+ = [0 +infin) N0and N denote the sets of

nonnegative integers and positive integers respectively

N119905= 119899 119899 isin N with 119899 ge 119905 forall119905 isin N

120573 = min 1198990minus 120591 inf ℎ

119897119899 119891119897119899 1 le 119897 le 119896 119899 isin N

1198990

isin N

119867119899= max ℎ2


1198990

119865119899= max 1198912


1198990

(8)

and 119897infin

120573represents the Banach space of all real sequences

on N120573with norm

119909 = sup119899isinN120573

1003816100381610038161003816100381610038161003816

119909119899

1198992

1003816100381610038161003816100381610038161003816lt +infin for each 119909 = 119909

119899119899isinN120573

isin 119897infin

120573

119860 (119873119872) = 119909 = 119909119899119899isinN120573

isin 119897infin

120573 119873 le

119909119899

1198992le 119872 119899 isin N

120573

for any 119872 gt 119873 gt 0

(9)

It is easy to see that 119860(119873119872) is a closed and convex subsetof 119897infin120573 By a solution of (6) wemean a sequence 119909

119899119899isinN120573

witha positive integer 119879 ge 119899

0+120591+120573 such that (6) holds for all 119899 ge

119879

Lemma 1 Let 119901119905119905isinN be a nonnegative sequence and 120591 isin N

(i) If lim119899rarrinfin

(11198992) suminfin

119905=119899+1205911199052119901119905

= 0


(11198992) suminfin

119894=1suminfin

119904=119899+119894120591suminfin

119905=119904119901119905= 0

(ii) If lim119899rarrinfin

(11198992) suminfin

119905=119899+1205911199053119901119905

= 0


(11198992) suminfin

119894=1suminfin

119906=119899+119894120591suminfin

119904=119906suminfin

119905=119904119901119905= 0

Proof Note that

0 le1

1198992

infin

sum

119894=1

infin

sum

119904=119899+119894120591

infin

sum

119905=119904

119901119905

=1

1198992

infin

sum

119894=1

(

infin

sum

119905=119899+119894120591

119901119905+

infin

sum

119905=119899+1+119894120591

119901119905+

infin

sum

119905=119899+2+119894120591

119901119905+ sdot sdot sdot )

=1

1198992

infin

sum

119894=1

infin

sum

119905=119899+119894120591

(1 + 119905 minus 119899 minus 119894120591) 119901119905le1

1198992

infin

sum

119894=1

infin

sum

119905=119899+119894120591

119905119901119905

=1

1198992(

infin

sum

119905=119899+120591

119905119901119905+

infin

sum

119905=119899+2120591

119905119901119905+

infin

sum

119905=119899+3120591

119905119901119905+ sdot sdot sdot )

le1

1198992

infin

sum

119905=119899+120591

(1 +119905 minus 119899 minus 120591

120591) 119905119901119905=1

1198992

infin

sum

119905=119899+120591

119905 minus 119899

120591119905119901119905

le1

1198992120591

infin

sum

119905=119899+120591


(10)

that is

lim119899rarrinfin

1

1198992

infin

sum

119894=1

infin

sum

119904=119899+119894120591

infin

sum

119905=119904

119901119905= 0 (11)

As in the proof of (10) we infer that

0 le1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

infin

sum

119905=119904

119901119905

=1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119905=119906

(1 + 119905 minus 119906) 119901119905

le1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119905=119906

119905119901119905le

1

1198992120591

infin

sum

119905=119899+120591


(12)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

infin

sum

119905=119904

119901119905= 0 (13)

This completes the proof

2 Uncountably Many Positive Solutions andMann Iterative Schemes

In this section using the Banach fixed point theoremand Mann iterative schemes we establish the existence ofuncountably many positive solutions of (6) prove conver-gence of the Mann iterative schemes relative to these positivesolutions and compute the error estimates between theManniterative schemes and the positive solutions

Theorem 2 Assume that there exist twoconstants 119872 and 119873 with 119872 gt 119873 gt 0 and four nonnegativesequences 119875

119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying1003816100381610038161003816119891 (119899 1199061 1199062 119906119896) minus 119891 (119899 1199061 1199062 119906119896)

1003816100381610038161003816

le 119875119899max 1003816100381610038161003816119906119897 minus 119906119897

1003816100381610038161003816 1 le 119897 le 119896

1003816100381610038161003816ℎ (119899 1199061 1199062 119906119896) minus ℎ (119899 1199061 1199062 119906119896)1003816100381610038161003816

le 119877119899max 1003816100381610038161003816119906119897 minus 119906119897

1003816100381610038161003816 1 le 119897 le 119896

forall (119899 119906119897 119906119897) isin N1198990

times (R+ 0)2

1 le 119897 le 119896

(14)

1003816100381610038161003816119891 (119899 1199061 1199062 119906119896)1003816100381610038161003816 le 119876119899

1003816100381610038161003816ℎ (119899 1199061 1199062 119906119896)1003816100381610038161003816 le 119882119899

forall (119899 119906119897) isin N1198990

times (R+ 0) 1 le 119897 le 119896

(15)

lim119899rarrinfin

1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (16)

lim119899rarrinfin

1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (17)

119887119899= minus1 eventually (18)




0= 119909

0119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+1205721198981198992119871

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871

+

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(19)


isin 119860(119873119872) of (6)with lim



119898

119894=01205721198941003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817 forall119898 isin N0 (20)

where 120572119898119898isinN0


sum

119898=0

120572119898= +infin (21)



1198990+ 120591 + 120573 satisfying

120579 =1

1198792

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (22)

1

1198792

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


(23)

119887119899= minus1 forall119899 ge 119879 (24)


infin

120573by

119878119871119909119899

=

1198992119871

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879 119878119871119909119879 120573 le 119899 lt 119879

(25)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872)

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[119877119904max ℎ2

119897119904 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 1198912

119897119905 1 le 119897 le 119896]

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198792

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus 119871

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816


+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816 ]

le1

1198792

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]



119878119871 (119860 (119873119872)) sube 119860 (119873119872)

1003817100381710038171003817119878119871119909 minus 1198781198711199101003817100381710038171003817 le 120579


(27)




isin 119860(119873119872) that is

119911119899= 1198992119871

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905]

forall119899 ge 119879

119911119899minus120591

= (119899 minus 120591)2119871

+

infin

sum

119894=1

infin

sum

119906=119899+(119894minus1)120591

infin

sum

119904=119906

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

)

minus 119888119905] forall119899 ge 119879 + 120591


= (2119899120591 minus 1205912) 119871

minus

infin

sum

119906=119899

infin

sum

119904=119906

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905]

forall119899 ge 119879 + 120591


= 2120591119871 +

infin

sum

119904=119899

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905]

forall119899 ge 119879 + 120591

Δ2(119911119899minus 119911119899minus120591)

= minusℎ (119899 119911ℎ1119899

119911ℎ2119899

119911ℎ119896119899

)

+

infin

sum

119905=119899

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905] forall119899 ge 119879 + 120591

(30)

which gives that

Δ3(119911119899minus 119911119899minus120591)

= minusΔℎ (119899 119911ℎ1119899

119911ℎ2119899

119911ℎ119896119899

)

minus 119891 (119905 1199111198911119899

1199111198912119899

119911119891119896119899

) + 119888119899 forall119899 ge 119879 + 120591

(31)



119873 le119911119899

1198992le 119872 forall119899 isin N

120573 (32)




1003816100381610038161003816100381610038161003816

119909119898+1119899

1198992minus119911119899

1198992

1003816100381610038161003816100381610038161003816

=1

1198992

1003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572119898) 119909119898119899

+ 1205721198981198992119871

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

) minus 119888119905)] minus 119911

119899

1003816100381610038161003816100381610038161003816100381610038161003816

le (1 minus 120572119898)

1003816100381610038161003816119909119898119899 minus 1199111198991003816100381610038161003816

1198992+ 120572119898

1003816100381610038161003816119878119871119909119898119899 minus 1198781198711199111198991003816100381610038161003816

1198992

le (1 minus 120572119898)1003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817 + 1205791205721198981003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817


1003817100381710038171003817 forall119898 isin N0 119899 ge 119879

(33)

which implies that


119898

119894=01205721198941003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817 forall119898 isin N0 (34)


119898rarrinfin119909119898= 119911


isin




119871119888

satisfying



119888



119911119888

119899119899isinN120573


119899rarrinfin119911119888


119911119888

119899= 1198992119871119888

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119911119888

ℎ1119904

119911119888

ℎ2119904

119911119888

ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 119911119888

1198911119905

119911119888

1198912119905

119911119888

119891119896119905

) minus 119888119905]

forall119899 ge 119879119888

(35)


max1198791 1198792

100381610038161003816100381610038161003816100381610038161003816

1199111

119899

1198992minus1199112

119899

1198992

100381610038161003816100381610038161003816100381610038161003816

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119911

1

ℎ1119904

1199111

ℎ2119904

1199111

ℎ119896119904

)

minus ℎ (119904 1199112

ℎ1119904

1199112

ℎ2119904

1199112

ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119911

1

1198911119905

1199111

1198912119905

1199111

119891119896119905

) minus 119891 (119905 1199112

1198911119905

1199112

1198912119905

1199112

119891119896119905

)10038161003816100381610038161003816

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119911

1

ℎ119897119904

minus 1199112

ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119911

1

119891119897119905

minus 1199112

119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

max 11987921 1198792

2

infin

sum

119894=1

infin

sum

119906=max1198791 1198792+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816 minusmax 1205791 1205792100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

(36)

which yields that

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817ge

10038161003816100381610038161198711 minus 11987121003816100381610038161003816

1 +max 1205791 1205792gt 0 (37)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990


lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (38)

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (39)

119887119899= 1 eventually (40)




0= 119909

0119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+1205721198981198992119871

minus

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

) minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871

minus

infin

sum

119894=1

119879+2119894120591minus1

sum

119906=119879+(2119894minus1)120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

) minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(41)


isin








0+ 120591 + 120573 satisfying

120579 =1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (42)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904


(43)

119887119899= 1 forall119899 ge 119879 (44)


infin

120573by

119878119871119909119899

=

1198992119871

minus

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus119888119905] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(45)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 = 119910119899119899isinN120573

isin

119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) = 120579

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus 119871

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)])


(46)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990




1199090

= 1199090119899119899isinN120573


119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+1205721198981198992119871 minus 119887119899119909119898119899minus120591

minus

infin

sum

119906=119899

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871 minus 119887119879119909119898119879minus120591

minus

infin

sum

119906=119879

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(48)


isin








0+ 120591 + 120573 satisfying

120579 = 119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


0 le 119887119899le 119887 lt 1 forall119899 ge 119879

(49)


infin

120573by

119878119871119909119899

=

1198992119871 minus 119887119899119909119899minus120591

minus

infin

sum

119906=119899

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(50)for each 119909 = 119909

119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 11987910038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le 119887119899

100381610038161003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

(119899 minus 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 minus 120591)2

1198992

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le 1198871003817100381710038171003817119909 minus 119910

1003817100381710038171003817

+

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max ℎ2

119897119904 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 1198912

119897119905 1 le 119897 le 119896]

le [119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 120579

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119878119871119909119899

1198992

ge 119871 minus 119887119872

minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus 119887119872 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


(51)



119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14)(15) (38) (39) and





any 1199090= 1199090119899119899isinN120573


119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin








0+ 120591 + 120573 satisfying

120579 = minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (53)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


(54)



infin



119910 = 119910

119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le [minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 minus 119887119872

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 minus 119887119872 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119878119871119909119899

1198992

ge 119871 minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


(56)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882

119899119899isinN1198990

satisfying (14) (15) (38) (39) and

119887119899ge 119887 gt 1 eventually (57)



0=

1199090119899119899isinN120573


= 119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 1205721198981198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(58)



isin







0+ 120591 +

120573 satisfying

120579 =1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (59)

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873

(60)

119887119899ge 119887 gt 1 forall119899 ge 119879 (61)


infin

120573by

119878119871119909119899

=

1198992119871 minus

119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[(119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus 119888119905)] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(62)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

+1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

+1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le 119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873 le 119872

119878119871119909119899

1198992

ge 119871 minus1

119887119872

minus1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

119887119872 minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

gt 119871 minus1


1

119887119872 minus119873 ge 119873

(63)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14) (15) (38) (39) and





0+ 120591 + 120573 such that for any 119909

0= 1199090119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 120572119898 minus 1198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+120572119898 minus 119879

2119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(65)


isin







0+ 120591 +

120573 satisfying

120579 = minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (66)

minus1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt min (1 + 1

119887)119872 minus 119871 119871 minus

1

119887119872 minus119873

(67)

119887119899ge 119887 gt 1 forall119899 ge 119879 (68)


infin

120573by

119878119871119909119899

=


119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(69)

for each 119909 = 119909119899119899isinN120573


119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le minus1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

minus1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le minus1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

minus1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992


119887minus

1

1198871198992

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]



119887minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887+min (1 + 1

119887)119872 + 119871 minus119871 minus 119873 le 119872

119878119871119909119899

1198992

ge minus119871

+1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge minus119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


119887)119872 + 119871 minus119871 minus 119873 ge 119873

(70)


3 Examples



Δ3(119909119899minus 119909119899minus120591) + Δ(

sin2119909119899minus3

1198997) +

1

(1198999 + 21198995 + 1) (1 + 1199092

1198992)

=1198992minus 2119899

1198998 + 1198993 + 1 forall119899 ge 4

(71)



119887119899= minus1 119888

119899=

1198992minus 2119899

1198998 + 1198993 + 1

119891 (119899 119906) =1

(1198999 + 21198995 + 1) (1 + 1199062)

ℎ (119899 119906) =sin21199061198997

1198911119899= 1198992 119865

119899= 1198994

ℎ1119899= 119899 minus 3 119867

119899= (119899 minus 3)

2 119875

119899=2119872

1198999

119876119899=1

1198999 119877

119899=2

1198997 119882

119899=1

1198997

forall (119899 119906) isin N1198990

timesR

(72)


1

1198992

infin

sum

119905=119899+120591

1199052max 119877

119905119867119905119882119905

=1

1198992

infin

sum

119905=119899+120591

1199052max2(119905 minus 3)

2

11990571

1199057

=

infin

sum

119905=119899+120591

2(119905 minus 3)2+ 1

1199055

le2

1198992

infin

sum

119905=119899+120591

1


1

1198992

infin

sum

119905=119899+120591

1199053max 119875

119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119905=119899+120591

1199053max2119872

11990551

1199059

100381610038161003816100381610038161199052minus 2119905

10038161003816100381610038161003816

1199058 + 1199053 + 1

lemax 1 2119872

1198992

infin

sum

119905=119899+120591

1


(73)


0+120591+120573 such



119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin 119860(119873119872) of(71) with lim


where 120572119898119898isinN0



Δ3(119909119899+ 119909119899minus120591) + Δ(

sin211990931198993+1

1198993 (1198996 + 2) (1 + 1199094

21198992minus3)

)


+ 119909(119899+1)(119899+2)

)

(11989913 + 1198995 + 1) (1 + 1199092


+ 1199092

(119899+1)(119899+2))

=1198992minus ln 119899

1198996 + 1198995 + 1 forall119899 ge 5

(74)



119887119899= 1 119888

119899=

1198992minus ln 119899

1198996 + 1198995 + 1

119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)

(11989913 + 1198995 + 1) (1 + 1199062 + V2)


ℎ (119899 119906 V) =sin2V

1198993 (1198996 + 2) (1 + 1199064) 119891

1119899= 1198992minus 119899 minus 1

1198912119899= (119899 + 1) (119899 + 2)

119865119899= (119899 + 1)

2(119899 + 2)

2 ℎ

1119899= 21198992minus 3

ℎ2119899= 31198993+ 1

119867119899= (3119899

3+ 1)2

119875119899= 119876119899=

4

11989910 119877

119899= 119882119899=10

1198999

forall (119899 119906 V) isin N1198990

timesR2

(75)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

10(31199043+ 1)2

119904910

1199049

le160

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le160

1198992

infin

sum

119904=119899

1


(76)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (77)

Observe that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max4(119905 + 1)2(119905 + 2)

2

119905104

119905101199052minus ln 119905

1199056 + 1199055 + 1

le196

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054=196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

119905 minus 119906 + 1

1199054

le196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le196

1198992

infin

sum

119905=119899

1


(78)

which yields that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)



119898119898isinN0

=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899

119909119899minus120591)

+ Δ(

(minus1)119899 sin (119890minus119899

2|11990951198992minus3|)

11989915 minus radic119899 + 3

+1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|11990921198993+1|)

+(minus1)119899

1198996 (1 + 1199092

119899minus3)minus

1

(1198997 + 21198994 minus 1) (1 + 1199092

119899+4)

=3(minus1)1198991198992

911989910ln3119899 forall119899 ge 7

(80)



119887119899=1 + 3 ln 1198992 + 4 ln 119899

119888119899=3(minus1)1198991198992

911989910ln3119899

119891 (119899 119906 V) =(minus1)119899

1198996 (1 + 1199062)minus

1

(1198997 + 21198994 minus 1) (1 + V2)


2|119906|)

11989915 minus radic119899 + 3+

1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|V|

1198911119899= 119899 minus 3 119891

2119899= 119899 + 4

119865119899= (119899 + 4)

2 ℎ

1119899= 51198992minus 3

ℎ2119899= 21198993+ 1 119867

119899= (2119899

3+ 1)2

119875119899= 119876119899=3

1198996 119877

119899= 119882119899=

2

11989910

forall (119899 119906 V) isin N1198990

timesR2

(81)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

2(21199043+ 1)2

119904102

11990410

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le18

1198992

infin

sum

119904=119899

1


(82)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (83)


Observe that1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max3(119905 + 4)2

11990563

1199056

100381610038161003816100381610038161003816100381610038161003816

3(minus1)1199051199052

911990510ln3119905

100381610038161003816100381610038161003816100381610038161003816

le12

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054

le12

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le12

1198992

infin

sum

119905=119899

1


(84)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 minus 5119899

3

2 + 61198993119909119899minus120591) + Δ(

21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199092

3119899minus7))

+

sin (119899211990931198992minus2)

(radic119899 + 14)22

=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9

(86)



119887119899=1 minus 5119899

3

2 + 61198993 119888

119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7

119891 (119899 119906) =

sin (1198992119906)

(radic119899 + 14)22

ℎ (119899 119906) =21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199062)

1198911119899= 31198992minus 2 119865

119899= (3119899

2minus 2)2

ℎ1119899= 3119899 minus 7

119867119899= (3119899 minus 7)

2 119875

119899= 119876119899=3

1198999 119877119899= 119882119899=1

1198995

forall (119899 119906) isin N1198990

timesR

(87)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max(3119904 minus 7)2

11990451

1199045

le9

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le9

1198992

infin

sum

119904=119899

1


(88)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (89)

Notice that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max

3(31199052minus 2)2

11990593

1199059

100381610038161003816100381610038161003816100381610038161003816

(minus1)1199051199053+ 51199052+ 4119905 minus 2

1199059 + 1199058 + 21199055 + 1199053 + 7

100381610038161003816100381610038161003816100381610038161003816

le27

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le27

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le27

1198992

infin

sum

119905=119899

1


(90)

which gives that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)




= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin







Δ3(119909119899+ (

120587

2+ 119899 sin 1

119899) 119909119899minus120591)

+ Δ((minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119909

2119899+1)))

+119899 sin (119899119909

119899minus2)

2 + (119899 + 5)16

=


forall119899 ge 3

(92)



119887119899=120587

2+ 119899 sin 1

119899 119888

119899=


119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)

16

ℎ (119899 119906) =(minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119906))

1198911119899= 119899 minus 2 119865

119899= (119899 minus 2)

2

ℎ1119899= 2119899 + 1 119867

119899= (2119899 + 1)

2

119875119899= 119876119899=

1

11989914 119877119899= 119882119899=2

1198999 forall (119899 119906) isin N

1198990

timesR

(93)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max2(2119904 + 1)2

11990492

1199049

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199047le18

1198992

infin

sum

119904=119899

1


(94)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


119905141

11990514

10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

11990514le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

11990513

le1

1198992

infin

sum

119905=119899

1


(95)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899minus21198995+ 91198992minus 1

1198995 + 31198992 + 2119909119899minus120591) + Δ(

cos ((minus1)119899119890119899)

(119899 + 7)6radic1 +

1003816100381610038161003816119909119899minus21003816100381610038161003816

)

+

sin (1198992119909119899minus1)

1198999 + 31198995 + 21198992 + 1


11989911 + 61198993 + 7119899 + 2 forall119899 ge 6

(97)



119887119899= minus

21198995+ 91198992minus 1

1198995 + 31198992 + 2


11989911 + 61198993 + 7119899 + 2

119891 (119899 119906) =

sin (1198992119906)1198999 + 31198995 + 21198992 + 1


ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816


11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)


0+ 120591 + 120573 such that


= 119909119898119899119899isinN120573

119898isinN0


119911119899119899isinN120573


119911119899= +infin and





Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0= 119909

0119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+1205721198981198992119871

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871

+

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(19)


isin 119860(119873119872) of (6)with lim



119898

119894=01205721198941003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817 forall119898 isin N0 (20)

where 120572119898119898isinN0


sum

119898=0

120572119898= +infin (21)



1198990+ 120591 + 120573 satisfying

120579 =1

1198792

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (22)

1

1198792

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


(23)

119887119899= minus1 forall119899 ge 119879 (24)


infin

120573by

119878119871119909119899

=

1198992119871

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879 119878119871119909119879 120573 le 119899 lt 119879

(25)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872)

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[119877119904max ℎ2

119897119904 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 1198912

119897119905 1 le 119897 le 119896]

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198792

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus 119871

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816


+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816 ]

le1

1198792

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]



119878119871 (119860 (119873119872)) sube 119860 (119873119872)

1003817100381710038171003817119878119871119909 minus 1198781198711199101003817100381710038171003817 le 120579


(27)




isin 119860(119873119872) that is

119911119899= 1198992119871

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905]

forall119899 ge 119879

119911119899minus120591

= (119899 minus 120591)2119871

+

infin

sum

119894=1

infin

sum

119906=119899+(119894minus1)120591

infin

sum

119904=119906

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

)

minus 119888119905] forall119899 ge 119879 + 120591


= (2119899120591 minus 1205912) 119871

minus

infin

sum

119906=119899

infin

sum

119904=119906

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905]

forall119899 ge 119879 + 120591


= 2120591119871 +

infin

sum

119904=119899

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905]

forall119899 ge 119879 + 120591

Δ2(119911119899minus 119911119899minus120591)

= minusℎ (119899 119911ℎ1119899

119911ℎ2119899

119911ℎ119896119899

)

+

infin

sum

119905=119899

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905] forall119899 ge 119879 + 120591

(30)

which gives that

Δ3(119911119899minus 119911119899minus120591)

= minusΔℎ (119899 119911ℎ1119899

119911ℎ2119899

119911ℎ119896119899

)

minus 119891 (119905 1199111198911119899

1199111198912119899

119911119891119896119899

) + 119888119899 forall119899 ge 119879 + 120591

(31)



119873 le119911119899

1198992le 119872 forall119899 isin N

120573 (32)




1003816100381610038161003816100381610038161003816

119909119898+1119899

1198992minus119911119899

1198992

1003816100381610038161003816100381610038161003816

=1

1198992

1003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572119898) 119909119898119899

+ 1205721198981198992119871

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

) minus 119888119905)] minus 119911

119899

1003816100381610038161003816100381610038161003816100381610038161003816

le (1 minus 120572119898)

1003816100381610038161003816119909119898119899 minus 1199111198991003816100381610038161003816

1198992+ 120572119898

1003816100381610038161003816119878119871119909119898119899 minus 1198781198711199111198991003816100381610038161003816

1198992

le (1 minus 120572119898)1003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817 + 1205791205721198981003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817


1003817100381710038171003817 forall119898 isin N0 119899 ge 119879

(33)

which implies that


119898

119894=01205721198941003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817 forall119898 isin N0 (34)


119898rarrinfin119909119898= 119911


isin




119871119888

satisfying



119888



119911119888

119899119899isinN120573


119899rarrinfin119911119888


119911119888

119899= 1198992119871119888

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119911119888

ℎ1119904

119911119888

ℎ2119904

119911119888

ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 119911119888

1198911119905

119911119888

1198912119905

119911119888

119891119896119905

) minus 119888119905]

forall119899 ge 119879119888

(35)


max1198791 1198792

100381610038161003816100381610038161003816100381610038161003816

1199111

119899

1198992minus1199112

119899

1198992

100381610038161003816100381610038161003816100381610038161003816

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119911

1

ℎ1119904

1199111

ℎ2119904

1199111

ℎ119896119904

)

minus ℎ (119904 1199112

ℎ1119904

1199112

ℎ2119904

1199112

ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119911

1

1198911119905

1199111

1198912119905

1199111

119891119896119905

) minus 119891 (119905 1199112

1198911119905

1199112

1198912119905

1199112

119891119896119905

)10038161003816100381610038161003816

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119911

1

ℎ119897119904

minus 1199112

ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119911

1

119891119897119905

minus 1199112

119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

max 11987921 1198792

2

infin

sum

119894=1

infin

sum

119906=max1198791 1198792+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816 minusmax 1205791 1205792100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

(36)

which yields that

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817ge

10038161003816100381610038161198711 minus 11987121003816100381610038161003816

1 +max 1205791 1205792gt 0 (37)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990


lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (38)

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (39)

119887119899= 1 eventually (40)




0= 119909

0119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+1205721198981198992119871

minus

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

) minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871

minus

infin

sum

119894=1

119879+2119894120591minus1

sum

119906=119879+(2119894minus1)120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

) minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(41)


isin








0+ 120591 + 120573 satisfying

120579 =1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (42)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904


(43)

119887119899= 1 forall119899 ge 119879 (44)


infin

120573by

119878119871119909119899

=

1198992119871

minus

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus119888119905] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(45)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 = 119910119899119899isinN120573

isin

119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) = 120579

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus 119871

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)])


(46)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990




1199090

= 1199090119899119899isinN120573


119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+1205721198981198992119871 minus 119887119899119909119898119899minus120591

minus

infin

sum

119906=119899

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871 minus 119887119879119909119898119879minus120591

minus

infin

sum

119906=119879

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(48)


isin








0+ 120591 + 120573 satisfying

120579 = 119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


0 le 119887119899le 119887 lt 1 forall119899 ge 119879

(49)


infin

120573by

119878119871119909119899

=

1198992119871 minus 119887119899119909119899minus120591

minus

infin

sum

119906=119899

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(50)for each 119909 = 119909

119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 11987910038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le 119887119899

100381610038161003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

(119899 minus 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 minus 120591)2

1198992

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le 1198871003817100381710038171003817119909 minus 119910

1003817100381710038171003817

+

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max ℎ2

119897119904 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 1198912

119897119905 1 le 119897 le 119896]

le [119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 120579

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119878119871119909119899

1198992

ge 119871 minus 119887119872

minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus 119887119872 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


(51)



119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14)(15) (38) (39) and





any 1199090= 1199090119899119899isinN120573


119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin








0+ 120591 + 120573 satisfying

120579 = minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (53)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


(54)



infin



119910 = 119910

119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le [minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 minus 119887119872

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 minus 119887119872 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119878119871119909119899

1198992

ge 119871 minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


(56)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882

119899119899isinN1198990

satisfying (14) (15) (38) (39) and

119887119899ge 119887 gt 1 eventually (57)



0=

1199090119899119899isinN120573


= 119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 1205721198981198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(58)



isin







0+ 120591 +

120573 satisfying

120579 =1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (59)

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873

(60)

119887119899ge 119887 gt 1 forall119899 ge 119879 (61)


infin

120573by

119878119871119909119899

=

1198992119871 minus

119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[(119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus 119888119905)] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(62)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

+1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

+1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le 119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873 le 119872

119878119871119909119899

1198992

ge 119871 minus1

119887119872

minus1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

119887119872 minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

gt 119871 minus1


1

119887119872 minus119873 ge 119873

(63)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14) (15) (38) (39) and





0+ 120591 + 120573 such that for any 119909

0= 1199090119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 120572119898 minus 1198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+120572119898 minus 119879

2119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(65)


isin







0+ 120591 +

120573 satisfying

120579 = minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (66)

minus1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt min (1 + 1

119887)119872 minus 119871 119871 minus

1

119887119872 minus119873

(67)

119887119899ge 119887 gt 1 forall119899 ge 119879 (68)


infin

120573by

119878119871119909119899

=


119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(69)

for each 119909 = 119909119899119899isinN120573


119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le minus1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

minus1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le minus1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

minus1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992


119887minus

1

1198871198992

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]



119887minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887+min (1 + 1

119887)119872 + 119871 minus119871 minus 119873 le 119872

119878119871119909119899

1198992

ge minus119871

+1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge minus119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


119887)119872 + 119871 minus119871 minus 119873 ge 119873

(70)


3 Examples



Δ3(119909119899minus 119909119899minus120591) + Δ(

sin2119909119899minus3

1198997) +

1

(1198999 + 21198995 + 1) (1 + 1199092

1198992)

=1198992minus 2119899

1198998 + 1198993 + 1 forall119899 ge 4

(71)



119887119899= minus1 119888

119899=

1198992minus 2119899

1198998 + 1198993 + 1

119891 (119899 119906) =1

(1198999 + 21198995 + 1) (1 + 1199062)

ℎ (119899 119906) =sin21199061198997

1198911119899= 1198992 119865

119899= 1198994

ℎ1119899= 119899 minus 3 119867

119899= (119899 minus 3)

2 119875

119899=2119872

1198999

119876119899=1

1198999 119877

119899=2

1198997 119882

119899=1

1198997

forall (119899 119906) isin N1198990

timesR

(72)


1

1198992

infin

sum

119905=119899+120591

1199052max 119877

119905119867119905119882119905

=1

1198992

infin

sum

119905=119899+120591

1199052max2(119905 minus 3)

2

11990571

1199057

=

infin

sum

119905=119899+120591

2(119905 minus 3)2+ 1

1199055

le2

1198992

infin

sum

119905=119899+120591

1


1

1198992

infin

sum

119905=119899+120591

1199053max 119875

119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119905=119899+120591

1199053max2119872

11990551

1199059

100381610038161003816100381610038161199052minus 2119905

10038161003816100381610038161003816

1199058 + 1199053 + 1

lemax 1 2119872

1198992

infin

sum

119905=119899+120591

1


(73)


0+120591+120573 such



119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin 119860(119873119872) of(71) with lim


where 120572119898119898isinN0



Δ3(119909119899+ 119909119899minus120591) + Δ(

sin211990931198993+1

1198993 (1198996 + 2) (1 + 1199094

21198992minus3)

)


+ 119909(119899+1)(119899+2)

)

(11989913 + 1198995 + 1) (1 + 1199092


+ 1199092

(119899+1)(119899+2))

=1198992minus ln 119899

1198996 + 1198995 + 1 forall119899 ge 5

(74)



119887119899= 1 119888

119899=

1198992minus ln 119899

1198996 + 1198995 + 1

119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)

(11989913 + 1198995 + 1) (1 + 1199062 + V2)


ℎ (119899 119906 V) =sin2V

1198993 (1198996 + 2) (1 + 1199064) 119891

1119899= 1198992minus 119899 minus 1

1198912119899= (119899 + 1) (119899 + 2)

119865119899= (119899 + 1)

2(119899 + 2)

2 ℎ

1119899= 21198992minus 3

ℎ2119899= 31198993+ 1

119867119899= (3119899

3+ 1)2

119875119899= 119876119899=

4

11989910 119877

119899= 119882119899=10

1198999

forall (119899 119906 V) isin N1198990

timesR2

(75)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

10(31199043+ 1)2

119904910

1199049

le160

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le160

1198992

infin

sum

119904=119899

1


(76)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (77)

Observe that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max4(119905 + 1)2(119905 + 2)

2

119905104

119905101199052minus ln 119905

1199056 + 1199055 + 1

le196

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054=196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

119905 minus 119906 + 1

1199054

le196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le196

1198992

infin

sum

119905=119899

1


(78)

which yields that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)



119898119898isinN0

=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899

119909119899minus120591)

+ Δ(

(minus1)119899 sin (119890minus119899

2|11990951198992minus3|)

11989915 minus radic119899 + 3

+1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|11990921198993+1|)

+(minus1)119899

1198996 (1 + 1199092

119899minus3)minus

1

(1198997 + 21198994 minus 1) (1 + 1199092

119899+4)

=3(minus1)1198991198992

911989910ln3119899 forall119899 ge 7

(80)



119887119899=1 + 3 ln 1198992 + 4 ln 119899

119888119899=3(minus1)1198991198992

911989910ln3119899

119891 (119899 119906 V) =(minus1)119899

1198996 (1 + 1199062)minus

1

(1198997 + 21198994 minus 1) (1 + V2)


2|119906|)

11989915 minus radic119899 + 3+

1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|V|

1198911119899= 119899 minus 3 119891

2119899= 119899 + 4

119865119899= (119899 + 4)

2 ℎ

1119899= 51198992minus 3

ℎ2119899= 21198993+ 1 119867

119899= (2119899

3+ 1)2

119875119899= 119876119899=3

1198996 119877

119899= 119882119899=

2

11989910

forall (119899 119906 V) isin N1198990

timesR2

(81)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

2(21199043+ 1)2

119904102

11990410

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le18

1198992

infin

sum

119904=119899

1


(82)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (83)


Observe that1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max3(119905 + 4)2

11990563

1199056

100381610038161003816100381610038161003816100381610038161003816

3(minus1)1199051199052

911990510ln3119905

100381610038161003816100381610038161003816100381610038161003816

le12

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054

le12

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le12

1198992

infin

sum

119905=119899

1


(84)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 minus 5119899

3

2 + 61198993119909119899minus120591) + Δ(

21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199092

3119899minus7))

+

sin (119899211990931198992minus2)

(radic119899 + 14)22

=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9

(86)



119887119899=1 minus 5119899

3

2 + 61198993 119888

119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7

119891 (119899 119906) =

sin (1198992119906)

(radic119899 + 14)22

ℎ (119899 119906) =21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199062)

1198911119899= 31198992minus 2 119865

119899= (3119899

2minus 2)2

ℎ1119899= 3119899 minus 7

119867119899= (3119899 minus 7)

2 119875

119899= 119876119899=3

1198999 119877119899= 119882119899=1

1198995

forall (119899 119906) isin N1198990

timesR

(87)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max(3119904 minus 7)2

11990451

1199045

le9

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le9

1198992

infin

sum

119904=119899

1


(88)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (89)

Notice that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max

3(31199052minus 2)2

11990593

1199059

100381610038161003816100381610038161003816100381610038161003816

(minus1)1199051199053+ 51199052+ 4119905 minus 2

1199059 + 1199058 + 21199055 + 1199053 + 7

100381610038161003816100381610038161003816100381610038161003816

le27

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le27

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le27

1198992

infin

sum

119905=119899

1


(90)

which gives that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)




= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin







Δ3(119909119899+ (

120587

2+ 119899 sin 1

119899) 119909119899minus120591)

+ Δ((minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119909

2119899+1)))

+119899 sin (119899119909

119899minus2)

2 + (119899 + 5)16

=


forall119899 ge 3

(92)



119887119899=120587

2+ 119899 sin 1

119899 119888

119899=


119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)

16

ℎ (119899 119906) =(minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119906))

1198911119899= 119899 minus 2 119865

119899= (119899 minus 2)

2

ℎ1119899= 2119899 + 1 119867

119899= (2119899 + 1)

2

119875119899= 119876119899=

1

11989914 119877119899= 119882119899=2

1198999 forall (119899 119906) isin N

1198990

timesR

(93)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max2(2119904 + 1)2

11990492

1199049

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199047le18

1198992

infin

sum

119904=119899

1


(94)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


119905141

11990514

10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

11990514le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

11990513

le1

1198992

infin

sum

119905=119899

1


(95)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899minus21198995+ 91198992minus 1

1198995 + 31198992 + 2119909119899minus120591) + Δ(

cos ((minus1)119899119890119899)

(119899 + 7)6radic1 +

1003816100381610038161003816119909119899minus21003816100381610038161003816

)

+

sin (1198992119909119899minus1)

1198999 + 31198995 + 21198992 + 1


11989911 + 61198993 + 7119899 + 2 forall119899 ge 6

(97)



119887119899= minus

21198995+ 91198992minus 1

1198995 + 31198992 + 2


11989911 + 61198993 + 7119899 + 2

119891 (119899 119906) =

sin (1198992119906)1198999 + 31198995 + 21198992 + 1


ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816


11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)


0+ 120591 + 120573 such that


= 119909119898119899119899isinN120573

119898isinN0


119911119899119899isinN120573


119911119899= +infin and





Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816 ]

le1

1198792

infin

sum

119894=1

infin

sum

119906=119879+119894120591

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]



119878119871 (119860 (119873119872)) sube 119860 (119873119872)

1003817100381710038171003817119878119871119909 minus 1198781198711199101003817100381710038171003817 le 120579


(27)




isin 119860(119873119872) that is

119911119899= 1198992119871

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905]

forall119899 ge 119879

119911119899minus120591

= (119899 minus 120591)2119871

+

infin

sum

119894=1

infin

sum

119906=119899+(119894minus1)120591

infin

sum

119904=119906

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

)

minus 119888119905] forall119899 ge 119879 + 120591


= (2119899120591 minus 1205912) 119871

minus

infin

sum

119906=119899

infin

sum

119904=119906

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905]

forall119899 ge 119879 + 120591


= 2120591119871 +

infin

sum

119904=119899

ℎ (119904 119911ℎ1119904

119911ℎ2119904

119911ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905]

forall119899 ge 119879 + 120591

Δ2(119911119899minus 119911119899minus120591)

= minusℎ (119899 119911ℎ1119899

119911ℎ2119899

119911ℎ119896119899

)

+

infin

sum

119905=119899

[119891 (119905 1199111198911119905

1199111198912119905

119911119891119896119905

) minus 119888119905] forall119899 ge 119879 + 120591

(30)

which gives that

Δ3(119911119899minus 119911119899minus120591)

= minusΔℎ (119899 119911ℎ1119899

119911ℎ2119899

119911ℎ119896119899

)

minus 119891 (119905 1199111198911119899

1199111198912119899

119911119891119896119899

) + 119888119899 forall119899 ge 119879 + 120591

(31)



119873 le119911119899

1198992le 119872 forall119899 isin N

120573 (32)




1003816100381610038161003816100381610038161003816

119909119898+1119899

1198992minus119911119899

1198992

1003816100381610038161003816100381610038161003816

=1

1198992

1003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572119898) 119909119898119899

+ 1205721198981198992119871

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

) minus 119888119905)] minus 119911

119899

1003816100381610038161003816100381610038161003816100381610038161003816

le (1 minus 120572119898)

1003816100381610038161003816119909119898119899 minus 1199111198991003816100381610038161003816

1198992+ 120572119898

1003816100381610038161003816119878119871119909119898119899 minus 1198781198711199111198991003816100381610038161003816

1198992

le (1 minus 120572119898)1003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817 + 1205791205721198981003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817


1003817100381710038171003817 forall119898 isin N0 119899 ge 119879

(33)

which implies that


119898

119894=01205721198941003817100381710038171003817119909119898 minus 119911

1003817100381710038171003817 forall119898 isin N0 (34)


119898rarrinfin119909119898= 119911


isin




119871119888

satisfying



119888



119911119888

119899119899isinN120573


119899rarrinfin119911119888


119911119888

119899= 1198992119871119888

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119911119888

ℎ1119904

119911119888

ℎ2119904

119911119888

ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 119911119888

1198911119905

119911119888

1198912119905

119911119888

119891119896119905

) minus 119888119905]

forall119899 ge 119879119888

(35)


max1198791 1198792

100381610038161003816100381610038161003816100381610038161003816

1199111

119899

1198992minus1199112

119899

1198992

100381610038161003816100381610038161003816100381610038161003816

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119911

1

ℎ1119904

1199111

ℎ2119904

1199111

ℎ119896119904

)

minus ℎ (119904 1199112

ℎ1119904

1199112

ℎ2119904

1199112

ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119911

1

1198911119905

1199111

1198912119905

1199111

119891119896119905

) minus 119891 (119905 1199112

1198911119905

1199112

1198912119905

1199112

119891119896119905

)10038161003816100381610038161003816

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119911

1

ℎ119897119904

minus 1199112

ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119911

1

119891119897119905

minus 1199112

119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

max 11987921 1198792

2

infin

sum

119894=1

infin

sum

119906=max1198791 1198792+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816 minusmax 1205791 1205792100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

(36)

which yields that

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817ge

10038161003816100381610038161198711 minus 11987121003816100381610038161003816

1 +max 1205791 1205792gt 0 (37)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990


lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (38)

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (39)

119887119899= 1 eventually (40)




0= 119909

0119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+1205721198981198992119871

minus

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

) minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871

minus

infin

sum

119894=1

119879+2119894120591minus1

sum

119906=119879+(2119894minus1)120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

) minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(41)


isin








0+ 120591 + 120573 satisfying

120579 =1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (42)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904


(43)

119887119899= 1 forall119899 ge 119879 (44)


infin

120573by

119878119871119909119899

=

1198992119871

minus

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus119888119905] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(45)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 = 119910119899119899isinN120573

isin

119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) = 120579

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus 119871

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)])


(46)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990




1199090

= 1199090119899119899isinN120573


119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+1205721198981198992119871 minus 119887119899119909119898119899minus120591

minus

infin

sum

119906=119899

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871 minus 119887119879119909119898119879minus120591

minus

infin

sum

119906=119879

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(48)


isin








0+ 120591 + 120573 satisfying

120579 = 119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


0 le 119887119899le 119887 lt 1 forall119899 ge 119879

(49)


infin

120573by

119878119871119909119899

=

1198992119871 minus 119887119899119909119899minus120591

minus

infin

sum

119906=119899

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(50)for each 119909 = 119909

119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 11987910038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le 119887119899

100381610038161003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

(119899 minus 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 minus 120591)2

1198992

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le 1198871003817100381710038171003817119909 minus 119910

1003817100381710038171003817

+

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max ℎ2

119897119904 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 1198912

119897119905 1 le 119897 le 119896]

le [119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 120579

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119878119871119909119899

1198992

ge 119871 minus 119887119872

minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus 119887119872 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


(51)



119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14)(15) (38) (39) and





any 1199090= 1199090119899119899isinN120573


119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin








0+ 120591 + 120573 satisfying

120579 = minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (53)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


(54)



infin



119910 = 119910

119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le [minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 minus 119887119872

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 minus 119887119872 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119878119871119909119899

1198992

ge 119871 minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


(56)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882

119899119899isinN1198990

satisfying (14) (15) (38) (39) and

119887119899ge 119887 gt 1 eventually (57)



0=

1199090119899119899isinN120573


= 119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 1205721198981198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(58)



isin







0+ 120591 +

120573 satisfying

120579 =1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (59)

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873

(60)

119887119899ge 119887 gt 1 forall119899 ge 119879 (61)


infin

120573by

119878119871119909119899

=

1198992119871 minus

119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[(119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus 119888119905)] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(62)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

+1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

+1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le 119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873 le 119872

119878119871119909119899

1198992

ge 119871 minus1

119887119872

minus1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

119887119872 minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

gt 119871 minus1


1

119887119872 minus119873 ge 119873

(63)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14) (15) (38) (39) and





0+ 120591 + 120573 such that for any 119909

0= 1199090119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 120572119898 minus 1198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+120572119898 minus 119879

2119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(65)


isin







0+ 120591 +

120573 satisfying

120579 = minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (66)

minus1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt min (1 + 1

119887)119872 minus 119871 119871 minus

1

119887119872 minus119873

(67)

119887119899ge 119887 gt 1 forall119899 ge 119879 (68)


infin

120573by

119878119871119909119899

=


119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(69)

for each 119909 = 119909119899119899isinN120573


119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le minus1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

minus1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le minus1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

minus1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992


119887minus

1

1198871198992

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]



119887minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887+min (1 + 1

119887)119872 + 119871 minus119871 minus 119873 le 119872

119878119871119909119899

1198992

ge minus119871

+1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge minus119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


119887)119872 + 119871 minus119871 minus 119873 ge 119873

(70)


3 Examples



Δ3(119909119899minus 119909119899minus120591) + Δ(

sin2119909119899minus3

1198997) +

1

(1198999 + 21198995 + 1) (1 + 1199092

1198992)

=1198992minus 2119899

1198998 + 1198993 + 1 forall119899 ge 4

(71)



119887119899= minus1 119888

119899=

1198992minus 2119899

1198998 + 1198993 + 1

119891 (119899 119906) =1

(1198999 + 21198995 + 1) (1 + 1199062)

ℎ (119899 119906) =sin21199061198997

1198911119899= 1198992 119865

119899= 1198994

ℎ1119899= 119899 minus 3 119867

119899= (119899 minus 3)

2 119875

119899=2119872

1198999

119876119899=1

1198999 119877

119899=2

1198997 119882

119899=1

1198997

forall (119899 119906) isin N1198990

timesR

(72)


1

1198992

infin

sum

119905=119899+120591

1199052max 119877

119905119867119905119882119905

=1

1198992

infin

sum

119905=119899+120591

1199052max2(119905 minus 3)

2

11990571

1199057

=

infin

sum

119905=119899+120591

2(119905 minus 3)2+ 1

1199055

le2

1198992

infin

sum

119905=119899+120591

1


1

1198992

infin

sum

119905=119899+120591

1199053max 119875

119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119905=119899+120591

1199053max2119872

11990551

1199059

100381610038161003816100381610038161199052minus 2119905

10038161003816100381610038161003816

1199058 + 1199053 + 1

lemax 1 2119872

1198992

infin

sum

119905=119899+120591

1


(73)


0+120591+120573 such



119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin 119860(119873119872) of(71) with lim


where 120572119898119898isinN0



Δ3(119909119899+ 119909119899minus120591) + Δ(

sin211990931198993+1

1198993 (1198996 + 2) (1 + 1199094

21198992minus3)

)


+ 119909(119899+1)(119899+2)

)

(11989913 + 1198995 + 1) (1 + 1199092


+ 1199092

(119899+1)(119899+2))

=1198992minus ln 119899

1198996 + 1198995 + 1 forall119899 ge 5

(74)



119887119899= 1 119888

119899=

1198992minus ln 119899

1198996 + 1198995 + 1

119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)

(11989913 + 1198995 + 1) (1 + 1199062 + V2)


ℎ (119899 119906 V) =sin2V

1198993 (1198996 + 2) (1 + 1199064) 119891

1119899= 1198992minus 119899 minus 1

1198912119899= (119899 + 1) (119899 + 2)

119865119899= (119899 + 1)

2(119899 + 2)

2 ℎ

1119899= 21198992minus 3

ℎ2119899= 31198993+ 1

119867119899= (3119899

3+ 1)2

119875119899= 119876119899=

4

11989910 119877

119899= 119882119899=10

1198999

forall (119899 119906 V) isin N1198990

timesR2

(75)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

10(31199043+ 1)2

119904910

1199049

le160

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le160

1198992

infin

sum

119904=119899

1


(76)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (77)

Observe that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max4(119905 + 1)2(119905 + 2)

2

119905104

119905101199052minus ln 119905

1199056 + 1199055 + 1

le196

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054=196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

119905 minus 119906 + 1

1199054

le196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le196

1198992

infin

sum

119905=119899

1


(78)

which yields that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)



119898119898isinN0

=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899

119909119899minus120591)

+ Δ(

(minus1)119899 sin (119890minus119899

2|11990951198992minus3|)

11989915 minus radic119899 + 3

+1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|11990921198993+1|)

+(minus1)119899

1198996 (1 + 1199092

119899minus3)minus

1

(1198997 + 21198994 minus 1) (1 + 1199092

119899+4)

=3(minus1)1198991198992

911989910ln3119899 forall119899 ge 7

(80)



119887119899=1 + 3 ln 1198992 + 4 ln 119899

119888119899=3(minus1)1198991198992

911989910ln3119899

119891 (119899 119906 V) =(minus1)119899

1198996 (1 + 1199062)minus

1

(1198997 + 21198994 minus 1) (1 + V2)


2|119906|)

11989915 minus radic119899 + 3+

1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|V|

1198911119899= 119899 minus 3 119891

2119899= 119899 + 4

119865119899= (119899 + 4)

2 ℎ

1119899= 51198992minus 3

ℎ2119899= 21198993+ 1 119867

119899= (2119899

3+ 1)2

119875119899= 119876119899=3

1198996 119877

119899= 119882119899=

2

11989910

forall (119899 119906 V) isin N1198990

timesR2

(81)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

2(21199043+ 1)2

119904102

11990410

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le18

1198992

infin

sum

119904=119899

1


(82)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (83)


Observe that1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max3(119905 + 4)2

11990563

1199056

100381610038161003816100381610038161003816100381610038161003816

3(minus1)1199051199052

911990510ln3119905

100381610038161003816100381610038161003816100381610038161003816

le12

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054

le12

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le12

1198992

infin

sum

119905=119899

1


(84)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 minus 5119899

3

2 + 61198993119909119899minus120591) + Δ(

21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199092

3119899minus7))

+

sin (119899211990931198992minus2)

(radic119899 + 14)22

=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9

(86)



119887119899=1 minus 5119899

3

2 + 61198993 119888

119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7

119891 (119899 119906) =

sin (1198992119906)

(radic119899 + 14)22

ℎ (119899 119906) =21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199062)

1198911119899= 31198992minus 2 119865

119899= (3119899

2minus 2)2

ℎ1119899= 3119899 minus 7

119867119899= (3119899 minus 7)

2 119875

119899= 119876119899=3

1198999 119877119899= 119882119899=1

1198995

forall (119899 119906) isin N1198990

timesR

(87)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max(3119904 minus 7)2

11990451

1199045

le9

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le9

1198992

infin

sum

119904=119899

1


(88)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (89)

Notice that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max

3(31199052minus 2)2

11990593

1199059

100381610038161003816100381610038161003816100381610038161003816

(minus1)1199051199053+ 51199052+ 4119905 minus 2

1199059 + 1199058 + 21199055 + 1199053 + 7

100381610038161003816100381610038161003816100381610038161003816

le27

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le27

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le27

1198992

infin

sum

119905=119899

1


(90)

which gives that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)




= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin







Δ3(119909119899+ (

120587

2+ 119899 sin 1

119899) 119909119899minus120591)

+ Δ((minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119909

2119899+1)))

+119899 sin (119899119909

119899minus2)

2 + (119899 + 5)16

=


forall119899 ge 3

(92)



119887119899=120587

2+ 119899 sin 1

119899 119888

119899=


119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)

16

ℎ (119899 119906) =(minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119906))

1198911119899= 119899 minus 2 119865

119899= (119899 minus 2)

2

ℎ1119899= 2119899 + 1 119867

119899= (2119899 + 1)

2

119875119899= 119876119899=

1

11989914 119877119899= 119882119899=2

1198999 forall (119899 119906) isin N

1198990

timesR

(93)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max2(2119904 + 1)2

11990492

1199049

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199047le18

1198992

infin

sum

119904=119899

1


(94)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


119905141

11990514

10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

11990514le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

11990513

le1

1198992

infin

sum

119905=119899

1


(95)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899minus21198995+ 91198992minus 1

1198995 + 31198992 + 2119909119899minus120591) + Δ(

cos ((minus1)119899119890119899)

(119899 + 7)6radic1 +

1003816100381610038161003816119909119899minus21003816100381610038161003816

)

+

sin (1198992119909119899minus1)

1198999 + 31198995 + 21198992 + 1


11989911 + 61198993 + 7119899 + 2 forall119899 ge 6

(97)



119887119899= minus

21198995+ 91198992minus 1

1198995 + 31198992 + 2


11989911 + 61198993 + 7119899 + 2

119891 (119899 119906) =

sin (1198992119906)1198999 + 31198995 + 21198992 + 1


ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816


11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)


0+ 120591 + 120573 such that


= 119909119898119899119899isinN120573

119898isinN0


119911119899119899isinN120573


119911119899= +infin and





Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119888



119911119888

119899119899isinN120573


119899rarrinfin119911119888


119911119888

119899= 1198992119871119888

+

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

ℎ (119904 119911119888

ℎ1119904

119911119888

ℎ2119904

119911119888

ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 119911119888

1198911119905

119911119888

1198912119905

119911119888

119891119896119905

) minus 119888119905]

forall119899 ge 119879119888

(35)


max1198791 1198792

100381610038161003816100381610038161003816100381610038161003816

1199111

119899

1198992minus1199112

119899

1198992

100381610038161003816100381610038161003816100381610038161003816

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119911

1

ℎ1119904

1199111

ℎ2119904

1199111

ℎ119896119904

)

minus ℎ (119904 1199112

ℎ1119904

1199112

ℎ2119904

1199112

ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119911

1

1198911119905

1199111

1198912119905

1199111

119891119896119905

) minus 119891 (119905 1199112

1198911119905

1199112

1198912119905

1199112

119891119896119905

)10038161003816100381610038161003816

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus1

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119911

1

ℎ119897119904

minus 1199112

ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119911

1

119891119897119905

minus 1199112

119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

1198992

infin

sum

119894=1

infin

sum

119906=119899+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816

minus

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

max 11987921 1198792

2

infin

sum

119894=1

infin

sum

119906=max1198791 1198792+119894120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

ge10038161003816100381610038161198711 minus 1198712

1003816100381610038161003816 minusmax 1205791 1205792100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817

(36)

which yields that

100381710038171003817100381710038171199111minus 119911210038171003817100381710038171003817ge

10038161003816100381610038161198711 minus 11987121003816100381610038161003816

1 +max 1205791 1205792gt 0 (37)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990


lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (38)

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (39)

119887119899= 1 eventually (40)




0= 119909

0119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+1205721198981198992119871

minus

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

) minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871

minus

infin

sum

119894=1

119879+2119894120591minus1

sum

119906=119879+(2119894minus1)120591

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

) minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(41)


isin








0+ 120591 + 120573 satisfying

120579 =1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (42)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904


(43)

119887119899= 1 forall119899 ge 119879 (44)


infin

120573by

119878119871119909119899

=

1198992119871

minus

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus119888119905] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(45)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 = 119910119899119899isinN120573

isin

119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) = 120579

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus 119871

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)])


(46)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990




1199090

= 1199090119899119899isinN120573


119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+1205721198981198992119871 minus 119887119899119909119898119899minus120591

minus

infin

sum

119906=119899

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871 minus 119887119879119909119898119879minus120591

minus

infin

sum

119906=119879

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(48)


isin








0+ 120591 + 120573 satisfying

120579 = 119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


0 le 119887119899le 119887 lt 1 forall119899 ge 119879

(49)


infin

120573by

119878119871119909119899

=

1198992119871 minus 119887119899119909119899minus120591

minus

infin

sum

119906=119899

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(50)for each 119909 = 119909

119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 11987910038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le 119887119899

100381610038161003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

(119899 minus 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 minus 120591)2

1198992

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le 1198871003817100381710038171003817119909 minus 119910

1003817100381710038171003817

+

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max ℎ2

119897119904 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 1198912

119897119905 1 le 119897 le 119896]

le [119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 120579

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119878119871119909119899

1198992

ge 119871 minus 119887119872

minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus 119887119872 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


(51)



119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14)(15) (38) (39) and





any 1199090= 1199090119899119899isinN120573


119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin








0+ 120591 + 120573 satisfying

120579 = minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (53)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


(54)



infin



119910 = 119910

119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le [minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 minus 119887119872

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 minus 119887119872 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119878119871119909119899

1198992

ge 119871 minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


(56)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882

119899119899isinN1198990

satisfying (14) (15) (38) (39) and

119887119899ge 119887 gt 1 eventually (57)



0=

1199090119899119899isinN120573


= 119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 1205721198981198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(58)



isin







0+ 120591 +

120573 satisfying

120579 =1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (59)

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873

(60)

119887119899ge 119887 gt 1 forall119899 ge 119879 (61)


infin

120573by

119878119871119909119899

=

1198992119871 minus

119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[(119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus 119888119905)] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(62)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

+1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

+1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le 119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873 le 119872

119878119871119909119899

1198992

ge 119871 minus1

119887119872

minus1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

119887119872 minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

gt 119871 minus1


1

119887119872 minus119873 ge 119873

(63)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14) (15) (38) (39) and





0+ 120591 + 120573 such that for any 119909

0= 1199090119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 120572119898 minus 1198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+120572119898 minus 119879

2119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(65)


isin







0+ 120591 +

120573 satisfying

120579 = minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (66)

minus1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt min (1 + 1

119887)119872 minus 119871 119871 minus

1

119887119872 minus119873

(67)

119887119899ge 119887 gt 1 forall119899 ge 119879 (68)


infin

120573by

119878119871119909119899

=


119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(69)

for each 119909 = 119909119899119899isinN120573


119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le minus1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

minus1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le minus1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

minus1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992


119887minus

1

1198871198992

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]



119887minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887+min (1 + 1

119887)119872 + 119871 minus119871 minus 119873 le 119872

119878119871119909119899

1198992

ge minus119871

+1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge minus119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


119887)119872 + 119871 minus119871 minus 119873 ge 119873

(70)


3 Examples



Δ3(119909119899minus 119909119899minus120591) + Δ(

sin2119909119899minus3

1198997) +

1

(1198999 + 21198995 + 1) (1 + 1199092

1198992)

=1198992minus 2119899

1198998 + 1198993 + 1 forall119899 ge 4

(71)



119887119899= minus1 119888

119899=

1198992minus 2119899

1198998 + 1198993 + 1

119891 (119899 119906) =1

(1198999 + 21198995 + 1) (1 + 1199062)

ℎ (119899 119906) =sin21199061198997

1198911119899= 1198992 119865

119899= 1198994

ℎ1119899= 119899 minus 3 119867

119899= (119899 minus 3)

2 119875

119899=2119872

1198999

119876119899=1

1198999 119877

119899=2

1198997 119882

119899=1

1198997

forall (119899 119906) isin N1198990

timesR

(72)


1

1198992

infin

sum

119905=119899+120591

1199052max 119877

119905119867119905119882119905

=1

1198992

infin

sum

119905=119899+120591

1199052max2(119905 minus 3)

2

11990571

1199057

=

infin

sum

119905=119899+120591

2(119905 minus 3)2+ 1

1199055

le2

1198992

infin

sum

119905=119899+120591

1


1

1198992

infin

sum

119905=119899+120591

1199053max 119875

119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119905=119899+120591

1199053max2119872

11990551

1199059

100381610038161003816100381610038161199052minus 2119905

10038161003816100381610038161003816

1199058 + 1199053 + 1

lemax 1 2119872

1198992

infin

sum

119905=119899+120591

1


(73)


0+120591+120573 such



119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin 119860(119873119872) of(71) with lim


where 120572119898119898isinN0



Δ3(119909119899+ 119909119899minus120591) + Δ(

sin211990931198993+1

1198993 (1198996 + 2) (1 + 1199094

21198992minus3)

)


+ 119909(119899+1)(119899+2)

)

(11989913 + 1198995 + 1) (1 + 1199092


+ 1199092

(119899+1)(119899+2))

=1198992minus ln 119899

1198996 + 1198995 + 1 forall119899 ge 5

(74)



119887119899= 1 119888

119899=

1198992minus ln 119899

1198996 + 1198995 + 1

119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)

(11989913 + 1198995 + 1) (1 + 1199062 + V2)


ℎ (119899 119906 V) =sin2V

1198993 (1198996 + 2) (1 + 1199064) 119891

1119899= 1198992minus 119899 minus 1

1198912119899= (119899 + 1) (119899 + 2)

119865119899= (119899 + 1)

2(119899 + 2)

2 ℎ

1119899= 21198992minus 3

ℎ2119899= 31198993+ 1

119867119899= (3119899

3+ 1)2

119875119899= 119876119899=

4

11989910 119877

119899= 119882119899=10

1198999

forall (119899 119906 V) isin N1198990

timesR2

(75)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

10(31199043+ 1)2

119904910

1199049

le160

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le160

1198992

infin

sum

119904=119899

1


(76)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (77)

Observe that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max4(119905 + 1)2(119905 + 2)

2

119905104

119905101199052minus ln 119905

1199056 + 1199055 + 1

le196

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054=196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

119905 minus 119906 + 1

1199054

le196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le196

1198992

infin

sum

119905=119899

1


(78)

which yields that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)



119898119898isinN0

=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899

119909119899minus120591)

+ Δ(

(minus1)119899 sin (119890minus119899

2|11990951198992minus3|)

11989915 minus radic119899 + 3

+1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|11990921198993+1|)

+(minus1)119899

1198996 (1 + 1199092

119899minus3)minus

1

(1198997 + 21198994 minus 1) (1 + 1199092

119899+4)

=3(minus1)1198991198992

911989910ln3119899 forall119899 ge 7

(80)



119887119899=1 + 3 ln 1198992 + 4 ln 119899

119888119899=3(minus1)1198991198992

911989910ln3119899

119891 (119899 119906 V) =(minus1)119899

1198996 (1 + 1199062)minus

1

(1198997 + 21198994 minus 1) (1 + V2)


2|119906|)

11989915 minus radic119899 + 3+

1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|V|

1198911119899= 119899 minus 3 119891

2119899= 119899 + 4

119865119899= (119899 + 4)

2 ℎ

1119899= 51198992minus 3

ℎ2119899= 21198993+ 1 119867

119899= (2119899

3+ 1)2

119875119899= 119876119899=3

1198996 119877

119899= 119882119899=

2

11989910

forall (119899 119906 V) isin N1198990

timesR2

(81)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

2(21199043+ 1)2

119904102

11990410

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le18

1198992

infin

sum

119904=119899

1


(82)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (83)


Observe that1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max3(119905 + 4)2

11990563

1199056

100381610038161003816100381610038161003816100381610038161003816

3(minus1)1199051199052

911990510ln3119905

100381610038161003816100381610038161003816100381610038161003816

le12

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054

le12

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le12

1198992

infin

sum

119905=119899

1


(84)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 minus 5119899

3

2 + 61198993119909119899minus120591) + Δ(

21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199092

3119899minus7))

+

sin (119899211990931198992minus2)

(radic119899 + 14)22

=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9

(86)



119887119899=1 minus 5119899

3

2 + 61198993 119888

119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7

119891 (119899 119906) =

sin (1198992119906)

(radic119899 + 14)22

ℎ (119899 119906) =21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199062)

1198911119899= 31198992minus 2 119865

119899= (3119899

2minus 2)2

ℎ1119899= 3119899 minus 7

119867119899= (3119899 minus 7)

2 119875

119899= 119876119899=3

1198999 119877119899= 119882119899=1

1198995

forall (119899 119906) isin N1198990

timesR

(87)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max(3119904 minus 7)2

11990451

1199045

le9

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le9

1198992

infin

sum

119904=119899

1


(88)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (89)

Notice that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max

3(31199052minus 2)2

11990593

1199059

100381610038161003816100381610038161003816100381610038161003816

(minus1)1199051199053+ 51199052+ 4119905 minus 2

1199059 + 1199058 + 21199055 + 1199053 + 7

100381610038161003816100381610038161003816100381610038161003816

le27

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le27

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le27

1198992

infin

sum

119905=119899

1


(90)

which gives that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)




= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin







Δ3(119909119899+ (

120587

2+ 119899 sin 1

119899) 119909119899minus120591)

+ Δ((minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119909

2119899+1)))

+119899 sin (119899119909

119899minus2)

2 + (119899 + 5)16

=


forall119899 ge 3

(92)



119887119899=120587

2+ 119899 sin 1

119899 119888

119899=


119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)

16

ℎ (119899 119906) =(minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119906))

1198911119899= 119899 minus 2 119865

119899= (119899 minus 2)

2

ℎ1119899= 2119899 + 1 119867

119899= (2119899 + 1)

2

119875119899= 119876119899=

1

11989914 119877119899= 119882119899=2

1198999 forall (119899 119906) isin N

1198990

timesR

(93)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max2(2119904 + 1)2

11990492

1199049

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199047le18

1198992

infin

sum

119904=119899

1


(94)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


119905141

11990514

10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

11990514le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

11990513

le1

1198992

infin

sum

119905=119899

1


(95)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899minus21198995+ 91198992minus 1

1198995 + 31198992 + 2119909119899minus120591) + Δ(

cos ((minus1)119899119890119899)

(119899 + 7)6radic1 +

1003816100381610038161003816119909119899minus21003816100381610038161003816

)

+

sin (1198992119909119899minus1)

1198999 + 31198995 + 21198992 + 1


11989911 + 61198993 + 7119899 + 2 forall119899 ge 6

(97)



119887119899= minus

21198995+ 91198992minus 1

1198995 + 31198992 + 2


11989911 + 61198993 + 7119899 + 2

119891 (119899 119906) =

sin (1198992119906)1198999 + 31198995 + 21198992 + 1


ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816


11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)


0+ 120591 + 120573 such that


= 119909119898119899119899isinN120573

119898isinN0


119911119899119899isinN120573


119911119899= +infin and





Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0+ 120591 + 120573 satisfying

120579 =1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (42)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904


(43)

119887119899= 1 forall119899 ge 119879 (44)


infin

120573by

119878119871119909119899

=

1198992119871

minus

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus119888119905] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(45)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 = 119910119899119899isinN120573

isin

119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) = 120579

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus 119871

10038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119894=1

119899+2119894120591minus1

sum

119906=119899+(2119894minus1)120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)])


(46)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990




1199090

= 1199090119899119899isinN120573


119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+1205721198981198992119871 minus 119887119899119909119898119899minus120591

minus

infin

sum

119906=119899

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871 minus 119887119879119909119898119879minus120591

minus

infin

sum

119906=119879

infin

sum

119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum

119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(48)


isin








0+ 120591 + 120573 satisfying

120579 = 119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


0 le 119887119899le 119887 lt 1 forall119899 ge 119879

(49)


infin

120573by

119878119871119909119899

=

1198992119871 minus 119887119899119909119899minus120591

minus

infin

sum

119906=119899

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(50)for each 119909 = 119909

119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 11987910038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le 119887119899

100381610038161003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

(119899 minus 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 minus 120591)2

1198992

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le 1198871003817100381710038171003817119909 minus 119910

1003817100381710038171003817

+

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max ℎ2

119897119904 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 1198912

119897119905 1 le 119897 le 119896]

le [119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 120579

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119878119871119909119899

1198992

ge 119871 minus 119887119872

minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus 119887119872 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


(51)



119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14)(15) (38) (39) and





any 1199090= 1199090119899119899isinN120573


119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin








0+ 120591 + 120573 satisfying

120579 = minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (53)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


(54)



infin



119910 = 119910

119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le [minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 minus 119887119872

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 minus 119887119872 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119878119871119909119899

1198992

ge 119871 minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


(56)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882

119899119899isinN1198990

satisfying (14) (15) (38) (39) and

119887119899ge 119887 gt 1 eventually (57)



0=

1199090119899119899isinN120573


= 119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 1205721198981198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(58)



isin







0+ 120591 +

120573 satisfying

120579 =1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (59)

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873

(60)

119887119899ge 119887 gt 1 forall119899 ge 119879 (61)


infin

120573by

119878119871119909119899

=

1198992119871 minus

119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[(119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus 119888119905)] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(62)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

+1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

+1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le 119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873 le 119872

119878119871119909119899

1198992

ge 119871 minus1

119887119872

minus1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

119887119872 minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

gt 119871 minus1


1

119887119872 minus119873 ge 119873

(63)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14) (15) (38) (39) and





0+ 120591 + 120573 such that for any 119909

0= 1199090119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 120572119898 minus 1198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+120572119898 minus 119879

2119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(65)


isin







0+ 120591 +

120573 satisfying

120579 = minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (66)

minus1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt min (1 + 1

119887)119872 minus 119871 119871 minus

1

119887119872 minus119873

(67)

119887119899ge 119887 gt 1 forall119899 ge 119879 (68)


infin

120573by

119878119871119909119899

=


119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(69)

for each 119909 = 119909119899119899isinN120573


119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le minus1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

minus1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le minus1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

minus1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992


119887minus

1

1198871198992

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]



119887minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887+min (1 + 1

119887)119872 + 119871 minus119871 minus 119873 le 119872

119878119871119909119899

1198992

ge minus119871

+1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge minus119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


119887)119872 + 119871 minus119871 minus 119873 ge 119873

(70)


3 Examples



Δ3(119909119899minus 119909119899minus120591) + Δ(

sin2119909119899minus3

1198997) +

1

(1198999 + 21198995 + 1) (1 + 1199092

1198992)

=1198992minus 2119899

1198998 + 1198993 + 1 forall119899 ge 4

(71)



119887119899= minus1 119888

119899=

1198992minus 2119899

1198998 + 1198993 + 1

119891 (119899 119906) =1

(1198999 + 21198995 + 1) (1 + 1199062)

ℎ (119899 119906) =sin21199061198997

1198911119899= 1198992 119865

119899= 1198994

ℎ1119899= 119899 minus 3 119867

119899= (119899 minus 3)

2 119875

119899=2119872

1198999

119876119899=1

1198999 119877

119899=2

1198997 119882

119899=1

1198997

forall (119899 119906) isin N1198990

timesR

(72)


1

1198992

infin

sum

119905=119899+120591

1199052max 119877

119905119867119905119882119905

=1

1198992

infin

sum

119905=119899+120591

1199052max2(119905 minus 3)

2

11990571

1199057

=

infin

sum

119905=119899+120591

2(119905 minus 3)2+ 1

1199055

le2

1198992

infin

sum

119905=119899+120591

1


1

1198992

infin

sum

119905=119899+120591

1199053max 119875

119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119905=119899+120591

1199053max2119872

11990551

1199059

100381610038161003816100381610038161199052minus 2119905

10038161003816100381610038161003816

1199058 + 1199053 + 1

lemax 1 2119872

1198992

infin

sum

119905=119899+120591

1


(73)


0+120591+120573 such



119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin 119860(119873119872) of(71) with lim


where 120572119898119898isinN0



Δ3(119909119899+ 119909119899minus120591) + Δ(

sin211990931198993+1

1198993 (1198996 + 2) (1 + 1199094

21198992minus3)

)


+ 119909(119899+1)(119899+2)

)

(11989913 + 1198995 + 1) (1 + 1199092


+ 1199092

(119899+1)(119899+2))

=1198992minus ln 119899

1198996 + 1198995 + 1 forall119899 ge 5

(74)



119887119899= 1 119888

119899=

1198992minus ln 119899

1198996 + 1198995 + 1

119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)

(11989913 + 1198995 + 1) (1 + 1199062 + V2)


ℎ (119899 119906 V) =sin2V

1198993 (1198996 + 2) (1 + 1199064) 119891

1119899= 1198992minus 119899 minus 1

1198912119899= (119899 + 1) (119899 + 2)

119865119899= (119899 + 1)

2(119899 + 2)

2 ℎ

1119899= 21198992minus 3

ℎ2119899= 31198993+ 1

119867119899= (3119899

3+ 1)2

119875119899= 119876119899=

4

11989910 119877

119899= 119882119899=10

1198999

forall (119899 119906 V) isin N1198990

timesR2

(75)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

10(31199043+ 1)2

119904910

1199049

le160

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le160

1198992

infin

sum

119904=119899

1


(76)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (77)

Observe that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max4(119905 + 1)2(119905 + 2)

2

119905104

119905101199052minus ln 119905

1199056 + 1199055 + 1

le196

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054=196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

119905 minus 119906 + 1

1199054

le196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le196

1198992

infin

sum

119905=119899

1


(78)

which yields that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)



119898119898isinN0

=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899

119909119899minus120591)

+ Δ(

(minus1)119899 sin (119890minus119899

2|11990951198992minus3|)

11989915 minus radic119899 + 3

+1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|11990921198993+1|)

+(minus1)119899

1198996 (1 + 1199092

119899minus3)minus

1

(1198997 + 21198994 minus 1) (1 + 1199092

119899+4)

=3(minus1)1198991198992

911989910ln3119899 forall119899 ge 7

(80)



119887119899=1 + 3 ln 1198992 + 4 ln 119899

119888119899=3(minus1)1198991198992

911989910ln3119899

119891 (119899 119906 V) =(minus1)119899

1198996 (1 + 1199062)minus

1

(1198997 + 21198994 minus 1) (1 + V2)


2|119906|)

11989915 minus radic119899 + 3+

1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|V|

1198911119899= 119899 minus 3 119891

2119899= 119899 + 4

119865119899= (119899 + 4)

2 ℎ

1119899= 51198992minus 3

ℎ2119899= 21198993+ 1 119867

119899= (2119899

3+ 1)2

119875119899= 119876119899=3

1198996 119877

119899= 119882119899=

2

11989910

forall (119899 119906 V) isin N1198990

timesR2

(81)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

2(21199043+ 1)2

119904102

11990410

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le18

1198992

infin

sum

119904=119899

1


(82)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (83)


Observe that1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max3(119905 + 4)2

11990563

1199056

100381610038161003816100381610038161003816100381610038161003816

3(minus1)1199051199052

911990510ln3119905

100381610038161003816100381610038161003816100381610038161003816

le12

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054

le12

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le12

1198992

infin

sum

119905=119899

1


(84)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 minus 5119899

3

2 + 61198993119909119899minus120591) + Δ(

21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199092

3119899minus7))

+

sin (119899211990931198992minus2)

(radic119899 + 14)22

=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9

(86)



119887119899=1 minus 5119899

3

2 + 61198993 119888

119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7

119891 (119899 119906) =

sin (1198992119906)

(radic119899 + 14)22

ℎ (119899 119906) =21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199062)

1198911119899= 31198992minus 2 119865

119899= (3119899

2minus 2)2

ℎ1119899= 3119899 minus 7

119867119899= (3119899 minus 7)

2 119875

119899= 119876119899=3

1198999 119877119899= 119882119899=1

1198995

forall (119899 119906) isin N1198990

timesR

(87)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max(3119904 minus 7)2

11990451

1199045

le9

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le9

1198992

infin

sum

119904=119899

1


(88)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (89)

Notice that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max

3(31199052minus 2)2

11990593

1199059

100381610038161003816100381610038161003816100381610038161003816

(minus1)1199051199053+ 51199052+ 4119905 minus 2

1199059 + 1199058 + 21199055 + 1199053 + 7

100381610038161003816100381610038161003816100381610038161003816

le27

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le27

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le27

1198992

infin

sum

119905=119899

1


(90)

which gives that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)




= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin







Δ3(119909119899+ (

120587

2+ 119899 sin 1

119899) 119909119899minus120591)

+ Δ((minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119909

2119899+1)))

+119899 sin (119899119909

119899minus2)

2 + (119899 + 5)16

=


forall119899 ge 3

(92)



119887119899=120587

2+ 119899 sin 1

119899 119888

119899=


119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)

16

ℎ (119899 119906) =(minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119906))

1198911119899= 119899 minus 2 119865

119899= (119899 minus 2)

2

ℎ1119899= 2119899 + 1 119867

119899= (2119899 + 1)

2

119875119899= 119876119899=

1

11989914 119877119899= 119882119899=2

1198999 forall (119899 119906) isin N

1198990

timesR

(93)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max2(2119904 + 1)2

11990492

1199049

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199047le18

1198992

infin

sum

119904=119899

1


(94)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


119905141

11990514

10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

11990514le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

11990513

le1

1198992

infin

sum

119905=119899

1


(95)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899minus21198995+ 91198992minus 1

1198995 + 31198992 + 2119909119899minus120591) + Δ(

cos ((minus1)119899119890119899)

(119899 + 7)6radic1 +

1003816100381610038161003816119909119899minus21003816100381610038161003816

)

+

sin (1198992119909119899minus1)

1198999 + 31198995 + 21198992 + 1


11989911 + 61198993 + 7119899 + 2 forall119899 ge 6

(97)



119887119899= minus

21198995+ 91198992minus 1

1198995 + 31198992 + 2


11989911 + 61198993 + 7119899 + 2

119891 (119899 119906) =

sin (1198992119906)1198999 + 31198995 + 21198992 + 1


ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816


11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)


0+ 120591 + 120573 such that


= 119909119898119899119899isinN120573

119898isinN0


119911119899119899isinN120573


119911119899= +infin and





Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














0+ 120591 + 120573 satisfying

120579 = 119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


0 le 119887119899le 119887 lt 1 forall119899 ge 119879

(49)


infin

120573by

119878119871119909119899

=

1198992119871 minus 119887119899119909119899minus120591

minus

infin

sum

119906=119899

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(50)for each 119909 = 119909

119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 11987910038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le 119887119899

100381610038161003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

(119899 minus 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 minus 120591)2

1198992

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le 1198871003817100381710038171003817119909 minus 119910

1003817100381710038171003817

+

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max ℎ2

119897119904 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 1198912

119897119905 1 le 119897 le 119896]

le [119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 120579

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119878119871119909119899

1198992

ge 119871 minus 119887119872

minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus 119887119872 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


(51)



119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14)(15) (38) (39) and





any 1199090= 1199090119899119899isinN120573


119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin








0+ 120591 + 120573 satisfying

120579 = minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (53)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


(54)



infin



119910 = 119910

119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le [minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 minus 119887119872

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 minus 119887119872 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119878119871119909119899

1198992

ge 119871 minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


(56)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882

119899119899isinN1198990

satisfying (14) (15) (38) (39) and

119887119899ge 119887 gt 1 eventually (57)



0=

1199090119899119899isinN120573


= 119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 1205721198981198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(58)



isin







0+ 120591 +

120573 satisfying

120579 =1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (59)

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873

(60)

119887119899ge 119887 gt 1 forall119899 ge 119879 (61)


infin

120573by

119878119871119909119899

=

1198992119871 minus

119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[(119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus 119888119905)] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(62)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

+1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

+1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le 119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873 le 119872

119878119871119909119899

1198992

ge 119871 minus1

119887119872

minus1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

119887119872 minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

gt 119871 minus1


1

119887119872 minus119873 ge 119873

(63)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14) (15) (38) (39) and





0+ 120591 + 120573 such that for any 119909

0= 1199090119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 120572119898 minus 1198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+120572119898 minus 119879

2119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(65)


isin







0+ 120591 +

120573 satisfying

120579 = minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (66)

minus1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt min (1 + 1

119887)119872 minus 119871 119871 minus

1

119887119872 minus119873

(67)

119887119899ge 119887 gt 1 forall119899 ge 119879 (68)


infin

120573by

119878119871119909119899

=


119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(69)

for each 119909 = 119909119899119899isinN120573


119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le minus1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

minus1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le minus1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

minus1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992


119887minus

1

1198871198992

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]



119887minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887+min (1 + 1

119887)119872 + 119871 minus119871 minus 119873 le 119872

119878119871119909119899

1198992

ge minus119871

+1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge minus119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


119887)119872 + 119871 minus119871 minus 119873 ge 119873

(70)


3 Examples



Δ3(119909119899minus 119909119899minus120591) + Δ(

sin2119909119899minus3

1198997) +

1

(1198999 + 21198995 + 1) (1 + 1199092

1198992)

=1198992minus 2119899

1198998 + 1198993 + 1 forall119899 ge 4

(71)



119887119899= minus1 119888

119899=

1198992minus 2119899

1198998 + 1198993 + 1

119891 (119899 119906) =1

(1198999 + 21198995 + 1) (1 + 1199062)

ℎ (119899 119906) =sin21199061198997

1198911119899= 1198992 119865

119899= 1198994

ℎ1119899= 119899 minus 3 119867

119899= (119899 minus 3)

2 119875

119899=2119872

1198999

119876119899=1

1198999 119877

119899=2

1198997 119882

119899=1

1198997

forall (119899 119906) isin N1198990

timesR

(72)


1

1198992

infin

sum

119905=119899+120591

1199052max 119877

119905119867119905119882119905

=1

1198992

infin

sum

119905=119899+120591

1199052max2(119905 minus 3)

2

11990571

1199057

=

infin

sum

119905=119899+120591

2(119905 minus 3)2+ 1

1199055

le2

1198992

infin

sum

119905=119899+120591

1


1

1198992

infin

sum

119905=119899+120591

1199053max 119875

119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119905=119899+120591

1199053max2119872

11990551

1199059

100381610038161003816100381610038161199052minus 2119905

10038161003816100381610038161003816

1199058 + 1199053 + 1

lemax 1 2119872

1198992

infin

sum

119905=119899+120591

1


(73)


0+120591+120573 such



119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin 119860(119873119872) of(71) with lim


where 120572119898119898isinN0



Δ3(119909119899+ 119909119899minus120591) + Δ(

sin211990931198993+1

1198993 (1198996 + 2) (1 + 1199094

21198992minus3)

)


+ 119909(119899+1)(119899+2)

)

(11989913 + 1198995 + 1) (1 + 1199092


+ 1199092

(119899+1)(119899+2))

=1198992minus ln 119899

1198996 + 1198995 + 1 forall119899 ge 5

(74)



119887119899= 1 119888

119899=

1198992minus ln 119899

1198996 + 1198995 + 1

119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)

(11989913 + 1198995 + 1) (1 + 1199062 + V2)


ℎ (119899 119906 V) =sin2V

1198993 (1198996 + 2) (1 + 1199064) 119891

1119899= 1198992minus 119899 minus 1

1198912119899= (119899 + 1) (119899 + 2)

119865119899= (119899 + 1)

2(119899 + 2)

2 ℎ

1119899= 21198992minus 3

ℎ2119899= 31198993+ 1

119867119899= (3119899

3+ 1)2

119875119899= 119876119899=

4

11989910 119877

119899= 119882119899=10

1198999

forall (119899 119906 V) isin N1198990

timesR2

(75)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

10(31199043+ 1)2

119904910

1199049

le160

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le160

1198992

infin

sum

119904=119899

1


(76)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (77)

Observe that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max4(119905 + 1)2(119905 + 2)

2

119905104

119905101199052minus ln 119905

1199056 + 1199055 + 1

le196

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054=196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

119905 minus 119906 + 1

1199054

le196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le196

1198992

infin

sum

119905=119899

1


(78)

which yields that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)



119898119898isinN0

=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899

119909119899minus120591)

+ Δ(

(minus1)119899 sin (119890minus119899

2|11990951198992minus3|)

11989915 minus radic119899 + 3

+1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|11990921198993+1|)

+(minus1)119899

1198996 (1 + 1199092

119899minus3)minus

1

(1198997 + 21198994 minus 1) (1 + 1199092

119899+4)

=3(minus1)1198991198992

911989910ln3119899 forall119899 ge 7

(80)



119887119899=1 + 3 ln 1198992 + 4 ln 119899

119888119899=3(minus1)1198991198992

911989910ln3119899

119891 (119899 119906 V) =(minus1)119899

1198996 (1 + 1199062)minus

1

(1198997 + 21198994 minus 1) (1 + V2)


2|119906|)

11989915 minus radic119899 + 3+

1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|V|

1198911119899= 119899 minus 3 119891

2119899= 119899 + 4

119865119899= (119899 + 4)

2 ℎ

1119899= 51198992minus 3

ℎ2119899= 21198993+ 1 119867

119899= (2119899

3+ 1)2

119875119899= 119876119899=3

1198996 119877

119899= 119882119899=

2

11989910

forall (119899 119906 V) isin N1198990

timesR2

(81)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

2(21199043+ 1)2

119904102

11990410

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le18

1198992

infin

sum

119904=119899

1


(82)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (83)


Observe that1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max3(119905 + 4)2

11990563

1199056

100381610038161003816100381610038161003816100381610038161003816

3(minus1)1199051199052

911990510ln3119905

100381610038161003816100381610038161003816100381610038161003816

le12

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054

le12

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le12

1198992

infin

sum

119905=119899

1


(84)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 minus 5119899

3

2 + 61198993119909119899minus120591) + Δ(

21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199092

3119899minus7))

+

sin (119899211990931198992minus2)

(radic119899 + 14)22

=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9

(86)



119887119899=1 minus 5119899

3

2 + 61198993 119888

119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7

119891 (119899 119906) =

sin (1198992119906)

(radic119899 + 14)22

ℎ (119899 119906) =21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199062)

1198911119899= 31198992minus 2 119865

119899= (3119899

2minus 2)2

ℎ1119899= 3119899 minus 7

119867119899= (3119899 minus 7)

2 119875

119899= 119876119899=3

1198999 119877119899= 119882119899=1

1198995

forall (119899 119906) isin N1198990

timesR

(87)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max(3119904 minus 7)2

11990451

1199045

le9

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le9

1198992

infin

sum

119904=119899

1


(88)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (89)

Notice that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max

3(31199052minus 2)2

11990593

1199059

100381610038161003816100381610038161003816100381610038161003816

(minus1)1199051199053+ 51199052+ 4119905 minus 2

1199059 + 1199058 + 21199055 + 1199053 + 7

100381610038161003816100381610038161003816100381610038161003816

le27

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le27

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le27

1198992

infin

sum

119905=119899

1


(90)

which gives that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)




= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin







Δ3(119909119899+ (

120587

2+ 119899 sin 1

119899) 119909119899minus120591)

+ Δ((minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119909

2119899+1)))

+119899 sin (119899119909

119899minus2)

2 + (119899 + 5)16

=


forall119899 ge 3

(92)



119887119899=120587

2+ 119899 sin 1

119899 119888

119899=


119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)

16

ℎ (119899 119906) =(minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119906))

1198911119899= 119899 minus 2 119865

119899= (119899 minus 2)

2

ℎ1119899= 2119899 + 1 119867

119899= (2119899 + 1)

2

119875119899= 119876119899=

1

11989914 119877119899= 119882119899=2

1198999 forall (119899 119906) isin N

1198990

timesR

(93)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max2(2119904 + 1)2

11990492

1199049

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199047le18

1198992

infin

sum

119904=119899

1


(94)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


119905141

11990514

10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

11990514le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

11990513

le1

1198992

infin

sum

119905=119899

1


(95)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899minus21198995+ 91198992minus 1

1198995 + 31198992 + 2119909119899minus120591) + Δ(

cos ((minus1)119899119890119899)

(119899 + 7)6radic1 +

1003816100381610038161003816119909119899minus21003816100381610038161003816

)

+

sin (1198992119909119899minus1)

1198999 + 31198995 + 21198992 + 1


11989911 + 61198993 + 7119899 + 2 forall119899 ge 6

(97)



119887119899= minus

21198995+ 91198992minus 1

1198995 + 31198992 + 2


11989911 + 61198993 + 7119899 + 2

119891 (119899 119906) =

sin (1198992119906)1198999 + 31198995 + 21198992 + 1


ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816


11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)


0+ 120591 + 120573 such that


= 119909119898119899119899isinN120573

119898isinN0


119911119899119899isinN120573


119911119899= +infin and





Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















0+ 120591 + 120573 satisfying

120579 = minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905) (53)

1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


(54)



infin



119910 = 119910

119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le 119887119899

1003816100381610038161003816100381610038161003816

119909119899minus120591

minus 119910119899minus120591

1198992

1003816100381610038161003816100381610038161003816

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le [minus119887 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 minus 119887119872

+1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

le 119871 minus 119887119872 +1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119878119871119909119899

1198992

ge 119871 minus1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


(56)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882

119899119899isinN1198990

satisfying (14) (15) (38) (39) and

119887119899ge 119887 gt 1 eventually (57)



0=

1199090119899119899isinN120573


= 119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 1205721198981198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+1205721198981198792119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(58)



isin







0+ 120591 +

120573 satisfying

120579 =1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (59)

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873

(60)

119887119899ge 119887 gt 1 forall119899 ge 119879 (61)


infin

120573by

119878119871119909119899

=

1198992119871 minus

119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[(119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus 119888119905)] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(62)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

+1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

+1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le 119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873 le 119872

119878119871119909119899

1198992

ge 119871 minus1

119887119872

minus1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

119887119872 minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

gt 119871 minus1


1

119887119872 minus119873 ge 119873

(63)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14) (15) (38) (39) and





0+ 120591 + 120573 such that for any 119909

0= 1199090119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 120572119898 minus 1198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+120572119898 minus 119879

2119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(65)


isin







0+ 120591 +

120573 satisfying

120579 = minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (66)

minus1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt min (1 + 1

119887)119872 minus 119871 119871 minus

1

119887119872 minus119873

(67)

119887119899ge 119887 gt 1 forall119899 ge 119879 (68)


infin

120573by

119878119871119909119899

=


119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(69)

for each 119909 = 119909119899119899isinN120573


119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le minus1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

minus1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le minus1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

minus1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992


119887minus

1

1198871198992

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]



119887minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887+min (1 + 1

119887)119872 + 119871 minus119871 minus 119873 le 119872

119878119871119909119899

1198992

ge minus119871

+1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge minus119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


119887)119872 + 119871 minus119871 minus 119873 ge 119873

(70)


3 Examples



Δ3(119909119899minus 119909119899minus120591) + Δ(

sin2119909119899minus3

1198997) +

1

(1198999 + 21198995 + 1) (1 + 1199092

1198992)

=1198992minus 2119899

1198998 + 1198993 + 1 forall119899 ge 4

(71)



119887119899= minus1 119888

119899=

1198992minus 2119899

1198998 + 1198993 + 1

119891 (119899 119906) =1

(1198999 + 21198995 + 1) (1 + 1199062)

ℎ (119899 119906) =sin21199061198997

1198911119899= 1198992 119865

119899= 1198994

ℎ1119899= 119899 minus 3 119867

119899= (119899 minus 3)

2 119875

119899=2119872

1198999

119876119899=1

1198999 119877

119899=2

1198997 119882

119899=1

1198997

forall (119899 119906) isin N1198990

timesR

(72)


1

1198992

infin

sum

119905=119899+120591

1199052max 119877

119905119867119905119882119905

=1

1198992

infin

sum

119905=119899+120591

1199052max2(119905 minus 3)

2

11990571

1199057

=

infin

sum

119905=119899+120591

2(119905 minus 3)2+ 1

1199055

le2

1198992

infin

sum

119905=119899+120591

1


1

1198992

infin

sum

119905=119899+120591

1199053max 119875

119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119905=119899+120591

1199053max2119872

11990551

1199059

100381610038161003816100381610038161199052minus 2119905

10038161003816100381610038161003816

1199058 + 1199053 + 1

lemax 1 2119872

1198992

infin

sum

119905=119899+120591

1


(73)


0+120591+120573 such



119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin 119860(119873119872) of(71) with lim


where 120572119898119898isinN0



Δ3(119909119899+ 119909119899minus120591) + Δ(

sin211990931198993+1

1198993 (1198996 + 2) (1 + 1199094

21198992minus3)

)


+ 119909(119899+1)(119899+2)

)

(11989913 + 1198995 + 1) (1 + 1199092


+ 1199092

(119899+1)(119899+2))

=1198992minus ln 119899

1198996 + 1198995 + 1 forall119899 ge 5

(74)



119887119899= 1 119888

119899=

1198992minus ln 119899

1198996 + 1198995 + 1

119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)

(11989913 + 1198995 + 1) (1 + 1199062 + V2)


ℎ (119899 119906 V) =sin2V

1198993 (1198996 + 2) (1 + 1199064) 119891

1119899= 1198992minus 119899 minus 1

1198912119899= (119899 + 1) (119899 + 2)

119865119899= (119899 + 1)

2(119899 + 2)

2 ℎ

1119899= 21198992minus 3

ℎ2119899= 31198993+ 1

119867119899= (3119899

3+ 1)2

119875119899= 119876119899=

4

11989910 119877

119899= 119882119899=10

1198999

forall (119899 119906 V) isin N1198990

timesR2

(75)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

10(31199043+ 1)2

119904910

1199049

le160

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le160

1198992

infin

sum

119904=119899

1


(76)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (77)

Observe that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max4(119905 + 1)2(119905 + 2)

2

119905104

119905101199052minus ln 119905

1199056 + 1199055 + 1

le196

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054=196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

119905 minus 119906 + 1

1199054

le196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le196

1198992

infin

sum

119905=119899

1


(78)

which yields that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)



119898119898isinN0

=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899

119909119899minus120591)

+ Δ(

(minus1)119899 sin (119890minus119899

2|11990951198992minus3|)

11989915 minus radic119899 + 3

+1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|11990921198993+1|)

+(minus1)119899

1198996 (1 + 1199092

119899minus3)minus

1

(1198997 + 21198994 minus 1) (1 + 1199092

119899+4)

=3(minus1)1198991198992

911989910ln3119899 forall119899 ge 7

(80)



119887119899=1 + 3 ln 1198992 + 4 ln 119899

119888119899=3(minus1)1198991198992

911989910ln3119899

119891 (119899 119906 V) =(minus1)119899

1198996 (1 + 1199062)minus

1

(1198997 + 21198994 minus 1) (1 + V2)


2|119906|)

11989915 minus radic119899 + 3+

1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|V|

1198911119899= 119899 minus 3 119891

2119899= 119899 + 4

119865119899= (119899 + 4)

2 ℎ

1119899= 51198992minus 3

ℎ2119899= 21198993+ 1 119867

119899= (2119899

3+ 1)2

119875119899= 119876119899=3

1198996 119877

119899= 119882119899=

2

11989910

forall (119899 119906 V) isin N1198990

timesR2

(81)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

2(21199043+ 1)2

119904102

11990410

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le18

1198992

infin

sum

119904=119899

1


(82)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (83)


Observe that1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max3(119905 + 4)2

11990563

1199056

100381610038161003816100381610038161003816100381610038161003816

3(minus1)1199051199052

911990510ln3119905

100381610038161003816100381610038161003816100381610038161003816

le12

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054

le12

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le12

1198992

infin

sum

119905=119899

1


(84)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 minus 5119899

3

2 + 61198993119909119899minus120591) + Δ(

21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199092

3119899minus7))

+

sin (119899211990931198992minus2)

(radic119899 + 14)22

=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9

(86)



119887119899=1 minus 5119899

3

2 + 61198993 119888

119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7

119891 (119899 119906) =

sin (1198992119906)

(radic119899 + 14)22

ℎ (119899 119906) =21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199062)

1198911119899= 31198992minus 2 119865

119899= (3119899

2minus 2)2

ℎ1119899= 3119899 minus 7

119867119899= (3119899 minus 7)

2 119875

119899= 119876119899=3

1198999 119877119899= 119882119899=1

1198995

forall (119899 119906) isin N1198990

timesR

(87)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max(3119904 minus 7)2

11990451

1199045

le9

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le9

1198992

infin

sum

119904=119899

1


(88)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (89)

Notice that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max

3(31199052minus 2)2

11990593

1199059

100381610038161003816100381610038161003816100381610038161003816

(minus1)1199051199053+ 51199052+ 4119905 minus 2

1199059 + 1199058 + 21199055 + 1199053 + 7

100381610038161003816100381610038161003816100381610038161003816

le27

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le27

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le27

1198992

infin

sum

119905=119899

1


(90)

which gives that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)




= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin







Δ3(119909119899+ (

120587

2+ 119899 sin 1

119899) 119909119899minus120591)

+ Δ((minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119909

2119899+1)))

+119899 sin (119899119909

119899minus2)

2 + (119899 + 5)16

=


forall119899 ge 3

(92)



119887119899=120587

2+ 119899 sin 1

119899 119888

119899=


119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)

16

ℎ (119899 119906) =(minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119906))

1198911119899= 119899 minus 2 119865

119899= (119899 minus 2)

2

ℎ1119899= 2119899 + 1 119867

119899= (2119899 + 1)

2

119875119899= 119876119899=

1

11989914 119877119899= 119882119899=2

1198999 forall (119899 119906) isin N

1198990

timesR

(93)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max2(2119904 + 1)2

11990492

1199049

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199047le18

1198992

infin

sum

119904=119899

1


(94)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


119905141

11990514

10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

11990514le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

11990513

le1

1198992

infin

sum

119905=119899

1


(95)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899minus21198995+ 91198992minus 1

1198995 + 31198992 + 2119909119899minus120591) + Δ(

cos ((minus1)119899119890119899)

(119899 + 7)6radic1 +

1003816100381610038161003816119909119899minus21003816100381610038161003816

)

+

sin (1198992119909119899minus1)

1198999 + 31198995 + 21198992 + 1


11989911 + 61198993 + 7119899 + 2 forall119899 ge 6

(97)



119887119899= minus

21198995+ 91198992minus 1

1198995 + 31198992 + 2


11989911 + 61198993 + 7119899 + 2

119891 (119899 119906) =

sin (1198992119906)1198999 + 31198995 + 21198992 + 1


ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816


11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)


0+ 120591 + 120573 such that


= 119909119898119899119899isinN120573

119898isinN0


119911119899119899isinN120573


119911119899= +infin and





Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











isin







0+ 120591 +

120573 satisfying

120579 =1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (59)

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873

(60)

119887119899ge 119887 gt 1 forall119899 ge 119879 (61)


infin

120573by

119878119871119909119899

=

1198992119871 minus

119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[(119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

)

minus 119888119905)] 119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(62)

for each 119909 = 119909119899119899isinN120573


119899119899isinN120573

119910 =

119910119899119899isinN120573

isin 119860(119873119872) and 119899 ge 119879

10038161003816100381610038161003816100381610038161003816

119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

+1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

+1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992

le 119871 +1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816

+10038161003816100381610038161198881199051003816100381610038161003816 ]

le 119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887119872 minus119873 le 119872

119878119871119909119899

1198992

ge 119871 minus1

119887119872

minus1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge 119871 minus1

119887119872 minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

gt 119871 minus1


1

119887119872 minus119873 ge 119873

(63)



119899119899isinN1198990

119876119899119899isinN1198990

119877119899119899isinN1198990

and 119882119899119899isinN1198990

satisfying (14) (15) (38) (39) and





0+ 120591 + 120573 such that for any 119909

0= 1199090119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 120572119898 minus 1198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+120572119898 minus 119879

2119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(65)


isin







0+ 120591 +

120573 satisfying

120579 = minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (66)

minus1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt min (1 + 1

119887)119872 minus 119871 119871 minus

1

119887119872 minus119873

(67)

119887119899ge 119887 gt 1 forall119899 ge 119879 (68)


infin

120573by

119878119871119909119899

=


119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(69)

for each 119909 = 119909119899119899isinN120573


119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le minus1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

minus1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le minus1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

minus1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992


119887minus

1

1198871198992

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]



119887minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887+min (1 + 1

119887)119872 + 119871 minus119871 minus 119873 le 119872

119878119871119909119899

1198992

ge minus119871

+1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge minus119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


119887)119872 + 119871 minus119871 minus 119873 ge 119873

(70)


3 Examples



Δ3(119909119899minus 119909119899minus120591) + Δ(

sin2119909119899minus3

1198997) +

1

(1198999 + 21198995 + 1) (1 + 1199092

1198992)

=1198992minus 2119899

1198998 + 1198993 + 1 forall119899 ge 4

(71)



119887119899= minus1 119888

119899=

1198992minus 2119899

1198998 + 1198993 + 1

119891 (119899 119906) =1

(1198999 + 21198995 + 1) (1 + 1199062)

ℎ (119899 119906) =sin21199061198997

1198911119899= 1198992 119865

119899= 1198994

ℎ1119899= 119899 minus 3 119867

119899= (119899 minus 3)

2 119875

119899=2119872

1198999

119876119899=1

1198999 119877

119899=2

1198997 119882

119899=1

1198997

forall (119899 119906) isin N1198990

timesR

(72)


1

1198992

infin

sum

119905=119899+120591

1199052max 119877

119905119867119905119882119905

=1

1198992

infin

sum

119905=119899+120591

1199052max2(119905 minus 3)

2

11990571

1199057

=

infin

sum

119905=119899+120591

2(119905 minus 3)2+ 1

1199055

le2

1198992

infin

sum

119905=119899+120591

1


1

1198992

infin

sum

119905=119899+120591

1199053max 119875

119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119905=119899+120591

1199053max2119872

11990551

1199059

100381610038161003816100381610038161199052minus 2119905

10038161003816100381610038161003816

1199058 + 1199053 + 1

lemax 1 2119872

1198992

infin

sum

119905=119899+120591

1


(73)


0+120591+120573 such



119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin 119860(119873119872) of(71) with lim


where 120572119898119898isinN0



Δ3(119909119899+ 119909119899minus120591) + Δ(

sin211990931198993+1

1198993 (1198996 + 2) (1 + 1199094

21198992minus3)

)


+ 119909(119899+1)(119899+2)

)

(11989913 + 1198995 + 1) (1 + 1199092


+ 1199092

(119899+1)(119899+2))

=1198992minus ln 119899

1198996 + 1198995 + 1 forall119899 ge 5

(74)



119887119899= 1 119888

119899=

1198992minus ln 119899

1198996 + 1198995 + 1

119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)

(11989913 + 1198995 + 1) (1 + 1199062 + V2)


ℎ (119899 119906 V) =sin2V

1198993 (1198996 + 2) (1 + 1199064) 119891

1119899= 1198992minus 119899 minus 1

1198912119899= (119899 + 1) (119899 + 2)

119865119899= (119899 + 1)

2(119899 + 2)

2 ℎ

1119899= 21198992minus 3

ℎ2119899= 31198993+ 1

119867119899= (3119899

3+ 1)2

119875119899= 119876119899=

4

11989910 119877

119899= 119882119899=10

1198999

forall (119899 119906 V) isin N1198990

timesR2

(75)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

10(31199043+ 1)2

119904910

1199049

le160

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le160

1198992

infin

sum

119904=119899

1


(76)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (77)

Observe that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max4(119905 + 1)2(119905 + 2)

2

119905104

119905101199052minus ln 119905

1199056 + 1199055 + 1

le196

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054=196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

119905 minus 119906 + 1

1199054

le196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le196

1198992

infin

sum

119905=119899

1


(78)

which yields that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)



119898119898isinN0

=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899

119909119899minus120591)

+ Δ(

(minus1)119899 sin (119890minus119899

2|11990951198992minus3|)

11989915 minus radic119899 + 3

+1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|11990921198993+1|)

+(minus1)119899

1198996 (1 + 1199092

119899minus3)minus

1

(1198997 + 21198994 minus 1) (1 + 1199092

119899+4)

=3(minus1)1198991198992

911989910ln3119899 forall119899 ge 7

(80)



119887119899=1 + 3 ln 1198992 + 4 ln 119899

119888119899=3(minus1)1198991198992

911989910ln3119899

119891 (119899 119906 V) =(minus1)119899

1198996 (1 + 1199062)minus

1

(1198997 + 21198994 minus 1) (1 + V2)


2|119906|)

11989915 minus radic119899 + 3+

1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|V|

1198911119899= 119899 minus 3 119891

2119899= 119899 + 4

119865119899= (119899 + 4)

2 ℎ

1119899= 51198992minus 3

ℎ2119899= 21198993+ 1 119867

119899= (2119899

3+ 1)2

119875119899= 119876119899=3

1198996 119877

119899= 119882119899=

2

11989910

forall (119899 119906 V) isin N1198990

timesR2

(81)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

2(21199043+ 1)2

119904102

11990410

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le18

1198992

infin

sum

119904=119899

1


(82)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (83)


Observe that1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max3(119905 + 4)2

11990563

1199056

100381610038161003816100381610038161003816100381610038161003816

3(minus1)1199051199052

911990510ln3119905

100381610038161003816100381610038161003816100381610038161003816

le12

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054

le12

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le12

1198992

infin

sum

119905=119899

1


(84)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 minus 5119899

3

2 + 61198993119909119899minus120591) + Δ(

21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199092

3119899minus7))

+

sin (119899211990931198992minus2)

(radic119899 + 14)22

=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9

(86)



119887119899=1 minus 5119899

3

2 + 61198993 119888

119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7

119891 (119899 119906) =

sin (1198992119906)

(radic119899 + 14)22

ℎ (119899 119906) =21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199062)

1198911119899= 31198992minus 2 119865

119899= (3119899

2minus 2)2

ℎ1119899= 3119899 minus 7

119867119899= (3119899 minus 7)

2 119875

119899= 119876119899=3

1198999 119877119899= 119882119899=1

1198995

forall (119899 119906) isin N1198990

timesR

(87)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max(3119904 minus 7)2

11990451

1199045

le9

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le9

1198992

infin

sum

119904=119899

1


(88)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (89)

Notice that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max

3(31199052minus 2)2

11990593

1199059

100381610038161003816100381610038161003816100381610038161003816

(minus1)1199051199053+ 51199052+ 4119905 minus 2

1199059 + 1199058 + 21199055 + 1199053 + 7

100381610038161003816100381610038161003816100381610038161003816

le27

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le27

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le27

1198992

infin

sum

119905=119899

1


(90)

which gives that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)




= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin







Δ3(119909119899+ (

120587

2+ 119899 sin 1

119899) 119909119899minus120591)

+ Δ((minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119909

2119899+1)))

+119899 sin (119899119909

119899minus2)

2 + (119899 + 5)16

=


forall119899 ge 3

(92)



119887119899=120587

2+ 119899 sin 1

119899 119888

119899=


119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)

16

ℎ (119899 119906) =(minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119906))

1198911119899= 119899 minus 2 119865

119899= (119899 minus 2)

2

ℎ1119899= 2119899 + 1 119867

119899= (2119899 + 1)

2

119875119899= 119876119899=

1

11989914 119877119899= 119882119899=2

1198999 forall (119899 119906) isin N

1198990

timesR

(93)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max2(2119904 + 1)2

11990492

1199049

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199047le18

1198992

infin

sum

119904=119899

1


(94)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


119905141

11990514

10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

11990514le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

11990513

le1

1198992

infin

sum

119905=119899

1


(95)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899minus21198995+ 91198992minus 1

1198995 + 31198992 + 2119909119899minus120591) + Δ(

cos ((minus1)119899119890119899)

(119899 + 7)6radic1 +

1003816100381610038161003816119909119899minus21003816100381610038161003816

)

+

sin (1198992119909119899minus1)

1198999 + 31198995 + 21198992 + 1


11989911 + 61198993 + 7119899 + 2 forall119899 ge 6

(97)



119887119899= minus

21198995+ 91198992minus 1

1198995 + 31198992 + 2


11989911 + 61198993 + 7119899 + 2

119891 (119899 119906) =

sin (1198992119906)1198999 + 31198995 + 21198992 + 1


ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816


11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)


0+ 120591 + 120573 such that


= 119909119898119899119899isinN120573

119898isinN0


119911119899119899isinN120573


119911119899= +infin and





Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0+ 120591 + 120573 such that for any 119909

0= 1199090119899119899isinN120573

isin


=

119909119898119899119899isinN120573

119898isinN0


119909119898+1119899

=

(1 minus 120572119898) 119909119898119899

+ 120572119898 minus 1198992119871 minus

119909119898119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum119906=119899+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

119899 ge 119879 119898 ge 0

(1 minus 120572119898) 119909119898119879

+120572119898 minus 119879

2119871 minus

119909119898119879+120591

119887119879+120591

minus

infin

sum

119906=119879+120591

infin

sum119904=119906

[ℎ (119904 119909119898ℎ1119904

119909119898ℎ2119904

119909119898ℎ119896119904

)

minus

infin

sum119905=119904

(119891 (119905 1199091198981198911119905

1199091198981198912119905

119909119898119891119896119905

)

minus 119888119905)]

120573 le 119899 lt 119879 119898 ge 0

(65)


isin







0+ 120591 +

120573 satisfying

120579 = minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)] (66)

minus1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))

lt min (1 + 1

119887)119872 minus 119871 119871 minus

1

119887119872 minus119873

(67)

119887119899ge 119887 gt 1 forall119899 ge 119879 (68)


infin

120573by

119878119871119909119899

=


119909119899+120591

119887119899+120591

minus1

119887119899+120591

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

ℎ (119904 119909ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus

infin

sum

119905=119904

[119891 (119905 1199091198911119905

1199091198912119905

119909119891119896119905

) minus 119888119905]

119899 ge 119879

119878119871119909119879 120573 le 119899 lt 119879

(69)

for each 119909 = 119909119899119899isinN120573


119878119871119909119899

1198992minus119878119871119910119899

1198992

10038161003816100381610038161003816100381610038161003816

le minus1

119887119899+120591

1003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

1198992

1003816100381610038161003816100381610038161003816

minus1

119887119899+1205911198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

[10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)

minus ℎ (119904 119910ℎ1119904

119910ℎ2119904

119910ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)

minus 119891 (119905 1199101198911119905

1199101198912119905

119910119891119896119905

)10038161003816100381610038161003816]

le minus1

119887119899+120591

100381610038161003816100381610038161003816100381610038161003816

119909119899+120591

minus 119910119899+120591

(119899 + 120591)2

100381610038161003816100381610038161003816100381610038161003816

(119899 + 120591)2

1198992

minus1

119887119899+1205911198992

infin

sum

119906=119899

infin

sum

119904=119906

[119877119904max 10038161003816100381610038161003816119909ℎ119897119904 minus 119910ℎ119897119904

10038161003816100381610038161003816 1 le 119897 le 119896

+

infin

sum

119905=119904

119875119905max 10038161003816100381610038161003816119909119891119897119905 minus 119910119891119897119905

10038161003816100381610038161003816 1 le 119897 le 119896]

le minus1

119887[(1 +

120591

119879)

2

+1

1198792

infin

sum

119906=119879

infin

sum

119904=119906

(119877119904119867119904+

infin

sum

119905=119904

119875119905119865119905)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= 1205791003817100381710038171003817119909 minus 119910

1003817100381710038171003817

119878119871119909119899

1198992


119887minus

1

1198871198992

times

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]



119887minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887+min (1 + 1

119887)119872 + 119871 minus119871 minus 119873 le 119872

119878119871119909119899

1198992

ge minus119871

+1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge minus119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


119887)119872 + 119871 minus119871 minus 119873 ge 119873

(70)


3 Examples



Δ3(119909119899minus 119909119899minus120591) + Δ(

sin2119909119899minus3

1198997) +

1

(1198999 + 21198995 + 1) (1 + 1199092

1198992)

=1198992minus 2119899

1198998 + 1198993 + 1 forall119899 ge 4

(71)



119887119899= minus1 119888

119899=

1198992minus 2119899

1198998 + 1198993 + 1

119891 (119899 119906) =1

(1198999 + 21198995 + 1) (1 + 1199062)

ℎ (119899 119906) =sin21199061198997

1198911119899= 1198992 119865

119899= 1198994

ℎ1119899= 119899 minus 3 119867

119899= (119899 minus 3)

2 119875

119899=2119872

1198999

119876119899=1

1198999 119877

119899=2

1198997 119882

119899=1

1198997

forall (119899 119906) isin N1198990

timesR

(72)


1

1198992

infin

sum

119905=119899+120591

1199052max 119877

119905119867119905119882119905

=1

1198992

infin

sum

119905=119899+120591

1199052max2(119905 minus 3)

2

11990571

1199057

=

infin

sum

119905=119899+120591

2(119905 minus 3)2+ 1

1199055

le2

1198992

infin

sum

119905=119899+120591

1


1

1198992

infin

sum

119905=119899+120591

1199053max 119875

119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119905=119899+120591

1199053max2119872

11990551

1199059

100381610038161003816100381610038161199052minus 2119905

10038161003816100381610038161003816

1199058 + 1199053 + 1

lemax 1 2119872

1198992

infin

sum

119905=119899+120591

1


(73)


0+120591+120573 such



119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin 119860(119873119872) of(71) with lim


where 120572119898119898isinN0



Δ3(119909119899+ 119909119899minus120591) + Δ(

sin211990931198993+1

1198993 (1198996 + 2) (1 + 1199094

21198992minus3)

)


+ 119909(119899+1)(119899+2)

)

(11989913 + 1198995 + 1) (1 + 1199092


+ 1199092

(119899+1)(119899+2))

=1198992minus ln 119899

1198996 + 1198995 + 1 forall119899 ge 5

(74)



119887119899= 1 119888

119899=

1198992minus ln 119899

1198996 + 1198995 + 1

119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)

(11989913 + 1198995 + 1) (1 + 1199062 + V2)


ℎ (119899 119906 V) =sin2V

1198993 (1198996 + 2) (1 + 1199064) 119891

1119899= 1198992minus 119899 minus 1

1198912119899= (119899 + 1) (119899 + 2)

119865119899= (119899 + 1)

2(119899 + 2)

2 ℎ

1119899= 21198992minus 3

ℎ2119899= 31198993+ 1

119867119899= (3119899

3+ 1)2

119875119899= 119876119899=

4

11989910 119877

119899= 119882119899=10

1198999

forall (119899 119906 V) isin N1198990

timesR2

(75)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

10(31199043+ 1)2

119904910

1199049

le160

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le160

1198992

infin

sum

119904=119899

1


(76)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (77)

Observe that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max4(119905 + 1)2(119905 + 2)

2

119905104

119905101199052minus ln 119905

1199056 + 1199055 + 1

le196

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054=196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

119905 minus 119906 + 1

1199054

le196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le196

1198992

infin

sum

119905=119899

1


(78)

which yields that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)



119898119898isinN0

=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899

119909119899minus120591)

+ Δ(

(minus1)119899 sin (119890minus119899

2|11990951198992minus3|)

11989915 minus radic119899 + 3

+1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|11990921198993+1|)

+(minus1)119899

1198996 (1 + 1199092

119899minus3)minus

1

(1198997 + 21198994 minus 1) (1 + 1199092

119899+4)

=3(minus1)1198991198992

911989910ln3119899 forall119899 ge 7

(80)



119887119899=1 + 3 ln 1198992 + 4 ln 119899

119888119899=3(minus1)1198991198992

911989910ln3119899

119891 (119899 119906 V) =(minus1)119899

1198996 (1 + 1199062)minus

1

(1198997 + 21198994 minus 1) (1 + V2)


2|119906|)

11989915 minus radic119899 + 3+

1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|V|

1198911119899= 119899 minus 3 119891

2119899= 119899 + 4

119865119899= (119899 + 4)

2 ℎ

1119899= 51198992minus 3

ℎ2119899= 21198993+ 1 119867

119899= (2119899

3+ 1)2

119875119899= 119876119899=3

1198996 119877

119899= 119882119899=

2

11989910

forall (119899 119906 V) isin N1198990

timesR2

(81)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

2(21199043+ 1)2

119904102

11990410

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le18

1198992

infin

sum

119904=119899

1


(82)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (83)


Observe that1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max3(119905 + 4)2

11990563

1199056

100381610038161003816100381610038161003816100381610038161003816

3(minus1)1199051199052

911990510ln3119905

100381610038161003816100381610038161003816100381610038161003816

le12

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054

le12

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le12

1198992

infin

sum

119905=119899

1


(84)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 minus 5119899

3

2 + 61198993119909119899minus120591) + Δ(

21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199092

3119899minus7))

+

sin (119899211990931198992minus2)

(radic119899 + 14)22

=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9

(86)



119887119899=1 minus 5119899

3

2 + 61198993 119888

119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7

119891 (119899 119906) =

sin (1198992119906)

(radic119899 + 14)22

ℎ (119899 119906) =21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199062)

1198911119899= 31198992minus 2 119865

119899= (3119899

2minus 2)2

ℎ1119899= 3119899 minus 7

119867119899= (3119899 minus 7)

2 119875

119899= 119876119899=3

1198999 119877119899= 119882119899=1

1198995

forall (119899 119906) isin N1198990

timesR

(87)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max(3119904 minus 7)2

11990451

1199045

le9

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le9

1198992

infin

sum

119904=119899

1


(88)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (89)

Notice that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max

3(31199052minus 2)2

11990593

1199059

100381610038161003816100381610038161003816100381610038161003816

(minus1)1199051199053+ 51199052+ 4119905 minus 2

1199059 + 1199058 + 21199055 + 1199053 + 7

100381610038161003816100381610038161003816100381610038161003816

le27

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le27

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le27

1198992

infin

sum

119905=119899

1


(90)

which gives that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)




= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin







Δ3(119909119899+ (

120587

2+ 119899 sin 1

119899) 119909119899minus120591)

+ Δ((minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119909

2119899+1)))

+119899 sin (119899119909

119899minus2)

2 + (119899 + 5)16

=


forall119899 ge 3

(92)



119887119899=120587

2+ 119899 sin 1

119899 119888

119899=


119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)

16

ℎ (119899 119906) =(minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119906))

1198911119899= 119899 minus 2 119865

119899= (119899 minus 2)

2

ℎ1119899= 2119899 + 1 119867

119899= (2119899 + 1)

2

119875119899= 119876119899=

1

11989914 119877119899= 119882119899=2

1198999 forall (119899 119906) isin N

1198990

timesR

(93)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max2(2119904 + 1)2

11990492

1199049

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199047le18

1198992

infin

sum

119904=119899

1


(94)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


119905141

11990514

10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

11990514le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

11990513

le1

1198992

infin

sum

119905=119899

1


(95)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899minus21198995+ 91198992minus 1

1198995 + 31198992 + 2119909119899minus120591) + Δ(

cos ((minus1)119899119890119899)

(119899 + 7)6radic1 +

1003816100381610038161003816119909119899minus21003816100381610038161003816

)

+

sin (1198992119909119899minus1)

1198999 + 31198995 + 21198992 + 1


11989911 + 61198993 + 7119899 + 2 forall119899 ge 6

(97)



119887119899= minus

21198995+ 91198992minus 1

1198995 + 31198992 + 2


11989911 + 61198993 + 7119899 + 2

119891 (119899 119906) =

sin (1198992119906)1198999 + 31198995 + 21198992 + 1


ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816


11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)


0+ 120591 + 120573 such that


= 119909119898119899119899isinN120573

119898isinN0


119911119899119899isinN120573


119911119899= +infin and





Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119887minus

1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

(119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816))


119887+min (1 + 1

119887)119872 + 119871 minus119871 minus 119873 le 119872

119878119871119909119899

1198992

ge minus119871

+1

1198871198992

infin

sum

119906=119899+120591

infin

sum

119904=119906

10038161003816100381610038161003816ℎ (119904 119909

ℎ1119904

119909ℎ2119904

119909ℎ119896119904

)10038161003816100381610038161003816

+

infin

sum

119905=119904

[10038161003816100381610038161003816119891 (119905 119909

1198911119905

1199091198912119905

119909119891119896119905

)10038161003816100381610038161003816+10038161003816100381610038161198881199051003816100381610038161003816]

ge minus119871 +1

1198871198792

infin

sum

119906=119879

infin

sum

119904=119906

[119882119904+

infin

sum

119905=119904

(119876119905+10038161003816100381610038161198881199051003816100381610038161003816)]


119887)119872 + 119871 minus119871 minus 119873 ge 119873

(70)


3 Examples



Δ3(119909119899minus 119909119899minus120591) + Δ(

sin2119909119899minus3

1198997) +

1

(1198999 + 21198995 + 1) (1 + 1199092

1198992)

=1198992minus 2119899

1198998 + 1198993 + 1 forall119899 ge 4

(71)



119887119899= minus1 119888

119899=

1198992minus 2119899

1198998 + 1198993 + 1

119891 (119899 119906) =1

(1198999 + 21198995 + 1) (1 + 1199062)

ℎ (119899 119906) =sin21199061198997

1198911119899= 1198992 119865

119899= 1198994

ℎ1119899= 119899 minus 3 119867

119899= (119899 minus 3)

2 119875

119899=2119872

1198999

119876119899=1

1198999 119877

119899=2

1198997 119882

119899=1

1198997

forall (119899 119906) isin N1198990

timesR

(72)


1

1198992

infin

sum

119905=119899+120591

1199052max 119877

119905119867119905119882119905

=1

1198992

infin

sum

119905=119899+120591

1199052max2(119905 minus 3)

2

11990571

1199057

=

infin

sum

119905=119899+120591

2(119905 minus 3)2+ 1

1199055

le2

1198992

infin

sum

119905=119899+120591

1


1

1198992

infin

sum

119905=119899+120591

1199053max 119875

119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119905=119899+120591

1199053max2119872

11990551

1199059

100381610038161003816100381610038161199052minus 2119905

10038161003816100381610038161003816

1199058 + 1199053 + 1

lemax 1 2119872

1198992

infin

sum

119905=119899+120591

1


(73)


0+120591+120573 such



119898119898isinN0

= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin 119860(119873119872) of(71) with lim


where 120572119898119898isinN0



Δ3(119909119899+ 119909119899minus120591) + Δ(

sin211990931198993+1

1198993 (1198996 + 2) (1 + 1199094

21198992minus3)

)


+ 119909(119899+1)(119899+2)

)

(11989913 + 1198995 + 1) (1 + 1199092


+ 1199092

(119899+1)(119899+2))

=1198992minus ln 119899

1198996 + 1198995 + 1 forall119899 ge 5

(74)



119887119899= 1 119888

119899=

1198992minus ln 119899

1198996 + 1198995 + 1

119891 (119899 119906 V) =(minus1)1198991198993(119906 + V)

(11989913 + 1198995 + 1) (1 + 1199062 + V2)


ℎ (119899 119906 V) =sin2V

1198993 (1198996 + 2) (1 + 1199064) 119891

1119899= 1198992minus 119899 minus 1

1198912119899= (119899 + 1) (119899 + 2)

119865119899= (119899 + 1)

2(119899 + 2)

2 ℎ

1119899= 21198992minus 3

ℎ2119899= 31198993+ 1

119867119899= (3119899

3+ 1)2

119875119899= 119876119899=

4

11989910 119877

119899= 119882119899=10

1198999

forall (119899 119906 V) isin N1198990

timesR2

(75)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

10(31199043+ 1)2

119904910

1199049

le160

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le160

1198992

infin

sum

119904=119899

1


(76)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (77)

Observe that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max4(119905 + 1)2(119905 + 2)

2

119905104

119905101199052minus ln 119905

1199056 + 1199055 + 1

le196

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054=196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

119905 minus 119906 + 1

1199054

le196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le196

1198992

infin

sum

119905=119899

1


(78)

which yields that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)



119898119898isinN0

=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899

119909119899minus120591)

+ Δ(

(minus1)119899 sin (119890minus119899

2|11990951198992minus3|)

11989915 minus radic119899 + 3

+1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|11990921198993+1|)

+(minus1)119899

1198996 (1 + 1199092

119899minus3)minus

1

(1198997 + 21198994 minus 1) (1 + 1199092

119899+4)

=3(minus1)1198991198992

911989910ln3119899 forall119899 ge 7

(80)



119887119899=1 + 3 ln 1198992 + 4 ln 119899

119888119899=3(minus1)1198991198992

911989910ln3119899

119891 (119899 119906 V) =(minus1)119899

1198996 (1 + 1199062)minus

1

(1198997 + 21198994 minus 1) (1 + V2)


2|119906|)

11989915 minus radic119899 + 3+

1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|V|

1198911119899= 119899 minus 3 119891

2119899= 119899 + 4

119865119899= (119899 + 4)

2 ℎ

1119899= 51198992minus 3

ℎ2119899= 21198993+ 1 119867

119899= (2119899

3+ 1)2

119875119899= 119876119899=3

1198996 119877

119899= 119882119899=

2

11989910

forall (119899 119906 V) isin N1198990

timesR2

(81)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

2(21199043+ 1)2

119904102

11990410

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le18

1198992

infin

sum

119904=119899

1


(82)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (83)


Observe that1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max3(119905 + 4)2

11990563

1199056

100381610038161003816100381610038161003816100381610038161003816

3(minus1)1199051199052

911990510ln3119905

100381610038161003816100381610038161003816100381610038161003816

le12

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054

le12

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le12

1198992

infin

sum

119905=119899

1


(84)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 minus 5119899

3

2 + 61198993119909119899minus120591) + Δ(

21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199092

3119899minus7))

+

sin (119899211990931198992minus2)

(radic119899 + 14)22

=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9

(86)



119887119899=1 minus 5119899

3

2 + 61198993 119888

119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7

119891 (119899 119906) =

sin (1198992119906)

(radic119899 + 14)22

ℎ (119899 119906) =21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199062)

1198911119899= 31198992minus 2 119865

119899= (3119899

2minus 2)2

ℎ1119899= 3119899 minus 7

119867119899= (3119899 minus 7)

2 119875

119899= 119876119899=3

1198999 119877119899= 119882119899=1

1198995

forall (119899 119906) isin N1198990

timesR

(87)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max(3119904 minus 7)2

11990451

1199045

le9

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le9

1198992

infin

sum

119904=119899

1


(88)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (89)

Notice that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max

3(31199052minus 2)2

11990593

1199059

100381610038161003816100381610038161003816100381610038161003816

(minus1)1199051199053+ 51199052+ 4119905 minus 2

1199059 + 1199058 + 21199055 + 1199053 + 7

100381610038161003816100381610038161003816100381610038161003816

le27

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le27

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le27

1198992

infin

sum

119905=119899

1


(90)

which gives that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)




= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin







Δ3(119909119899+ (

120587

2+ 119899 sin 1

119899) 119909119899minus120591)

+ Δ((minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119909

2119899+1)))

+119899 sin (119899119909

119899minus2)

2 + (119899 + 5)16

=


forall119899 ge 3

(92)



119887119899=120587

2+ 119899 sin 1

119899 119888

119899=


119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)

16

ℎ (119899 119906) =(minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119906))

1198911119899= 119899 minus 2 119865

119899= (119899 minus 2)

2

ℎ1119899= 2119899 + 1 119867

119899= (2119899 + 1)

2

119875119899= 119876119899=

1

11989914 119877119899= 119882119899=2

1198999 forall (119899 119906) isin N

1198990

timesR

(93)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max2(2119904 + 1)2

11990492

1199049

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199047le18

1198992

infin

sum

119904=119899

1


(94)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


119905141

11990514

10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

11990514le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

11990513

le1

1198992

infin

sum

119905=119899

1


(95)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899minus21198995+ 91198992minus 1

1198995 + 31198992 + 2119909119899minus120591) + Δ(

cos ((minus1)119899119890119899)

(119899 + 7)6radic1 +

1003816100381610038161003816119909119899minus21003816100381610038161003816

)

+

sin (1198992119909119899minus1)

1198999 + 31198995 + 21198992 + 1


11989911 + 61198993 + 7119899 + 2 forall119899 ge 6

(97)



119887119899= minus

21198995+ 91198992minus 1

1198995 + 31198992 + 2


11989911 + 61198993 + 7119899 + 2

119891 (119899 119906) =

sin (1198992119906)1198999 + 31198995 + 21198992 + 1


ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816


11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)


0+ 120591 + 120573 such that


= 119909119898119899119899isinN120573

119898isinN0


119911119899119899isinN120573


119911119899= +infin and





Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ℎ (119899 119906 V) =sin2V

1198993 (1198996 + 2) (1 + 1199064) 119891

1119899= 1198992minus 119899 minus 1

1198912119899= (119899 + 1) (119899 + 2)

119865119899= (119899 + 1)

2(119899 + 2)

2 ℎ

1119899= 21198992minus 3

ℎ2119899= 31198993+ 1

119867119899= (3119899

3+ 1)2

119875119899= 119876119899=

4

11989910 119877

119899= 119882119899=10

1198999

forall (119899 119906 V) isin N1198990

timesR2

(75)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

10(31199043+ 1)2

119904910

1199049

le160

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le160

1198992

infin

sum

119904=119899

1


(76)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (77)

Observe that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max4(119905 + 1)2(119905 + 2)

2

119905104

119905101199052minus ln 119905

1199056 + 1199055 + 1

le196

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054=196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

119905 minus 119906 + 1

1199054

le196

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le196

1198992

infin

sum

119905=119899

1


(78)

which yields that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (79)



119898119898isinN0

=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 + 3 ln 1198992 + 4 ln 119899

119909119899minus120591)

+ Δ(

(minus1)119899 sin (119890minus119899

2|11990951198992minus3|)

11989915 minus radic119899 + 3

+1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|11990921198993+1|)

+(minus1)119899

1198996 (1 + 1199092

119899minus3)minus

1

(1198997 + 21198994 minus 1) (1 + 1199092

119899+4)

=3(minus1)1198991198992

911989910ln3119899 forall119899 ge 7

(80)



119887119899=1 + 3 ln 1198992 + 4 ln 119899

119888119899=3(minus1)1198991198992

911989910ln3119899

119891 (119899 119906 V) =(minus1)119899

1198996 (1 + 1199062)minus

1

(1198997 + 21198994 minus 1) (1 + V2)


2|119906|)

11989915 minus radic119899 + 3+

1198992+ (minus1)

119899(119899+1)2

(11989912 + 611989910 + 7) 119890|V|

1198911119899= 119899 minus 3 119891

2119899= 119899 + 4

119865119899= (119899 + 4)

2 ℎ

1119899= 51198992minus 3

ℎ2119899= 21198993+ 1 119867

119899= (2119899

3+ 1)2

119875119899= 119876119899=3

1198996 119877

119899= 119882119899=

2

11989910

forall (119899 119906 V) isin N1198990

timesR2

(81)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max

2(21199043+ 1)2

119904102

11990410

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le18

1198992

infin

sum

119904=119899

1


(82)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (83)


Observe that1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max3(119905 + 4)2

11990563

1199056

100381610038161003816100381610038161003816100381610038161003816

3(minus1)1199051199052

911990510ln3119905

100381610038161003816100381610038161003816100381610038161003816

le12

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054

le12

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le12

1198992

infin

sum

119905=119899

1


(84)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 minus 5119899

3

2 + 61198993119909119899minus120591) + Δ(

21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199092

3119899minus7))

+

sin (119899211990931198992minus2)

(radic119899 + 14)22

=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9

(86)



119887119899=1 minus 5119899

3

2 + 61198993 119888

119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7

119891 (119899 119906) =

sin (1198992119906)

(radic119899 + 14)22

ℎ (119899 119906) =21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199062)

1198911119899= 31198992minus 2 119865

119899= (3119899

2minus 2)2

ℎ1119899= 3119899 minus 7

119867119899= (3119899 minus 7)

2 119875

119899= 119876119899=3

1198999 119877119899= 119882119899=1

1198995

forall (119899 119906) isin N1198990

timesR

(87)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max(3119904 minus 7)2

11990451

1199045

le9

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le9

1198992

infin

sum

119904=119899

1


(88)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (89)

Notice that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max

3(31199052minus 2)2

11990593

1199059

100381610038161003816100381610038161003816100381610038161003816

(minus1)1199051199053+ 51199052+ 4119905 minus 2

1199059 + 1199058 + 21199055 + 1199053 + 7

100381610038161003816100381610038161003816100381610038161003816

le27

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le27

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le27

1198992

infin

sum

119905=119899

1


(90)

which gives that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)




= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin







Δ3(119909119899+ (

120587

2+ 119899 sin 1

119899) 119909119899minus120591)

+ Δ((minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119909

2119899+1)))

+119899 sin (119899119909

119899minus2)

2 + (119899 + 5)16

=


forall119899 ge 3

(92)



119887119899=120587

2+ 119899 sin 1

119899 119888

119899=


119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)

16

ℎ (119899 119906) =(minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119906))

1198911119899= 119899 minus 2 119865

119899= (119899 minus 2)

2

ℎ1119899= 2119899 + 1 119867

119899= (2119899 + 1)

2

119875119899= 119876119899=

1

11989914 119877119899= 119882119899=2

1198999 forall (119899 119906) isin N

1198990

timesR

(93)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max2(2119904 + 1)2

11990492

1199049

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199047le18

1198992

infin

sum

119904=119899

1


(94)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


119905141

11990514

10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

11990514le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

11990513

le1

1198992

infin

sum

119905=119899

1


(95)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899minus21198995+ 91198992minus 1

1198995 + 31198992 + 2119909119899minus120591) + Δ(

cos ((minus1)119899119890119899)

(119899 + 7)6radic1 +

1003816100381610038161003816119909119899minus21003816100381610038161003816

)

+

sin (1198992119909119899minus1)

1198999 + 31198995 + 21198992 + 1


11989911 + 61198993 + 7119899 + 2 forall119899 ge 6

(97)



119887119899= minus

21198995+ 91198992minus 1

1198995 + 31198992 + 2


11989911 + 61198993 + 7119899 + 2

119891 (119899 119906) =

sin (1198992119906)1198999 + 31198995 + 21198992 + 1


ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816


11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)


0+ 120591 + 120573 such that


= 119909119898119899119899isinN120573

119898isinN0


119911119899119899isinN120573


119911119899= +infin and





Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Observe that1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max3(119905 + 4)2

11990563

1199056

100381610038161003816100381610038161003816100381610038161003816

3(minus1)1199051199052

911990510ln3119905

100381610038161003816100381610038161003816100381610038161003816

le12

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199054

le12

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199053le12

1198992

infin

sum

119905=119899

1


(84)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (85)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899+1 minus 5119899

3

2 + 61198993119909119899minus120591) + Δ(

21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199092

3119899minus7))

+

sin (119899211990931198992minus2)

(radic119899 + 14)22

=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7 forall119899 ge 9

(86)



119887119899=1 minus 5119899

3

2 + 61198993 119888

119899=(minus1)1198991198993+ 51198992+ 4119899 minus 2

1198999 + 1198998 + 21198995 + 1198993 + 7

119891 (119899 119906) =

sin (1198992119906)

(radic119899 + 14)22

ℎ (119899 119906) =21198992+ 119899 minus 1

(1198998 + 31198996 + 2) (1 + 1199062)

1198911119899= 31198992minus 2 119865

119899= (3119899

2minus 2)2

ℎ1119899= 3119899 minus 7

119867119899= (3119899 minus 7)

2 119875

119899= 119876119899=3

1198999 119877119899= 119882119899=1

1198995

forall (119899 119906) isin N1198990

timesR

(87)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max(3119904 minus 7)2

11990451

1199045

le9

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199043le9

1198992

infin

sum

119904=119899

1


(88)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (89)

Notice that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max

3(31199052minus 2)2

11990593

1199059

100381610038161003816100381610038161003816100381610038161003816

(minus1)1199051199053+ 51199052+ 4119905 minus 2

1199059 + 1199058 + 21199055 + 1199053 + 7

100381610038161003816100381610038161003816100381610038161003816

le27

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le27

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le27

1198992

infin

sum

119905=119899

1


(90)

which gives that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (91)




= 119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573

isin







Δ3(119909119899+ (

120587

2+ 119899 sin 1

119899) 119909119899minus120591)

+ Δ((minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119909

2119899+1)))

+119899 sin (119899119909

119899minus2)

2 + (119899 + 5)16

=


forall119899 ge 3

(92)



119887119899=120587

2+ 119899 sin 1

119899 119888

119899=


119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)

16

ℎ (119899 119906) =(minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119906))

1198911119899= 119899 minus 2 119865

119899= (119899 minus 2)

2

ℎ1119899= 2119899 + 1 119867

119899= (2119899 + 1)

2

119875119899= 119876119899=

1

11989914 119877119899= 119882119899=2

1198999 forall (119899 119906) isin N

1198990

timesR

(93)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max2(2119904 + 1)2

11990492

1199049

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199047le18

1198992

infin

sum

119904=119899

1


(94)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


119905141

11990514

10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

11990514le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

11990513

le1

1198992

infin

sum

119905=119899

1


(95)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899minus21198995+ 91198992minus 1

1198995 + 31198992 + 2119909119899minus120591) + Δ(

cos ((minus1)119899119890119899)

(119899 + 7)6radic1 +

1003816100381610038161003816119909119899minus21003816100381610038161003816

)

+

sin (1198992119909119899minus1)

1198999 + 31198995 + 21198992 + 1


11989911 + 61198993 + 7119899 + 2 forall119899 ge 6

(97)



119887119899= minus

21198995+ 91198992minus 1

1198995 + 31198992 + 2


11989911 + 61198993 + 7119899 + 2

119891 (119899 119906) =

sin (1198992119906)1198999 + 31198995 + 21198992 + 1


ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816


11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)


0+ 120591 + 120573 such that


= 119909119898119899119899isinN120573

119898isinN0


119911119899119899isinN120573


119911119899= +infin and





Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Δ3(119909119899+ (

120587

2+ 119899 sin 1

119899) 119909119899minus120591)

+ Δ((minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119909

2119899+1)))

+119899 sin (119899119909

119899minus2)

2 + (119899 + 5)16

=


forall119899 ge 3

(92)



119887119899=120587

2+ 119899 sin 1

119899 119888

119899=


119891 (119899 119906) =119899 sin (119899119906)2 + (119899 + 5)

16

ℎ (119899 119906) =(minus1)119899(119899+1)2

(119899 + 4)8(119899 + 5)

3(1 + cos (1198992119906))

1198911119899= 119899 minus 2 119865

119899= (119899 minus 2)

2

ℎ1119899= 2119899 + 1 119867

119899= (2119899 + 1)

2

119875119899= 119876119899=

1

11989914 119877119899= 119882119899=2

1198999 forall (119899 119906) isin N

1198990

timesR

(93)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max2(2119904 + 1)2

11990492

1199049

le18

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199047le18

1198992

infin

sum

119904=119899

1


(94)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


119905141

11990514

10038161003816100381610038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

11990514le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

11990513

le1

1198992

infin

sum

119905=119899

1


(95)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (96)




=

119909119898119899119899isinN120573

119898isinN0


119899119899isinN120573


119911119899=




Δ3(119909119899minus21198995+ 91198992minus 1

1198995 + 31198992 + 2119909119899minus120591) + Δ(

cos ((minus1)119899119890119899)

(119899 + 7)6radic1 +

1003816100381610038161003816119909119899minus21003816100381610038161003816

)

+

sin (1198992119909119899minus1)

1198999 + 31198995 + 21198992 + 1


11989911 + 61198993 + 7119899 + 2 forall119899 ge 6

(97)



119887119899= minus

21198995+ 91198992minus 1

1198995 + 31198992 + 2


11989911 + 61198993 + 7119899 + 2

119891 (119899 119906) =

sin (1198992119906)1198999 + 31198995 + 21198992 + 1


ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816


11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)


0+ 120591 + 120573 such that


= 119909119898119899119899isinN120573

119898isinN0


119911119899119899isinN120573


119911119899= +infin and





Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ℎ (119899 119906) =cos ((minus1)119899119890119899)

(119899 + 7)6radic1 + |119906|

1198911119899= 119899 minus 1 119865

119899= (119899 minus 1)

2

ℎ1119899= 119899 minus 2 119867

119899= (119899 minus 2)

2

119875119899= 119876119899=1

1198997 119877119899= 119882119899=1

1198996

forall (119899 119906) isin N1198990

timesR

(98)


1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906


11990461

1199046

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

1

1199044le1

1198992

infin

sum

119904=119899

1


(99)

which means that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

max 119877119904119867119904119882119904 = 0 (100)

It is clear that

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816

=1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904


11990571

1199057

100381610038161003816100381610038161003816100381610038161003816


11990511 + 61199053 + 7119905 + 2

100381610038161003816100381610038161003816100381610038161003816

le1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

1

1199055le1

1198992

infin

sum

119906=119899

infin

sum

119905=119906

1

1199054

le1

1198992

infin

sum

119905=119899

1


(101)

which implies that

lim119899rarrinfin

1

1198992

infin

sum

119906=119899

infin

sum

119904=119906

infin

sum

119905=119904

max 119875119905119865119905 11987611990510038161003816100381610038161198881199051003816100381610038161003816 = 0 (102)


0+ 120591 + 120573 such that


= 119909119898119899119899isinN120573

119898isinN0


119911119899119899isinN120573


119911119899= +infin and





Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article On Positive Solutions and Mann Iterative …downloads.hindawi.com/journals/aaa/2014/470181.pdf · e existence of uncountably many positive solutions and converg ence