Research Article Existence of Positive Periodic Solutions ...downloads.hindawi.com/journals/ijde/2013/890281.pdf · Existence of Positive Periodic Solutions for Periodic Neutral Lotka-Volterra

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2013 Article ID 890281 13 pageshttpdxdoiorg1011552013890281

Research ArticleExistence of Positive Periodic Solutions for Periodic NeutralLotka-Volterra System with Distributed Delays and Impulses

Zhenguo Luo12 and Liping Luo2

1 Department of Mathematics National University of Defense Technology Changsha 410073 China2Department of Mathematics Hengyang Normal University Hengyang Hunan 421008 China

Correspondence should be addressed to Zhenguo Luo robert186163com

Received 20 April 2013 Accepted 13 May 2013

Academic Editor Norio Yoshida

Copyright copy 2013 Z Luo and L Luo 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

By using a fixed-point theorem of strict-set-contraction we investigate the existence of positive periodic solutions for aclass of the following impulsive neutral Lotka-Volterra system with distributed delays 1199091015840

119894(119905) = 119909

119894(119905)[119903119894(119905) minus sum

119899

119895=1119886119894119895(119905)119909119895(119905) minus

sum119899

119895=1119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585)119909119895(119905+120585)119889120585minussum

119899

119895=1119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585)1199091015840

119895(119905+120585)119889120585] Δ119909

119894(119905119896) = minus119868

119894119896(119909119894(119905119896)) 119894 = 1 2 119899 119896 = 1 2 Some verifiable

criteria are established easily

1 Introduction

It is well known that natural environments are physicallyhighly variable and in response birth rates death ratesand other vital rates of populations vary greatly in timeTheoretical evidence suggests that many population andcommunity patterns represent intricate interactions betweenbiology and variation in the physical environment (see [1ndash4]) Thus the focus in theoretical models of populationand community dynamics must be not only on how pop-ulations depend on their own population densities or thepopulation densities of other organisms but also on howpopulations change in response to the physical environmentIt is reasonable to study the models of population withperiodic coefficients In addition to the theoretical andpractical significance the Lotka-Volterra model is one of thefamous models for dynamics of population therefore it hasbeen studied extensively [5ndash9] In view of the above effectsby applying a fixed-point theoremof strict-set-contraction Li[10] established criteria to guarantee the existence of positiveperiodic solutions of the following neutral Lotka-Volterrasystem with distributed delays

1199091015840

119894(119905) = 119909

119894(119905)[

[

119886119894(119905) minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus119879119894119895

119870119894119895(120579) 119909119895(119905 + 120579) 119889120579

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus119894119895

119894119895(120579) 1199091015840

119895(119905 + 120579) 119889120579

]

]

119894 = 1 2 119899

(1)

where 119886119894 119887119894119895 119888119894119895

isin 119862(119877 119877+) (119894 119895 = 1 2 119899) are 120596-

periodic functions and 119879119894119895119879119894119895isin (0 +infin) (119894 119895 = 1 2 119899)

and 119870119894119895 119894119895

isin (119877 119877+) satisfying int

0

minus119879119894119895

119870119894119895(120579)119889120579 = 1

int

0

minus119894119895

119894119895(120579)119889120579 119894 119895 = 1 2 119899 On the other side birth

of many species is an annual birth pulse or harvesting Tohave a more accurate description of many mathematicalecology systems we need to consider the use of impulsivedifferential equations [11ndash13] Some qualitative propertiessuch as oscillation periodicity asymptotic behavior andstability properties have been investigated extensively bymany authors over the past few years [14ndash18] However toour knowledge there are few published papers discussingthe existence of periodic solutions for neutral Lotka-Volterrasystem with distributed delays and impulses In this paperwe are concerned with the following neutral Lotka-Volterrasystem with distributed delays and impulses

2 International Journal of Differential Equations

1199091015840

119894(119905) = 119909

119894(119905)[

[

119903119894(119905) minus

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

119894 = 1 2 119899 119905 = 119905119896

Δ119909119894(119905119896)=minus119868119894119896(119909119894(119905119896)) 119894=1 2 119899 119896=1 2

(2)

where 119903119894 119886119894119895 119887119894119895 119888119894119895

isin 119862(119877 119877+) (119894 119895 = 1 2 119899) are 120596-

periodic functions and 120591119894119895 120590119894119895isin (0 +infin) (119894 119895 = 1 2 119899)

and 119891119894119895 119892119894119895

isin 119862(119877 119877+) satisfying int

0

minus120591119894119895

119891119894119895(120585)119889120585 = 1

int

0

minus120590119894119895

119892119894119895(120585)119889120585 = 1 119894 119895 = 1 2 119899 Moreover Δ119909(119905

119896) =

119909(119905+

119896) minus 119909(119905

119896) (here 119909(119905+

119896) represents the right limit of 119909(119905)

at the point 119905119896) 119868119894119896

isin 119862(119877+ 119877+) that is 119909 changes

decreasingly suddenly at times 119905119896 120596 gt 0 is a constant 119877 =

(minusinfin +infin) 119877+= [0 +infin) We assume that there exists an

integer 119902 gt 0 such that 119905119896+119902

= 119905119896+ 120596 119868

119894(119896+119902)= 119868119894119896 where

0 lt 1199051lt 1199052lt sdot sdot sdot lt 119905

119902lt 120596

The main purpose of this paper is by using a fixed-point theorem of strict-set-contraction [19 20] to establishnew criteria to guarantee the existence of positive periodicsolutions of the system (2)

For convenience we introduce the notation

119891119872= max119905isin[0120596]

1003816100381610038161003816119891 (119905)

1003816100381610038161003816 119891

119871= min119905isin[0120596]

1003816100381610038161003816119891 (119905)

1003816100381610038161003816

120574119894= lim119906rarr0

sup sum

119905le119905119896le119905+120596

119868119894119896(119906)

119906

119894 = 1 2 119899

120575119894= 119890minusint120596

0119903119894(119905)119889119905

119894 = 1 2 119899

120572119894119895= int

120596

0

[120575119895119886119894119895(119905) minus 119887

119894119895(119905) minus 119888

119894119895(119905)] 119889119905

120573119894119895= int

120596

0

[119886119894119895(119905) + 119887

119894119895(119905) + 119888

119894119895(119905)] 119889119905 119894 119895 = 1 2 119899

(3)

where 119891 is a continuous 120596-periodic functionThroughout this paper we assume that

(1198601) 120575119894= 119890minusint120596

0119903119894(119905)119889119905

lt 1 119894 = 1 2 119899(1198602) 120575119895119886119894119895(119905) minus 119887

119894119895(119905) minus 119888

119894119895(119905) ge 0 119894 119895 = 1 2 119899

(1198603) (1 + 119903

119871

119894)(1205752

119894(1 minus 120575

119894))120572119894119895ge max

119905isin[0120596]119886119894119895(119905) + 119887

119894119895(119905) +

119888119894119895(119905) 119894 119895 = 1 2 119899

(1198604) (119903119872

119894minus1)(1120575

119894(1 minus 120575119894))120573119894119895le min

119905isin[0120596]120575119895119886119894119895(119905) minus 119887

119894119895(119905) minus

119888119894119895(119905) 119894 119895 = 1 2 119899

(1198605) max

1le119894le119899sum119899

119895=1119888119872

119894119895 le min

1le119894le119899(1205752

119894(1 minus 120575

119894))

min1le119895le119899

120572119894119895

The paper is organized as follows In Section 2 we givesome definitions and lemmas to prove themain results of thispaper In Section 3 by using a fixed-point theorem of strict-set-contraction we established some criteria to guarantee theexistence of at least one positive periodic solution of system(2) Finally in Section 4 we give an example to show thevalidity of our result

2 Preliminaries

In order to obtain the existence of a periodic solution ofsystem (2) we first introduce some definitions and lemmas

Definition 1 (see [13]) A function 119909119894 119877 rarr (0 +infin) is said

to be a positive solution of (2) if the following conditions aresatisfied

(a) 119909119894(119905) is absolutely continuous on each (119905

119896 119905119896+1)

(b) for each 119896 isin 119885+ 119909119894(119905+

119896) and 119909

119894(119905minus

119896) exist and 119909

119894(119905minus

119896) =

119909119894(119905119896)

(c) 119909119894(119905) satisfies the first equation of (2) for almost every-

where in 119877 and 119909119894(119905119896) satisfies the second equation of

(2) at impulsive point 119905119896 119896 isin 119885

+

Definition 2 (see [21]) Let 119883 be a real Banach space and 119864 aclosed nonempty subset of119883 119864 is a cone provided

(i) 120572119909 + 120573119910 isin 119864 for all 119909 119910 isin 119864 and all 120572 120573 ge 0(ii) 119909 minus119909 isin 119864 imply 119909 = 0

Definition 3 (see [21]) Let 119860 be a bounded subset in 119883Define

120572119883(119860) = inf 120575 gt 0 there is a finite number

of subsets 119860119894sub 119860 such that

119860 = ⋃

119894

119860119894and diam (119860

119894) le 120575

(4)

where diam(119860119894) denotes the diameter of the set 119860

119894 obvi-

ously 0 le 120572119883(119860) lt infin So 120572

119883(119860) is called the Kuratowski

measure of noncompactness of 119883

Definition 4 (see [21]) Let 119883 119884 be two Banach spaces and119863 sub 119883 a continuous and boundedmap 119879 119863 rarr 119884 is called119896-set contractive if for any bounded set 119878 sub 119863 we have

120572119884(119879 (119878)) le 119896120572

119883(119878) (5)

120601 is called strict-set-contractive if it is 119896-set-contractive forsome 0 le 119896 lt 1

Definition 5 (see [22]) The set 119865 isin 119875119862120596is said to be quasi-

equicontinuous in [0 120596] if for any 120598 gt 0 there exists 120575 gt 0

such that if 119909 isin 119865 119896 isin 119873+ 1199051 1199052isin (119905119896minus1 119905119896)cap[0 120596] and |119905

1minus

1199052| lt 120575 then |119909(119905

1) minus 119909(119905

2)| lt 120598

International Journal of Differential Equations 3

Lemma 6 (see [22]) The set 119865 sub 119875119862120596is relatively compact if

and only if

(1) 119865 is bounded that is 119909 le 119872 for each 119909 isin 119865 andsome119872 gt 0

(2) 119865 is quasi-equicontinuous in [0 120596]

Lemma 7 119909119894(119905) is an 120596-periodic solution of (2) is equivalent

to 119909119894(119905) is an 120596-periodic solution of the following equation

119909119894(119905) = int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

(6)

where

119866119894(119905 119904) =

119890minusint119904

119905119903119894(120579)119889120579

1 minus 119890minusint120596

0119903119894(120579)119889120579

119904 isin [119905 119905 + 120596] 119894 = 1 2 119899

(7)

Proof Assume that 119909119894(119905) isin 119883 119894 = 1 2 119899 is a periodic

solution of (2) Then we have

119889

119889119905

[119909119894(119905) 119890minusint119905

0119903119894(120579)119889120579

]

= 119890minusint119905

0119903119894(120579)119889120579

119909119894(119905)

times[

[

minus

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

119905 = 119905119896 119894 = 1 2 119899

(8)

Integrating the previous equation with [119905 119905 + 120596] we can have

119909119894(119904) 119890minusint119904

0119903119894(120579)119889120579

100381610038161003816100381610038161003816

1199051198981+119899120596

119905

+ 119909119894(119904) 119890minusint119904

0119903119894(120579)119889120579

100381610038161003816100381610038161003816

1199051198982+119899120596

1199051198981+119899120596

+ sdot sdot sdot + 119909119894(119904) 119890minusint119904

0119903119894(120579)119889120579

100381610038161003816100381610038161003816

119905+120596

119905119898119902+119899120596

= int

119905+120596

119905


0119903119894(120579)119889120579

119909119894(119904)

times[

[

minus

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)minus

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

(9)

where 119905119898119896

+ 119899120596 isin (119905 119905 + 120596) 119898119896isin 1 2 119902 119896 =

1 2 119902 119899 isin 119885+ Therefore

119909119894(119905) 119890minusint119905

0119903119894(120579)119889120579

[1 minus 119890minusint119905+120596

119905119903119894(120579)119889120579

]

+ sum

119905le119905119896lt119905+120596

Δ119909119894(119905119898119896) 119890minusint

119905119898119896+119899120596

0119903119894(120579)119889120579

= int

119905+120596

119905


0119903119894(120579)119889120579

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

(10)

which can be transformed into

119909119894(119905) = int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896)) 119894 = 1 2 119899

(11)


Thus 119909119894is a periodic solution of (6) If 119909

119894(119905) isin 119864 119894 =

1 2 119899 is a periodic solution of (6) for any 119905 = 119905119896 from

(6) we have

1199091015840

(119905) =

119889

119889119905

int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

=[

[

119866119894(119905 119905 + 120596) 119909

119894(119905 + 120596)

times (

119899

sum

119895=1

119886119894119895(119905 + 120596) 119909

119895(119905 + 120596)

+

119899

sum

119895=1

119887119894119895(119905 + 120596) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905 + 120596) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120596 + 120585) 119889120585)

minus 119866119894(119905 119905) 119909

119894(119905)

times (

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905+120585) 119889120585)

]

]

+119903119894(119905) 119909119894(119905)

= 119909119894(119905)[

[

119903119894(119905) minus

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909j (119905 minus 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

(12)

For any 119905 = 119905119895 119895 isin 119885

+ we have from (6) that

119909119894(119905+

119895) minus 119909119894(119905119895)

= int

119905119895+120596

119905119895

[119866119894(119905+

119895 119904) minus 119866

119894(119905119895 119904)] 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905+

119895le119905119896lt119905119895+120596

119866119894(119905+

119895 119905119896) 119868119894119896(119909119894(119905119896))

minus sum

119905119895le119905119896lt119905119895+120596

119866119894(119905119895 119905119896) 119868119894119896(119909119894(119905k))

= minus119868119894119896(119909 (119905119896))

(13)

Hence 119909119894(119905) is a positive 120596-periodic solution of (2) Thus

we complete the proof of Lemma 7

Lemma 8 (see [20 23]) Let 119864 be a cone of the real Banachspace 119883 and 119864

119903119877= 119909 isin 119864 119903 le 119909 le 119877 with 0 lt 119903 lt 119877

Assume that 119860 119864119903119877

rarr 119864 is strict-set-contractive such thatone of the following two conditions is satisfied

(a) 119860119909 ≰ 119909 for all 119909 isin 119864 119909 = 119903 and 119860119909 ge

119909 for all 119909 isin 119864 119909 = 119877

(b) 119860119909 ge 119909 for all 119909 isin 119864 119909 = 119903 and 119860119909 ≰

119909 for all 119909 isin 119864 119909 = 119877

Then 119860 has at least one fixed point in 119864119903119877

In order to apply Lemma 7 to system (1) one sets

119875119862 (119877) = 119909 = (1199091 1199092 119909

119899)119879

119877 997888rarr 119877 | 119909 isin 119862 ((119905119896 119905119896+1) 119877)

exist119909 (119905minus

119896) = 119909 (119905

119896) 119909 (119905

+

119896) 119896 isin 119873

1198751198621

(119877) = 119909 = (1199091 1199092 sdot sdot sdot 119909

119899)119879

119877 997888rarr 119877 | 119909 isin 1198621((119905119896 119905119896+1) 119877)

exist1199091015840(119905minus

119896) = 1199091015840(119905119896) 119909 (119905

+

119896) 119896 isin 119873 119905 isin 119877

(14)

Define

119883 = 119909 = (1199091 1199092 119909

119899)119879

119909 isin 119875119862 (119877) | 119909 (119905 + 120596) = 119909 (119905)

(15)


with the norm defined by 1199090= sum119899

119894=1|119909119894|0 where |119909

119894|0=

sup119905isin[0120596]

|119909119894(119905)| 119894 = 1 2 119899 and

119884=119909=(1199091 1199092 119909

119899)119879

119909 isin 1198751198621

(119877) | 119909 (119905+120596)

=119909 (119905) 119905isin119877

(16)


119894=1|119909119894|1 where |119909

119894|1=

max|119909119894|0 |1199091015840

119894|0Then119883 and119884 are both Banach spaces Define

the cone 119864 in 119884 by

119864=119909=(1199091 1199092 119909

119899)119879

119909isin1198751198621

(119877) | 119909119894(119905)=120575

119895

1003816100381610038161003816119909119894(119905)10038161003816100381610038161

119905isin[0 120596] 119894 119895=1 2 119899

(17)

Let the map 119879 be defined by

(119879119909) (119905) = ((1198791119909) (119905) (119879

2119909) (119905) (119879

119899119909) (119905))

119879

(18)

where 119909 isin 119864 119905 isin 119877

(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904)int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

(19)

where

119866119894(119905 119904) =


119905119903119894(120579)119889120579


0119903119894(120579)119889120579

119904 isin [119905 119905 + 120596] 119894 = 1 2 119899

(20)

It is obvious to see that119866119894(119905+120596 119904+120596) = 119866

119894(119905 119904) 120597119866

119894(119905 119904)120597119905 =

119903119894(119905)119866119894(119905 119904) 119866

119894(119905 119905 + 120596) minus 119866

119894(119905 119905) = minus1 and

120575119894

1 minus 120575119894

le 119866119894(119905 119904) le

1

1 minus 120575119894

119904 isin [119905 119905 + 120596] 119894 = 1 2 119899

(21)

In what follows we will give some lemmas concerning 119864and 119860 defined by (17) and (18) respectively

Lemma 9 Assume that (1198601)ndash(1198603) hold

(i) Ifmax119903119872119894 119894 = 1 2 119899 le 1 then 119879 119864 rarr 119864 is well

defined

(ii) If (1198604) holds and max119903119872

119894 119894 = 1 2 119899 gt 1 then

119879 119864 rarr 119864 is well defined

Proof For any 119909 isin 119864 it is clear that 119879119909 isin 1198751198621(119877) From (18)for 119905 isin [0 120596] we have

(119879119894119909) (119905 + 120596)

= int

119905+2120596

119905+120596

119866119894(119905 + 120596 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905+120596le119905119896lt119905+2120596

119866119894(119905 + 120596 119905

119896) 119868119894119896(119909119894(119905119896))

= int

119905+120596

119905

119866119894(119905 + 120596 119906 + 120596) 119909

119894(119906 + 120596)

times[

[

119899

sum

119895=1

119886119894119895(119906 + 120596) 119909

119895(119906 + 120596) +

119899

sum

119895=1

119887119894119895(119906 + 120596)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119906 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119906 + 120596)int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119906 + 120596 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= int

119905+120596

119905

119866119894(119905 119906) 119909

119894(119906)

times[

[

119899

sum

119895=1

119886119894119895(119906) 119909119895(119906) +

119899

sum

119895=1

119887119894119895(119906) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119906+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119906) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119906+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= (119879119894119909) (119905) 119894 = 1 2 119899

(22)

That is (119879119894119909)(119905 + 120596) = (119879

119894119909)(119905) 119905 isin [0 120596] So 119879119909 isin 119884 In

view of (1198602) for 119909 isin 119864 119905 isin [0 120596] we have


119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

ge

119899

sum

119895=1

119886119894119895(119905) 120575119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161minus

119899

sum

119895=1

119887119894119895(119905)

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161minus

119899

sum

119895=1

119888119894119895(119905)

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

119899

sum

119895=1

[119886119894119895(119905) 120575119895minus119887119894119895(119905)minus119888119894119895(119905)]

100381610038161003816100381610038161199091015840

119895

100381610038161003816100381610038161ge 0 119894=1 2 119899

(23)

Therefore for 119909 isin 119864 119905 isin [0 120596] we find

100381610038161003816100381611987911989411990910038161003816100381610038160

le

1

1 minus 120575119894

int

120596

0

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)+

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

(24)

(119879119894119909) (119905)

ge

120575119894

1 minus 120575119894

int

119905+120596

119905

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

=

120575119894

1 minus 120575119894

times int

120596

0

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

ge 120575119894

100381610038161003816100381611987911989411990910038161003816100381610038160 119894 = 1 2 119899

(25)

Now we show that (119879119894119909)(119905) ge 120575

119894|119879119894119909|1 119894 = 1 2 119899 119905 isin

[0 120596] From (18) we obtain

(119879119894119909)1015840

(119905)

= 119866119894(119905 119905 + 120596) 119909

119894(119905 + 120596)

times[

[

119899

sum

119895=1

119886119894119895(119905 + 120596) 119909

119895(119905 + 120596)

+

119899

sum

119895=1

119887119894119895(119905 + 120596) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905 + 120596) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120596 + 120585) 119889120585

]

]

minus 119866119894(119905 119905) 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)+

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119903119894(119905) (119879119894119909) (119905)

= 119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)


times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905+120585) 119889120585

]

]

119894 = 1 2 119899

(26)

It follows from (23) and (25) that if (119879119894119909)1015840(119905) ge 0 119894 =

1 2 119899 then

(119879119894119909)1015840

(119905) le 119903119894(119905) (119879119894119909) (119905) le 119903

119872

119894(119879119894119909) (119905) le (119879

119894119909) (119905)

119894 = 1 2 119899

(27)

On the other hand from (25) and (1198603) if (119879

119894119909)1015840(119905) lt 0 119894 =

1 2 119899 thenminus (119879119894119909)1015840

(119905)

= minus119903119894(119905) (119879119894119909) (119905) + 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

le1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

[119886119894119895(119905) + 119887

119894119895(119905) + 119888

119894119895(119905)]

1003816100381610038161003816119909119894

10038161003816100381610038161minus 119903119871

119894(119879119894119909) (119905)

le (1 + 119903119871

119894)

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)]

1003816100381610038161003816119909119894

10038161003816100381610038161119889119904

minus 119903119871

119894(119879119894119909) (119905)

= (1 + 119903119871

119894)int

119905+120596

119905

120575119894

1 minus 120575119894

120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus c

119894119895(119904)]

1003816100381610038161003816119909119894

10038161003816100381610038161119889119904 minus 119903

119871

119894(119879119894119909) (119905)

le (1 + 119903119871

119894)int

119905+120596

119905

int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896)) minus 119903

119871

119894(119879119894119909) (119905)

= (1 + 119903119871

119894) (119879119894119909) (119905) minus 119903

119871

119894(119879119894119909) (119905)

= (119879119894119909) (119905) 119894 = 1 2 119899

(28)

It follows from (27) and (28) that |(119879119894119909)1015840|0le |119879119894119909|0 119894 =

1 2 119899 So |119879119894119909|1= |119879119894x|0 119894 = 1 2 119899 By (25) we have

(119879119894119909)(119905) ge 120575

119894|119879119894119909|1 119894 = 1 2 119899 Hence 119879119909 isin 119864 This

completes the proof of (i) In view of the proof of (ii) we onlyneed to prove that (119879

119894119909)1015840(119905) ge 0 119894 = 1 2 119899 implies

(119879119894119909)1015840

(119905) le (119879119894119909) (119905) 119894 = 1 2 119899 (29)

From (23) (26) (1198602) and (119860

4) we have

(119879119894119909)1015840

(119905)

= 119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

le 119903119894(119905) (119879119894119909) (119905) minus 120575

119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[120575119895119886119894119895(119905) minus 119887

119894119895(119905) minus 119888

119894119895(119905)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

le 119903119872

119894(119879119894119909) (119905) minus 120575

119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119903119872

119894minus 1

120575119894(1 minus 120575

119894)

times

119899

sum

119895=1

int

120596

0

[119886119894119895(119904) + 119887

119894119895(119904) + 119888

119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1)int

119905+120596

119905

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[119886119894119895(119904) + 119887

119894119895(119904) + 119888

119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904

le 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1)


times int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1) (119879

119894119909) (119905) = (119879

119894119909) (119905)

119894 = 1 2 119899

(30)

The proof of (ii) is complete Thus we complete the proof ofLemma 9

Lemma 10 Assume that (1198601)ndash(1198603) hold and

119877max1le119895le119899

sum119899

119895=1119887119872

119894119895 lt 1

(i) Ifmax119903119872119894 119894 = 1 2 119899 le 1 then 119879 119864⋂Ω

119877rarr 119864

is strict-set-contractive

(ii) If (1198604) holds and min119903119872

119894 119894 = 1 2 119899 gt 1 then

119879 119864⋂Ω119877rarr 119864 is strict-set-contractive where Ω

119877=

119909 isin 119884 |119909|1lt 119877

Proof We only need to prove (i) since the proof of (ii) issimilar It is easy to see that 119879 is continuous and boundedNow we prove that a 120572

119884(119879(119878)) le 119877max

1le119895le119899sum119899

119895=1119887119872

119894119895120572119884(119878)

for any bounded set 119878 isin Ω119877 Let 120578 = 120572

119884(119878) Then for any

positive number 120598 lt 119877max1le119895le119899

sum119899

119895=1119887119872

119894119895120578 there is a finite

family of subsets 119878119894 satisfying 119878 = ⋃

119894119878119894with diam(119878

119894) le 120578+120598

Therefore

1003816100381610038161003816119909 minus 119910

10038161003816100381610038161le 120578 + 120598 for any 119909 119910 isin 119878

119894 (31)

As 119878 and 119878119894are precompact in119883 it follows that there is a finite

family of subsets 119878119894119895 of 119878119894such that 119878

119894= ⋃119895119878119894119895and

1003816100381610038161003816119909 minus 119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895 (32)

In addition for any 119909 isin 119878 and 119905 isin [0 120596] we have

(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1198772

1 minus 120575119894

int

120596

0

[

[

119899

sum

119895=1

119886119894119895(119904) +

119899

sum

119895=1

119887119894119895(119904) +

119899

sum

119895=1

119888119894119895(119904)]

]

119889119904

+

1

1 minus 120575119894

sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896)) = 119861

119894 119894 = 1 2 119899

(33)10038161003816100381610038161003816(119879119894119909)1015840

(119905)

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899

(34)

Hence

(119879119909)0le

119899

sum

119894=1

119861119894

10038171003817100381710038171003817(119879119909)1015840100381710038171003817100381710038170le

119899

sum

119894=1

[

[

119861119894119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895)]

]

(35)

Applying the Arzela-Ascoli theorem we know that 119879(119878) isprecompact in119883Then there is a finite family of subsets 119878

119894119895119896

of 119878119894119895such that 119878

119894119895= ⋃119896119878119894119895119896

and

1003816100381610038161003816119879119909 minus 119879119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895119896 (36)


From (23) (26) (31)ndash(34) and (1198602) for any 119909 119910 isin 119878

119894119895119896 we

have

10038161003816100381610038161003816(119879119894119909)1015840

minus (119879119894119910)1015840100381610038161003816100381610038160

= max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119903

119894(119905) (119879119894119910) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119910119894(119905)[

[

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le max119905isin[0120596]

1003816100381610038161003816119903119894(119905) [(119879

119894119909) (119905) minus (119879

119894119910) (119905)]

1003816100381610038161003816

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119894[

[

(

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585)

minus(

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585 +

119899

sum

119895=1

119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ max119905isin[0120596]

10038161003816100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905) minus 119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894

1003816100381610038161003816(119879119894119909) minus (119879

119894119910)10038161003816100381610038160

+ 119877max119905isin[0120596]

119899

sum

119895=1

[

[

119886119894119895(119905)

10038161003816100381610038161003816119909119895(119905)minus119910

119895(119905)

10038161003816100381610038161003816+119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585)

times1003816100381610038161003816119909119894(119905+120585)minus119910

119894(119905+120585)

1003816100381610038161003816119889120585+119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585)

times

100381610038161003816100381610038161199091015840

119895(119905+120585) minus119910

1015840

(119905minus120585)

10038161003816100381610038161003816119889120585]

]

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905)minus119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905)

timesint

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585)

100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894120598+119877120598

119899

sum

119895=1

119886119872

119894119895+119877120598

119899

sum

119895=1

119887119872

119894119895+1003816100381610038161003816119909119894

10038161003816100381610038160(120578+120598)

times

119899

sum

119895=1

119888119872

119894119895+119877120598

119899

sum

119895=1

(119886119872

119894119895+119887119872

119894119895+119888119872

119894119895)

=1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895120578 + 119861119894120598

(37)

where

119861119894= 119903119872

119894+ 2119877

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899 (38)


From (36) and (37) we obtain

1003817100381710038171003817119879119909 minus 119879119910

10038171003817100381710038171le (

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895)120578 + 120598

119899

sum

119894=1

119861119894

le 119877max1le119894le119899

119899

sum

119895=1

119888119872

119894119895

120578 + 120598

119899

sum

119894=1

119861119894 for any 119909 119910 isin 119878

119894119895119896

(39)

As 120598 is arbitrarily small it follows that

120572119884(119879 (119878)) le 119877max

1le119895le119899

119899

sum

119895=1

119888119872

119894119895

120572119884(119878) (40)

Therefore 119879 is strict-set-contractive The proof of Lemma 10is complete

3 Main Results

Our main result of this paper is as follows

Theorem 11 Assume that (1198601)ndash(1198603) (1198605) hold

(i) If max119903119872119894 119894 = 1 2 119899 le 1 then system (2) has at

least one positive 120596-periodic solution


119894 119894 = 1 2 119899 gt 1 then

system (2) has at least one positive 120596-periodic solution

Proof We only need to prove (i) since the proof of (ii) issimilar Let

119877 = (min119894isin[1119899]

1205752

119894

1 minus 120575119894

min119895isin[1119899]

120572119894119895)

minus1

0 lt 119903 lt min119894isin[1119899]

120575119894(1 minus 120575

119894) minus 120574119894

max119895isin[1119899]

120573119894119895

(41)

Then it is easy to see that 0 lt 119903 lt 119877 From Lemmas 9 and10 we know that 119879 is strict-set-contractive on 119864

119903119877 In view

of (26) we see that if there exists 119909lowast isin 119864 such that 119879119909lowast =119909lowast then 119909lowast is one positive 120596-periodic solution of system (2)

Nowwewill prove that condition (ii) of Lemma 7holds Firstwe prove that 119879119909 ge 119909 for all 119909 isin 119864 119909

1lt 119903 Otherwise

there exists 119909 isin 119864 1199091lt 119903 such that 119879119909 ge 119909 So 119909

0gt 0

and 119879119909 minus 119909 ge 0 which implies that

119879119894119909 (119905) minus 119909

119894(119905) ge 120575

119894

1003816100381610038161003816119879119894119909 minus 119909119894

10038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(42)

Moreover for 119905 isin [0 120596] we have(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

int

120596

0

[119886119894119895(119904)+119887119894119895(119904)+119888119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904+120574119894

le

120573119894119895119903 + 120574119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160le 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160 119894 = 1 2 119899

(43)In view of (42) and (43) we obtain

1199090le 119879119909

0le max1le119894le119899

120575119894 1199090lt 119909

0 (44)

which is a contradiction Finally we prove that 119879119909 ≰

119909 for all 119909 isin 119864 and 1199091= 119877 also hold For this case

suppose for the sake of contradiction that there exist 119909 isin

119864 and 1199091= 119877 such that 119879119909 le 119909 Furthermore for any

119905 isin [0 120596] we have119909119894(119905) minus 119879

119894119909 (119905) ge 120575

119894

1003816100381610038161003816119909119894minus 11987911989411990910038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(45)

In addition for any 119905 isin [0 120596] we find(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

gt

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)] 119889119904

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161) 119894 = 1 2 119899

(46)


Thus we have

1198791199090=

119899

sum

119894=1

100381610038161003816100381611987911989411990910038161003816100381610038160gt

119899

sum

119894=1

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161)

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

1198772= 119877

(47)


1199091ge 119879119909

1ge 119879119909

0gt 119877 (48)

which is a contradictionTherefore condition (ii) of Lemma 7holds By Lemma 7 we see that 119879 has at least one nonzerofixed point in 119864 Thus the system (6) has at least one positive120596-periodic solutionTherefore it follows from Lemma 6 thatsystem (2) has a positive 120596-periodic solution The proof ofTheorem 11 is complete

Remark 12 If 119868119894119896(119909(119905119896)) = 0 we can easily derive the

corresponding results in [10] Sowe extend the correspondingresults in [10]

4 Example

In this section we give an example to show the effectivenessof our result

Example 1 Consider the following nonimpulsive system

1199091015840

1(119905)

= 1199091(119905) [

1 minus sin 11990524

minus (10 + sin 119905) 1199091(119905)

minus (9 minus cos 119905) 1199092(119905) minus (3 minus 2 cos 119905)

times int

0

minus12059111

11989111(120585) 1199091(119905 + 120585) 119889120585 minus

5 + sin 11990515

times int

0

minus12059112

11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)

times int

0

minus12059011

11989211(120585) 1199091015840

1(119905 + 120585) 119889120585 minus

4 minus 3 cos 1199056

timesint

0

minus12059012

11989212(120585) 1199091015840

2(119905 + 120585) 119889120585]

1199091015840

2(119905)

= 1199092(119905) [

1 + cos 1199058120587

minus (12 + 3 cos 119905) 1199091(119905)

minus (7 minus sin 119905) 1199092(119905) minus (2 + 3 sin 119905)

times int

0

minus12059121

11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)

times int

0

minus12059122

11989122(120585) 1199092(119905 + 120585) 119889120585 minus

4 + sin 11990512

times int

0

minus12059021

11989221(120585) 1199091015840

2(119905 + 120585) 119889120585 minus

5 minus cos 11990510

timesint

0

minus12059022

11989222(120585) 1199091015840

2(119905 + 120585) 119889120585]

(49)

where 120591119894119895 120590119894119895

isin (0 +infin) (119894 119895 = 1 2) and 119891119894119895 119892119894119895

isin

119862(119877 119877+) satisfying int0

minus120591119894119895

119891119894119895(120585)119889120585 = 1 int

0

minus120590119894119895

119892119894119895(120585)119889120585 119894 119895 = 1 2

Obviously

1199031(119905) =


1199032=

1 + cos 1199058120587

11988611(119905) = 10 + sin 119905

11988612(119905) = 9 minus cos 119905 119886

21(119905) = 12 + 3 cos 119905

11988622(119905) = 7 minus sin 119905 119887

11(119905) = 3 minus 2 cos 119905

11988712(119905) =

5 + sin 11990515

11988721(119905) = 2 + 3 sin 119905 119887

22(119905) = 1 minus cos 119905

11988811(119905) = 1 + sin 119905 119888

12(119905) =

4 minus 3 cos 1199056

11988821(119905) =

4 + sin 11990512

11988822(119905) =


119903119871

1= 119903119871

2= 0

(50)

Furthermore we obtain

1205751= 119890minus(12058712)

lt 1 1205752= 119890minus(14)

lt 1

12057211= 20120587120575

1minus 8120587 asymp 238007

12057212= 18120587120575

2minus 2120587 asymp 372423

12057221= 24120587120575

1minus

14120587

3

asymp 440594

12057222= 14120587120575

2minus 3120587 asymp 244284


min1le119895le2

1205721119895 = 12057211= 20120587120575

1minus 8120587 asymp 238007

min1le119895le2

1205722119895 = 12057222= 14120587120575

2minus 3120587 asymp 244284

120575111988611(119905) minus 119887

11(119905) minus 119888

11(119905) gt 0009 gt 0

120575211988612(119905) minus 119887

12(119905) minus 119888

12(119905) gt 1 gt 0

120575111988621(119905) minus 119887

21(119905) minus 119888

21(119905) gt 4 gt 0

120575211988622(119905) minus 119887

22(119905) minus 119888

22(119905) gt 2 gt 0

(1 + 119903119871

1)

1205752

112057211

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 8120587)

1 minus 119890minus(12058712)

asymp 652614 gt 18

ge max0le119905le2120587

11988611(119905) + 119887

11(119905) + 119888

11(119905)

(1 + 119903119871

1)

1205752

112057212

1 minus 1205751

=

119890minus(1205876)

(181205871205752minus 2120587)

1 minus 119890minus(12058712)

asymp 1021184 gt

347

30

ge max0le119905le2120587

11988612(119905) + 119887

12(119905) + 119888

12(119905)

(1 + 119903119871

2)

1205752

212057221

1 minus 1205752

=

119890minus(12)

(241205871205751minus (141205873))

1 minus 119890minus(14)

asymp 1133411 gt

245

12

gt max0le119905le2120587

11988621(119905) + 119887

21(119905) + 119888

21(119905)

(1 + 119903119871

2)

1205752

212057222

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411 gt

53

5

gt max0le119905le2120587

11988622(119905) + 119887

22(119905) + 119888

22(119905)

1205752

1min1le119895le2

1205721119895

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 6120587)

1 minus 119890minus(12058712)

asymp 652614

1205752

2min1le119895le2

1205722119895

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411

min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

=

1205752

2min1le119895le2

1205722119895

1 minus 1205751

asymp 628411

119888119872

11+ 119888119872

12=

19

6

119888119872

21+ 119888119872

22=

61

60

max1le119894le2

2

sum

119895=1

119889119872

119894119895

=

19

6

(51)

Therefore

19

6

= max1le119894le2

2

sum

119894=1

119889119872

119894119895 lt min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

asymp 628411

(52)

Hence (1198601)ndash(1198603) (1198605) hold and 119886

119872

119894le 1 119894 = 1 2

According toTheorem 11 system (49) has at least one positive2120587-periodic solution

Acknowledgments

The authors are thankful to the referees and editor for theimprovement of the paper This work was supported bythe Construct Program of the Key Discipline in HunanProvince Research was supported by the National NaturalScience Foundation of China (10971229 11161015) the ChinaPostdoctoral Science Foundation (2012M512162) and HunanProvincial Natural Science-Hengyang United Foundation ofChina (11JJ9002)

References

[1] P Chesson ldquoUnderstanding the role of environmental variationin population and community dynamicsrdquo Theoretical Popula-tion Biology vol 64 no 3 pp 253ndash254 2003

[2] J Zhen and Z E Ma ldquoPeriodic solutions for delay differentialequations model of plankton allelopathyrdquo Computers amp Mathe-matics with Applications vol 44 no 3-4 pp 491ndash500 2002

[3] X Liu and G Ballinger ldquoBoundedness for impulsive delaydifferential equations and applications to population growthmodelsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 53 no 7-8 pp 1041ndash1062 2003

[4] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press New York NY USA1993

[5] Y K Li and Y Kuang ldquoPeriodic solutions of periodic delayLotka-Volterra equations and systemsrdquo Journal of MathematicalAnalysis and Applications vol 255 no 1 pp 260ndash280 2001

[6] Y K Li and L F Zhu ldquoExistence of periodic solutions of discreteLotka-Volterra systems with delaysrdquo Bulletin of the Institute ofMathematics Academia Sinica vol 33 no 4 pp 369ndash380 2005

[7] H Zhao and N Ding ldquoExistence and global attractivity ofpositive periodic solution for competition-predator systemwithvariable delaysrdquoChaos SolitonsampFractals vol 29 no 1 pp 162ndash170 2006

[8] Y Song M Han and Y Peng ldquoStability and Hopf bifurcationsin a competitive Lotka-Volterra system with two delaysrdquo ChaosSolitons amp Fractals vol 22 no 5 pp 1139ndash1148 2004

[9] Z Yang and J Cao ldquoPositive periodic solutions of neutral Lotka-Volterra system with periodic delaysrdquo Applied Mathematics andComputation vol 149 no 3 pp 661ndash687 2004

[10] Y K Li ldquoPositive periodic solutions of periodic neutral Lotka-Volterra system with distributed delaysrdquo Chaos Solitons ampFractals vol 37 no 1 pp 288ndash298 2008

[11] A M Samoikleno and N A Perestyuk Impulsive DifferentialEquations World Scientific Singapore 1995

[12] S T Zavalishchin and A N SesekinDynamic Impulse SystemsTheory and Applications Kluwer Academic Publishers Dor-drecht The Netherlands 1997


[13] V Lakshmikantham D D Bainov and P S Simenov Theoryof Impulsive Differential Equations World Science Singapore1989

[14] J R Yan A P Zhao and J J Nieto ldquoExistence and globalattractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systemsrdquo Mathematical andComputer Modelling vol 40 no 5-6 pp 509ndash518 2004


[16] W Zhang and M Fan ldquoPeriodicity in a generalized ecologicalcompetition system governed by impulsive differential equa-tions with delaysrdquo Mathematical and Computer Modelling vol39 no 4-5 pp 479ndash493 2004

[17] S L Sun and L S Chen ldquoExistence of positive periodic solutionof an impulsive delay logistic modelrdquo Applied Mathematics andComputation vol 184 no 2 pp 617ndash623 2007

[18] X Li X Lin D Jiang and X Zhang ldquoExistence andmultiplicityof positive periodic solutions to functional differential equa-tions with impulse effectsrdquoNonlinear Analysis Theory Methodsamp Applications vol 62 no 4 pp 683ndash701 2005

[19] R E Gaines and J L Mawhin Coincidence Degree Theory andNonlinear Differential Equations Springer Berlin Germany1977

[20] N P Cac and J A Gatica ldquoFixed point theorems for mappingsin orderedBanach spacesrdquo Journal ofMathematical Analysis andApplications vol 71 no 2 pp 547ndash557 1979

[21] D J GuoNonlinear Functional Analysis in Chinese ShanDongScience and Technology Press 2001

[22] D Baınov and P Simeonov Impulsive Differential EquationsPeriodic Solutions and Applications vol 66 of Pitman Mono-graphs and Surveys in Pure and Applied Mathematics 1993

[23] D J Guo ldquoPositive solutions to nonlinear operator equa-tions and their applications to nonlinear integral equationsrdquoAdvances in Mathematics vol 13 no 4 pp 294ndash310 1984(Chinese)

Submit your manuscripts athttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


1199091015840

119894(119905) = 119909

119894(119905)[

[

119903119894(119905) minus

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

119894 = 1 2 119899 119905 = 119905119896

Δ119909119894(119905119896)=minus119868119894119896(119909119894(119905119896)) 119894=1 2 119899 119896=1 2

(2)

where 119903119894 119886119894119895 119887119894119895 119888119894119895

isin 119862(119877 119877+) (119894 119895 = 1 2 119899) are 120596-

periodic functions and 120591119894119895 120590119894119895isin (0 +infin) (119894 119895 = 1 2 119899)

and 119891119894119895 119892119894119895

isin 119862(119877 119877+) satisfying int

0

minus120591119894119895

119891119894119895(120585)119889120585 = 1

int

0

minus120590119894119895

119892119894119895(120585)119889120585 = 1 119894 119895 = 1 2 119899 Moreover Δ119909(119905

119896) =

119909(119905+

119896) minus 119909(119905

119896) (here 119909(119905+

119896) represents the right limit of 119909(119905)

at the point 119905119896) 119868119894119896

isin 119862(119877+ 119877+) that is 119909 changes

decreasingly suddenly at times 119905119896 120596 gt 0 is a constant 119877 =

(minusinfin +infin) 119877+= [0 +infin) We assume that there exists an

integer 119902 gt 0 such that 119905119896+119902

= 119905119896+ 120596 119868

119894(119896+119902)= 119868119894119896 where

0 lt 1199051lt 1199052lt sdot sdot sdot lt 119905

119902lt 120596

The main purpose of this paper is by using a fixed-point theorem of strict-set-contraction [19 20] to establishnew criteria to guarantee the existence of positive periodicsolutions of the system (2)

For convenience we introduce the notation

119891119872= max119905isin[0120596]

1003816100381610038161003816119891 (119905)

1003816100381610038161003816 119891

119871= min119905isin[0120596]

1003816100381610038161003816119891 (119905)

1003816100381610038161003816

120574119894= lim119906rarr0

sup sum

119905le119905119896le119905+120596

119868119894119896(119906)

119906

119894 = 1 2 119899

120575119894= 119890minusint120596

0119903119894(119905)119889119905

119894 = 1 2 119899

120572119894119895= int

120596

0

[120575119895119886119894119895(119905) minus 119887

119894119895(119905) minus 119888

119894119895(119905)] 119889119905

120573119894119895= int

120596

0

[119886119894119895(119905) + 119887

119894119895(119905) + 119888

119894119895(119905)] 119889119905 119894 119895 = 1 2 119899

(3)

where 119891 is a continuous 120596-periodic functionThroughout this paper we assume that

(1198601) 120575119894= 119890minusint120596

0119903119894(119905)119889119905

lt 1 119894 = 1 2 119899(1198602) 120575119895119886119894119895(119905) minus 119887

119894119895(119905) minus 119888

119894119895(119905) ge 0 119894 119895 = 1 2 119899

(1198603) (1 + 119903

119871

119894)(1205752

119894(1 minus 120575

119894))120572119894119895ge max

119905isin[0120596]119886119894119895(119905) + 119887

119894119895(119905) +

119888119894119895(119905) 119894 119895 = 1 2 119899

(1198604) (119903119872

119894minus1)(1120575

119894(1 minus 120575119894))120573119894119895le min

119905isin[0120596]120575119895119886119894119895(119905) minus 119887

119894119895(119905) minus

119888119894119895(119905) 119894 119895 = 1 2 119899

(1198605) max

1le119894le119899sum119899

119895=1119888119872

119894119895 le min

1le119894le119899(1205752

119894(1 minus 120575

119894))

min1le119895le119899

120572119894119895

The paper is organized as follows In Section 2 we givesome definitions and lemmas to prove themain results of thispaper In Section 3 by using a fixed-point theorem of strict-set-contraction we established some criteria to guarantee theexistence of at least one positive periodic solution of system(2) Finally in Section 4 we give an example to show thevalidity of our result

2 Preliminaries

In order to obtain the existence of a periodic solution ofsystem (2) we first introduce some definitions and lemmas

Definition 1 (see [13]) A function 119909119894 119877 rarr (0 +infin) is said

to be a positive solution of (2) if the following conditions aresatisfied

(a) 119909119894(119905) is absolutely continuous on each (119905

119896 119905119896+1)

(b) for each 119896 isin 119885+ 119909119894(119905+

119896) and 119909

119894(119905minus

119896) exist and 119909

119894(119905minus

119896) =

119909119894(119905119896)

(c) 119909119894(119905) satisfies the first equation of (2) for almost every-

where in 119877 and 119909119894(119905119896) satisfies the second equation of

(2) at impulsive point 119905119896 119896 isin 119885

+

Definition 2 (see [21]) Let 119883 be a real Banach space and 119864 aclosed nonempty subset of119883 119864 is a cone provided

(i) 120572119909 + 120573119910 isin 119864 for all 119909 119910 isin 119864 and all 120572 120573 ge 0(ii) 119909 minus119909 isin 119864 imply 119909 = 0

Definition 3 (see [21]) Let 119860 be a bounded subset in 119883Define

120572119883(119860) = inf 120575 gt 0 there is a finite number

of subsets 119860119894sub 119860 such that

119860 = ⋃

119894

119860119894and diam (119860

119894) le 120575

(4)

where diam(119860119894) denotes the diameter of the set 119860

119894 obvi-

ously 0 le 120572119883(119860) lt infin So 120572

119883(119860) is called the Kuratowski

measure of noncompactness of 119883

Definition 4 (see [21]) Let 119883 119884 be two Banach spaces and119863 sub 119883 a continuous and boundedmap 119879 119863 rarr 119884 is called119896-set contractive if for any bounded set 119878 sub 119863 we have

120572119884(119879 (119878)) le 119896120572

119883(119878) (5)

120601 is called strict-set-contractive if it is 119896-set-contractive forsome 0 le 119896 lt 1

Definition 5 (see [22]) The set 119865 isin 119875119862120596is said to be quasi-

equicontinuous in [0 120596] if for any 120598 gt 0 there exists 120575 gt 0

such that if 119909 isin 119865 119896 isin 119873+ 1199051 1199052isin (119905119896minus1 119905119896)cap[0 120596] and |119905

1minus

1199052| lt 120575 then |119909(119905

1) minus 119909(119905

2)| lt 120598



and only if





119909119894(119905) = int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

(6)

where

119866119894(119905 119904) =


119905119903119894(120579)119889120579


0119903119894(120579)119889120579

119904 isin [119905 119905 + 120596] 119894 = 1 2 119899

(7)



119889

119889119905

[119909119894(119905) 119890minusint119905

0119903119894(120579)119889120579

]

= 119890minusint119905

0119903119894(120579)119889120579

119909119894(119905)

times[

[

minus

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

119905 = 119905119896 119894 = 1 2 119899

(8)


119909119894(119904) 119890minusint119904

0119903119894(120579)119889120579

100381610038161003816100381610038161003816

1199051198981+119899120596

119905

+ 119909119894(119904) 119890minusint119904

0119903119894(120579)119889120579

100381610038161003816100381610038161003816

1199051198982+119899120596

1199051198981+119899120596


0119903119894(120579)119889120579

100381610038161003816100381610038161003816

119905+120596

119905119898119902+119899120596

= int

119905+120596

119905


0119903119894(120579)119889120579

119909119894(119904)

times[

[

minus

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)minus

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

(9)

where 119905119898119896

+ 119899120596 isin (119905 119905 + 120596) 119898119896isin 1 2 119902 119896 =

1 2 119902 119899 isin 119885+ Therefore

119909119894(119905) 119890minusint119905

0119903119894(120579)119889120579

[1 minus 119890minusint119905+120596

119905119903119894(120579)119889120579

]

+ sum

119905le119905119896lt119905+120596

Δ119909119894(119905119898119896) 119890minusint

119905119898119896+119899120596

0119903119894(120579)119889120579

= int

119905+120596

119905


0119903119894(120579)119889120579

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

(10)


119909119894(119905) = int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896)) 119894 = 1 2 119899

(11)



119894(119905) isin 119864 119894 =


(6) we have

1199091015840

(119905) =

119889

119889119905

int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

=[

[

119866119894(119905 119905 + 120596) 119909

119894(119905 + 120596)

times (

119899

sum

119895=1

119886119894119895(119905 + 120596) 119909

119895(119905 + 120596)

+

119899

sum

119895=1

119887119894119895(119905 + 120596) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905 + 120596) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120596 + 120585) 119889120585)

minus 119866119894(119905 119905) 119909

119894(119905)

times (

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905+120585) 119889120585)

]

]

+119903119894(119905) 119909119894(119905)

= 119909119894(119905)[

[

119903119894(119905) minus

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909j (119905 minus 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

(12)

For any 119905 = 119905119895 119895 isin 119885


119909119894(119905+

119895) minus 119909119894(119905119895)

= int

119905119895+120596

119905119895

[119866119894(119905+

119895 119904) minus 119866

119894(119905119895 119904)] 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905+

119895le119905119896lt119905119895+120596

119866119894(119905+

119895 119905119896) 119868119894119896(119909119894(119905119896))

minus sum

119905119895le119905119896lt119905119895+120596

119866119894(119905119895 119905119896) 119868119894119896(119909119894(119905k))

= minus119868119894119896(119909 (119905119896))

(13)





Assume that 119860 119864119903119877



119909 for all 119909 isin 119864 119909 = 119877


119909 for all 119909 isin 119864 119909 = 119877



119875119862 (119877) = 119909 = (1199091 1199092 119909

119899)119879

119877 997888rarr 119877 | 119909 isin 119862 ((119905119896 119905119896+1) 119877)


119896) = 119909 (119905

119896) 119909 (119905

+

119896) 119896 isin 119873

1198751198621

(119877) = 119909 = (1199091 1199092 sdot sdot sdot 119909

119899)119879

119877 997888rarr 119877 | 119909 isin 1198621((119905119896 119905119896+1) 119877)

exist1199091015840(119905minus

119896) = 1199091015840(119905119896) 119909 (119905

+

119896) 119896 isin 119873 119905 isin 119877

(14)

Define

119883 = 119909 = (1199091 1199092 119909

119899)119879

119909 isin 119875119862 (119877) | 119909 (119905 + 120596) = 119909 (119905)

(15)



119894=1|119909119894|0 where |119909

119894|0=

sup119905isin[0120596]

|119909119894(119905)| 119894 = 1 2 119899 and

119884=119909=(1199091 1199092 119909

119899)119879

119909 isin 1198751198621

(119877) | 119909 (119905+120596)

=119909 (119905) 119905isin119877

(16)


119894=1|119909119894|1 where |119909

119894|1=

max|119909119894|0 |1199091015840


the cone 119864 in 119884 by

119864=119909=(1199091 1199092 119909

119899)119879

119909isin1198751198621

(119877) | 119909119894(119905)=120575

119895

1003816100381610038161003816119909119894(119905)10038161003816100381610038161

119905isin[0 120596] 119894 119895=1 2 119899

(17)


(119879119909) (119905) = ((1198791119909) (119905) (119879

2119909) (119905) (119879

119899119909) (119905))

119879

(18)

where 119909 isin 119864 119905 isin 119877

(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904)int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

(19)

where

119866119894(119905 119904) =


119905119903119894(120579)119889120579


0119903119894(120579)119889120579

119904 isin [119905 119905 + 120596] 119894 = 1 2 119899

(20)


119894(119905 119904) 120597119866

119894(119905 119904)120597119905 =

119903119894(119905)119866119894(119905 119904) 119866

119894(119905 119905 + 120596) minus 119866

119894(119905 119905) = minus1 and

120575119894

1 minus 120575119894

le 119866119894(119905 119904) le

1

1 minus 120575119894

119904 isin [119905 119905 + 120596] 119894 = 1 2 119899

(21)




defined


119894 119894 = 1 2 119899 gt 1 then



(119879119894119909) (119905 + 120596)

= int

119905+2120596

119905+120596

119866119894(119905 + 120596 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905+120596le119905119896lt119905+2120596

119866119894(119905 + 120596 119905

119896) 119868119894119896(119909119894(119905119896))

= int

119905+120596

119905

119866119894(119905 + 120596 119906 + 120596) 119909

119894(119906 + 120596)

times[

[

119899

sum

119895=1

119886119894119895(119906 + 120596) 119909

119895(119906 + 120596) +

119899

sum

119895=1

119887119894119895(119906 + 120596)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119906 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119906 + 120596)int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119906 + 120596 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= int

119905+120596

119905

119866119894(119905 119906) 119909

119894(119906)

times[

[

119899

sum

119895=1

119886119894119895(119906) 119909119895(119906) +

119899

sum

119895=1

119887119894119895(119906) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119906+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119906) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119906+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= (119879119894119909) (119905) 119894 = 1 2 119899

(22)

That is (119879119894119909)(119905 + 120596) = (119879

119894119909)(119905) 119905 isin [0 120596] So 119879119909 isin 119884 In



119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

ge

119899

sum

119895=1

119886119894119895(119905) 120575119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161minus

119899

sum

119895=1

119887119894119895(119905)

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161minus

119899

sum

119895=1

119888119894119895(119905)

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

119899

sum

119895=1

[119886119894119895(119905) 120575119895minus119887119894119895(119905)minus119888119894119895(119905)]

100381610038161003816100381610038161199091015840

119895

100381610038161003816100381610038161ge 0 119894=1 2 119899

(23)


100381610038161003816100381611987911989411990910038161003816100381610038160

le

1

1 minus 120575119894

int

120596

0

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)+

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

(24)

(119879119894119909) (119905)

ge

120575119894

1 minus 120575119894

int

119905+120596

119905

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

=

120575119894

1 minus 120575119894

times int

120596

0

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

ge 120575119894

100381610038161003816100381611987911989411990910038161003816100381610038160 119894 = 1 2 119899

(25)


119894|119879119894119909|1 119894 = 1 2 119899 119905 isin


(119879119894119909)1015840

(119905)

= 119866119894(119905 119905 + 120596) 119909

119894(119905 + 120596)

times[

[

119899

sum

119895=1

119886119894119895(119905 + 120596) 119909

119895(119905 + 120596)

+

119899

sum

119895=1

119887119894119895(119905 + 120596) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905 + 120596) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120596 + 120585) 119889120585

]

]

minus 119866119894(119905 119905) 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)+

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119903119894(119905) (119879119894119909) (119905)

= 119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)


times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905+120585) 119889120585

]

]

119894 = 1 2 119899

(26)


1 2 119899 then

(119879119894119909)1015840

(119905) le 119903119894(119905) (119879119894119909) (119905) le 119903

119872

119894(119879119894119909) (119905) le (119879

119894119909) (119905)

119894 = 1 2 119899

(27)


119894119909)1015840(119905) lt 0 119894 =

1 2 119899 thenminus (119879119894119909)1015840

(119905)

= minus119903119894(119905) (119879119894119909) (119905) + 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

le1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

[119886119894119895(119905) + 119887

119894119895(119905) + 119888

119894119895(119905)]

1003816100381610038161003816119909119894

10038161003816100381610038161minus 119903119871

119894(119879119894119909) (119905)

le (1 + 119903119871

119894)

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)]

1003816100381610038161003816119909119894

10038161003816100381610038161119889119904

minus 119903119871

119894(119879119894119909) (119905)

= (1 + 119903119871

119894)int

119905+120596

119905

120575119894

1 minus 120575119894

120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus c

119894119895(119904)]

1003816100381610038161003816119909119894

10038161003816100381610038161119889119904 minus 119903

119871

119894(119879119894119909) (119905)

le (1 + 119903119871

119894)int

119905+120596

119905

int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896)) minus 119903

119871

119894(119879119894119909) (119905)

= (1 + 119903119871

119894) (119879119894119909) (119905) minus 119903

119871

119894(119879119894119909) (119905)

= (119879119894119909) (119905) 119894 = 1 2 119899

(28)


1 2 119899 So |119879119894119909|1= |119879119894x|0 119894 = 1 2 119899 By (25) we have

(119879119894119909)(119905) ge 120575

119894|119879119894119909|1 119894 = 1 2 119899 Hence 119879119909 isin 119864 This


119894119909)1015840(119905) ge 0 119894 = 1 2 119899 implies

(119879119894119909)1015840

(119905) le (119879119894119909) (119905) 119894 = 1 2 119899 (29)

From (23) (26) (1198602) and (119860

4) we have

(119879119894119909)1015840

(119905)

= 119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

le 119903119894(119905) (119879119894119909) (119905) minus 120575

119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[120575119895119886119894119895(119905) minus 119887

119894119895(119905) minus 119888

119894119895(119905)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

le 119903119872

119894(119879119894119909) (119905) minus 120575

119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119903119872

119894minus 1

120575119894(1 minus 120575

119894)

times

119899

sum

119895=1

int

120596

0

[119886119894119895(119904) + 119887

119894119895(119904) + 119888

119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1)int

119905+120596

119905

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[119886119894119895(119904) + 119887

119894119895(119904) + 119888

119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904

le 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1)


times int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1) (119879

119894119909) (119905) = (119879

119894119909) (119905)

119894 = 1 2 119899

(30)



119877max1le119895le119899

sum119899

119895=1119887119872

119894119895 lt 1

(i) Ifmax119903119872119894 119894 = 1 2 119899 le 1 then 119879 119864⋂Ω

119877rarr 119864



119894 119894 = 1 2 119899 gt 1 then


119877=

119909 isin 119884 |119909|1lt 119877


119884(119879(119878)) le 119877max

1le119895le119899sum119899

119895=1119887119872

119894119895120572119884(119878)


119884(119878) Then for any


sum119899

119895=1119887119872

119894119895120578 there is a finite


119894119878119894with diam(119878

119894) le 120578+120598

Therefore

1003816100381610038161003816119909 minus 119910

10038161003816100381610038161le 120578 + 120598 for any 119909 119910 isin 119878

119894 (31)



119894= ⋃119895119878119894119895and

1003816100381610038161003816119909 minus 119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895 (32)


(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1198772

1 minus 120575119894

int

120596

0

[

[

119899

sum

119895=1

119886119894119895(119904) +

119899

sum

119895=1

119887119894119895(119904) +

119899

sum

119895=1

119888119894119895(119904)]

]

119889119904

+

1

1 minus 120575119894

sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896)) = 119861

119894 119894 = 1 2 119899

(33)10038161003816100381610038161003816(119879119894119909)1015840

(119905)

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899

(34)

Hence

(119879119909)0le

119899

sum

119894=1

119861119894

10038171003817100381710038171003817(119879119909)1015840100381710038171003817100381710038170le

119899

sum

119894=1

[

[

119861119894119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895)]

]

(35)


119894119895119896

of 119878119894119895such that 119878

119894119895= ⋃119896119878119894119895119896

and

1003816100381610038161003816119879119909 minus 119879119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895119896 (36)



119894119895119896 we

have

10038161003816100381610038161003816(119879119894119909)1015840

minus (119879119894119910)1015840100381610038161003816100381610038160

= max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119903

119894(119905) (119879119894119910) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119910119894(119905)[

[

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le max119905isin[0120596]

1003816100381610038161003816119903119894(119905) [(119879

119894119909) (119905) minus (119879

119894119910) (119905)]

1003816100381610038161003816

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119894[

[

(

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585)

minus(

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585 +

119899

sum

119895=1

119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ max119905isin[0120596]

10038161003816100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905) minus 119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894

1003816100381610038161003816(119879119894119909) minus (119879

119894119910)10038161003816100381610038160

+ 119877max119905isin[0120596]

119899

sum

119895=1

[

[

119886119894119895(119905)

10038161003816100381610038161003816119909119895(119905)minus119910

119895(119905)

10038161003816100381610038161003816+119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585)

times1003816100381610038161003816119909119894(119905+120585)minus119910

119894(119905+120585)

1003816100381610038161003816119889120585+119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585)

times

100381610038161003816100381610038161199091015840

119895(119905+120585) minus119910

1015840

(119905minus120585)

10038161003816100381610038161003816119889120585]

]

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905)minus119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905)

timesint

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585)

100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894120598+119877120598

119899

sum

119895=1

119886119872

119894119895+119877120598

119899

sum

119895=1

119887119872

119894119895+1003816100381610038161003816119909119894

10038161003816100381610038160(120578+120598)

times

119899

sum

119895=1

119888119872

119894119895+119877120598

119899

sum

119895=1

(119886119872

119894119895+119887119872

119894119895+119888119872

119894119895)

=1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895120578 + 119861119894120598

(37)

where

119861119894= 119903119872

119894+ 2119877

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899 (38)



1003817100381710038171003817119879119909 minus 119879119910

10038171003817100381710038171le (

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895)120578 + 120598

119899

sum

119894=1

119861119894

le 119877max1le119894le119899

119899

sum

119895=1

119888119872

119894119895

120578 + 120598

119899

sum

119894=1

119861119894 for any 119909 119910 isin 119878

119894119895119896

(39)


120572119884(119879 (119878)) le 119877max

1le119895le119899

119899

sum

119895=1

119888119872

119894119895

120572119884(119878) (40)


3 Main Results






119894 119894 = 1 2 119899 gt 1 then



119877 = (min119894isin[1119899]

1205752

119894

1 minus 120575119894

min119895isin[1119899]

120572119894119895)

minus1

0 lt 119903 lt min119894isin[1119899]

120575119894(1 minus 120575

119894) minus 120574119894

max119895isin[1119899]

120573119894119895

(41)


119903119877 In view





0gt 0


119879119894119909 (119905) minus 119909

119894(119905) ge 120575

119894

1003816100381610038161003816119879119894119909 minus 119909119894

10038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(42)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

int

120596

0

[119886119894119895(119904)+119887119894119895(119904)+119888119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904+120574119894

le

120573119894119895119903 + 120574119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160le 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160 119894 = 1 2 119899


1199090le 119879119909

0le max1le119894le119899

120575119894 1199090lt 119909

0 (44)






119894119909 (119905) ge 120575

119894

1003816100381610038161003816119909119894minus 11987911989411990910038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(45)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

gt

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)] 119889119904

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161) 119894 = 1 2 119899

(46)


Thus we have

1198791199090=

119899

sum

119894=1

100381610038161003816100381611987911989411990910038161003816100381610038160gt

119899

sum

119894=1

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161)

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

1198772= 119877

(47)


1199091ge 119879119909

1ge 119879119909

0gt 119877 (48)




4 Example



1199091015840

1(119905)

= 1199091(119905) [


minus (10 + sin 119905) 1199091(119905)


times int

0

minus12059111

11989111(120585) 1199091(119905 + 120585) 119889120585 minus

5 + sin 11990515

times int

0

minus12059112

11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)

times int

0

minus12059011

11989211(120585) 1199091015840

1(119905 + 120585) 119889120585 minus

4 minus 3 cos 1199056

timesint

0

minus12059012

11989212(120585) 1199091015840

2(119905 + 120585) 119889120585]

1199091015840

2(119905)

= 1199092(119905) [

1 + cos 1199058120587

minus (12 + 3 cos 119905) 1199091(119905)


times int

0

minus12059121

11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)

times int

0

minus12059122

11989122(120585) 1199092(119905 + 120585) 119889120585 minus

4 + sin 11990512

times int

0

minus12059021

11989221(120585) 1199091015840

2(119905 + 120585) 119889120585 minus


timesint

0

minus12059022

11989222(120585) 1199091015840

2(119905 + 120585) 119889120585]

(49)

where 120591119894119895 120590119894119895

isin (0 +infin) (119894 119895 = 1 2) and 119891119894119895 119892119894119895

isin

119862(119877 119877+) satisfying int0

minus120591119894119895

119891119894119895(120585)119889120585 = 1 int

0

minus120590119894119895

119892119894119895(120585)119889120585 119894 119895 = 1 2

Obviously

1199031(119905) =


1199032=

1 + cos 1199058120587

11988611(119905) = 10 + sin 119905

11988612(119905) = 9 minus cos 119905 119886

21(119905) = 12 + 3 cos 119905

11988622(119905) = 7 minus sin 119905 119887

11(119905) = 3 minus 2 cos 119905

11988712(119905) =

5 + sin 11990515

11988721(119905) = 2 + 3 sin 119905 119887

22(119905) = 1 minus cos 119905

11988811(119905) = 1 + sin 119905 119888

12(119905) =

4 minus 3 cos 1199056

11988821(119905) =

4 + sin 11990512

11988822(119905) =


119903119871

1= 119903119871

2= 0

(50)


1205751= 119890minus(12058712)

lt 1 1205752= 119890minus(14)

lt 1

12057211= 20120587120575

1minus 8120587 asymp 238007

12057212= 18120587120575

2minus 2120587 asymp 372423

12057221= 24120587120575

1minus

14120587

3

asymp 440594

12057222= 14120587120575

2minus 3120587 asymp 244284


min1le119895le2

1205721119895 = 12057211= 20120587120575

1minus 8120587 asymp 238007

min1le119895le2

1205722119895 = 12057222= 14120587120575

2minus 3120587 asymp 244284

120575111988611(119905) minus 119887

11(119905) minus 119888

11(119905) gt 0009 gt 0

120575211988612(119905) minus 119887

12(119905) minus 119888

12(119905) gt 1 gt 0

120575111988621(119905) minus 119887

21(119905) minus 119888

21(119905) gt 4 gt 0

120575211988622(119905) minus 119887

22(119905) minus 119888

22(119905) gt 2 gt 0

(1 + 119903119871

1)

1205752

112057211

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 8120587)

1 minus 119890minus(12058712)

asymp 652614 gt 18

ge max0le119905le2120587

11988611(119905) + 119887

11(119905) + 119888

11(119905)

(1 + 119903119871

1)

1205752

112057212

1 minus 1205751

=

119890minus(1205876)

(181205871205752minus 2120587)

1 minus 119890minus(12058712)

asymp 1021184 gt

347

30

ge max0le119905le2120587

11988612(119905) + 119887

12(119905) + 119888

12(119905)

(1 + 119903119871

2)

1205752

212057221

1 minus 1205752

=

119890minus(12)

(241205871205751minus (141205873))


asymp 1133411 gt

245

12

gt max0le119905le2120587

11988621(119905) + 119887

21(119905) + 119888

21(119905)

(1 + 119903119871

2)

1205752

212057222

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411 gt

53

5

gt max0le119905le2120587

11988622(119905) + 119887

22(119905) + 119888

22(119905)

1205752

1min1le119895le2

1205721119895

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 6120587)

1 minus 119890minus(12058712)

asymp 652614

1205752

2min1le119895le2

1205722119895

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411

min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

=

1205752

2min1le119895le2

1205722119895

1 minus 1205751

asymp 628411

119888119872

11+ 119888119872

12=

19

6

119888119872

21+ 119888119872

22=

61

60

max1le119894le2

2

sum

119895=1

119889119872

119894119895

=

19

6

(51)

Therefore

19

6

= max1le119894le2

2

sum

119894=1

119889119872

119894119895 lt min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

asymp 628411

(52)


119872

119894le 1 119894 = 1 2


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











and only if





119909119894(119905) = int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

(6)

where

119866119894(119905 119904) =


119905119903119894(120579)119889120579


0119903119894(120579)119889120579

119904 isin [119905 119905 + 120596] 119894 = 1 2 119899

(7)



119889

119889119905

[119909119894(119905) 119890minusint119905

0119903119894(120579)119889120579

]

= 119890minusint119905

0119903119894(120579)119889120579

119909119894(119905)

times[

[

minus

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

119905 = 119905119896 119894 = 1 2 119899

(8)


119909119894(119904) 119890minusint119904

0119903119894(120579)119889120579

100381610038161003816100381610038161003816

1199051198981+119899120596

119905

+ 119909119894(119904) 119890minusint119904

0119903119894(120579)119889120579

100381610038161003816100381610038161003816

1199051198982+119899120596

1199051198981+119899120596


0119903119894(120579)119889120579

100381610038161003816100381610038161003816

119905+120596

119905119898119902+119899120596

= int

119905+120596

119905


0119903119894(120579)119889120579

119909119894(119904)

times[

[

minus

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)minus

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

(9)

where 119905119898119896

+ 119899120596 isin (119905 119905 + 120596) 119898119896isin 1 2 119902 119896 =

1 2 119902 119899 isin 119885+ Therefore

119909119894(119905) 119890minusint119905

0119903119894(120579)119889120579

[1 minus 119890minusint119905+120596

119905119903119894(120579)119889120579

]

+ sum

119905le119905119896lt119905+120596

Δ119909119894(119905119898119896) 119890minusint

119905119898119896+119899120596

0119903119894(120579)119889120579

= int

119905+120596

119905


0119903119894(120579)119889120579

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

(10)


119909119894(119905) = int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896)) 119894 = 1 2 119899

(11)



119894(119905) isin 119864 119894 =


(6) we have

1199091015840

(119905) =

119889

119889119905

int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

=[

[

119866119894(119905 119905 + 120596) 119909

119894(119905 + 120596)

times (

119899

sum

119895=1

119886119894119895(119905 + 120596) 119909

119895(119905 + 120596)

+

119899

sum

119895=1

119887119894119895(119905 + 120596) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905 + 120596) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120596 + 120585) 119889120585)

minus 119866119894(119905 119905) 119909

119894(119905)

times (

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905+120585) 119889120585)

]

]

+119903119894(119905) 119909119894(119905)

= 119909119894(119905)[

[

119903119894(119905) minus

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909j (119905 minus 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

(12)

For any 119905 = 119905119895 119895 isin 119885


119909119894(119905+

119895) minus 119909119894(119905119895)

= int

119905119895+120596

119905119895

[119866119894(119905+

119895 119904) minus 119866

119894(119905119895 119904)] 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905+

119895le119905119896lt119905119895+120596

119866119894(119905+

119895 119905119896) 119868119894119896(119909119894(119905119896))

minus sum

119905119895le119905119896lt119905119895+120596

119866119894(119905119895 119905119896) 119868119894119896(119909119894(119905k))

= minus119868119894119896(119909 (119905119896))

(13)





Assume that 119860 119864119903119877



119909 for all 119909 isin 119864 119909 = 119877


119909 for all 119909 isin 119864 119909 = 119877



119875119862 (119877) = 119909 = (1199091 1199092 119909

119899)119879

119877 997888rarr 119877 | 119909 isin 119862 ((119905119896 119905119896+1) 119877)


119896) = 119909 (119905

119896) 119909 (119905

+

119896) 119896 isin 119873

1198751198621

(119877) = 119909 = (1199091 1199092 sdot sdot sdot 119909

119899)119879

119877 997888rarr 119877 | 119909 isin 1198621((119905119896 119905119896+1) 119877)

exist1199091015840(119905minus

119896) = 1199091015840(119905119896) 119909 (119905

+

119896) 119896 isin 119873 119905 isin 119877

(14)

Define

119883 = 119909 = (1199091 1199092 119909

119899)119879

119909 isin 119875119862 (119877) | 119909 (119905 + 120596) = 119909 (119905)

(15)



119894=1|119909119894|0 where |119909

119894|0=

sup119905isin[0120596]

|119909119894(119905)| 119894 = 1 2 119899 and

119884=119909=(1199091 1199092 119909

119899)119879

119909 isin 1198751198621

(119877) | 119909 (119905+120596)

=119909 (119905) 119905isin119877

(16)


119894=1|119909119894|1 where |119909

119894|1=

max|119909119894|0 |1199091015840


the cone 119864 in 119884 by

119864=119909=(1199091 1199092 119909

119899)119879

119909isin1198751198621

(119877) | 119909119894(119905)=120575

119895

1003816100381610038161003816119909119894(119905)10038161003816100381610038161

119905isin[0 120596] 119894 119895=1 2 119899

(17)


(119879119909) (119905) = ((1198791119909) (119905) (119879

2119909) (119905) (119879

119899119909) (119905))

119879

(18)

where 119909 isin 119864 119905 isin 119877

(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904)int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

(19)

where

119866119894(119905 119904) =


119905119903119894(120579)119889120579


0119903119894(120579)119889120579

119904 isin [119905 119905 + 120596] 119894 = 1 2 119899

(20)


119894(119905 119904) 120597119866

119894(119905 119904)120597119905 =

119903119894(119905)119866119894(119905 119904) 119866

119894(119905 119905 + 120596) minus 119866

119894(119905 119905) = minus1 and

120575119894

1 minus 120575119894

le 119866119894(119905 119904) le

1

1 minus 120575119894

119904 isin [119905 119905 + 120596] 119894 = 1 2 119899

(21)




defined


119894 119894 = 1 2 119899 gt 1 then



(119879119894119909) (119905 + 120596)

= int

119905+2120596

119905+120596

119866119894(119905 + 120596 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905+120596le119905119896lt119905+2120596

119866119894(119905 + 120596 119905

119896) 119868119894119896(119909119894(119905119896))

= int

119905+120596

119905

119866119894(119905 + 120596 119906 + 120596) 119909

119894(119906 + 120596)

times[

[

119899

sum

119895=1

119886119894119895(119906 + 120596) 119909

119895(119906 + 120596) +

119899

sum

119895=1

119887119894119895(119906 + 120596)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119906 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119906 + 120596)int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119906 + 120596 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= int

119905+120596

119905

119866119894(119905 119906) 119909

119894(119906)

times[

[

119899

sum

119895=1

119886119894119895(119906) 119909119895(119906) +

119899

sum

119895=1

119887119894119895(119906) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119906+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119906) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119906+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= (119879119894119909) (119905) 119894 = 1 2 119899

(22)

That is (119879119894119909)(119905 + 120596) = (119879

119894119909)(119905) 119905 isin [0 120596] So 119879119909 isin 119884 In



119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

ge

119899

sum

119895=1

119886119894119895(119905) 120575119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161minus

119899

sum

119895=1

119887119894119895(119905)

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161minus

119899

sum

119895=1

119888119894119895(119905)

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

119899

sum

119895=1

[119886119894119895(119905) 120575119895minus119887119894119895(119905)minus119888119894119895(119905)]

100381610038161003816100381610038161199091015840

119895

100381610038161003816100381610038161ge 0 119894=1 2 119899

(23)


100381610038161003816100381611987911989411990910038161003816100381610038160

le

1

1 minus 120575119894

int

120596

0

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)+

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

(24)

(119879119894119909) (119905)

ge

120575119894

1 minus 120575119894

int

119905+120596

119905

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

=

120575119894

1 minus 120575119894

times int

120596

0

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

ge 120575119894

100381610038161003816100381611987911989411990910038161003816100381610038160 119894 = 1 2 119899

(25)


119894|119879119894119909|1 119894 = 1 2 119899 119905 isin


(119879119894119909)1015840

(119905)

= 119866119894(119905 119905 + 120596) 119909

119894(119905 + 120596)

times[

[

119899

sum

119895=1

119886119894119895(119905 + 120596) 119909

119895(119905 + 120596)

+

119899

sum

119895=1

119887119894119895(119905 + 120596) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905 + 120596) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120596 + 120585) 119889120585

]

]

minus 119866119894(119905 119905) 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)+

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119903119894(119905) (119879119894119909) (119905)

= 119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)


times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905+120585) 119889120585

]

]

119894 = 1 2 119899

(26)


1 2 119899 then

(119879119894119909)1015840

(119905) le 119903119894(119905) (119879119894119909) (119905) le 119903

119872

119894(119879119894119909) (119905) le (119879

119894119909) (119905)

119894 = 1 2 119899

(27)


119894119909)1015840(119905) lt 0 119894 =

1 2 119899 thenminus (119879119894119909)1015840

(119905)

= minus119903119894(119905) (119879119894119909) (119905) + 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

le1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

[119886119894119895(119905) + 119887

119894119895(119905) + 119888

119894119895(119905)]

1003816100381610038161003816119909119894

10038161003816100381610038161minus 119903119871

119894(119879119894119909) (119905)

le (1 + 119903119871

119894)

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)]

1003816100381610038161003816119909119894

10038161003816100381610038161119889119904

minus 119903119871

119894(119879119894119909) (119905)

= (1 + 119903119871

119894)int

119905+120596

119905

120575119894

1 minus 120575119894

120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus c

119894119895(119904)]

1003816100381610038161003816119909119894

10038161003816100381610038161119889119904 minus 119903

119871

119894(119879119894119909) (119905)

le (1 + 119903119871

119894)int

119905+120596

119905

int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896)) minus 119903

119871

119894(119879119894119909) (119905)

= (1 + 119903119871

119894) (119879119894119909) (119905) minus 119903

119871

119894(119879119894119909) (119905)

= (119879119894119909) (119905) 119894 = 1 2 119899

(28)


1 2 119899 So |119879119894119909|1= |119879119894x|0 119894 = 1 2 119899 By (25) we have

(119879119894119909)(119905) ge 120575

119894|119879119894119909|1 119894 = 1 2 119899 Hence 119879119909 isin 119864 This


119894119909)1015840(119905) ge 0 119894 = 1 2 119899 implies

(119879119894119909)1015840

(119905) le (119879119894119909) (119905) 119894 = 1 2 119899 (29)

From (23) (26) (1198602) and (119860

4) we have

(119879119894119909)1015840

(119905)

= 119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

le 119903119894(119905) (119879119894119909) (119905) minus 120575

119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[120575119895119886119894119895(119905) minus 119887

119894119895(119905) minus 119888

119894119895(119905)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

le 119903119872

119894(119879119894119909) (119905) minus 120575

119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119903119872

119894minus 1

120575119894(1 minus 120575

119894)

times

119899

sum

119895=1

int

120596

0

[119886119894119895(119904) + 119887

119894119895(119904) + 119888

119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1)int

119905+120596

119905

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[119886119894119895(119904) + 119887

119894119895(119904) + 119888

119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904

le 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1)


times int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1) (119879

119894119909) (119905) = (119879

119894119909) (119905)

119894 = 1 2 119899

(30)



119877max1le119895le119899

sum119899

119895=1119887119872

119894119895 lt 1

(i) Ifmax119903119872119894 119894 = 1 2 119899 le 1 then 119879 119864⋂Ω

119877rarr 119864



119894 119894 = 1 2 119899 gt 1 then


119877=

119909 isin 119884 |119909|1lt 119877


119884(119879(119878)) le 119877max

1le119895le119899sum119899

119895=1119887119872

119894119895120572119884(119878)


119884(119878) Then for any


sum119899

119895=1119887119872

119894119895120578 there is a finite


119894119878119894with diam(119878

119894) le 120578+120598

Therefore

1003816100381610038161003816119909 minus 119910

10038161003816100381610038161le 120578 + 120598 for any 119909 119910 isin 119878

119894 (31)



119894= ⋃119895119878119894119895and

1003816100381610038161003816119909 minus 119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895 (32)


(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1198772

1 minus 120575119894

int

120596

0

[

[

119899

sum

119895=1

119886119894119895(119904) +

119899

sum

119895=1

119887119894119895(119904) +

119899

sum

119895=1

119888119894119895(119904)]

]

119889119904

+

1

1 minus 120575119894

sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896)) = 119861

119894 119894 = 1 2 119899

(33)10038161003816100381610038161003816(119879119894119909)1015840

(119905)

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899

(34)

Hence

(119879119909)0le

119899

sum

119894=1

119861119894

10038171003817100381710038171003817(119879119909)1015840100381710038171003817100381710038170le

119899

sum

119894=1

[

[

119861119894119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895)]

]

(35)


119894119895119896

of 119878119894119895such that 119878

119894119895= ⋃119896119878119894119895119896

and

1003816100381610038161003816119879119909 minus 119879119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895119896 (36)



119894119895119896 we

have

10038161003816100381610038161003816(119879119894119909)1015840

minus (119879119894119910)1015840100381610038161003816100381610038160

= max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119903

119894(119905) (119879119894119910) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119910119894(119905)[

[

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le max119905isin[0120596]

1003816100381610038161003816119903119894(119905) [(119879

119894119909) (119905) minus (119879

119894119910) (119905)]

1003816100381610038161003816

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119894[

[

(

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585)

minus(

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585 +

119899

sum

119895=1

119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ max119905isin[0120596]

10038161003816100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905) minus 119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894

1003816100381610038161003816(119879119894119909) minus (119879

119894119910)10038161003816100381610038160

+ 119877max119905isin[0120596]

119899

sum

119895=1

[

[

119886119894119895(119905)

10038161003816100381610038161003816119909119895(119905)minus119910

119895(119905)

10038161003816100381610038161003816+119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585)

times1003816100381610038161003816119909119894(119905+120585)minus119910

119894(119905+120585)

1003816100381610038161003816119889120585+119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585)

times

100381610038161003816100381610038161199091015840

119895(119905+120585) minus119910

1015840

(119905minus120585)

10038161003816100381610038161003816119889120585]

]

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905)minus119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905)

timesint

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585)

100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894120598+119877120598

119899

sum

119895=1

119886119872

119894119895+119877120598

119899

sum

119895=1

119887119872

119894119895+1003816100381610038161003816119909119894

10038161003816100381610038160(120578+120598)

times

119899

sum

119895=1

119888119872

119894119895+119877120598

119899

sum

119895=1

(119886119872

119894119895+119887119872

119894119895+119888119872

119894119895)

=1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895120578 + 119861119894120598

(37)

where

119861119894= 119903119872

119894+ 2119877

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899 (38)



1003817100381710038171003817119879119909 minus 119879119910

10038171003817100381710038171le (

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895)120578 + 120598

119899

sum

119894=1

119861119894

le 119877max1le119894le119899

119899

sum

119895=1

119888119872

119894119895

120578 + 120598

119899

sum

119894=1

119861119894 for any 119909 119910 isin 119878

119894119895119896

(39)


120572119884(119879 (119878)) le 119877max

1le119895le119899

119899

sum

119895=1

119888119872

119894119895

120572119884(119878) (40)


3 Main Results






119894 119894 = 1 2 119899 gt 1 then



119877 = (min119894isin[1119899]

1205752

119894

1 minus 120575119894

min119895isin[1119899]

120572119894119895)

minus1

0 lt 119903 lt min119894isin[1119899]

120575119894(1 minus 120575

119894) minus 120574119894

max119895isin[1119899]

120573119894119895

(41)


119903119877 In view





0gt 0


119879119894119909 (119905) minus 119909

119894(119905) ge 120575

119894

1003816100381610038161003816119879119894119909 minus 119909119894

10038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(42)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

int

120596

0

[119886119894119895(119904)+119887119894119895(119904)+119888119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904+120574119894

le

120573119894119895119903 + 120574119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160le 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160 119894 = 1 2 119899


1199090le 119879119909

0le max1le119894le119899

120575119894 1199090lt 119909

0 (44)






119894119909 (119905) ge 120575

119894

1003816100381610038161003816119909119894minus 11987911989411990910038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(45)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

gt

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)] 119889119904

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161) 119894 = 1 2 119899

(46)


Thus we have

1198791199090=

119899

sum

119894=1

100381610038161003816100381611987911989411990910038161003816100381610038160gt

119899

sum

119894=1

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161)

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

1198772= 119877

(47)


1199091ge 119879119909

1ge 119879119909

0gt 119877 (48)




4 Example



1199091015840

1(119905)

= 1199091(119905) [


minus (10 + sin 119905) 1199091(119905)


times int

0

minus12059111

11989111(120585) 1199091(119905 + 120585) 119889120585 minus

5 + sin 11990515

times int

0

minus12059112

11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)

times int

0

minus12059011

11989211(120585) 1199091015840

1(119905 + 120585) 119889120585 minus

4 minus 3 cos 1199056

timesint

0

minus12059012

11989212(120585) 1199091015840

2(119905 + 120585) 119889120585]

1199091015840

2(119905)

= 1199092(119905) [

1 + cos 1199058120587

minus (12 + 3 cos 119905) 1199091(119905)


times int

0

minus12059121

11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)

times int

0

minus12059122

11989122(120585) 1199092(119905 + 120585) 119889120585 minus

4 + sin 11990512

times int

0

minus12059021

11989221(120585) 1199091015840

2(119905 + 120585) 119889120585 minus


timesint

0

minus12059022

11989222(120585) 1199091015840

2(119905 + 120585) 119889120585]

(49)

where 120591119894119895 120590119894119895

isin (0 +infin) (119894 119895 = 1 2) and 119891119894119895 119892119894119895

isin

119862(119877 119877+) satisfying int0

minus120591119894119895

119891119894119895(120585)119889120585 = 1 int

0

minus120590119894119895

119892119894119895(120585)119889120585 119894 119895 = 1 2

Obviously

1199031(119905) =


1199032=

1 + cos 1199058120587

11988611(119905) = 10 + sin 119905

11988612(119905) = 9 minus cos 119905 119886

21(119905) = 12 + 3 cos 119905

11988622(119905) = 7 minus sin 119905 119887

11(119905) = 3 minus 2 cos 119905

11988712(119905) =

5 + sin 11990515

11988721(119905) = 2 + 3 sin 119905 119887

22(119905) = 1 minus cos 119905

11988811(119905) = 1 + sin 119905 119888

12(119905) =

4 minus 3 cos 1199056

11988821(119905) =

4 + sin 11990512

11988822(119905) =


119903119871

1= 119903119871

2= 0

(50)


1205751= 119890minus(12058712)

lt 1 1205752= 119890minus(14)

lt 1

12057211= 20120587120575

1minus 8120587 asymp 238007

12057212= 18120587120575

2minus 2120587 asymp 372423

12057221= 24120587120575

1minus

14120587

3

asymp 440594

12057222= 14120587120575

2minus 3120587 asymp 244284


min1le119895le2

1205721119895 = 12057211= 20120587120575

1minus 8120587 asymp 238007

min1le119895le2

1205722119895 = 12057222= 14120587120575

2minus 3120587 asymp 244284

120575111988611(119905) minus 119887

11(119905) minus 119888

11(119905) gt 0009 gt 0

120575211988612(119905) minus 119887

12(119905) minus 119888

12(119905) gt 1 gt 0

120575111988621(119905) minus 119887

21(119905) minus 119888

21(119905) gt 4 gt 0

120575211988622(119905) minus 119887

22(119905) minus 119888

22(119905) gt 2 gt 0

(1 + 119903119871

1)

1205752

112057211

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 8120587)

1 minus 119890minus(12058712)

asymp 652614 gt 18

ge max0le119905le2120587

11988611(119905) + 119887

11(119905) + 119888

11(119905)

(1 + 119903119871

1)

1205752

112057212

1 minus 1205751

=

119890minus(1205876)

(181205871205752minus 2120587)

1 minus 119890minus(12058712)

asymp 1021184 gt

347

30

ge max0le119905le2120587

11988612(119905) + 119887

12(119905) + 119888

12(119905)

(1 + 119903119871

2)

1205752

212057221

1 minus 1205752

=

119890minus(12)

(241205871205751minus (141205873))


asymp 1133411 gt

245

12

gt max0le119905le2120587

11988621(119905) + 119887

21(119905) + 119888

21(119905)

(1 + 119903119871

2)

1205752

212057222

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411 gt

53

5

gt max0le119905le2120587

11988622(119905) + 119887

22(119905) + 119888

22(119905)

1205752

1min1le119895le2

1205721119895

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 6120587)

1 minus 119890minus(12058712)

asymp 652614

1205752

2min1le119895le2

1205722119895

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411

min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

=

1205752

2min1le119895le2

1205722119895

1 minus 1205751

asymp 628411

119888119872

11+ 119888119872

12=

19

6

119888119872

21+ 119888119872

22=

61

60

max1le119894le2

2

sum

119895=1

119889119872

119894119895

=

19

6

(51)

Therefore

19

6

= max1le119894le2

2

sum

119894=1

119889119872

119894119895 lt min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

asymp 628411

(52)


119872

119894le 1 119894 = 1 2


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894(119905) isin 119864 119894 =


(6) we have

1199091015840

(119905) =

119889

119889119905

int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

=[

[

119866119894(119905 119905 + 120596) 119909

119894(119905 + 120596)

times (

119899

sum

119895=1

119886119894119895(119905 + 120596) 119909

119895(119905 + 120596)

+

119899

sum

119895=1

119887119894119895(119905 + 120596) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905 + 120596) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120596 + 120585) 119889120585)

minus 119866119894(119905 119905) 119909

119894(119905)

times (

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905+120585) 119889120585)

]

]

+119903119894(119905) 119909119894(119905)

= 119909119894(119905)[

[

119903119894(119905) minus

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909j (119905 minus 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

(12)

For any 119905 = 119905119895 119895 isin 119885


119909119894(119905+

119895) minus 119909119894(119905119895)

= int

119905119895+120596

119905119895

[119866119894(119905+

119895 119904) minus 119866

119894(119905119895 119904)] 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905+

119895le119905119896lt119905119895+120596

119866119894(119905+

119895 119905119896) 119868119894119896(119909119894(119905119896))

minus sum

119905119895le119905119896lt119905119895+120596

119866119894(119905119895 119905119896) 119868119894119896(119909119894(119905k))

= minus119868119894119896(119909 (119905119896))

(13)





Assume that 119860 119864119903119877



119909 for all 119909 isin 119864 119909 = 119877


119909 for all 119909 isin 119864 119909 = 119877



119875119862 (119877) = 119909 = (1199091 1199092 119909

119899)119879

119877 997888rarr 119877 | 119909 isin 119862 ((119905119896 119905119896+1) 119877)


119896) = 119909 (119905

119896) 119909 (119905

+

119896) 119896 isin 119873

1198751198621

(119877) = 119909 = (1199091 1199092 sdot sdot sdot 119909

119899)119879

119877 997888rarr 119877 | 119909 isin 1198621((119905119896 119905119896+1) 119877)

exist1199091015840(119905minus

119896) = 1199091015840(119905119896) 119909 (119905

+

119896) 119896 isin 119873 119905 isin 119877

(14)

Define

119883 = 119909 = (1199091 1199092 119909

119899)119879

119909 isin 119875119862 (119877) | 119909 (119905 + 120596) = 119909 (119905)

(15)



119894=1|119909119894|0 where |119909

119894|0=

sup119905isin[0120596]

|119909119894(119905)| 119894 = 1 2 119899 and

119884=119909=(1199091 1199092 119909

119899)119879

119909 isin 1198751198621

(119877) | 119909 (119905+120596)

=119909 (119905) 119905isin119877

(16)


119894=1|119909119894|1 where |119909

119894|1=

max|119909119894|0 |1199091015840


the cone 119864 in 119884 by

119864=119909=(1199091 1199092 119909

119899)119879

119909isin1198751198621

(119877) | 119909119894(119905)=120575

119895

1003816100381610038161003816119909119894(119905)10038161003816100381610038161

119905isin[0 120596] 119894 119895=1 2 119899

(17)


(119879119909) (119905) = ((1198791119909) (119905) (119879

2119909) (119905) (119879

119899119909) (119905))

119879

(18)

where 119909 isin 119864 119905 isin 119877

(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904)int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

(19)

where

119866119894(119905 119904) =


119905119903119894(120579)119889120579


0119903119894(120579)119889120579

119904 isin [119905 119905 + 120596] 119894 = 1 2 119899

(20)


119894(119905 119904) 120597119866

119894(119905 119904)120597119905 =

119903119894(119905)119866119894(119905 119904) 119866

119894(119905 119905 + 120596) minus 119866

119894(119905 119905) = minus1 and

120575119894

1 minus 120575119894

le 119866119894(119905 119904) le

1

1 minus 120575119894

119904 isin [119905 119905 + 120596] 119894 = 1 2 119899

(21)




defined


119894 119894 = 1 2 119899 gt 1 then



(119879119894119909) (119905 + 120596)

= int

119905+2120596

119905+120596

119866119894(119905 + 120596 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905+120596le119905119896lt119905+2120596

119866119894(119905 + 120596 119905

119896) 119868119894119896(119909119894(119905119896))

= int

119905+120596

119905

119866119894(119905 + 120596 119906 + 120596) 119909

119894(119906 + 120596)

times[

[

119899

sum

119895=1

119886119894119895(119906 + 120596) 119909

119895(119906 + 120596) +

119899

sum

119895=1

119887119894119895(119906 + 120596)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119906 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119906 + 120596)int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119906 + 120596 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= int

119905+120596

119905

119866119894(119905 119906) 119909

119894(119906)

times[

[

119899

sum

119895=1

119886119894119895(119906) 119909119895(119906) +

119899

sum

119895=1

119887119894119895(119906) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119906+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119906) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119906+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= (119879119894119909) (119905) 119894 = 1 2 119899

(22)

That is (119879119894119909)(119905 + 120596) = (119879

119894119909)(119905) 119905 isin [0 120596] So 119879119909 isin 119884 In



119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

ge

119899

sum

119895=1

119886119894119895(119905) 120575119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161minus

119899

sum

119895=1

119887119894119895(119905)

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161minus

119899

sum

119895=1

119888119894119895(119905)

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

119899

sum

119895=1

[119886119894119895(119905) 120575119895minus119887119894119895(119905)minus119888119894119895(119905)]

100381610038161003816100381610038161199091015840

119895

100381610038161003816100381610038161ge 0 119894=1 2 119899

(23)


100381610038161003816100381611987911989411990910038161003816100381610038160

le

1

1 minus 120575119894

int

120596

0

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)+

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

(24)

(119879119894119909) (119905)

ge

120575119894

1 minus 120575119894

int

119905+120596

119905

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

=

120575119894

1 minus 120575119894

times int

120596

0

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

ge 120575119894

100381610038161003816100381611987911989411990910038161003816100381610038160 119894 = 1 2 119899

(25)


119894|119879119894119909|1 119894 = 1 2 119899 119905 isin


(119879119894119909)1015840

(119905)

= 119866119894(119905 119905 + 120596) 119909

119894(119905 + 120596)

times[

[

119899

sum

119895=1

119886119894119895(119905 + 120596) 119909

119895(119905 + 120596)

+

119899

sum

119895=1

119887119894119895(119905 + 120596) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905 + 120596) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120596 + 120585) 119889120585

]

]

minus 119866119894(119905 119905) 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)+

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119903119894(119905) (119879119894119909) (119905)

= 119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)


times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905+120585) 119889120585

]

]

119894 = 1 2 119899

(26)


1 2 119899 then

(119879119894119909)1015840

(119905) le 119903119894(119905) (119879119894119909) (119905) le 119903

119872

119894(119879119894119909) (119905) le (119879

119894119909) (119905)

119894 = 1 2 119899

(27)


119894119909)1015840(119905) lt 0 119894 =

1 2 119899 thenminus (119879119894119909)1015840

(119905)

= minus119903119894(119905) (119879119894119909) (119905) + 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

le1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

[119886119894119895(119905) + 119887

119894119895(119905) + 119888

119894119895(119905)]

1003816100381610038161003816119909119894

10038161003816100381610038161minus 119903119871

119894(119879119894119909) (119905)

le (1 + 119903119871

119894)

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)]

1003816100381610038161003816119909119894

10038161003816100381610038161119889119904

minus 119903119871

119894(119879119894119909) (119905)

= (1 + 119903119871

119894)int

119905+120596

119905

120575119894

1 minus 120575119894

120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus c

119894119895(119904)]

1003816100381610038161003816119909119894

10038161003816100381610038161119889119904 minus 119903

119871

119894(119879119894119909) (119905)

le (1 + 119903119871

119894)int

119905+120596

119905

int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896)) minus 119903

119871

119894(119879119894119909) (119905)

= (1 + 119903119871

119894) (119879119894119909) (119905) minus 119903

119871

119894(119879119894119909) (119905)

= (119879119894119909) (119905) 119894 = 1 2 119899

(28)


1 2 119899 So |119879119894119909|1= |119879119894x|0 119894 = 1 2 119899 By (25) we have

(119879119894119909)(119905) ge 120575

119894|119879119894119909|1 119894 = 1 2 119899 Hence 119879119909 isin 119864 This


119894119909)1015840(119905) ge 0 119894 = 1 2 119899 implies

(119879119894119909)1015840

(119905) le (119879119894119909) (119905) 119894 = 1 2 119899 (29)

From (23) (26) (1198602) and (119860

4) we have

(119879119894119909)1015840

(119905)

= 119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

le 119903119894(119905) (119879119894119909) (119905) minus 120575

119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[120575119895119886119894119895(119905) minus 119887

119894119895(119905) minus 119888

119894119895(119905)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

le 119903119872

119894(119879119894119909) (119905) minus 120575

119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119903119872

119894minus 1

120575119894(1 minus 120575

119894)

times

119899

sum

119895=1

int

120596

0

[119886119894119895(119904) + 119887

119894119895(119904) + 119888

119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1)int

119905+120596

119905

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[119886119894119895(119904) + 119887

119894119895(119904) + 119888

119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904

le 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1)


times int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1) (119879

119894119909) (119905) = (119879

119894119909) (119905)

119894 = 1 2 119899

(30)



119877max1le119895le119899

sum119899

119895=1119887119872

119894119895 lt 1

(i) Ifmax119903119872119894 119894 = 1 2 119899 le 1 then 119879 119864⋂Ω

119877rarr 119864



119894 119894 = 1 2 119899 gt 1 then


119877=

119909 isin 119884 |119909|1lt 119877


119884(119879(119878)) le 119877max

1le119895le119899sum119899

119895=1119887119872

119894119895120572119884(119878)


119884(119878) Then for any


sum119899

119895=1119887119872

119894119895120578 there is a finite


119894119878119894with diam(119878

119894) le 120578+120598

Therefore

1003816100381610038161003816119909 minus 119910

10038161003816100381610038161le 120578 + 120598 for any 119909 119910 isin 119878

119894 (31)



119894= ⋃119895119878119894119895and

1003816100381610038161003816119909 minus 119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895 (32)


(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1198772

1 minus 120575119894

int

120596

0

[

[

119899

sum

119895=1

119886119894119895(119904) +

119899

sum

119895=1

119887119894119895(119904) +

119899

sum

119895=1

119888119894119895(119904)]

]

119889119904

+

1

1 minus 120575119894

sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896)) = 119861

119894 119894 = 1 2 119899

(33)10038161003816100381610038161003816(119879119894119909)1015840

(119905)

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899

(34)

Hence

(119879119909)0le

119899

sum

119894=1

119861119894

10038171003817100381710038171003817(119879119909)1015840100381710038171003817100381710038170le

119899

sum

119894=1

[

[

119861119894119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895)]

]

(35)


119894119895119896

of 119878119894119895such that 119878

119894119895= ⋃119896119878119894119895119896

and

1003816100381610038161003816119879119909 minus 119879119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895119896 (36)



119894119895119896 we

have

10038161003816100381610038161003816(119879119894119909)1015840

minus (119879119894119910)1015840100381610038161003816100381610038160

= max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119903

119894(119905) (119879119894119910) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119910119894(119905)[

[

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le max119905isin[0120596]

1003816100381610038161003816119903119894(119905) [(119879

119894119909) (119905) minus (119879

119894119910) (119905)]

1003816100381610038161003816

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119894[

[

(

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585)

minus(

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585 +

119899

sum

119895=1

119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ max119905isin[0120596]

10038161003816100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905) minus 119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894

1003816100381610038161003816(119879119894119909) minus (119879

119894119910)10038161003816100381610038160

+ 119877max119905isin[0120596]

119899

sum

119895=1

[

[

119886119894119895(119905)

10038161003816100381610038161003816119909119895(119905)minus119910

119895(119905)

10038161003816100381610038161003816+119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585)

times1003816100381610038161003816119909119894(119905+120585)minus119910

119894(119905+120585)

1003816100381610038161003816119889120585+119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585)

times

100381610038161003816100381610038161199091015840

119895(119905+120585) minus119910

1015840

(119905minus120585)

10038161003816100381610038161003816119889120585]

]

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905)minus119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905)

timesint

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585)

100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894120598+119877120598

119899

sum

119895=1

119886119872

119894119895+119877120598

119899

sum

119895=1

119887119872

119894119895+1003816100381610038161003816119909119894

10038161003816100381610038160(120578+120598)

times

119899

sum

119895=1

119888119872

119894119895+119877120598

119899

sum

119895=1

(119886119872

119894119895+119887119872

119894119895+119888119872

119894119895)

=1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895120578 + 119861119894120598

(37)

where

119861119894= 119903119872

119894+ 2119877

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899 (38)



1003817100381710038171003817119879119909 minus 119879119910

10038171003817100381710038171le (

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895)120578 + 120598

119899

sum

119894=1

119861119894

le 119877max1le119894le119899

119899

sum

119895=1

119888119872

119894119895

120578 + 120598

119899

sum

119894=1

119861119894 for any 119909 119910 isin 119878

119894119895119896

(39)


120572119884(119879 (119878)) le 119877max

1le119895le119899

119899

sum

119895=1

119888119872

119894119895

120572119884(119878) (40)


3 Main Results






119894 119894 = 1 2 119899 gt 1 then



119877 = (min119894isin[1119899]

1205752

119894

1 minus 120575119894

min119895isin[1119899]

120572119894119895)

minus1

0 lt 119903 lt min119894isin[1119899]

120575119894(1 minus 120575

119894) minus 120574119894

max119895isin[1119899]

120573119894119895

(41)


119903119877 In view





0gt 0


119879119894119909 (119905) minus 119909

119894(119905) ge 120575

119894

1003816100381610038161003816119879119894119909 minus 119909119894

10038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(42)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

int

120596

0

[119886119894119895(119904)+119887119894119895(119904)+119888119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904+120574119894

le

120573119894119895119903 + 120574119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160le 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160 119894 = 1 2 119899


1199090le 119879119909

0le max1le119894le119899

120575119894 1199090lt 119909

0 (44)






119894119909 (119905) ge 120575

119894

1003816100381610038161003816119909119894minus 11987911989411990910038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(45)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

gt

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)] 119889119904

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161) 119894 = 1 2 119899

(46)


Thus we have

1198791199090=

119899

sum

119894=1

100381610038161003816100381611987911989411990910038161003816100381610038160gt

119899

sum

119894=1

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161)

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

1198772= 119877

(47)


1199091ge 119879119909

1ge 119879119909

0gt 119877 (48)




4 Example



1199091015840

1(119905)

= 1199091(119905) [


minus (10 + sin 119905) 1199091(119905)


times int

0

minus12059111

11989111(120585) 1199091(119905 + 120585) 119889120585 minus

5 + sin 11990515

times int

0

minus12059112

11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)

times int

0

minus12059011

11989211(120585) 1199091015840

1(119905 + 120585) 119889120585 minus

4 minus 3 cos 1199056

timesint

0

minus12059012

11989212(120585) 1199091015840

2(119905 + 120585) 119889120585]

1199091015840

2(119905)

= 1199092(119905) [

1 + cos 1199058120587

minus (12 + 3 cos 119905) 1199091(119905)


times int

0

minus12059121

11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)

times int

0

minus12059122

11989122(120585) 1199092(119905 + 120585) 119889120585 minus

4 + sin 11990512

times int

0

minus12059021

11989221(120585) 1199091015840

2(119905 + 120585) 119889120585 minus


timesint

0

minus12059022

11989222(120585) 1199091015840

2(119905 + 120585) 119889120585]

(49)

where 120591119894119895 120590119894119895

isin (0 +infin) (119894 119895 = 1 2) and 119891119894119895 119892119894119895

isin

119862(119877 119877+) satisfying int0

minus120591119894119895

119891119894119895(120585)119889120585 = 1 int

0

minus120590119894119895

119892119894119895(120585)119889120585 119894 119895 = 1 2

Obviously

1199031(119905) =


1199032=

1 + cos 1199058120587

11988611(119905) = 10 + sin 119905

11988612(119905) = 9 minus cos 119905 119886

21(119905) = 12 + 3 cos 119905

11988622(119905) = 7 minus sin 119905 119887

11(119905) = 3 minus 2 cos 119905

11988712(119905) =

5 + sin 11990515

11988721(119905) = 2 + 3 sin 119905 119887

22(119905) = 1 minus cos 119905

11988811(119905) = 1 + sin 119905 119888

12(119905) =

4 minus 3 cos 1199056

11988821(119905) =

4 + sin 11990512

11988822(119905) =


119903119871

1= 119903119871

2= 0

(50)


1205751= 119890minus(12058712)

lt 1 1205752= 119890minus(14)

lt 1

12057211= 20120587120575

1minus 8120587 asymp 238007

12057212= 18120587120575

2minus 2120587 asymp 372423

12057221= 24120587120575

1minus

14120587

3

asymp 440594

12057222= 14120587120575

2minus 3120587 asymp 244284


min1le119895le2

1205721119895 = 12057211= 20120587120575

1minus 8120587 asymp 238007

min1le119895le2

1205722119895 = 12057222= 14120587120575

2minus 3120587 asymp 244284

120575111988611(119905) minus 119887

11(119905) minus 119888

11(119905) gt 0009 gt 0

120575211988612(119905) minus 119887

12(119905) minus 119888

12(119905) gt 1 gt 0

120575111988621(119905) minus 119887

21(119905) minus 119888

21(119905) gt 4 gt 0

120575211988622(119905) minus 119887

22(119905) minus 119888

22(119905) gt 2 gt 0

(1 + 119903119871

1)

1205752

112057211

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 8120587)

1 minus 119890minus(12058712)

asymp 652614 gt 18

ge max0le119905le2120587

11988611(119905) + 119887

11(119905) + 119888

11(119905)

(1 + 119903119871

1)

1205752

112057212

1 minus 1205751

=

119890minus(1205876)

(181205871205752minus 2120587)

1 minus 119890minus(12058712)

asymp 1021184 gt

347

30

ge max0le119905le2120587

11988612(119905) + 119887

12(119905) + 119888

12(119905)

(1 + 119903119871

2)

1205752

212057221

1 minus 1205752

=

119890minus(12)

(241205871205751minus (141205873))


asymp 1133411 gt

245

12

gt max0le119905le2120587

11988621(119905) + 119887

21(119905) + 119888

21(119905)

(1 + 119903119871

2)

1205752

212057222

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411 gt

53

5

gt max0le119905le2120587

11988622(119905) + 119887

22(119905) + 119888

22(119905)

1205752

1min1le119895le2

1205721119895

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 6120587)

1 minus 119890minus(12058712)

asymp 652614

1205752

2min1le119895le2

1205722119895

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411

min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

=

1205752

2min1le119895le2

1205722119895

1 minus 1205751

asymp 628411

119888119872

11+ 119888119872

12=

19

6

119888119872

21+ 119888119872

22=

61

60

max1le119894le2

2

sum

119895=1

119889119872

119894119895

=

19

6

(51)

Therefore

19

6

= max1le119894le2

2

sum

119894=1

119889119872

119894119895 lt min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

asymp 628411

(52)


119872

119894le 1 119894 = 1 2


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894=1|119909119894|0 where |119909

119894|0=

sup119905isin[0120596]

|119909119894(119905)| 119894 = 1 2 119899 and

119884=119909=(1199091 1199092 119909

119899)119879

119909 isin 1198751198621

(119877) | 119909 (119905+120596)

=119909 (119905) 119905isin119877

(16)


119894=1|119909119894|1 where |119909

119894|1=

max|119909119894|0 |1199091015840


the cone 119864 in 119884 by

119864=119909=(1199091 1199092 119909

119899)119879

119909isin1198751198621

(119877) | 119909119894(119905)=120575

119895

1003816100381610038161003816119909119894(119905)10038161003816100381610038161

119905isin[0 120596] 119894 119895=1 2 119899

(17)


(119879119909) (119905) = ((1198791119909) (119905) (119879

2119909) (119905) (119879

119899119909) (119905))

119879

(18)

where 119909 isin 119864 119905 isin 119877

(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904)int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

(19)

where

119866119894(119905 119904) =


119905119903119894(120579)119889120579


0119903119894(120579)119889120579

119904 isin [119905 119905 + 120596] 119894 = 1 2 119899

(20)


119894(119905 119904) 120597119866

119894(119905 119904)120597119905 =

119903119894(119905)119866119894(119905 119904) 119866

119894(119905 119905 + 120596) minus 119866

119894(119905 119905) = minus1 and

120575119894

1 minus 120575119894

le 119866119894(119905 119904) le

1

1 minus 120575119894

119904 isin [119905 119905 + 120596] 119894 = 1 2 119899

(21)




defined


119894 119894 = 1 2 119899 gt 1 then



(119879119894119909) (119905 + 120596)

= int

119905+2120596

119905+120596

119866119894(119905 + 120596 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905+120596le119905119896lt119905+2120596

119866119894(119905 + 120596 119905

119896) 119868119894119896(119909119894(119905119896))

= int

119905+120596

119905

119866119894(119905 + 120596 119906 + 120596) 119909

119894(119906 + 120596)

times[

[

119899

sum

119895=1

119886119894119895(119906 + 120596) 119909

119895(119906 + 120596) +

119899

sum

119895=1

119887119894119895(119906 + 120596)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119906 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119906 + 120596)int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119906 + 120596 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= int

119905+120596

119905

119866119894(119905 119906) 119909

119894(119906)

times[

[

119899

sum

119895=1

119886119894119895(119906) 119909119895(119906) +

119899

sum

119895=1

119887119894119895(119906) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119906+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119906) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119906+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= (119879119894119909) (119905) 119894 = 1 2 119899

(22)

That is (119879119894119909)(119905 + 120596) = (119879

119894119909)(119905) 119905 isin [0 120596] So 119879119909 isin 119884 In



119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

ge

119899

sum

119895=1

119886119894119895(119905) 120575119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161minus

119899

sum

119895=1

119887119894119895(119905)

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161minus

119899

sum

119895=1

119888119894119895(119905)

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

119899

sum

119895=1

[119886119894119895(119905) 120575119895minus119887119894119895(119905)minus119888119894119895(119905)]

100381610038161003816100381610038161199091015840

119895

100381610038161003816100381610038161ge 0 119894=1 2 119899

(23)


100381610038161003816100381611987911989411990910038161003816100381610038160

le

1

1 minus 120575119894

int

120596

0

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)+

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

(24)

(119879119894119909) (119905)

ge

120575119894

1 minus 120575119894

int

119905+120596

119905

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

=

120575119894

1 minus 120575119894

times int

120596

0

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

ge 120575119894

100381610038161003816100381611987911989411990910038161003816100381610038160 119894 = 1 2 119899

(25)


119894|119879119894119909|1 119894 = 1 2 119899 119905 isin


(119879119894119909)1015840

(119905)

= 119866119894(119905 119905 + 120596) 119909

119894(119905 + 120596)

times[

[

119899

sum

119895=1

119886119894119895(119905 + 120596) 119909

119895(119905 + 120596)

+

119899

sum

119895=1

119887119894119895(119905 + 120596) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905 + 120596) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120596 + 120585) 119889120585

]

]

minus 119866119894(119905 119905) 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)+

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119903119894(119905) (119879119894119909) (119905)

= 119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)


times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905+120585) 119889120585

]

]

119894 = 1 2 119899

(26)


1 2 119899 then

(119879119894119909)1015840

(119905) le 119903119894(119905) (119879119894119909) (119905) le 119903

119872

119894(119879119894119909) (119905) le (119879

119894119909) (119905)

119894 = 1 2 119899

(27)


119894119909)1015840(119905) lt 0 119894 =

1 2 119899 thenminus (119879119894119909)1015840

(119905)

= minus119903119894(119905) (119879119894119909) (119905) + 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

le1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

[119886119894119895(119905) + 119887

119894119895(119905) + 119888

119894119895(119905)]

1003816100381610038161003816119909119894

10038161003816100381610038161minus 119903119871

119894(119879119894119909) (119905)

le (1 + 119903119871

119894)

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)]

1003816100381610038161003816119909119894

10038161003816100381610038161119889119904

minus 119903119871

119894(119879119894119909) (119905)

= (1 + 119903119871

119894)int

119905+120596

119905

120575119894

1 minus 120575119894

120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus c

119894119895(119904)]

1003816100381610038161003816119909119894

10038161003816100381610038161119889119904 minus 119903

119871

119894(119879119894119909) (119905)

le (1 + 119903119871

119894)int

119905+120596

119905

int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896)) minus 119903

119871

119894(119879119894119909) (119905)

= (1 + 119903119871

119894) (119879119894119909) (119905) minus 119903

119871

119894(119879119894119909) (119905)

= (119879119894119909) (119905) 119894 = 1 2 119899

(28)


1 2 119899 So |119879119894119909|1= |119879119894x|0 119894 = 1 2 119899 By (25) we have

(119879119894119909)(119905) ge 120575

119894|119879119894119909|1 119894 = 1 2 119899 Hence 119879119909 isin 119864 This


119894119909)1015840(119905) ge 0 119894 = 1 2 119899 implies

(119879119894119909)1015840

(119905) le (119879119894119909) (119905) 119894 = 1 2 119899 (29)

From (23) (26) (1198602) and (119860

4) we have

(119879119894119909)1015840

(119905)

= 119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

le 119903119894(119905) (119879119894119909) (119905) minus 120575

119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[120575119895119886119894119895(119905) minus 119887

119894119895(119905) minus 119888

119894119895(119905)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

le 119903119872

119894(119879119894119909) (119905) minus 120575

119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119903119872

119894minus 1

120575119894(1 minus 120575

119894)

times

119899

sum

119895=1

int

120596

0

[119886119894119895(119904) + 119887

119894119895(119904) + 119888

119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1)int

119905+120596

119905

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[119886119894119895(119904) + 119887

119894119895(119904) + 119888

119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904

le 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1)


times int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1) (119879

119894119909) (119905) = (119879

119894119909) (119905)

119894 = 1 2 119899

(30)



119877max1le119895le119899

sum119899

119895=1119887119872

119894119895 lt 1

(i) Ifmax119903119872119894 119894 = 1 2 119899 le 1 then 119879 119864⋂Ω

119877rarr 119864



119894 119894 = 1 2 119899 gt 1 then


119877=

119909 isin 119884 |119909|1lt 119877


119884(119879(119878)) le 119877max

1le119895le119899sum119899

119895=1119887119872

119894119895120572119884(119878)


119884(119878) Then for any


sum119899

119895=1119887119872

119894119895120578 there is a finite


119894119878119894with diam(119878

119894) le 120578+120598

Therefore

1003816100381610038161003816119909 minus 119910

10038161003816100381610038161le 120578 + 120598 for any 119909 119910 isin 119878

119894 (31)



119894= ⋃119895119878119894119895and

1003816100381610038161003816119909 minus 119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895 (32)


(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1198772

1 minus 120575119894

int

120596

0

[

[

119899

sum

119895=1

119886119894119895(119904) +

119899

sum

119895=1

119887119894119895(119904) +

119899

sum

119895=1

119888119894119895(119904)]

]

119889119904

+

1

1 minus 120575119894

sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896)) = 119861

119894 119894 = 1 2 119899

(33)10038161003816100381610038161003816(119879119894119909)1015840

(119905)

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899

(34)

Hence

(119879119909)0le

119899

sum

119894=1

119861119894

10038171003817100381710038171003817(119879119909)1015840100381710038171003817100381710038170le

119899

sum

119894=1

[

[

119861119894119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895)]

]

(35)


119894119895119896

of 119878119894119895such that 119878

119894119895= ⋃119896119878119894119895119896

and

1003816100381610038161003816119879119909 minus 119879119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895119896 (36)



119894119895119896 we

have

10038161003816100381610038161003816(119879119894119909)1015840

minus (119879119894119910)1015840100381610038161003816100381610038160

= max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119903

119894(119905) (119879119894119910) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119910119894(119905)[

[

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le max119905isin[0120596]

1003816100381610038161003816119903119894(119905) [(119879

119894119909) (119905) minus (119879

119894119910) (119905)]

1003816100381610038161003816

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119894[

[

(

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585)

minus(

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585 +

119899

sum

119895=1

119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ max119905isin[0120596]

10038161003816100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905) minus 119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894

1003816100381610038161003816(119879119894119909) minus (119879

119894119910)10038161003816100381610038160

+ 119877max119905isin[0120596]

119899

sum

119895=1

[

[

119886119894119895(119905)

10038161003816100381610038161003816119909119895(119905)minus119910

119895(119905)

10038161003816100381610038161003816+119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585)

times1003816100381610038161003816119909119894(119905+120585)minus119910

119894(119905+120585)

1003816100381610038161003816119889120585+119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585)

times

100381610038161003816100381610038161199091015840

119895(119905+120585) minus119910

1015840

(119905minus120585)

10038161003816100381610038161003816119889120585]

]

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905)minus119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905)

timesint

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585)

100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894120598+119877120598

119899

sum

119895=1

119886119872

119894119895+119877120598

119899

sum

119895=1

119887119872

119894119895+1003816100381610038161003816119909119894

10038161003816100381610038160(120578+120598)

times

119899

sum

119895=1

119888119872

119894119895+119877120598

119899

sum

119895=1

(119886119872

119894119895+119887119872

119894119895+119888119872

119894119895)

=1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895120578 + 119861119894120598

(37)

where

119861119894= 119903119872

119894+ 2119877

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899 (38)



1003817100381710038171003817119879119909 minus 119879119910

10038171003817100381710038171le (

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895)120578 + 120598

119899

sum

119894=1

119861119894

le 119877max1le119894le119899

119899

sum

119895=1

119888119872

119894119895

120578 + 120598

119899

sum

119894=1

119861119894 for any 119909 119910 isin 119878

119894119895119896

(39)


120572119884(119879 (119878)) le 119877max

1le119895le119899

119899

sum

119895=1

119888119872

119894119895

120572119884(119878) (40)


3 Main Results






119894 119894 = 1 2 119899 gt 1 then



119877 = (min119894isin[1119899]

1205752

119894

1 minus 120575119894

min119895isin[1119899]

120572119894119895)

minus1

0 lt 119903 lt min119894isin[1119899]

120575119894(1 minus 120575

119894) minus 120574119894

max119895isin[1119899]

120573119894119895

(41)


119903119877 In view





0gt 0


119879119894119909 (119905) minus 119909

119894(119905) ge 120575

119894

1003816100381610038161003816119879119894119909 minus 119909119894

10038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(42)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

int

120596

0

[119886119894119895(119904)+119887119894119895(119904)+119888119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904+120574119894

le

120573119894119895119903 + 120574119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160le 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160 119894 = 1 2 119899


1199090le 119879119909

0le max1le119894le119899

120575119894 1199090lt 119909

0 (44)






119894119909 (119905) ge 120575

119894

1003816100381610038161003816119909119894minus 11987911989411990910038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(45)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

gt

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)] 119889119904

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161) 119894 = 1 2 119899

(46)


Thus we have

1198791199090=

119899

sum

119894=1

100381610038161003816100381611987911989411990910038161003816100381610038160gt

119899

sum

119894=1

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161)

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

1198772= 119877

(47)


1199091ge 119879119909

1ge 119879119909

0gt 119877 (48)




4 Example



1199091015840

1(119905)

= 1199091(119905) [


minus (10 + sin 119905) 1199091(119905)


times int

0

minus12059111

11989111(120585) 1199091(119905 + 120585) 119889120585 minus

5 + sin 11990515

times int

0

minus12059112

11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)

times int

0

minus12059011

11989211(120585) 1199091015840

1(119905 + 120585) 119889120585 minus

4 minus 3 cos 1199056

timesint

0

minus12059012

11989212(120585) 1199091015840

2(119905 + 120585) 119889120585]

1199091015840

2(119905)

= 1199092(119905) [

1 + cos 1199058120587

minus (12 + 3 cos 119905) 1199091(119905)


times int

0

minus12059121

11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)

times int

0

minus12059122

11989122(120585) 1199092(119905 + 120585) 119889120585 minus

4 + sin 11990512

times int

0

minus12059021

11989221(120585) 1199091015840

2(119905 + 120585) 119889120585 minus


timesint

0

minus12059022

11989222(120585) 1199091015840

2(119905 + 120585) 119889120585]

(49)

where 120591119894119895 120590119894119895

isin (0 +infin) (119894 119895 = 1 2) and 119891119894119895 119892119894119895

isin

119862(119877 119877+) satisfying int0

minus120591119894119895

119891119894119895(120585)119889120585 = 1 int

0

minus120590119894119895

119892119894119895(120585)119889120585 119894 119895 = 1 2

Obviously

1199031(119905) =


1199032=

1 + cos 1199058120587

11988611(119905) = 10 + sin 119905

11988612(119905) = 9 minus cos 119905 119886

21(119905) = 12 + 3 cos 119905

11988622(119905) = 7 minus sin 119905 119887

11(119905) = 3 minus 2 cos 119905

11988712(119905) =

5 + sin 11990515

11988721(119905) = 2 + 3 sin 119905 119887

22(119905) = 1 minus cos 119905

11988811(119905) = 1 + sin 119905 119888

12(119905) =

4 minus 3 cos 1199056

11988821(119905) =

4 + sin 11990512

11988822(119905) =


119903119871

1= 119903119871

2= 0

(50)


1205751= 119890minus(12058712)

lt 1 1205752= 119890minus(14)

lt 1

12057211= 20120587120575

1minus 8120587 asymp 238007

12057212= 18120587120575

2minus 2120587 asymp 372423

12057221= 24120587120575

1minus

14120587

3

asymp 440594

12057222= 14120587120575

2minus 3120587 asymp 244284


min1le119895le2

1205721119895 = 12057211= 20120587120575

1minus 8120587 asymp 238007

min1le119895le2

1205722119895 = 12057222= 14120587120575

2minus 3120587 asymp 244284

120575111988611(119905) minus 119887

11(119905) minus 119888

11(119905) gt 0009 gt 0

120575211988612(119905) minus 119887

12(119905) minus 119888

12(119905) gt 1 gt 0

120575111988621(119905) minus 119887

21(119905) minus 119888

21(119905) gt 4 gt 0

120575211988622(119905) minus 119887

22(119905) minus 119888

22(119905) gt 2 gt 0

(1 + 119903119871

1)

1205752

112057211

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 8120587)

1 minus 119890minus(12058712)

asymp 652614 gt 18

ge max0le119905le2120587

11988611(119905) + 119887

11(119905) + 119888

11(119905)

(1 + 119903119871

1)

1205752

112057212

1 minus 1205751

=

119890minus(1205876)

(181205871205752minus 2120587)

1 minus 119890minus(12058712)

asymp 1021184 gt

347

30

ge max0le119905le2120587

11988612(119905) + 119887

12(119905) + 119888

12(119905)

(1 + 119903119871

2)

1205752

212057221

1 minus 1205752

=

119890minus(12)

(241205871205751minus (141205873))


asymp 1133411 gt

245

12

gt max0le119905le2120587

11988621(119905) + 119887

21(119905) + 119888

21(119905)

(1 + 119903119871

2)

1205752

212057222

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411 gt

53

5

gt max0le119905le2120587

11988622(119905) + 119887

22(119905) + 119888

22(119905)

1205752

1min1le119895le2

1205721119895

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 6120587)

1 minus 119890minus(12058712)

asymp 652614

1205752

2min1le119895le2

1205722119895

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411

min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

=

1205752

2min1le119895le2

1205722119895

1 minus 1205751

asymp 628411

119888119872

11+ 119888119872

12=

19

6

119888119872

21+ 119888119872

22=

61

60

max1le119894le2

2

sum

119895=1

119889119872

119894119895

=

19

6

(51)

Therefore

19

6

= max1le119894le2

2

sum

119894=1

119889119872

119894119895 lt min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

asymp 628411

(52)


119872

119894le 1 119894 = 1 2


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

ge

119899

sum

119895=1

119886119894119895(119905) 120575119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161minus

119899

sum

119895=1

119887119894119895(119905)

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161minus

119899

sum

119895=1

119888119894119895(119905)

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

119899

sum

119895=1

[119886119894119895(119905) 120575119895minus119887119894119895(119905)minus119888119894119895(119905)]

100381610038161003816100381610038161199091015840

119895

100381610038161003816100381610038161ge 0 119894=1 2 119899

(23)


100381610038161003816100381611987911989411990910038161003816100381610038160

le

1

1 minus 120575119894

int

120596

0

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)+

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

(24)

(119879119894119909) (119905)

ge

120575119894

1 minus 120575119894

int

119905+120596

119905

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

=

120575119894

1 minus 120575119894

times int

120596

0

119909119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904)

+

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896))

ge 120575119894

100381610038161003816100381611987911989411990910038161003816100381610038160 119894 = 1 2 119899

(25)


119894|119879119894119909|1 119894 = 1 2 119899 119905 isin


(119879119894119909)1015840

(119905)

= 119866119894(119905 119905 + 120596) 119909

119894(119905 + 120596)

times[

[

119899

sum

119895=1

119886119894119895(119905 + 120596) 119909

119895(119905 + 120596)

+

119899

sum

119895=1

119887119894119895(119905 + 120596) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120596 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905 + 120596) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120596 + 120585) 119889120585

]

]

minus 119866119894(119905 119905) 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)+

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119903119894(119905) (119879119894119909) (119905)

= 119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)


times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905+120585) 119889120585

]

]

119894 = 1 2 119899

(26)


1 2 119899 then

(119879119894119909)1015840

(119905) le 119903119894(119905) (119879119894119909) (119905) le 119903

119872

119894(119879119894119909) (119905) le (119879

119894119909) (119905)

119894 = 1 2 119899

(27)


119894119909)1015840(119905) lt 0 119894 =

1 2 119899 thenminus (119879119894119909)1015840

(119905)

= minus119903119894(119905) (119879119894119909) (119905) + 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

le1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

[119886119894119895(119905) + 119887

119894119895(119905) + 119888

119894119895(119905)]

1003816100381610038161003816119909119894

10038161003816100381610038161minus 119903119871

119894(119879119894119909) (119905)

le (1 + 119903119871

119894)

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)]

1003816100381610038161003816119909119894

10038161003816100381610038161119889119904

minus 119903119871

119894(119879119894119909) (119905)

= (1 + 119903119871

119894)int

119905+120596

119905

120575119894

1 minus 120575119894

120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus c

119894119895(119904)]

1003816100381610038161003816119909119894

10038161003816100381610038161119889119904 minus 119903

119871

119894(119879119894119909) (119905)

le (1 + 119903119871

119894)int

119905+120596

119905

int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896)) minus 119903

119871

119894(119879119894119909) (119905)

= (1 + 119903119871

119894) (119879119894119909) (119905) minus 119903

119871

119894(119879119894119909) (119905)

= (119879119894119909) (119905) 119894 = 1 2 119899

(28)


1 2 119899 So |119879119894119909|1= |119879119894x|0 119894 = 1 2 119899 By (25) we have

(119879119894119909)(119905) ge 120575

119894|119879119894119909|1 119894 = 1 2 119899 Hence 119879119909 isin 119864 This


119894119909)1015840(119905) ge 0 119894 = 1 2 119899 implies

(119879119894119909)1015840

(119905) le (119879119894119909) (119905) 119894 = 1 2 119899 (29)

From (23) (26) (1198602) and (119860

4) we have

(119879119894119909)1015840

(119905)

= 119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

le 119903119894(119905) (119879119894119909) (119905) minus 120575

119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[120575119895119886119894119895(119905) minus 119887

119894119895(119905) minus 119888

119894119895(119905)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

le 119903119872

119894(119879119894119909) (119905) minus 120575

119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119903119872

119894minus 1

120575119894(1 minus 120575

119894)

times

119899

sum

119895=1

int

120596

0

[119886119894119895(119904) + 119887

119894119895(119904) + 119888

119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1)int

119905+120596

119905

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[119886119894119895(119904) + 119887

119894119895(119904) + 119888

119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904

le 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1)


times int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1) (119879

119894119909) (119905) = (119879

119894119909) (119905)

119894 = 1 2 119899

(30)



119877max1le119895le119899

sum119899

119895=1119887119872

119894119895 lt 1

(i) Ifmax119903119872119894 119894 = 1 2 119899 le 1 then 119879 119864⋂Ω

119877rarr 119864



119894 119894 = 1 2 119899 gt 1 then


119877=

119909 isin 119884 |119909|1lt 119877


119884(119879(119878)) le 119877max

1le119895le119899sum119899

119895=1119887119872

119894119895120572119884(119878)


119884(119878) Then for any


sum119899

119895=1119887119872

119894119895120578 there is a finite


119894119878119894with diam(119878

119894) le 120578+120598

Therefore

1003816100381610038161003816119909 minus 119910

10038161003816100381610038161le 120578 + 120598 for any 119909 119910 isin 119878

119894 (31)



119894= ⋃119895119878119894119895and

1003816100381610038161003816119909 minus 119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895 (32)


(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1198772

1 minus 120575119894

int

120596

0

[

[

119899

sum

119895=1

119886119894119895(119904) +

119899

sum

119895=1

119887119894119895(119904) +

119899

sum

119895=1

119888119894119895(119904)]

]

119889119904

+

1

1 minus 120575119894

sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896)) = 119861

119894 119894 = 1 2 119899

(33)10038161003816100381610038161003816(119879119894119909)1015840

(119905)

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899

(34)

Hence

(119879119909)0le

119899

sum

119894=1

119861119894

10038171003817100381710038171003817(119879119909)1015840100381710038171003817100381710038170le

119899

sum

119894=1

[

[

119861119894119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895)]

]

(35)


119894119895119896

of 119878119894119895such that 119878

119894119895= ⋃119896119878119894119895119896

and

1003816100381610038161003816119879119909 minus 119879119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895119896 (36)



119894119895119896 we

have

10038161003816100381610038161003816(119879119894119909)1015840

minus (119879119894119910)1015840100381610038161003816100381610038160

= max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119903

119894(119905) (119879119894119910) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119910119894(119905)[

[

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le max119905isin[0120596]

1003816100381610038161003816119903119894(119905) [(119879

119894119909) (119905) minus (119879

119894119910) (119905)]

1003816100381610038161003816

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119894[

[

(

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585)

minus(

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585 +

119899

sum

119895=1

119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ max119905isin[0120596]

10038161003816100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905) minus 119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894

1003816100381610038161003816(119879119894119909) minus (119879

119894119910)10038161003816100381610038160

+ 119877max119905isin[0120596]

119899

sum

119895=1

[

[

119886119894119895(119905)

10038161003816100381610038161003816119909119895(119905)minus119910

119895(119905)

10038161003816100381610038161003816+119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585)

times1003816100381610038161003816119909119894(119905+120585)minus119910

119894(119905+120585)

1003816100381610038161003816119889120585+119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585)

times

100381610038161003816100381610038161199091015840

119895(119905+120585) minus119910

1015840

(119905minus120585)

10038161003816100381610038161003816119889120585]

]

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905)minus119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905)

timesint

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585)

100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894120598+119877120598

119899

sum

119895=1

119886119872

119894119895+119877120598

119899

sum

119895=1

119887119872

119894119895+1003816100381610038161003816119909119894

10038161003816100381610038160(120578+120598)

times

119899

sum

119895=1

119888119872

119894119895+119877120598

119899

sum

119895=1

(119886119872

119894119895+119887119872

119894119895+119888119872

119894119895)

=1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895120578 + 119861119894120598

(37)

where

119861119894= 119903119872

119894+ 2119877

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899 (38)



1003817100381710038171003817119879119909 minus 119879119910

10038171003817100381710038171le (

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895)120578 + 120598

119899

sum

119894=1

119861119894

le 119877max1le119894le119899

119899

sum

119895=1

119888119872

119894119895

120578 + 120598

119899

sum

119894=1

119861119894 for any 119909 119910 isin 119878

119894119895119896

(39)


120572119884(119879 (119878)) le 119877max

1le119895le119899

119899

sum

119895=1

119888119872

119894119895

120572119884(119878) (40)


3 Main Results






119894 119894 = 1 2 119899 gt 1 then



119877 = (min119894isin[1119899]

1205752

119894

1 minus 120575119894

min119895isin[1119899]

120572119894119895)

minus1

0 lt 119903 lt min119894isin[1119899]

120575119894(1 minus 120575

119894) minus 120574119894

max119895isin[1119899]

120573119894119895

(41)


119903119877 In view





0gt 0


119879119894119909 (119905) minus 119909

119894(119905) ge 120575

119894

1003816100381610038161003816119879119894119909 minus 119909119894

10038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(42)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

int

120596

0

[119886119894119895(119904)+119887119894119895(119904)+119888119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904+120574119894

le

120573119894119895119903 + 120574119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160le 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160 119894 = 1 2 119899


1199090le 119879119909

0le max1le119894le119899

120575119894 1199090lt 119909

0 (44)






119894119909 (119905) ge 120575

119894

1003816100381610038161003816119909119894minus 11987911989411990910038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(45)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

gt

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)] 119889119904

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161) 119894 = 1 2 119899

(46)


Thus we have

1198791199090=

119899

sum

119894=1

100381610038161003816100381611987911989411990910038161003816100381610038160gt

119899

sum

119894=1

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161)

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

1198772= 119877

(47)


1199091ge 119879119909

1ge 119879119909

0gt 119877 (48)




4 Example



1199091015840

1(119905)

= 1199091(119905) [


minus (10 + sin 119905) 1199091(119905)


times int

0

minus12059111

11989111(120585) 1199091(119905 + 120585) 119889120585 minus

5 + sin 11990515

times int

0

minus12059112

11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)

times int

0

minus12059011

11989211(120585) 1199091015840

1(119905 + 120585) 119889120585 minus

4 minus 3 cos 1199056

timesint

0

minus12059012

11989212(120585) 1199091015840

2(119905 + 120585) 119889120585]

1199091015840

2(119905)

= 1199092(119905) [

1 + cos 1199058120587

minus (12 + 3 cos 119905) 1199091(119905)


times int

0

minus12059121

11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)

times int

0

minus12059122

11989122(120585) 1199092(119905 + 120585) 119889120585 minus

4 + sin 11990512

times int

0

minus12059021

11989221(120585) 1199091015840

2(119905 + 120585) 119889120585 minus


timesint

0

minus12059022

11989222(120585) 1199091015840

2(119905 + 120585) 119889120585]

(49)

where 120591119894119895 120590119894119895

isin (0 +infin) (119894 119895 = 1 2) and 119891119894119895 119892119894119895

isin

119862(119877 119877+) satisfying int0

minus120591119894119895

119891119894119895(120585)119889120585 = 1 int

0

minus120590119894119895

119892119894119895(120585)119889120585 119894 119895 = 1 2

Obviously

1199031(119905) =


1199032=

1 + cos 1199058120587

11988611(119905) = 10 + sin 119905

11988612(119905) = 9 minus cos 119905 119886

21(119905) = 12 + 3 cos 119905

11988622(119905) = 7 minus sin 119905 119887

11(119905) = 3 minus 2 cos 119905

11988712(119905) =

5 + sin 11990515

11988721(119905) = 2 + 3 sin 119905 119887

22(119905) = 1 minus cos 119905

11988811(119905) = 1 + sin 119905 119888

12(119905) =

4 minus 3 cos 1199056

11988821(119905) =

4 + sin 11990512

11988822(119905) =


119903119871

1= 119903119871

2= 0

(50)


1205751= 119890minus(12058712)

lt 1 1205752= 119890minus(14)

lt 1

12057211= 20120587120575

1minus 8120587 asymp 238007

12057212= 18120587120575

2minus 2120587 asymp 372423

12057221= 24120587120575

1minus

14120587

3

asymp 440594

12057222= 14120587120575

2minus 3120587 asymp 244284


min1le119895le2

1205721119895 = 12057211= 20120587120575

1minus 8120587 asymp 238007

min1le119895le2

1205722119895 = 12057222= 14120587120575

2minus 3120587 asymp 244284

120575111988611(119905) minus 119887

11(119905) minus 119888

11(119905) gt 0009 gt 0

120575211988612(119905) minus 119887

12(119905) minus 119888

12(119905) gt 1 gt 0

120575111988621(119905) minus 119887

21(119905) minus 119888

21(119905) gt 4 gt 0

120575211988622(119905) minus 119887

22(119905) minus 119888

22(119905) gt 2 gt 0

(1 + 119903119871

1)

1205752

112057211

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 8120587)

1 minus 119890minus(12058712)

asymp 652614 gt 18

ge max0le119905le2120587

11988611(119905) + 119887

11(119905) + 119888

11(119905)

(1 + 119903119871

1)

1205752

112057212

1 minus 1205751

=

119890minus(1205876)

(181205871205752minus 2120587)

1 minus 119890minus(12058712)

asymp 1021184 gt

347

30

ge max0le119905le2120587

11988612(119905) + 119887

12(119905) + 119888

12(119905)

(1 + 119903119871

2)

1205752

212057221

1 minus 1205752

=

119890minus(12)

(241205871205751minus (141205873))


asymp 1133411 gt

245

12

gt max0le119905le2120587

11988621(119905) + 119887

21(119905) + 119888

21(119905)

(1 + 119903119871

2)

1205752

212057222

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411 gt

53

5

gt max0le119905le2120587

11988622(119905) + 119887

22(119905) + 119888

22(119905)

1205752

1min1le119895le2

1205721119895

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 6120587)

1 minus 119890minus(12058712)

asymp 652614

1205752

2min1le119895le2

1205722119895

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411

min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

=

1205752

2min1le119895le2

1205722119895

1 minus 1205751

asymp 628411

119888119872

11+ 119888119872

12=

19

6

119888119872

21+ 119888119872

22=

61

60

max1le119894le2

2

sum

119895=1

119889119872

119894119895

=

19

6

(51)

Therefore

19

6

= max1le119894le2

2

sum

119894=1

119889119872

119894119895 lt min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

asymp 628411

(52)


119872

119894le 1 119894 = 1 2


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905+120585) 119889120585

]

]

119894 = 1 2 119899

(26)


1 2 119899 then

(119879119894119909)1015840

(119905) le 119903119894(119905) (119879119894119909) (119905) le 119903

119872

119894(119879119894119909) (119905) le (119879

119894119909) (119905)

119894 = 1 2 119899

(27)


119894119909)1015840(119905) lt 0 119894 =

1 2 119899 thenminus (119879119894119909)1015840

(119905)

= minus119903119894(119905) (119879119894119909) (119905) + 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

le1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

[119886119894119895(119905) + 119887

119894119895(119905) + 119888

119894119895(119905)]

1003816100381610038161003816119909119894

10038161003816100381610038161minus 119903119871

119894(119879119894119909) (119905)

le (1 + 119903119871

119894)

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)]

1003816100381610038161003816119909119894

10038161003816100381610038161119889119904

minus 119903119871

119894(119879119894119909) (119905)

= (1 + 119903119871

119894)int

119905+120596

119905

120575119894

1 minus 120575119894

120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus c

119894119895(119904)]

1003816100381610038161003816119909119894

10038161003816100381610038161119889119904 minus 119903

119871

119894(119879119894119909) (119905)

le (1 + 119903119871

119894)int

119905+120596

119905

int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896)) minus 119903

119871

119894(119879119894119909) (119905)

= (1 + 119903119871

119894) (119879119894119909) (119905) minus 119903

119871

119894(119879119894119909) (119905)

= (119879119894119909) (119905) 119894 = 1 2 119899

(28)


1 2 119899 So |119879119894119909|1= |119879119894x|0 119894 = 1 2 119899 By (25) we have

(119879119894119909)(119905) ge 120575

119894|119879119894119909|1 119894 = 1 2 119899 Hence 119879119909 isin 119864 This


119894119909)1015840(119905) ge 0 119894 = 1 2 119899 implies

(119879119894119909)1015840

(119905) le (119879119894119909) (119905) 119894 = 1 2 119899 (29)

From (23) (26) (1198602) and (119860

4) we have

(119879119894119909)1015840

(119905)

= 119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) minus

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

minus

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

le 119903119894(119905) (119879119894119909) (119905) minus 120575

119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[120575119895119886119894119895(119905) minus 119887

119894119895(119905) minus 119888

119894119895(119905)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

le 119903119872

119894(119879119894119909) (119905) minus 120575

119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119903119872

119894minus 1

120575119894(1 minus 120575

119894)

times

119899

sum

119895=1

int

120596

0

[119886119894119895(119904) + 119887

119894119895(119904) + 119888

119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1)int

119905+120596

119905

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

times

119899

sum

119895=1

[119886119894119895(119904) + 119887

119894119895(119904) + 119888

119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904

le 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1)


times int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1) (119879

119894119909) (119905) = (119879

119894119909) (119905)

119894 = 1 2 119899

(30)



119877max1le119895le119899

sum119899

119895=1119887119872

119894119895 lt 1

(i) Ifmax119903119872119894 119894 = 1 2 119899 le 1 then 119879 119864⋂Ω

119877rarr 119864



119894 119894 = 1 2 119899 gt 1 then


119877=

119909 isin 119884 |119909|1lt 119877


119884(119879(119878)) le 119877max

1le119895le119899sum119899

119895=1119887119872

119894119895120572119884(119878)


119884(119878) Then for any


sum119899

119895=1119887119872

119894119895120578 there is a finite


119894119878119894with diam(119878

119894) le 120578+120598

Therefore

1003816100381610038161003816119909 minus 119910

10038161003816100381610038161le 120578 + 120598 for any 119909 119910 isin 119878

119894 (31)



119894= ⋃119895119878119894119895and

1003816100381610038161003816119909 minus 119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895 (32)


(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1198772

1 minus 120575119894

int

120596

0

[

[

119899

sum

119895=1

119886119894119895(119904) +

119899

sum

119895=1

119887119894119895(119904) +

119899

sum

119895=1

119888119894119895(119904)]

]

119889119904

+

1

1 minus 120575119894

sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896)) = 119861

119894 119894 = 1 2 119899

(33)10038161003816100381610038161003816(119879119894119909)1015840

(119905)

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899

(34)

Hence

(119879119909)0le

119899

sum

119894=1

119861119894

10038171003817100381710038171003817(119879119909)1015840100381710038171003817100381710038170le

119899

sum

119894=1

[

[

119861119894119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895)]

]

(35)


119894119895119896

of 119878119894119895such that 119878

119894119895= ⋃119896119878119894119895119896

and

1003816100381610038161003816119879119909 minus 119879119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895119896 (36)



119894119895119896 we

have

10038161003816100381610038161003816(119879119894119909)1015840

minus (119879119894119910)1015840100381610038161003816100381610038160

= max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119903

119894(119905) (119879119894119910) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119910119894(119905)[

[

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le max119905isin[0120596]

1003816100381610038161003816119903119894(119905) [(119879

119894119909) (119905) minus (119879

119894119910) (119905)]

1003816100381610038161003816

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119894[

[

(

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585)

minus(

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585 +

119899

sum

119895=1

119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ max119905isin[0120596]

10038161003816100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905) minus 119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894

1003816100381610038161003816(119879119894119909) minus (119879

119894119910)10038161003816100381610038160

+ 119877max119905isin[0120596]

119899

sum

119895=1

[

[

119886119894119895(119905)

10038161003816100381610038161003816119909119895(119905)minus119910

119895(119905)

10038161003816100381610038161003816+119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585)

times1003816100381610038161003816119909119894(119905+120585)minus119910

119894(119905+120585)

1003816100381610038161003816119889120585+119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585)

times

100381610038161003816100381610038161199091015840

119895(119905+120585) minus119910

1015840

(119905minus120585)

10038161003816100381610038161003816119889120585]

]

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905)minus119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905)

timesint

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585)

100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894120598+119877120598

119899

sum

119895=1

119886119872

119894119895+119877120598

119899

sum

119895=1

119887119872

119894119895+1003816100381610038161003816119909119894

10038161003816100381610038160(120578+120598)

times

119899

sum

119895=1

119888119872

119894119895+119877120598

119899

sum

119895=1

(119886119872

119894119895+119887119872

119894119895+119888119872

119894119895)

=1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895120578 + 119861119894120598

(37)

where

119861119894= 119903119872

119894+ 2119877

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899 (38)



1003817100381710038171003817119879119909 minus 119879119910

10038171003817100381710038171le (

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895)120578 + 120598

119899

sum

119894=1

119861119894

le 119877max1le119894le119899

119899

sum

119895=1

119888119872

119894119895

120578 + 120598

119899

sum

119894=1

119861119894 for any 119909 119910 isin 119878

119894119895119896

(39)


120572119884(119879 (119878)) le 119877max

1le119895le119899

119899

sum

119895=1

119888119872

119894119895

120572119884(119878) (40)


3 Main Results






119894 119894 = 1 2 119899 gt 1 then



119877 = (min119894isin[1119899]

1205752

119894

1 minus 120575119894

min119895isin[1119899]

120572119894119895)

minus1

0 lt 119903 lt min119894isin[1119899]

120575119894(1 minus 120575

119894) minus 120574119894

max119895isin[1119899]

120573119894119895

(41)


119903119877 In view





0gt 0


119879119894119909 (119905) minus 119909

119894(119905) ge 120575

119894

1003816100381610038161003816119879119894119909 minus 119909119894

10038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(42)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

int

120596

0

[119886119894119895(119904)+119887119894119895(119904)+119888119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904+120574119894

le

120573119894119895119903 + 120574119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160le 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160 119894 = 1 2 119899


1199090le 119879119909

0le max1le119894le119899

120575119894 1199090lt 119909

0 (44)






119894119909 (119905) ge 120575

119894

1003816100381610038161003816119909119894minus 11987911989411990910038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(45)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

gt

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)] 119889119904

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161) 119894 = 1 2 119899

(46)


Thus we have

1198791199090=

119899

sum

119894=1

100381610038161003816100381611987911989411990910038161003816100381610038160gt

119899

sum

119894=1

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161)

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

1198772= 119877

(47)


1199091ge 119879119909

1ge 119879119909

0gt 119877 (48)




4 Example



1199091015840

1(119905)

= 1199091(119905) [


minus (10 + sin 119905) 1199091(119905)


times int

0

minus12059111

11989111(120585) 1199091(119905 + 120585) 119889120585 minus

5 + sin 11990515

times int

0

minus12059112

11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)

times int

0

minus12059011

11989211(120585) 1199091015840

1(119905 + 120585) 119889120585 minus

4 minus 3 cos 1199056

timesint

0

minus12059012

11989212(120585) 1199091015840

2(119905 + 120585) 119889120585]

1199091015840

2(119905)

= 1199092(119905) [

1 + cos 1199058120587

minus (12 + 3 cos 119905) 1199091(119905)


times int

0

minus12059121

11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)

times int

0

minus12059122

11989122(120585) 1199092(119905 + 120585) 119889120585 minus

4 + sin 11990512

times int

0

minus12059021

11989221(120585) 1199091015840

2(119905 + 120585) 119889120585 minus


timesint

0

minus12059022

11989222(120585) 1199091015840

2(119905 + 120585) 119889120585]

(49)

where 120591119894119895 120590119894119895

isin (0 +infin) (119894 119895 = 1 2) and 119891119894119895 119892119894119895

isin

119862(119877 119877+) satisfying int0

minus120591119894119895

119891119894119895(120585)119889120585 = 1 int

0

minus120590119894119895

119892119894119895(120585)119889120585 119894 119895 = 1 2

Obviously

1199031(119905) =


1199032=

1 + cos 1199058120587

11988611(119905) = 10 + sin 119905

11988612(119905) = 9 minus cos 119905 119886

21(119905) = 12 + 3 cos 119905

11988622(119905) = 7 minus sin 119905 119887

11(119905) = 3 minus 2 cos 119905

11988712(119905) =

5 + sin 11990515

11988721(119905) = 2 + 3 sin 119905 119887

22(119905) = 1 minus cos 119905

11988811(119905) = 1 + sin 119905 119888

12(119905) =

4 minus 3 cos 1199056

11988821(119905) =

4 + sin 11990512

11988822(119905) =


119903119871

1= 119903119871

2= 0

(50)


1205751= 119890minus(12058712)

lt 1 1205752= 119890minus(14)

lt 1

12057211= 20120587120575

1minus 8120587 asymp 238007

12057212= 18120587120575

2minus 2120587 asymp 372423

12057221= 24120587120575

1minus

14120587

3

asymp 440594

12057222= 14120587120575

2minus 3120587 asymp 244284


min1le119895le2

1205721119895 = 12057211= 20120587120575

1minus 8120587 asymp 238007

min1le119895le2

1205722119895 = 12057222= 14120587120575

2minus 3120587 asymp 244284

120575111988611(119905) minus 119887

11(119905) minus 119888

11(119905) gt 0009 gt 0

120575211988612(119905) minus 119887

12(119905) minus 119888

12(119905) gt 1 gt 0

120575111988621(119905) minus 119887

21(119905) minus 119888

21(119905) gt 4 gt 0

120575211988622(119905) minus 119887

22(119905) minus 119888

22(119905) gt 2 gt 0

(1 + 119903119871

1)

1205752

112057211

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 8120587)

1 minus 119890minus(12058712)

asymp 652614 gt 18

ge max0le119905le2120587

11988611(119905) + 119887

11(119905) + 119888

11(119905)

(1 + 119903119871

1)

1205752

112057212

1 minus 1205751

=

119890minus(1205876)

(181205871205752minus 2120587)

1 minus 119890minus(12058712)

asymp 1021184 gt

347

30

ge max0le119905le2120587

11988612(119905) + 119887

12(119905) + 119888

12(119905)

(1 + 119903119871

2)

1205752

212057221

1 minus 1205752

=

119890minus(12)

(241205871205751minus (141205873))


asymp 1133411 gt

245

12

gt max0le119905le2120587

11988621(119905) + 119887

21(119905) + 119888

21(119905)

(1 + 119903119871

2)

1205752

212057222

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411 gt

53

5

gt max0le119905le2120587

11988622(119905) + 119887

22(119905) + 119888

22(119905)

1205752

1min1le119895le2

1205721119895

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 6120587)

1 minus 119890minus(12058712)

asymp 652614

1205752

2min1le119895le2

1205722119895

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411

min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

=

1205752

2min1le119895le2

1205722119895

1 minus 1205751

asymp 628411

119888119872

11+ 119888119872

12=

19

6

119888119872

21+ 119888119872

22=

61

60

max1le119894le2

2

sum

119895=1

119889119872

119894119895

=

19

6

(51)

Therefore

19

6

= max1le119894le2

2

sum

119894=1

119889119872

119894119895 lt min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

asymp 628411

(52)


119872

119894le 1 119894 = 1 2


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904)

times int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

= 119903119872

119894(119879119894119909) (119905) minus (119903

119872

119894minus 1) (119879

119894119909) (119905) = (119879

119894119909) (119905)

119894 = 1 2 119899

(30)



119877max1le119895le119899

sum119899

119895=1119887119872

119894119895 lt 1

(i) Ifmax119903119872119894 119894 = 1 2 119899 le 1 then 119879 119864⋂Ω

119877rarr 119864



119894 119894 = 1 2 119899 gt 1 then


119877=

119909 isin 119884 |119909|1lt 119877


119884(119879(119878)) le 119877max

1le119895le119899sum119899

119895=1119887119872

119894119895120572119884(119878)


119884(119878) Then for any


sum119899

119895=1119887119872

119894119895120578 there is a finite


119894119878119894with diam(119878

119894) le 120578+120598

Therefore

1003816100381610038161003816119909 minus 119910

10038161003816100381610038161le 120578 + 120598 for any 119909 119910 isin 119878

119894 (31)



119894= ⋃119895119878119894119895and

1003816100381610038161003816119909 minus 119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895 (32)


(119879119894119909) (119905)

= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1198772

1 minus 120575119894

int

120596

0

[

[

119899

sum

119895=1

119886119894119895(119904) +

119899

sum

119895=1

119887119894119895(119904) +

119899

sum

119895=1

119888119894119895(119904)]

]

119889119904

+

1

1 minus 120575119894

sum

119905le119905119896lt119905+120596

119868119894119896(119909119894(119905119896)) = 119861

119894 119894 = 1 2 119899

(33)10038161003816100381610038161003816(119879119894119909)1015840

(119905)

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905) +

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899

(34)

Hence

(119879119909)0le

119899

sum

119894=1

119861119894

10038171003817100381710038171003817(119879119909)1015840100381710038171003817100381710038170le

119899

sum

119894=1

[

[

119861119894119903119872

119894119861119894+ 1198772

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895)]

]

(35)


119894119895119896

of 119878119894119895such that 119878

119894119895= ⋃119896119878119894119895119896

and

1003816100381610038161003816119879119909 minus 119879119910

10038161003816100381610038160le 120598 for any 119909 119910 isin 119878

119894119895119896 (36)



119894119895119896 we

have

10038161003816100381610038161003816(119879119894119909)1015840

minus (119879119894119910)1015840100381610038161003816100381610038160

= max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119903

119894(119905) (119879119894119910) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119910119894(119905)[

[

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le max119905isin[0120596]

1003816100381610038161003816119903119894(119905) [(119879

119894119909) (119905) minus (119879

119894119910) (119905)]

1003816100381610038161003816

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119894[

[

(

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585)

minus(

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585 +

119899

sum

119895=1

119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ max119905isin[0120596]

10038161003816100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905) minus 119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894

1003816100381610038161003816(119879119894119909) minus (119879

119894119910)10038161003816100381610038160

+ 119877max119905isin[0120596]

119899

sum

119895=1

[

[

119886119894119895(119905)

10038161003816100381610038161003816119909119895(119905)minus119910

119895(119905)

10038161003816100381610038161003816+119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585)

times1003816100381610038161003816119909119894(119905+120585)minus119910

119894(119905+120585)

1003816100381610038161003816119889120585+119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585)

times

100381610038161003816100381610038161199091015840

119895(119905+120585) minus119910

1015840

(119905minus120585)

10038161003816100381610038161003816119889120585]

]

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905)minus119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905)

timesint

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585)

100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894120598+119877120598

119899

sum

119895=1

119886119872

119894119895+119877120598

119899

sum

119895=1

119887119872

119894119895+1003816100381610038161003816119909119894

10038161003816100381610038160(120578+120598)

times

119899

sum

119895=1

119888119872

119894119895+119877120598

119899

sum

119895=1

(119886119872

119894119895+119887119872

119894119895+119888119872

119894119895)

=1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895120578 + 119861119894120598

(37)

where

119861119894= 119903119872

119894+ 2119877

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899 (38)



1003817100381710038171003817119879119909 minus 119879119910

10038171003817100381710038171le (

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895)120578 + 120598

119899

sum

119894=1

119861119894

le 119877max1le119894le119899

119899

sum

119895=1

119888119872

119894119895

120578 + 120598

119899

sum

119894=1

119861119894 for any 119909 119910 isin 119878

119894119895119896

(39)


120572119884(119879 (119878)) le 119877max

1le119895le119899

119899

sum

119895=1

119888119872

119894119895

120572119884(119878) (40)


3 Main Results






119894 119894 = 1 2 119899 gt 1 then



119877 = (min119894isin[1119899]

1205752

119894

1 minus 120575119894

min119895isin[1119899]

120572119894119895)

minus1

0 lt 119903 lt min119894isin[1119899]

120575119894(1 minus 120575

119894) minus 120574119894

max119895isin[1119899]

120573119894119895

(41)


119903119877 In view





0gt 0


119879119894119909 (119905) minus 119909

119894(119905) ge 120575

119894

1003816100381610038161003816119879119894119909 minus 119909119894

10038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(42)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

int

120596

0

[119886119894119895(119904)+119887119894119895(119904)+119888119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904+120574119894

le

120573119894119895119903 + 120574119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160le 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160 119894 = 1 2 119899


1199090le 119879119909

0le max1le119894le119899

120575119894 1199090lt 119909

0 (44)






119894119909 (119905) ge 120575

119894

1003816100381610038161003816119909119894minus 11987911989411990910038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(45)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

gt

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)] 119889119904

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161) 119894 = 1 2 119899

(46)


Thus we have

1198791199090=

119899

sum

119894=1

100381610038161003816100381611987911989411990910038161003816100381610038160gt

119899

sum

119894=1

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161)

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

1198772= 119877

(47)


1199091ge 119879119909

1ge 119879119909

0gt 119877 (48)




4 Example



1199091015840

1(119905)

= 1199091(119905) [


minus (10 + sin 119905) 1199091(119905)


times int

0

minus12059111

11989111(120585) 1199091(119905 + 120585) 119889120585 minus

5 + sin 11990515

times int

0

minus12059112

11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)

times int

0

minus12059011

11989211(120585) 1199091015840

1(119905 + 120585) 119889120585 minus

4 minus 3 cos 1199056

timesint

0

minus12059012

11989212(120585) 1199091015840

2(119905 + 120585) 119889120585]

1199091015840

2(119905)

= 1199092(119905) [

1 + cos 1199058120587

minus (12 + 3 cos 119905) 1199091(119905)


times int

0

minus12059121

11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)

times int

0

minus12059122

11989122(120585) 1199092(119905 + 120585) 119889120585 minus

4 + sin 11990512

times int

0

minus12059021

11989221(120585) 1199091015840

2(119905 + 120585) 119889120585 minus


timesint

0

minus12059022

11989222(120585) 1199091015840

2(119905 + 120585) 119889120585]

(49)

where 120591119894119895 120590119894119895

isin (0 +infin) (119894 119895 = 1 2) and 119891119894119895 119892119894119895

isin

119862(119877 119877+) satisfying int0

minus120591119894119895

119891119894119895(120585)119889120585 = 1 int

0

minus120590119894119895

119892119894119895(120585)119889120585 119894 119895 = 1 2

Obviously

1199031(119905) =


1199032=

1 + cos 1199058120587

11988611(119905) = 10 + sin 119905

11988612(119905) = 9 minus cos 119905 119886

21(119905) = 12 + 3 cos 119905

11988622(119905) = 7 minus sin 119905 119887

11(119905) = 3 minus 2 cos 119905

11988712(119905) =

5 + sin 11990515

11988721(119905) = 2 + 3 sin 119905 119887

22(119905) = 1 minus cos 119905

11988811(119905) = 1 + sin 119905 119888

12(119905) =

4 minus 3 cos 1199056

11988821(119905) =

4 + sin 11990512

11988822(119905) =


119903119871

1= 119903119871

2= 0

(50)


1205751= 119890minus(12058712)

lt 1 1205752= 119890minus(14)

lt 1

12057211= 20120587120575

1minus 8120587 asymp 238007

12057212= 18120587120575

2minus 2120587 asymp 372423

12057221= 24120587120575

1minus

14120587

3

asymp 440594

12057222= 14120587120575

2minus 3120587 asymp 244284


min1le119895le2

1205721119895 = 12057211= 20120587120575

1minus 8120587 asymp 238007

min1le119895le2

1205722119895 = 12057222= 14120587120575

2minus 3120587 asymp 244284

120575111988611(119905) minus 119887

11(119905) minus 119888

11(119905) gt 0009 gt 0

120575211988612(119905) minus 119887

12(119905) minus 119888

12(119905) gt 1 gt 0

120575111988621(119905) minus 119887

21(119905) minus 119888

21(119905) gt 4 gt 0

120575211988622(119905) minus 119887

22(119905) minus 119888

22(119905) gt 2 gt 0

(1 + 119903119871

1)

1205752

112057211

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 8120587)

1 minus 119890minus(12058712)

asymp 652614 gt 18

ge max0le119905le2120587

11988611(119905) + 119887

11(119905) + 119888

11(119905)

(1 + 119903119871

1)

1205752

112057212

1 minus 1205751

=

119890minus(1205876)

(181205871205752minus 2120587)

1 minus 119890minus(12058712)

asymp 1021184 gt

347

30

ge max0le119905le2120587

11988612(119905) + 119887

12(119905) + 119888

12(119905)

(1 + 119903119871

2)

1205752

212057221

1 minus 1205752

=

119890minus(12)

(241205871205751minus (141205873))


asymp 1133411 gt

245

12

gt max0le119905le2120587

11988621(119905) + 119887

21(119905) + 119888

21(119905)

(1 + 119903119871

2)

1205752

212057222

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411 gt

53

5

gt max0le119905le2120587

11988622(119905) + 119887

22(119905) + 119888

22(119905)

1205752

1min1le119895le2

1205721119895

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 6120587)

1 minus 119890minus(12058712)

asymp 652614

1205752

2min1le119895le2

1205722119895

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411

min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

=

1205752

2min1le119895le2

1205722119895

1 minus 1205751

asymp 628411

119888119872

11+ 119888119872

12=

19

6

119888119872

21+ 119888119872

22=

61

60

max1le119894le2

2

sum

119895=1

119889119872

119894119895

=

19

6

(51)

Therefore

19

6

= max1le119894le2

2

sum

119894=1

119889119872

119894119895 lt min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

asymp 628411

(52)


119872

119894le 1 119894 = 1 2


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894119895119896 we

have

10038161003816100381610038161003816(119879119894119909)1015840

minus (119879119894119910)1015840100381610038161003816100381610038160

= max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119903119894(119905) (119879119894119909) (119905) minus 119903

119894(119905) (119879119894119910) (119905) minus 119909

119894(119905)

times[

[

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585

]

]

+ 119910119894(119905)[

[

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le max119905isin[0120596]

1003816100381610038161003816119903119894(119905) [(119879

119894119909) (119905) minus (119879

119894119910) (119905)]

1003816100381610038161003816

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119894[

[

(

119899

sum

119895=1

119886119894119895(119905) 119909119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119905 + 120585) 119889120585)

minus(

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585 +

119899

sum

119895=1

119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

]

]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ max119905isin[0120596]

10038161003816100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905) minus 119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905) +

119899

sum

119895=1

119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905) int

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905+120585) 119889120585)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894

1003816100381610038161003816(119879119894119909) minus (119879

119894119910)10038161003816100381610038160

+ 119877max119905isin[0120596]

119899

sum

119895=1

[

[

119886119894119895(119905)

10038161003816100381610038161003816119909119895(119905)minus119910

119895(119905)

10038161003816100381610038161003816+119887119894119895(119905)

times int

0

minus120591119894119895

119891119894119895(120585)

times1003816100381610038161003816119909119894(119905+120585)minus119910

119894(119905+120585)

1003816100381610038161003816119889120585+119888119894119895(119905)

times int

0

minus120590119894119895

119892119894119895(120585)

times

100381610038161003816100381610038161199091015840

119895(119905+120585) minus119910

1015840

(119905minus120585)

10038161003816100381610038161003816119889120585]

]

+ max119905isin[0120596]

100381610038161003816100381610038161003816100381610038161003816

[119909119894(119905)minus119910

119894(119905)]

times (

119899

sum

119895=1

119886119894119895(119905) 119910119895(119905)

+

119899

sum

119895=1

119887119894119895(119905) int

0

minus120591119894119895

119891119894119895(120585) 119910119895(119905+120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119905)

timesint

0

minus120590119894119895

119892119894119895(120585) 1199101015840

119895(119905 + 120585) 119889120585)

100381610038161003816100381610038161003816100381610038161003816

le 119903119872

119894120598+119877120598

119899

sum

119895=1

119886119872

119894119895+119877120598

119899

sum

119895=1

119887119872

119894119895+1003816100381610038161003816119909119894

10038161003816100381610038160(120578+120598)

times

119899

sum

119895=1

119888119872

119894119895+119877120598

119899

sum

119895=1

(119886119872

119894119895+119887119872

119894119895+119888119872

119894119895)

=1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895120578 + 119861119894120598

(37)

where

119861119894= 119903119872

119894+ 2119877

119899

sum

119895=1

(119886119872

119894119895+ 119887119872

119894119895+ 119888119872

119894119895) 119894 = 1 2 119899 (38)



1003817100381710038171003817119879119909 minus 119879119910

10038171003817100381710038171le (

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895)120578 + 120598

119899

sum

119894=1

119861119894

le 119877max1le119894le119899

119899

sum

119895=1

119888119872

119894119895

120578 + 120598

119899

sum

119894=1

119861119894 for any 119909 119910 isin 119878

119894119895119896

(39)


120572119884(119879 (119878)) le 119877max

1le119895le119899

119899

sum

119895=1

119888119872

119894119895

120572119884(119878) (40)


3 Main Results






119894 119894 = 1 2 119899 gt 1 then



119877 = (min119894isin[1119899]

1205752

119894

1 minus 120575119894

min119895isin[1119899]

120572119894119895)

minus1

0 lt 119903 lt min119894isin[1119899]

120575119894(1 minus 120575

119894) minus 120574119894

max119895isin[1119899]

120573119894119895

(41)


119903119877 In view





0gt 0


119879119894119909 (119905) minus 119909

119894(119905) ge 120575

119894

1003816100381610038161003816119879119894119909 minus 119909119894

10038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(42)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

int

120596

0

[119886119894119895(119904)+119887119894119895(119904)+119888119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904+120574119894

le

120573119894119895119903 + 120574119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160le 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160 119894 = 1 2 119899


1199090le 119879119909

0le max1le119894le119899

120575119894 1199090lt 119909

0 (44)






119894119909 (119905) ge 120575

119894

1003816100381610038161003816119909119894minus 11987911989411990910038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(45)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

gt

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)] 119889119904

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161) 119894 = 1 2 119899

(46)


Thus we have

1198791199090=

119899

sum

119894=1

100381610038161003816100381611987911989411990910038161003816100381610038160gt

119899

sum

119894=1

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161)

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

1198772= 119877

(47)


1199091ge 119879119909

1ge 119879119909

0gt 119877 (48)




4 Example



1199091015840

1(119905)

= 1199091(119905) [


minus (10 + sin 119905) 1199091(119905)


times int

0

minus12059111

11989111(120585) 1199091(119905 + 120585) 119889120585 minus

5 + sin 11990515

times int

0

minus12059112

11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)

times int

0

minus12059011

11989211(120585) 1199091015840

1(119905 + 120585) 119889120585 minus

4 minus 3 cos 1199056

timesint

0

minus12059012

11989212(120585) 1199091015840

2(119905 + 120585) 119889120585]

1199091015840

2(119905)

= 1199092(119905) [

1 + cos 1199058120587

minus (12 + 3 cos 119905) 1199091(119905)


times int

0

minus12059121

11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)

times int

0

minus12059122

11989122(120585) 1199092(119905 + 120585) 119889120585 minus

4 + sin 11990512

times int

0

minus12059021

11989221(120585) 1199091015840

2(119905 + 120585) 119889120585 minus


timesint

0

minus12059022

11989222(120585) 1199091015840

2(119905 + 120585) 119889120585]

(49)

where 120591119894119895 120590119894119895

isin (0 +infin) (119894 119895 = 1 2) and 119891119894119895 119892119894119895

isin

119862(119877 119877+) satisfying int0

minus120591119894119895

119891119894119895(120585)119889120585 = 1 int

0

minus120590119894119895

119892119894119895(120585)119889120585 119894 119895 = 1 2

Obviously

1199031(119905) =


1199032=

1 + cos 1199058120587

11988611(119905) = 10 + sin 119905

11988612(119905) = 9 minus cos 119905 119886

21(119905) = 12 + 3 cos 119905

11988622(119905) = 7 minus sin 119905 119887

11(119905) = 3 minus 2 cos 119905

11988712(119905) =

5 + sin 11990515

11988721(119905) = 2 + 3 sin 119905 119887

22(119905) = 1 minus cos 119905

11988811(119905) = 1 + sin 119905 119888

12(119905) =

4 minus 3 cos 1199056

11988821(119905) =

4 + sin 11990512

11988822(119905) =


119903119871

1= 119903119871

2= 0

(50)


1205751= 119890minus(12058712)

lt 1 1205752= 119890minus(14)

lt 1

12057211= 20120587120575

1minus 8120587 asymp 238007

12057212= 18120587120575

2minus 2120587 asymp 372423

12057221= 24120587120575

1minus

14120587

3

asymp 440594

12057222= 14120587120575

2minus 3120587 asymp 244284


min1le119895le2

1205721119895 = 12057211= 20120587120575

1minus 8120587 asymp 238007

min1le119895le2

1205722119895 = 12057222= 14120587120575

2minus 3120587 asymp 244284

120575111988611(119905) minus 119887

11(119905) minus 119888

11(119905) gt 0009 gt 0

120575211988612(119905) minus 119887

12(119905) minus 119888

12(119905) gt 1 gt 0

120575111988621(119905) minus 119887

21(119905) minus 119888

21(119905) gt 4 gt 0

120575211988622(119905) minus 119887

22(119905) minus 119888

22(119905) gt 2 gt 0

(1 + 119903119871

1)

1205752

112057211

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 8120587)

1 minus 119890minus(12058712)

asymp 652614 gt 18

ge max0le119905le2120587

11988611(119905) + 119887

11(119905) + 119888

11(119905)

(1 + 119903119871

1)

1205752

112057212

1 minus 1205751

=

119890minus(1205876)

(181205871205752minus 2120587)

1 minus 119890minus(12058712)

asymp 1021184 gt

347

30

ge max0le119905le2120587

11988612(119905) + 119887

12(119905) + 119888

12(119905)

(1 + 119903119871

2)

1205752

212057221

1 minus 1205752

=

119890minus(12)

(241205871205751minus (141205873))


asymp 1133411 gt

245

12

gt max0le119905le2120587

11988621(119905) + 119887

21(119905) + 119888

21(119905)

(1 + 119903119871

2)

1205752

212057222

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411 gt

53

5

gt max0le119905le2120587

11988622(119905) + 119887

22(119905) + 119888

22(119905)

1205752

1min1le119895le2

1205721119895

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 6120587)

1 minus 119890minus(12058712)

asymp 652614

1205752

2min1le119895le2

1205722119895

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411

min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

=

1205752

2min1le119895le2

1205722119895

1 minus 1205751

asymp 628411

119888119872

11+ 119888119872

12=

19

6

119888119872

21+ 119888119872

22=

61

60

max1le119894le2

2

sum

119895=1

119889119872

119894119895

=

19

6

(51)

Therefore

19

6

= max1le119894le2

2

sum

119894=1

119889119872

119894119895 lt min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

asymp 628411

(52)


119872

119894le 1 119894 = 1 2


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003817100381710038171003817119879119909 minus 119879119910

10038171003817100381710038171le (

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

119888119872

119894119895)120578 + 120598

119899

sum

119894=1

119861119894

le 119877max1le119894le119899

119899

sum

119895=1

119888119872

119894119895

120578 + 120598

119899

sum

119894=1

119861119894 for any 119909 119910 isin 119878

119894119895119896

(39)


120572119884(119879 (119878)) le 119877max

1le119895le119899

119899

sum

119895=1

119888119872

119894119895

120572119884(119878) (40)


3 Main Results






119894 119894 = 1 2 119899 gt 1 then



119877 = (min119894isin[1119899]

1205752

119894

1 minus 120575119894

min119895isin[1119899]

120572119894119895)

minus1

0 lt 119903 lt min119894isin[1119899]

120575119894(1 minus 120575

119894) minus 120574119894

max119895isin[1119899]

120573119894119895

(41)


119903119877 In view





0gt 0


119879119894119909 (119905) minus 119909

119894(119905) ge 120575

119894

1003816100381610038161003816119879119894119909 minus 119909119894

10038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(42)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904+120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

le

1

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160

119899

sum

119895=1

int

120596

0

[119886119894119895(119904)+119887119894119895(119904)+119888119894119895(119904)]

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161119889119904+120574119894

le

120573119894119895119903 + 120574119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160le 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038160 119894 = 1 2 119899


1199090le 119879119909

0le max1le119894le119899

120575119894 1199090lt 119909

0 (44)






119894119909 (119905) ge 120575

119894

1003816100381610038161003816119909119894minus 11987911989411990910038161003816100381610038161ge 0

for any 119905 isin [0 120596] 119894 = 1 2 119899(45)


= int

119905+120596

119905

119866119894(119905 119904) 119909

119894(119904)

times[

[

119899

sum

119895=1

119886119894119895(119904) 119909119895(119904) +

119899

sum

119895=1

119887119894119895(119904) int

0

minus120591119894119895

119891119894119895(120585) 119909119895(119904 + 120585) 119889120585

+

119899

sum

119895=1

119888119894119895(119904) int

0

minus120590119894119895

119892119894119895(120585) 1199091015840

119895(119904 + 120585) 119889120585

]

]

119889119904

+ sum

119905le119905119896lt119905+120596

119866119894(119905 119905119896) 119868119894119896(119909119894(119905119896))

gt

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

int

120596

0

[120575119895119886119894119895(119904) minus 119887

119894119895(119904) minus 119888

119894119895(119904)] 119889119904

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

=

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161) 119894 = 1 2 119899

(46)


Thus we have

1198791199090=

119899

sum

119894=1

100381610038161003816100381611987911989411990910038161003816100381610038160gt

119899

sum

119894=1

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161)

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

1198772= 119877

(47)


1199091ge 119879119909

1ge 119879119909

0gt 119877 (48)




4 Example



1199091015840

1(119905)

= 1199091(119905) [


minus (10 + sin 119905) 1199091(119905)


times int

0

minus12059111

11989111(120585) 1199091(119905 + 120585) 119889120585 minus

5 + sin 11990515

times int

0

minus12059112

11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)

times int

0

minus12059011

11989211(120585) 1199091015840

1(119905 + 120585) 119889120585 minus

4 minus 3 cos 1199056

timesint

0

minus12059012

11989212(120585) 1199091015840

2(119905 + 120585) 119889120585]

1199091015840

2(119905)

= 1199092(119905) [

1 + cos 1199058120587

minus (12 + 3 cos 119905) 1199091(119905)


times int

0

minus12059121

11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)

times int

0

minus12059122

11989122(120585) 1199092(119905 + 120585) 119889120585 minus

4 + sin 11990512

times int

0

minus12059021

11989221(120585) 1199091015840

2(119905 + 120585) 119889120585 minus


timesint

0

minus12059022

11989222(120585) 1199091015840

2(119905 + 120585) 119889120585]

(49)

where 120591119894119895 120590119894119895

isin (0 +infin) (119894 119895 = 1 2) and 119891119894119895 119892119894119895

isin

119862(119877 119877+) satisfying int0

minus120591119894119895

119891119894119895(120585)119889120585 = 1 int

0

minus120590119894119895

119892119894119895(120585)119889120585 119894 119895 = 1 2

Obviously

1199031(119905) =


1199032=

1 + cos 1199058120587

11988611(119905) = 10 + sin 119905

11988612(119905) = 9 minus cos 119905 119886

21(119905) = 12 + 3 cos 119905

11988622(119905) = 7 minus sin 119905 119887

11(119905) = 3 minus 2 cos 119905

11988712(119905) =

5 + sin 11990515

11988721(119905) = 2 + 3 sin 119905 119887

22(119905) = 1 minus cos 119905

11988811(119905) = 1 + sin 119905 119888

12(119905) =

4 minus 3 cos 1199056

11988821(119905) =

4 + sin 11990512

11988822(119905) =


119903119871

1= 119903119871

2= 0

(50)


1205751= 119890minus(12058712)

lt 1 1205752= 119890minus(14)

lt 1

12057211= 20120587120575

1minus 8120587 asymp 238007

12057212= 18120587120575

2minus 2120587 asymp 372423

12057221= 24120587120575

1minus

14120587

3

asymp 440594

12057222= 14120587120575

2minus 3120587 asymp 244284


min1le119895le2

1205721119895 = 12057211= 20120587120575

1minus 8120587 asymp 238007

min1le119895le2

1205722119895 = 12057222= 14120587120575

2minus 3120587 asymp 244284

120575111988611(119905) minus 119887

11(119905) minus 119888

11(119905) gt 0009 gt 0

120575211988612(119905) minus 119887

12(119905) minus 119888

12(119905) gt 1 gt 0

120575111988621(119905) minus 119887

21(119905) minus 119888

21(119905) gt 4 gt 0

120575211988622(119905) minus 119887

22(119905) minus 119888

22(119905) gt 2 gt 0

(1 + 119903119871

1)

1205752

112057211

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 8120587)

1 minus 119890minus(12058712)

asymp 652614 gt 18

ge max0le119905le2120587

11988611(119905) + 119887

11(119905) + 119888

11(119905)

(1 + 119903119871

1)

1205752

112057212

1 minus 1205751

=

119890minus(1205876)

(181205871205752minus 2120587)

1 minus 119890minus(12058712)

asymp 1021184 gt

347

30

ge max0le119905le2120587

11988612(119905) + 119887

12(119905) + 119888

12(119905)

(1 + 119903119871

2)

1205752

212057221

1 minus 1205752

=

119890minus(12)

(241205871205751minus (141205873))


asymp 1133411 gt

245

12

gt max0le119905le2120587

11988621(119905) + 119887

21(119905) + 119888

21(119905)

(1 + 119903119871

2)

1205752

212057222

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411 gt

53

5

gt max0le119905le2120587

11988622(119905) + 119887

22(119905) + 119888

22(119905)

1205752

1min1le119895le2

1205721119895

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 6120587)

1 minus 119890minus(12058712)

asymp 652614

1205752

2min1le119895le2

1205722119895

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411

min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

=

1205752

2min1le119895le2

1205722119895

1 minus 1205751

asymp 628411

119888119872

11+ 119888119872

12=

19

6

119888119872

21+ 119888119872

22=

61

60

max1le119894le2

2

sum

119895=1

119889119872

119894119895

=

19

6

(51)

Therefore

19

6

= max1le119894le2

2

sum

119894=1

119889119872

119894119895 lt min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

asymp 628411

(52)


119872

119894le 1 119894 = 1 2


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Thus we have

1198791199090=

119899

sum

119894=1

100381610038161003816100381611987911989411990910038161003816100381610038160gt

119899

sum

119894=1

1205752

119894

1 minus 120575119894

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

(120572119894119895

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161)

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

119899

sum

119894=1

1003816100381610038161003816119909119894

10038161003816100381610038161

119899

sum

119895=1

10038161003816100381610038161003816119909119895

100381610038161003816100381610038161

ge min1le119894le119899

1205752

119894min1le119895le119899

120572119894119895

1 minus 120575119894

1198772= 119877

(47)


1199091ge 119879119909

1ge 119879119909

0gt 119877 (48)




4 Example



1199091015840

1(119905)

= 1199091(119905) [


minus (10 + sin 119905) 1199091(119905)


times int

0

minus12059111

11989111(120585) 1199091(119905 + 120585) 119889120585 minus

5 + sin 11990515

times int

0

minus12059112

11989112(120585) 1199092(119905 + 120585) 119889120585 minus (1 + sin 119905)

times int

0

minus12059011

11989211(120585) 1199091015840

1(119905 + 120585) 119889120585 minus

4 minus 3 cos 1199056

timesint

0

minus12059012

11989212(120585) 1199091015840

2(119905 + 120585) 119889120585]

1199091015840

2(119905)

= 1199092(119905) [

1 + cos 1199058120587

minus (12 + 3 cos 119905) 1199091(119905)


times int

0

minus12059121

11989121(120585) 1199092(119905 + 120585) 119889120585 minus (1 minus cos 119905)

times int

0

minus12059122

11989122(120585) 1199092(119905 + 120585) 119889120585 minus

4 + sin 11990512

times int

0

minus12059021

11989221(120585) 1199091015840

2(119905 + 120585) 119889120585 minus


timesint

0

minus12059022

11989222(120585) 1199091015840

2(119905 + 120585) 119889120585]

(49)

where 120591119894119895 120590119894119895

isin (0 +infin) (119894 119895 = 1 2) and 119891119894119895 119892119894119895

isin

119862(119877 119877+) satisfying int0

minus120591119894119895

119891119894119895(120585)119889120585 = 1 int

0

minus120590119894119895

119892119894119895(120585)119889120585 119894 119895 = 1 2

Obviously

1199031(119905) =


1199032=

1 + cos 1199058120587

11988611(119905) = 10 + sin 119905

11988612(119905) = 9 minus cos 119905 119886

21(119905) = 12 + 3 cos 119905

11988622(119905) = 7 minus sin 119905 119887

11(119905) = 3 minus 2 cos 119905

11988712(119905) =

5 + sin 11990515

11988721(119905) = 2 + 3 sin 119905 119887

22(119905) = 1 minus cos 119905

11988811(119905) = 1 + sin 119905 119888

12(119905) =

4 minus 3 cos 1199056

11988821(119905) =

4 + sin 11990512

11988822(119905) =


119903119871

1= 119903119871

2= 0

(50)


1205751= 119890minus(12058712)

lt 1 1205752= 119890minus(14)

lt 1

12057211= 20120587120575

1minus 8120587 asymp 238007

12057212= 18120587120575

2minus 2120587 asymp 372423

12057221= 24120587120575

1minus

14120587

3

asymp 440594

12057222= 14120587120575

2minus 3120587 asymp 244284


min1le119895le2

1205721119895 = 12057211= 20120587120575

1minus 8120587 asymp 238007

min1le119895le2

1205722119895 = 12057222= 14120587120575

2minus 3120587 asymp 244284

120575111988611(119905) minus 119887

11(119905) minus 119888

11(119905) gt 0009 gt 0

120575211988612(119905) minus 119887

12(119905) minus 119888

12(119905) gt 1 gt 0

120575111988621(119905) minus 119887

21(119905) minus 119888

21(119905) gt 4 gt 0

120575211988622(119905) minus 119887

22(119905) minus 119888

22(119905) gt 2 gt 0

(1 + 119903119871

1)

1205752

112057211

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 8120587)

1 minus 119890minus(12058712)

asymp 652614 gt 18

ge max0le119905le2120587

11988611(119905) + 119887

11(119905) + 119888

11(119905)

(1 + 119903119871

1)

1205752

112057212

1 minus 1205751

=

119890minus(1205876)

(181205871205752minus 2120587)

1 minus 119890minus(12058712)

asymp 1021184 gt

347

30

ge max0le119905le2120587

11988612(119905) + 119887

12(119905) + 119888

12(119905)

(1 + 119903119871

2)

1205752

212057221

1 minus 1205752

=

119890minus(12)

(241205871205751minus (141205873))


asymp 1133411 gt

245

12

gt max0le119905le2120587

11988621(119905) + 119887

21(119905) + 119888

21(119905)

(1 + 119903119871

2)

1205752

212057222

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411 gt

53

5

gt max0le119905le2120587

11988622(119905) + 119887

22(119905) + 119888

22(119905)

1205752

1min1le119895le2

1205721119895

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 6120587)

1 minus 119890minus(12058712)

asymp 652614

1205752

2min1le119895le2

1205722119895

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411

min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

=

1205752

2min1le119895le2

1205722119895

1 minus 1205751

asymp 628411

119888119872

11+ 119888119872

12=

19

6

119888119872

21+ 119888119872

22=

61

60

max1le119894le2

2

sum

119895=1

119889119872

119894119895

=

19

6

(51)

Therefore

19

6

= max1le119894le2

2

sum

119894=1

119889119872

119894119895 lt min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

asymp 628411

(52)


119872

119894le 1 119894 = 1 2


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










min1le119895le2

1205721119895 = 12057211= 20120587120575

1minus 8120587 asymp 238007

min1le119895le2

1205722119895 = 12057222= 14120587120575

2minus 3120587 asymp 244284

120575111988611(119905) minus 119887

11(119905) minus 119888

11(119905) gt 0009 gt 0

120575211988612(119905) minus 119887

12(119905) minus 119888

12(119905) gt 1 gt 0

120575111988621(119905) minus 119887

21(119905) minus 119888

21(119905) gt 4 gt 0

120575211988622(119905) minus 119887

22(119905) minus 119888

22(119905) gt 2 gt 0

(1 + 119903119871

1)

1205752

112057211

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 8120587)

1 minus 119890minus(12058712)

asymp 652614 gt 18

ge max0le119905le2120587

11988611(119905) + 119887

11(119905) + 119888

11(119905)

(1 + 119903119871

1)

1205752

112057212

1 minus 1205751

=

119890minus(1205876)

(181205871205752minus 2120587)

1 minus 119890minus(12058712)

asymp 1021184 gt

347

30

ge max0le119905le2120587

11988612(119905) + 119887

12(119905) + 119888

12(119905)

(1 + 119903119871

2)

1205752

212057221

1 minus 1205752

=

119890minus(12)

(241205871205751minus (141205873))


asymp 1133411 gt

245

12

gt max0le119905le2120587

11988621(119905) + 119887

21(119905) + 119888

21(119905)

(1 + 119903119871

2)

1205752

212057222

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411 gt

53

5

gt max0le119905le2120587

11988622(119905) + 119887

22(119905) + 119888

22(119905)

1205752

1min1le119895le2

1205721119895

1 minus 1205751

=

119890minus(1205876)

(201205871205751minus 6120587)

1 minus 119890minus(12058712)

asymp 652614

1205752

2min1le119895le2

1205722119895

1 minus 1205752

=

119890minus(12)

(141205871205752minus 3120587)


asymp 628411

min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

=

1205752

2min1le119895le2

1205722119895

1 minus 1205751

asymp 628411

119888119872

11+ 119888119872

12=

19

6

119888119872

21+ 119888119872

22=

61

60

max1le119894le2

2

sum

119895=1

119889119872

119894119895

=

19

6

(51)

Therefore

19

6

= max1le119894le2

2

sum

119894=1

119889119872

119894119895 lt min1le119894le2

1205752

119894min1le119895le2

120572119894119895

1 minus 120575119894

asymp 628411

(52)


119872

119894le 1 119894 = 1 2


Acknowledgments


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 Existence of Positive Periodic Solutions ...downloads.hindawi.com/journals/ijde/2013/890281.pdf · Existence of Positive Periodic Solutions for Periodic Neutral Lotka-Volterra