Pullback-Forward Dynamics for Damped Schrödinger Equations ...downloads.hindawi.com/journals/ddns/2018/2139792.pdf · ResearchArticle Pullback-Forward Dynamics for Damped Schrödinger

Research ArticlePullback-Forward Dynamics for Damped Schroumldinger Equationswith Time-Dependent Forcing

Lianbing She 12 Yangrong Li 1 and Renhai Wang1

1School of Mathematics and Statistics Southwest University Chongqing 400715 China2School of Mathematics and Information Engineering Liupanshui Normal College Liupanshui Guizhou 553004 China

Correspondence should be addressed to Yangrong Li liyrswueducn

Received 27 September 2017 Accepted 19 December 2017 Published 17 January 2018

Academic Editor Xiaohua Ding

Copyright copy 2018 Lianbing She et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with pullback dynamics for the weakly damped Schrodinger equationwith time-dependent forcing An increasingbounded and pullback absorbing set is obtained if the forcing and its time-derivative are backward uniformly integrable Alsowe obtain the forward absorption which is only used to deduce the backward compact-decay decomposition according tohigh and low frequencies Based on a new existence theorem of a backward compact pullback attractor we show that thenonautonomous Schrodinger equation has a pullback attractor which is compact in the past The method of energy high-lowfrequency decomposition Sobolev embedding and interpolation are quite involved in calculating a priori pullback or forwardbound

1 Introduction

This paper is concerned with the backward compact dynam-ics of space-periodic solutions for the nonautonomouscomplex-valued Schrodinger equation in R

119906119905 + 119894119906119909119909 + 119894 |119906|2 119906 + 120572119906 minus 119894119906 = 119891 (119905 119909) (1)

The above equation for 120572 119891 = 0 was introduced in [1] as amodel for the propagation of solitons and laser beams In sucha case (without damping and forcing) it is easy to prove theenergy conservation law that is 119906(119905) = 119906(0) (see eg[2]) So no attractors exist To obtain an attractor we have toassume that the equation has a positive damping parameter120572 gt 0

The dynamical behavior of the damped Schrodingerequation was widely investigated by many physicists andmathematicians (see eg [3ndash11]) but restricted in theautonomous case that is the force 119891 is time-independent(only space dependent)

This paper deals with dynamics for the nonautonomousSchrodinger equation that is the force 119891 is time-dependentTo the best of our knowledge there is no literature treatingnonautonomous dynamics (including random dynamics) for

the Schrodinger equation even in the simple case for theexistence of a pullback attractor although the theory andapplication of pullback attractors had been widely developedfor many other PDEs (see [12ndash16]) and for pullback randomattractors see for example [17ndash20]

When one tries to look for a bounded pullback absorbingset for (1) (see Lemma 7) it seems to be assumed that 119891 isbackward bounded in 1198712 that is

sup119904⩽119905

1003817100381710038171003817119891 (119904)1003817100381710038171003817 lt infin forall119905 isin R (2)

rather than the ordinary tempered integrable conditionAnother difficulty is that we must treat the time-derivative119891119905 which is assumed to be backward tempered All thoseassumptions are different from dealing with the pullbackdynamics for other nonautonomous dissipative equations(see eg [17 21ndash28])

On the other hand the backward condition (2) permitsus to consider further properties of the pullback attractor forexample backward compactness as considered in [29ndash31]where a pullback attractor is called backward compact if theunion over the past is precompact

So in Section 2 we establish a new abstract theorem ona backward compact pullback attractor for a decomposable

HindawiDiscrete Dynamics in Nature and SocietyVolume 2018 Article ID 2139792 14 pageshttpsdoiorg10115520182139792

2 Discrete Dynamics in Nature and Society

evolution process that is it has a backward compact-decaydecomposition For such a decomposable process we showthat the existence of a backward compact attractor is equiva-lent to the existence of an increasing bounded and pullbackabsorbing set (see Theorem 4)

We then apply the abstract result to the nonautonomousSchrodinger equation In Section 3 the increasingly pullbackabsorption is verified if119891 is assumed to be backward boundedand the time-derivative 119891119905 is backward tempered

The difficulty arises from verifying the compact-decaydecomposition according to the high and low frequency ofthe Fourier series In this case the pullback absorption maynot be suitable for verifying such a decomposable propertySo in Section 4 we have to give an auxiliary result on theforward absorption It may be possible to deduce a forwardattractor (cf [32 33]) but we do not pursue this forwardattractor in the present paper

In Section 5 we present some techniques of splittingthe solutions of (1) into high and low frequency parts andestablish a new equation with initial value zero in the high-frequency part Then the forward absorption obtained inSection 4 can be applied to prove that the new equation hasa forward uniformly bounded solution in 1198672 which furtherproves that the component system is backward asymptoticallycompact in1198671 Also we prove that the difference of solutionsfrom both equations in the high-frequency part is backwardexponential decay and so obtain the compact-decay decom-position as required

The final existence result of a backward compact attractoris given in Theorem 14 It is worth pointing out that thepullback-forward method (involving the high-low frequencydecomposition) may be special for the nonautonomousSchrodinger equation which is different from treating thepullback dynamics for other nonautonomous dissipativeequations

2 Backward Compact Dynamics forDecomposable Systems

Let (119883 sdot 119883) be a Banach space equipped with the class Bof all bounded subset in 119883 We consider a nonautonomousprocess 119878 on 119883 which means 119878(119905 119904) 119883 rarr 119883 is a continuousnonlinear mapping such that 119878(119904 119904) = id119883 and 119878(119905 119904) =119878(119905 119903)119878(119903 119904) for all 119905 ⩾ 119903 ⩾ 119904 with 119904 isin R

We assume that the process is decomposable in thefollowing sense

Definition 1 A nonautonomous process 119878 is said to have abackward compact-decay decomposition 119878(119905 119904) = 1198781(119905 119904) +1198782(119905 119904) (119905 ⩾ 119904) if

(i) 1198781 is backward asymptotically compact that is thesequence 1198781(119904119899 119904119899minus120591119899)119909119899infin119899=1 is precompact whenever 119904119899 ⩽ 119905with fixed 119905 isin R 120591119899 rarr +infin and 119909119899infin119899=1 is bounded in119883

(ii) 1198782 is backward exponential decay that is for each 119905 isin R

and 119861 isin B there exist two positive constants 1198881 = 1198881(119905 119861) and1198882 = 1198882(119905 119861) such that for some 1205910 ⩾ 0sup119904⩽119905

sup119909isin119861

10038171003817100381710038171198782 (119904 119904 minus 120591) 1199091003817100381710038171003817119883 ⩽ 1198881119890minus1198882(120591minus1205910) forall120591 ⩾ 1205910 (3)

A nonautonomous set D = D(119905)119905isinR in 119883 means aset-valued mapping D(sdot) R rarr 2119883 0 which is saidto have some topological properties (such as compactnessboundedness and closedness) if the component setD(119905) hasthe corresponding properties

Definition 2 A nonautonomous setD = D(119905)119905isinR is called(i) backward (resp forward or globally) compact if D(119905)

and cup119904⩽119905D(119904) (resp cup119903⩾119905D(119903) or cup119903isinRD(119903)) are compact foreach 119905 isin R

(ii) increasing (resp decreasing) if D(119904) sub D(119905) (respD(119904) sup D(119905)) for all 119904 ⩽ 119905Definition 3 A backward compact set A = A(119905)119905isinR in 119883is called a backward compact attractor for a nonautonomousprocess 119878 ifA is invariant that is 119878(119905 119904)A(119904) = A(119905) for 119905 ⩾ 119904and pullback attracting that is for any 119905 isin R and 119861 isin B

lim120591rarr+infin

dist119883 (119878 (119905 119905 minus 120591) 119861A (119905)) = 0 (4)

where the Hausdorff semi-distance dist119883(119860 119861) =sup119886isin119860inf119887isin119861119886 minus 119887119883

A backward compact attractor must be unique andminimal where the minimality meansA sub K for any closedattracting setK (see [30])

For the purpose of applying to the Schrodinger equationwe need to establish a new existence theorem of a backwardcompact attractor for a backward compact-decay processalthough other existence criteria were established in [29 30]

Recall that a nonautonomous set K is called pullbackabsorbing for 119878 if for each 119905 isin R and 119861 isin B there is a1205910 = 1205910(119905 119861) such that

119878 (119905 119905 minus 120591) 119861 sub K (119905) forall120591 ⩾ 1205910 (5)

Theorem 4 Suppose a nonautonomous process 119878 has a back-ward compact-decay decomposition 119878(119905 119904) = 1198781(119905 119904) + 1198782(119905 119904)for 119905 ⩾ 119904 Then the following statements are equivalent

(i) There exists an increasing bounded and pullbackabsorbing setK = K(119905)119905isinR

(ii) There is a backward compact attractor A = A(119905)119905isinRgiven by

A (119905) = 120596 (K (119905) 119905) fl ⋂1205910gt0

⋃120591⩾1205910

119878 (119905 119905 minus 120591)K (119905)119905 isin R

(6)

Proof The necessity is easily proved by setting K(119905) =1198731(cup119904⩽119905A(119904)) the 1-neighborhood of cup119904⩽119905A(119904) for each 119905 isinR Since cup119904⩽119905A(119904) is obviously an increasing family in 119905 isin Rit follows thatK is increasing Since cup119904⩽119905A(119904) is precompactit follows thatK(119905) is bounded (not necessarily precompact)Finally it is easy to deduce the pullback absorption ofK fromthe pullback attraction ofA

Conversely suppose (i) is true we show that A(119905) fl120596(K(119905) 119905) is a backward compact attractor in three steps

Step 1 We show the invariance As usual (cf [12]) it is easyto prove the positive invariance 119878(119905 119904)A(119904) sub A(119905) On the

Discrete Dynamics in Nature and Society 3

other hand let 119910 isin A(119905) = 120596(K(119905) 119905) We choose 119910119899 isin K(119905)and 120591119899 rarr +infin such that

lim119899rarrinfin

119878 (119905 119905 minus 120591119899) 119910119899 = 119910 (7)

Let 119904 ⩽ 119905 and 120591119899 = 119904 minus 119905 + 120591119899 rarr +infin Then 120591119899 gt 0 if119899 is large enough By the compact-decay decomposition weknow that 1198781(119904 119904 minus 120591119899)119910119899 is precompact and 1198782(119904 119904 minus 120591119899)119910119899is exponential decay Then passing to subsequences there isan 119911 isin 119883 such that

1198781 (119904 119904 minus 120591119899119896) 119910119899119896 997888rarr 1199111198782 (119904 119904 minus 120591119899119896) 119910119899119896 997888rarr 0 (8)

where 120591119899119896 gt 0 for all 119896 isin N Hence it further implies

119878 (119904 119904 minus 120591119899119896) 119910119899119896 = 1198781 (119904 119904 minus 120591119899119896) 119910119899119896+ 1198782 (119904 119904 minus 120591119899119896) 119910119899119896 997888rarr 119911 (9)

SinceK is pullback absorbing and increasing for each 119894 isin 119873we can choose a 119896119894 in 119899119896 such that 120591119896119894 minus 119894 is large enough itfollows that

119911119894 fl 119878 (119904 minus 119894 119904 minus 119894 minus (120591119896119894 minus 119894)) 119910119896119894 isin K (119904 minus 119894)sub K (119904) (10)

By the process property as 119894 rarr infin

119878 (119904 119904 minus 119894) 119911119894 = 119878 (119904 119904 minus 120591119896119894) 119910119896119894 997888rarr 119911 (11)

By (10)-(11) we have 119911 isin 120596(K(119904) 119904) = A(119904) By the continuityof 119878 it follows from (7) and (9) that

119910 = lim119894rarrinfin

119878 (119905 119905 minus 120591119896119894) 119910119896119894 = lim119896rarrinfin

119878 (119905 119904) 119878 (119904 119904 minus 120591119896119894) 119910119896119894= 119878 (119905 119904) 119911 isin 119878 (119905 119904)A (119904) (12)

which proves the negative invariance

Step 2We show the attraction ofA If it is not true then thereare 120575 gt 0 119905 isin R 120591119899 rarr +infin as 119899 rarr infin and 119909119899 isin 119861 with 119861 isin Bsuch that

dist119883 (119878 (119905 119905 minus 120591119899) 119909119899A (119905)) ⩾ 120575 for each 119899 isin N (13)

By using the compact-decay decomposition we can takesubsequences such that

lim119896rarrinfin

1198781 (119905 119905 minus 120591119899119896) 119909119899119896 = 119909lim119896rarrinfin

1198782 (119905 119905 minus 120591119899119896) 119909119899119896 = 0 (14)

where 119909 isin 119883 Therefore

lim119896rarrinfin


1198781 (119905 119905 minus 120591119899119896) 119909119899119896+ lim119896rarrinfin

1198782 (119905 119905 minus 120591119899119896) 119909119899119896 = 119909 (15)

Like we did in Step 1 we choose a subsequence 119896119894 of 119899119896such that


and by (15)

lim119896rarrinfin


119878 (119905 119905 minus 120591119896119894) 119909119896119894 = 119909 (17)

The above two formulations imply 119909 isin A(119905) which contra-dicts with (13)

Step 3 It remains to show the precompact of 119861119905 fl cup119904⩽119905A(119904)with fixed 119905 isin R Let 119904 ⩽ 119905 SinceK is pullback absorbing andincreasing there is a 1205910(119904) such that

⋃120591⩾1205910

119878 (119904 119904 minus 120591)K (119904) sub K (119904) sub K (119905) (18)

and so 120596(K(119904) 119904) sub K(119905) which further implies

119861119905 = ⋃119904⩽119905

A (119904) = ⋃119904⩽119905

120596 (K (119904) 119904) sub K (119905) (19)

Hence 119861119905 is at least bounded To prove the precompactnessof 119861119905 we take a sequence 119911119899 isin 119861119905 = cup119904⩽119905A(119904) and then 119911119899 isinA(119904119899) with 119904119899 ⩽ 119905 Let 0 ⩽ 120591119899 rarr infin Then invariance of Aimplies

119911119899 isin A (119904119899) = 119878 (119904119899 119904119899 minus 120591119899)A (119904119899 minus 120591119899)sub 119878 (119904119899 119904119899 minus 120591119899) 119861119905 (20)

Since119861119905 is proved to be bounded there is a bounded sequence119909119899 such that 119911119899 = 119878(119904119899 119904119899 minus 120591119899)119909119899 By the backwardcompact-decay decomposition we know 1198781 is backwardasymptotically compact on the bounded set 119861119905Then passingto a subsequence there is a 119909 isin 119883 such that

1198781 (119904119899119896 119904119899119896 minus 120591119899119896) 119909119899119896 997888rarr 119909 (21)

By using the decay property of 1198782 we know100381710038171003817100381710038171198782 (119904119899119896 119904119899119896 minus 120591119899119896) 11990911989911989610038171003817100381710038171003817 ⩽ 1198881119890minus1198882(120591119899119896minus1205910) 997888rarr 0 (22)

The above limits imply

119911119899119896 = 119878 (119904119899119896 119904119899119896 minus 120591119899119896) 119909119899119896= 1198781 (119904119899119896 119904119899119896 minus 120591119899119896) 119909119899119896 + 1198782 (119904119899119896 119904119899119896 minus 120591119899119896) 119909119899119896997888rarr 119909

(23)

Hence 119861119905 is precompact In particular A(119905) is precompactBut A(119905) is obviously closed and so it is compact The proofis complete


3 Pullback Absorption inSchroumldinger Equations

We come back to the Schrodinger equation with time-dependent force as follows

119906119905 + 119894119906119909119909 + 119894 |119906|2 119906 + 120572119906 minus 119894119906 = 119891 (119905 119909) 119906 (119905 0) = 119906 (119905 1) 119905 ⩾ 119904119906 (119904 119909) = 1199060 (119909)

119904 isin R 119909 isin Ω = (0 1) (24)

where 120572 gt 0 119906(119905 119909) is an unknown complex-valued function

31 Hypotheses and Existence of Solutions Let 1198712(Ω) bethe space of complex-valued 1198712-functions whose norm isdenoted by sdot Let1198671 be the space of all one-periodic1198671-function with the norm 11990621198671 = 1199062 + 1199061199092 The norm in119871119901(Ω) is denoted by sdot 119901 We then give some hypotheses onthe time-space dependent force 119891 = 119891(119905 119909)Assumption A1 119891 is continuous and backward bounded

1198651 (119905) fl sup119904⩽119905

1003817100381710038171003817119891 (119904 sdot)10038171003817100381710038172 lt +infin forall119905 isin R (25)

Assumption A2 The time-derivative 119891119905 is backward tempered

1198652 (119905) fl sup119904⩽119905

int119904

minusinfin119890120572(119903minus119904) 1003817100381710038171003817119891119903 (119903 sdot)10038171003817100381710038172 119889119903 lt +infin

forall119905 isin R(26)

where 120572 gt 0 as given in (24)Assumption A1 implies 119891 isin 1198712loc(R 1198712(Ω)) and the

finiteness of 1198653(119905) where1198653 (119905) fl sup

119904⩽119905int119904

minusinfin119890120572(119903minus119904) 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 119889119903 ⩽ 1198651 (119905)1 minus 119890minus120572 (27)

The last inequality can be proved as follows

1198653 (119905) = sup119904⩽119905

infinsum119898=0

int119904minus119898

119904minus(119898+1)119890120572(119903minus119904) 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 119889119903

⩽ sup119904⩽119905

infinsum119898=0

119890minus119898120572 int119904minus119898

119904minus(119898+1)

1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 119889119903

⩽ infinsum119898=0

119890minus119898120572sup119903⩽119905

1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 = 1198651 (119905)1 minus 119890minus120572

(28)

Note that all functions 119865119894(sdot) (119894 = 1 2 3) are finite nonnega-tive and increasing

We will repeatedly use the following two energy inequal-ities

Lemma 5 Let 119906(119905) = 119906(119905 119904 1199060) 119905 isin [119904 +infin) be the approxi-mation solution of (24) Then

119889119889119905 1199062 + 120572 1199062 ⩽ 1120572 1003817100381710038171003817119891 (119905 sdot)10038171003817100381710038172 (29)

119889119889119905Φ1 (119906) + 120572Φ1 (119906) ⩽ 9120572 1199066 + 3120572 1199064+ 1120572 1003817100381710038171003817119891119905 (119905 sdot)10038171003817100381710038172

(30)

where

Φ1 (119906) = 100381710038171003817100381711990611990910038171003817100381710038172 + 1199062 minus 12 11990644 + 2 ImintΩ119891119906119889119909 (31)

Proof Equation (29) is easily obtained by multiplying (24)with 119906 (the complex conjugate of 119906) and taking the real partof the final equation

To prove (30) wemultiply (24) by minus(119906119905+120572119906) and take theimaginary part after some complex-valued calculations weobtain

119889119889119905Φ1 (119906) + 2120572Φ2 (119906) minus 2ReintΩ119891119905119906 119889119909 = 0 (32)

where Φ1(119906) is given by (31) and

Φ2 (119906) = 100381710038171003817100381711990611990910038171003817100381710038172 + 1199062 minus 11990644 + ImintΩ119891119906119889119909 (33)

By the Agmon inequality 1199062infin ⩽ 2119906(119906119909 + 119906) we have11990644 ⩽ 21199063(119906119909 + 119906) HenceΦ1 (119906) minus 2Φ2 (119906) = minus 100381710038171003817100381711990611990910038171003817100381710038172 minus 1199062 + 32 11990644

⩽ minus 100381710038171003817100381711990611990910038171003817100381710038172 minus 1199062+ 3 1199063 (10038171003817100381710038171199061199091003817100381710038171003817 + 119906)

⩽ 9 1199066 + 3 1199064 minus 1199062

(34)

We then rewrite the energy equation (32) as follows

119889119889119905Φ1 (119906) + 120572Φ1 (119906) = 120572 (Φ1 (119906) minus 2Φ2 (119906))+ 2 Imint

Ω119891119905119906 119889119909

⩽ 9120572 1199066 + 3120572 1199064 + 1120572 100381710038171003817100381711989111990510038171003817100381710038172

(35)

which is just the needed energy inequality

Based on the above energy inequalities one can obtain apriori estimate (for absorption see the next subsection)Thenit is similar as the autonomous case (see [5 8]) to prove that(24) is well posed in1198671 Namely for each 1199060 isin 1198671 and 119904 isin R(24) has a unique solution 119906(sdot 119904 1199060) isin 119862([119904 +infin)1198671) and thesolution 119906(119905 119904 sdot) 1198671 rarr 1198671 is continuous This well-posedproperty permits us to define an evolution process 119878 on 1198671

by

119878 (119905 119904) 1199060 = 119906 (119905 119904 1199060) for 119905 ⩾ 119904 1199060 isin 1198671 (36)


32 Increasing Bounded and Pullback Absorbing Sets Inorder to use the results of Theorem 4 we need to look foran increasing bounded and pullback absorbing set

Lemma 6 Let A1 be satisfied Suppose 119861119877 be a ball in 1198671 (119877is the radius) and 119905 isin R then there are 1205910 = 1205910(119877) such thatfor all 1199060 isin 119861119877 and 120591 ⩾ 1205910sup119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)10038171003817100381710038172 ⩽ 1 + 1198653 (119905)120572 ⩽ 1 + 1198651 (119905)120572 (1 minus 119890minus120572) (37)

sup119904⩽119905

int119904

119904minus120591119890120572(119903minus119904) 1003817100381710038171003817119906 (119903 119904 minus 120591 1199060)10038171003817100381710038174 119889119903 ⩽ 1198881 (1 + 11986521 (119905)) (38)

sup119904⩽119905

int119904


where 1198881 1198882 are positive constants and1198651 1198653 are functions givenin A1

Proof Let 119905 isin R and 120591 ⩾ 0 We apply the Gronwall inequalityon (29) over the interval [119904 minus 120591 119903] with 119904 minus 120591 ⩽ 119903 ⩽ 119904 ⩽ 119905 andthen find

1003817100381710038171003817119906 (119903 119904 minus 120591 1199060)10038171003817100381710038172 ⩽ 119890minus120572(119903minus119904+120591) 1003817100381710038171003817119906010038171003817100381710038172+ 1120572 int119903

119904minus120591119890120572(119903minus119903) 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 119889119903

(40)

Letting 119903 = 119904 we have1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)10038171003817100381710038172 ⩽ 119890minus120572120591 10038171003817100381710038171199060100381710038171003817100381721198671 + 11205721198653 (119905) (41)

which yields (37) if we take 1205910 = 2 ln(119877 + 1)120572 On the otherhand by (40) again

1003817100381710038171003817119906 (119903 119904 minus 120591 1199060)10038171003817100381710038172 ⩽ 119890minus120572(119903minus119904+120591) 1003817100381710038171003817119906010038171003817100381710038172 + 112057221198651 (119905) (42)

where we use the fact that 119891(119903 sdot)2 ⩽ 1198651(119905) for 119903 ⩽ 119903 ⩽ 119905 Wethen consider the sixth power to obtain


which yields

int119904

119904minus120591119890120572(119903minus119904) 1003817100381710038171003817119906 (119903 119904 minus 120591 1199060)10038171003817100381710038176 119889119903

⩽ 8 1003817100381710038171003817119906010038171003817100381710038176 int119904

119904minus120591119890120572(119903minus119904)119890minus3120572(119903minus119904+120591)119889119903

+ 11988811986531 (119905) int119904

119904minus120591119890120572(119903minus119904)119889119903

⩽ 8119890minus3120572120591 1003817100381710038171003817119906010038171003817100381710038176 int0

minus120591119890minus2120572119903119889119903 + 11988811986531 (119905) int119904

minusinfin119890120572(119903minus119904)119889119903

⩽ 4120572119890minus120572120591 1003817100381710038171003817119906010038171003817100381710038176 + 11988811986531 (119905)

(44)

From this it is easy to deduce (39) with 1205910(119877) = (2 ln 2 +6 ln119877 minus ln120572)120572 and similarly obtain (38)

We then consider the backward bound of solutions in1198671

as follows

Lemma 7 LetA1A2 be satisfiedThen for each ball 119861119877 in1198671

and 119905 isin R there exists 1205911 = 1205911(119877 119905) such that for all 120591 ⩾ 1205911and 1199060 isin 119861119877

sup119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)100381710038171003817100381721198671 ⩽ 1198721 (119905)fl 1198883 (1 + 11986531 (119905) + 1198652 (119905))

(45)

where 1198883 fl 1198883(120572) is a constant and thus 1198721(sdot) is finite andincreasing

Proof Let 119904 ⩽ 119905 and 120591 ⩾ 0 Applying the Gronwall inequalityon (30) over [119904 minus 120591 119904] we get

Φ1 (119906 (119904 119904 minus 120591 1199060))⩽ 119890minus120572120591Φ1 (1199060) + 9120572int119904

119904minus120591119890120572(119903minus119904) 119906 (119903 119904 minus 120591)6 119889119903

+ 3120572int119904


+ 1120572 int119904

119904minus120591119890120572(119903minus119904) 1003817100381710038171003817119891119903 (119903 sdot)10038171003817100381710038172 119889119903

(46)

By Lemma 6 and A2 there is a 1205911 = 1205911(119877) such that for all120591 ⩾ 1205911 and 1199060 isin 119861119877sup119904⩽119905

Φ1 (119906 (119904 119904 minus 120591 1199060))⩽ 119890minus120572120591Φ1 (1199060) + 119888 (1 + 11986531 (119905) + 11986521 (119905) + 1198652 (119905))

(47)

On the other hand by the definition of Φ1 given in (31) wehave

Φ1 (119906 (119904 minus 120591 119904 minus 120591 1199060)) ⩽ 2 10038171003817100381710038171199060100381710038171003817100381721198671 + 1003817100381710038171003817119891 (119904 minus 120591 sdot)10038171003817100381710038172⩽ 2 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198651 (119905)

(48)

In particular at the initial value we have if 120591 is large enoughthen

119890minus120572120591Φ1 (1199060) ⩽ 2119890minus120572120591 (10038171003817100381710038171199060100381710038171003817100381721198671 + 1198651 (119905)) ⩽ 1 (49)

Both (47) and (49) imply that for all 1199060 isin 119861119877 and 120591 ⩾ 1205910 (witha larger 1205910)

sup119904⩽119905

Φ1 (119906 (119904 119904 minus 120591 1199060)) ⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905)) (50)

By the Agmon inequality again it follows from (31) that

Φ1 (119906) ⩾ 100381710038171003817100381711990611990910038171003817100381710038172 + 1199062 minus 1199063 (10038171003817100381710038171199061199091003817100381710038171003817 + 119906)+ 2 Imint

Ω119891119906119889119909

⩾ 12 11990621198671 minus 2 1199066 minus 1199064 minus 2 1003817100381710038171003817119891 (119904 sdot)10038171003817100381710038172 (51)


which together with (50) and Lemma 6 imply that for all 1199060 isin119861119877 and 120591 ⩾ 1205910 (with a larger 1205910)sup119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)100381710038171003817100381721198671⩽ 2 sup

119904⩽119905Φ1 (119906 (119904 119904 minus 120591 1199060))

+ 4 sup119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)10038171003817100381710038176+ 2 sup

119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)10038171003817100381710038174 + 4 sup119904⩽119905

1003817100381710038171003817119891 (119904 sdot)10038171003817100381710038172⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905)) + 4 (1 + 1198881198651 (119905))3

+ 2 (1 + 1198881198651 (119905))2 + 41198651 (119905)⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905))

(52)

which shows the needed result

Under the light of Lemma 7 we have the followingincreasing absorption

Theorem 8 Let A1 A2 be true Then the nonautonomousSchrodinger equation possesses an increasing bounded andpullback absorbing setK = K(119905)119905isinR in1198671 given by

K (119905) = 119908 isin 1198671 11990821198671 ⩽ 1198721 (119905)fl 1198883 (1 + 11986531 (119905) + 1198652 (119905)) 119905 isin R (53)

Proof By A1 A2 we can see that1198721(119905) is an increasing andfinite function with respect to 119905 this fact along with Lemma 7shows that the nonautonomous setK is increasing boundedand pullback absorbing in1198671 In fact by (45) the absorptionis backward uniform

Remark 9 It seems not to prove that the nonautonomousSchrodinger equation has a bounded absorbing set in 1198671

if the assumption A1 is replaced by the weaker assumptionthat 1198653(119905) lt infin although this weak assumption is enoughfor reaction-diffusion systems (see [29]) BBM equations (see[30]) and Navier-Stokes equations (see [31])

4 Forward Absorption inSchroumldinger Equations

This section establishes the forward absorption which willbe useful to deduce the compact-decay decomposition in thenext section

In this case we need to strengthen Assumptions A1 andA2 as follows

Assumption A11015840 119891 is continuous such that 1205881 fllim119905rarr+infin1198651(119905) lt infin

Assumption A21015840 119891119905 exists such that 1205882 fl lim119905rarr+infin1198652(119905) lt infin

Assumption A3 119891119905119909 exists such that 119891119905119909 isin 1198712loc(R 119867minus1(Ω))

Lemma 10 Let A11015840 A21015840 be satisfied Then for each ball 119861119877 in1198671 there exists 1199051 = 1199051(119877) gt 0 such thatsup119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038171198671 ⩽ 1205883 (54)

sup119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119906119905 (119904 + 119905 119904 1199060)1003817100381710038171003817119867minus1 ⩽ 1205884 (55)

where 1205883 1205884 are constants depending on 120572 and 119891Proof Let 119904 ⩽ 0 and 119905 ⩾ 0 We apply the Gronwall inequalityon (29) over [119904 119903] with 119903 isin [119904 119904 + 119905] and then find

1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038172 ⩽ 119890minus120572(119903minus119904) 1003817100381710038171003817119906010038171003817100381710038172 + 11205721198653 (119903) (56)

Taking 119903 = 119904 + 119905 in (56) we have

1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038172 ⩽ 119890minus120572119905 10038171003817100381710038171199060100381710038171003817100381721198671 + 11205721198653 (119904 + 119905)⩽ 119890minus120572119905 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198881198651 (119905)

(57)

Let 1199051 = 2 ln(119877 + 1)120572 gt 0 ByA11015840 we know for all 119905 ⩾ 1199051 and1199060 isin 1198611198771003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038172 ⩽ 1 + 1198881205881 (58)

We then consider the third power of (56) to obtain

1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 ⩽ 8119890minus3120572(119903minus119904) 1003817100381710038171003817119906010038171003817100381710038176 + 11988812058831 119904 ⩽ 119903 ⩽ 119904 + 119905 (59)

which yields

int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 119889119903

⩽ 8119890minus120572119905 1003817100381710038171003817119906010038171003817100381710038176 int119904+119905

119904119890minus2120572(119903minus119904)119889119903

+ 11988812058831 int119904+119905

119904119890120572(119903minus119904minus119905)119889119903

⩽ 4120572119890minus120572119905 1003817100381710038171003817119906010038171003817100381710038176 (1 minus 119890minus2120572119905) + 11988812058831⩽ 4120572119890minus120572119905 1003817100381710038171003817119906010038171003817100381710038176 + 11988812058831

(60)

Rewriting 1199051 by 1199051(119877) = (2 ln 2 + 6 ln(119877 + 1) minus ln120572)120572 wededuce for all 119905 ⩾ 1199051

int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 119889119903 ⩽ 1 + 11988812058831 (61)

and similarly we have

int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038174 119889119903 ⩽ 1 + 11988812058821 (62)


To prove (54) we apply the Gronwall inequality on (30) over[119904 119904 + 119905] to getΦ1 (119906 (119904 + 119905 119904 1199060)) ⩽ 119890minus120572119905Φ1 (1199060)

+ 9120572int119904+119905

119904119890120572(119903minus119904minus119905) 119906 (119903 119904)6 119889119903

+ 3120572int119904+119905

119904119890120572(119903minus119904minus119905) 119906 (119903 119904)4

+ 1120572 int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119891119903 (119903 sdot)10038171003817100381710038172 119889119903

(63)

By (61)-(62) and Assumption A2 we easily deduce that for119905 ⩾ 1199051Φ1 (119906 (119904 + 119905 s 1199060)) ⩽ 119890minus120572119905Φ1 (1199060) + 119888 (12058831 + 12058821)

+ 1198881198652 (119904 + 119905) (64)

By the definition ofΦ1 given in (31) we have

Φ1 (119906 (119903 119904 1199060)) ⩽ 2 11990621198671 + 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172⩽ 2 11990621198671 + 1198651 (119903) 119903 ⩾ 119904 (65)

In particular if 119905 is large enough then119890minus120572119905Φ1 (119906 (119904 119904 1199060)) ⩽ 119890minus120572119905 (2 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198651 (119904))

⩽ 119890minus120572119905 (2 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198651 (0)) ⩽ 1 (66)

By (61)-(62) and AssumptionA21015840 for 119905 ⩾ 1199051 (with a larger 1199051)Φ1 (119906 (119904 + 119905 119904 1199060)) ⩽ 1 + 119888 (12058831 + 12058821) + 1198881205882 ⩽ 119862 (67)

By the Agmon inequality again it is similar as (51) to provethat

Φ1 (119906 (119904 + 119905 119904 1199060)) ⩾ 12 11990621198671 minus 2 1199066 minus 1199064minus 2 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172

(68)

which together with (67) and (58) imply that for all 1199060 isin 119861119877119904 ⩽ 0 and 119905 ⩾ 1199051 (with a larger 1205910)1003817100381710038171003817119906 (119904 + 119905 119904 1199060)100381710038171003817100381721198671 ⩽ 2119862 + 4 1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038176

+ 2 1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038174+ 4 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172

⩽ 119888 (1 + 12058831 + 12058821 + 1198651 (119905))⩽ 119888 (1 + 12058831 + 12058821 + 1205881)

(69)

which shows the needed result (54)

To prove (55) we multiply (24) by a test function 120593 with1205931198671 = 1 then by the Agmon inequality

⟨119906119905 120593⟩ = minus119894 int 119906119909119909120593 minus 119894 int |119906|2 119906120593 + (119894 minus 120572) int 119906120593+ int119891120593

⩽ 100381710038171003817100381711990611990910038171003817100381710038172 + 100381710038171003817100381712059311990910038171003817100381710038172 + 11990644 + 1003817100381710038171003817120593100381710038171003817100381744+ 119888 (1199062 + 100381710038171003817100381712059310038171003817100381710038172) + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172

⩽ 119888 100381710038171003817100381711990611990910038171003817100381710038172 + 119862 + 1198651 (119905) ⩽ 119888 100381710038171003817100381711990611990910038171003817100381710038172 + 119862

(70)

Then (55) follows from (54)

5 Compact-Decay Decomposition

51 High-Low FrequencyDecomposition Weexpand 119906(119905) (thesolution of (24)) into its Fourier series

119906 (119905) = sum119896isinZ

119906119896 (119905) 1198902119896120587119894119909 (71)

and split 119906(119905) into low frequency part and high-frequencypart 119906(119905) = 119875119873119906(119905) + 119876119873119906(119905) with

119875119873119906 (119905) = sum|119896|⩽119873

119906119896 (119905) 1198902119896120587119894119909119876119873119906 (119905) = sum

|119896|gt119873

119906119896 (119905) 1198902119896120587119894119909(72)

Let 119904 ⩽ 0 be arbitrary but fixed and take the initial value1199060in a ball 119861119877 of1198671 Then we are concerned with two functionsof 119905 ⩾ 0

119910 = 119910 (119905) = 119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) 119911 = 119911 (119905) = 119911 (119905 119904) = 119876119873119906 (119904 + 119905 119904 1199060) (73)

By the forward absorption given in Lemma 10 we know thereis a 1199051 gt 0 (depending on the radius of initial ball) such that

sup119905⩾1199051

sup119904⩽0

(1003817100381710038171003817119910 (119905 119904)100381710038171003817100381721198671 + 119911 (119905 119904)21198671) ⩽ 119862 (74)

The high-frequency part 119911(119905) satisfies the following equation119911119905 + 120572119911 + 119894119911119909119909 minus 119894119911 + 119894119876119873 (1003816100381610038161003816119910 + 11991110038161003816100381610038162 (119910 + 119911))

= 119876119873119891 (119904 + 119905) 119905 ⩾ 1199051119911 (1199051) = 119876119873119906 (119904 + 1199051 119904 1199060)

(75)

Then we consider the following equation on 1198761198731198671 with theinitial value zero

119885119905 + 120572119885 + 119894119885119909119909 minus 119894119885 + 119894119876119873 (1003816100381610038161003816119910 + 11988510038161003816100381610038162 (119910 + 119885))= 119876119873119891 (119904 + 119905) 119905 ⩾ 1199051

119885 (1199051) = 119885 (1199051 119904) = 0(76)


where 119885 fl 119885(119905) (if exists) is actually a function wrt 119905 ⩾ 1199051119904 ⩽ 0 1199060 isin 119861119877 and 119873 isin N Sometimes we write it as 119885 =119885(119905 119904) We can decompose the evolution process 119878 by119906 (119904 + 119905 119904 1199060) = V (119905) + 119908 (119905)

where V fl 119910 + 119885 119908 fl 119911 minus 119885 (77)

for all 119905 ⩾ 1199051 119904 ⩽ 0 and1199060 isin 1198671 In the sequel themain task isto prove the asymptotic compactness of V and the exponentialdecay of 119908 in 1198761198731198671 for large119873

52 The Uniformly Bounded Estimate for 119885 To prove theexistence of solutions for (76) we need to consider theapproximation 119885119898 which is the solution for the projectionof (76) on the subspace

119875119898119876N1198671 = 119885119898 (119905) 119885119898 (119905) = sum119873lt|119896|⩽119898

119906119896 (119905) 1198902119896120587119894119909 119898 gt 119873

(78)

Then by the standard Galerkin method and a priori estimate(see the next proposition) one can prove the existence of 119885in 1198761198731198671 if119873 is large enough

In the following proposition we actually prove the resultfor 119885119898 However for the sake of simplicity we omit thesubscript 119898 and also omit the proof of convergence as 119898 rarrinfin

Proposition 11 Suppose A11015840 A21015840 are satisfied Then thereexists 1198730 = 1198730(120572 119891) such that for 119873 ⩾ 1198730 (76) has auniformly bounded solution 119885 in 1198761198731198671 such that

sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198671 ⩽ 119862 (79)

where 119862 is a constant and 1199051 is the forward absorbing enteringtime

Proof Multiplying (76) by minus119885119905 minus 120572119885 taking the imaginarypart after some computations we obtain an energy equation

12 119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) = Ψ2 (119885) 119905 ⩾ 1199051 (80)

with

Ψ1 (119885) = 11988521198671 + 2 ImintΩ119891119885119889119909 minus 12 11988544

minus 2ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus 2Reint

Ω|119885|2 119885119910119889119909 minus 2int

Ω

100381610038161003816100381611991010038161003816100381610038162 |119885|2 119889119909minus 2int

Ω(Re (119910119885))2 119889119909

Ψ2 (119885) = 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909 + 1205722 11988544

minus ReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909

minus 120572ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus int

ΩRe (119910119905119885) |119885|2 119889119909

+ 120572intΩRe (119910119885) |119885|2 119889119909

minus ReintΩ119910119910119905 |119885|2 119889119909

minus 2intΩRe (119910119905119885)Re (119910119885) 119889119909

(81)

We now consider the upper bound of Ψ2(119885) by AssumptionA11015840 we have

1198681 fl 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909

⩽ 12057216 1198852 + 119888 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172⩽ 12057216 11988521198671 + 1198881205881 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(82)

By the classical interpolation and the Poincare inequality on1198761198731198671

1198854 ⩽ 119888 11988534 119885141198671

119885 ⩽ 119888119873 1198851198671 (83)

the second term of Ψ2(119885) can be bounded by

1198682 fl 1205722 11988544 ⩽ 119888 1198853 1198851198671 ⩽ 119888119873311988541198671 (84)

Note that 1198671 forms a Banach multiplicative algebra in one-dimension that is

119906V1199081198671 ⩽ 119888 1199061198671 V1198671 1199081198671 (85)

Then by Lemma 10 and (74) the third term of Ψ2(119885) can bebounded by

1198683 fl minusReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909

⩽ 10038171003817100381710038171199101199051003817100381710038171003817119867minus1 10038171003817100381710038171003817100381610038161003816100381611991010038161003816100381610038162 119885100381710038171003817100381710038171198671 + 2 10038171003817100381710038171199101199051003817100381710038171003817119867minus1 1003817100381710038171003817119910Re (119910119885)10038171003817100381710038171198671⩽ 3 10038171003817100381710038171199101199051003817100381710038171003817119867minus1 1003817100381710038171003817119910100381710038171003817100381721198671 1198851198671 ⩽ 119862 1198851198671⩽ 12057232 11988521198671 + 119862

(86)


By (74) and the embedding 1198716 997893rarr 1198671 the fourth term isbounded by

1198684 fl minus120572ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909 ⩽ 12057232 1198852 + 119888 1003817100381710038171003817119910100381710038171003817100381766⩽ 12057232 1198852 + 119862

(87)

By the Agmon inequality and (83) we have

119885infin ⩽ 119888 11988512 119885121198671

⩽ 119888radic119873 1198851198671 (88)

Then the fifth term of Ψ2(119885) is bounded by

1198685 fl minusintΩRe (119910119905119885) |119885|2 119889119909 ⩽ 119888 10038171003817100381710038171199101199051003817100381710038171003817119867minus1 1198852infin 1198851198671

⩽ 119888119873 11988531198671 ⩽ 12057232 11988521198671 + 119888119873211988541198671

(89)

Similarly by (83) and the Holder inequality the sixth term ofΨ2(119885) is bounded by

1198686 fl 120572intΩRe (119910119885) |119885|2 119889119909 ⩽ 120572 100381710038171003817100381711991010038171003817100381710038174 1003817100381710038171003817100381711988531003817100381710038171003817100381743

= 120572 100381710038171003817100381711991010038171003817100381710038174 11988534 ⩽ 119888 100381710038171003817100381711991010038171003817100381710038171198671 11988594 119885341198671

⩽ 1198881198739411988531198671 ⩽ 12057232 11988521198671 + 11988811987392

11988541198671 (90)

By (88) the rest terms can be bounded by

1198687 fl minusReintΩ119910119910119905 |119885|2 119889119909 minus 2int

ΩRe (119910119905119885)Re (119910119885) 119889119909

⩽ 3 119885infin intΩ

10038161003816100381610038161199101199051199101198851003816100381610038161003816 119889119909 ⩽ 3 119885infin 10038171003817100381710038171199101199051003817100381710038171003817119867minus1 100381710038171003817100381711991011988510038171003817100381710038171198671⩽ 119888 119885infin 10038171003817100381710038171199101199051003817100381710038171003817119867minus1 100381710038171003817100381711991010038171003817100381710038171198671 1198851198671 ⩽ 1198881radic119873 Z21198671

(91)

By the above estimates we know that Ψ2(119885) can be boundedby

Ψ2 (119885) = 7sum119894=1

119868119894⩽ (312057216 + 1198881radic119873)11988521198671 + 1198881198732

11988541198671 + 1198881205881+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(92)

Letting 11987310158400 = [25611988811205722] + 1 where [sdot] denotes the integer-

valued function we have for119873 ⩾ 11987310158400

Ψ2 (119885) ⩽ 1205724 11988521198671 + 119888119873211988541198671 + 1198881205881

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (93)

On the other hand we similarly obtain a lower bound ofΨ1(119885)Ψ1 (119885) ⩾ 11988521198671 minus 119888 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172 minus 12 11988521198671 minus 12 11988544

minus 119888 (1003817100381710038171003817119910100381710038171003817100381734 1198854 + 100381710038171003817100381711991010038171003817100381710038174 11988534 + 1003817100381710038171003817119910100381710038171003817100381724 11988524)⩾ 12 11988521198671 minus 119862 minus 11988544 minus 119892 (100381710038171003817100381711991010038171003817100381710038174)⩾ 12 11988521198671 minus 1198881198733

11988541198671 minus 119862⩾ 12 11988521198671 minus 1198881198732

11988541198671 minus 119862

(94)

where 119892(1199104) is uniformly bounded in view of (74) By (93)-(94) we have

2Ψ2 (119885) minus 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (95)

Then it follows from (80) that

119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (96)

Since 119885(1199051) = 0 and Ψ1(119885(1199051)) = Ψ1(0) = 0 it follows fromthe Gronwall inequality on [1199051 119905] that

Ψ1 (119885 (119905)) ⩽ 1198881198732int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903+ 119862int119905

1199051

119890120572(119903minus119905)119889119903+ 119888int119905

1199051

119890120572(119903minus119905) 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172 119889119903⩽ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862120572+ 1198881198652 (119904 + 119905)

(97)

which along with (94) and AssumptionA21015840 imply that for all119905 ⩾ 1199051119885 (119905)21198671 ⩽ 1198881198732

119885 (119905)41198671+ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862 (98)


Hence 120585(119905) fl sup119903isin[1199051 119905]119885(119903)21198671 satisfies the following twiceinequality

120585 (119905) ⩽ 119888211987321205852 (119905) + 1198622

ie 11988821205852 (119905) minus 1198732120585 (119905) + 11986221198732 ⩾ 0(99)

where 11988821198622 are positive constants Let119873101584010158400 = [2radic11988821198622]+1 and1198730 = max(1198731015840

0 119873101584010158400 ) Then for all119873 ⩾ 1198730 the discriminant of

twice inequality is positive

Δ = 1198734 minus 4119888211986221198732 = 1198732 (1198732 minus 411988821198622) gt 0 (100)

In this case the equation 11988821205852 minus 1198732120585 + 11986221198732 = 0 has twopositive roots

1205721 = 1198732 minus 119873radic1198732 minus 411988821198622

21198882

1205722 = 1198732 + 119873radic1198732 minus 411988821198622

21198882 (101)

We show that 1205721 ⩽ 21198622 Indeed

1205721 ⩽ 21198622 lArrrArr1198732 minus 411988821198622 ⩽ 119873radic1198732 minus 411988821198622 lArrrArr

radic1198732 minus 411988821198622 ⩽ 119873(102)

where the last inequality is obviously true Now the twiceinequality (99) has the solution

120585 (119905) ⩽ 1205721or 120585 (119905) ⩾ 1205722

119905 ⩾ 1199051(103)

Since 120585(1199051) = 0 and the mapping 119905 rarr 120585(119905) is continuous on[1199051 +infin) it follows that120585 (119905) ⩽ 1205721 ⩽ 21198622 119905 ⩾ 1199051 (104)

which proves the required bound

53 Further Regularity Estimates We further show a regular-ity result for 119885 in1198672

Proposition 12 Suppose A11015840 A21015840 and A3 are satisfied Thenthere exists 1198731 = 1198731(120572 119891) ⩾ 1198730 such that for 119873 ⩾ 1198731 thesolution 119885 of (76) is uniformly bounded in 1198761198731198672

sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198672 ⩽ 119862 (119873 + 1) (105)


Proof Just like we did in the above proposition we needto firstly show the bound of 119885119898

119909 in 1198671 and then let 119898 rarr+infin to obtain the bound of 119885119909 in 1198671 Thus for the sake ofconvenience we omit the detail of letting119898 rarr +infin and alsodrop the superscript 119898 of 119885119898 and write 119885 = 119885119898 V = V119898 =119910 + 119885119898 119876119873 = 119875119898119876119873 in this proof

The first thing we shall do is to obtain the followingequation by differentiating (76)

119885119909119905 + 120572119885119909 + 119876119873 (119860 minus 1198651015840 (V)) 119885119909 = 119866 119905 ⩾ 1199051 (106)

with initial value 119885119909(1199051) = 0 where 119860 1198651015840 and 119866 are given by

119860119885119909 = 119894119885119909119909119909 minus 1198941198851199091198651015840 (V) 119885119909 = minus119894 |V|2 119885119909 minus 2119894Re (V119885119909) V

119866 = 119876119873 (119891119909 (119904 + 119905 sdot) + 1198651015840 (V) 119910119909) (107)

In order to show the needed result (105) we divide it intothree steps

Step 1 We show 119866 isin 119862119887([1199051 +infin)119867minus1) Let 120593 be a testfunction taken from 1198671 such that 1205931198671 = 1 Then byAssumption A11015840

⟨120593 119891119909 (119904 + 119905)⟩ = int120593119891119909 (119904 + 119905)⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172)⩽ 119888 (1003817100381710038171003817120593100381710038171003817100381721198671 + 1198651 (119904 + 119905)) ⩽ 119888 (1 + 1205881)

(108)

which implies that 119891119909(119904 + 119905)119867minus1 ⩽ 119862 Since119876119873 is a boundedoperator from119867minus1 to119867minus1 it follows that 119876119873119891119909(119904 + 119905)119867minus1 ⩽119862 for all 119905 ⩾ 1199051 and 119904 ⩽ 0

The rest term of 119866 can be rewritten as

1198761198731198651015840 (V) 119910119909 = minus119894119876119873 |V|2 119910119909 minus 2119894119876119873 Re (V119910119909) V (109)

Note that we have the inverse inequality 1199101199091198671 ⩽ 21205871198731199101198671on 1198751198731198671 By Lemma 10 and Proposition 11 we know 119910 V isin119862119887([1199051 +infin)1198671) where V = 119910+119885Then by themultiplicativealgebra (see (85)) we have for all 119905 ⩾ 119905110038171003817100381710038171003817minus119894119876119873 |V (119905)|2 119910119909 (119905)100381710038171003817100381710038171198671 ⩽ 119888 1003817100381710038171003817VV11991011990910038171003817100381710038171198671

⩽ 119888 V21198671 100381710038171003817100381711991011990910038171003817100381710038171198671 ⩽ 119888119873 100381710038171003817100381711991010038171003817100381710038171198671 ⩽ 1198621198731003817100381710038171003817minus2119894119876119873 Re (V (119905) 119910119909 (119905)) V (119905)10038171003817100381710038171198671 ⩽ 119888119873 V21198671 100381710038171003817100381711991010038171003817100381710038171198671

⩽ 119862119873(110)

Therefore 1198761198731198651015840(V)119910119909 isin 119862119887([1199051 +infin)1198671) and so 119866 isin119862119887([1199051 +infin)119867minus1) with 119866(119905)119867minus1 ⩽ 119862(119873 + 1)Step 2 We give the estimates of 119866119905119867minus1 where it is easy toobtain

119866119905 = 119876119873119891119909119905 (119904 + 119905 sdot) minus 2119894119876119873 Re (VV119905) 119910119909minus 119894119876119873 |V|2 119910119909119905 minus 2119894119876119873 (Re (V119910119909) V)119905 (111)


By Assumption A3 for the test function 1205931198671 = 1 we have⟨120593 119891119909119905 (119904 + 119905)⟩ = int120593119891119909119905 (119904 + 119905)

⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891119905 (119904 + 119905)10038171003817100381710038172) (112)

Hence 119876119873119891119909119905(119904 + 119905)119867minus1 ⩽ 119888(1 + 119891119905(119904 + 119905 sdot)2) for all 119905 ⩾ 1199051and 119904 ⩽ 0

To estimate the rest terms we note thatsup119905⩾1199051119885119905(119905)119867minus1 ⩽ 119862 which can be obtained from(76) and the uniform bound of 119885 By Lemma 10 119910119905119867minus1 ⩽ 119862and so V119905119867minus1 ⩽ 119862 For the second term of 119866119905 by themultiplicative algebra (85) again we have

1003816100381610038161003816⟨minus2119894Re (VV119905) 119910119909 120593⟩1003816100381610038161003816 ⩽ int 1003816100381610038161003816V119905V1199101199091205931003816100381610038161003816 119889119909⩽ 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 1003817100381710038171003817V11991011990912059310038171003817100381710038171198671⩽ 119862 V1198671 100381710038171003817100381712059310038171003817100381710038171198671 100381710038171003817100381711991011990910038171003817100381710038171198671⩽ 119862 100381710038171003817100381711991011990910038171003817100381710038171198671 ⩽ 119862119873100381710038171003817100381711991010038171003817100381710038171198671⩽ 119862119873

(113)

where 1205931198671 = 1 Hence Re (VV119905)119910119909119867minus1 ⩽ 119862119873 and so119876119873 Re(VV119905)119910119909119867minus1 ⩽ 119862119873 in view of the boundedness ofthe operator 119876119873 from 119867minus1 to 119867minus1 It is similar to obtain theestimates for the rest terms in 119866119905 and so1003817100381710038171003817119866t (119905 119904)1003817100381710038171003817119867minus1 ⩽ 119888 (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119905sdot)10038171003817100381710038172)

119905 ⩾ 1199051 119904 ⩽ 0 (114)

Step 3 Finally we bound 1198851199091198671 Multiplying (106) byminus119885119909119905minus120572119885119909 taking the imaginary part of the resulting equation andthen integrating overΩ we obtain an energy equation

119889119889119905Λ 1 (119885119909) + 2120572Λ 1 (119885119909) = 2Λ 2 (119885119909) (115)

where

Λ 1 (119885119909) = 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus int

Ω|V|2 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119885119909)2 119889119909

+ 2intΩIm (119866119885119909) 119889119909

(116)

Λ 2 (119885119909) = minusintΩRe (VV119905) 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119905119885119909) sdot Re (V119885119909) 119889119909

+ intΩIm (119866119905119885119909) 119889119909

+ 120572intΩIm (119866119885119909) 119889119909

(117)

By the bound of V in1198671 and V119905 in119867minus1 and by the inequalityVinfin ⩽ (119888radic119873)V1198671 on the space1198761198731198671 (see (88)) we havefor119873 ⩾ 1198730

Λ 2 (119885119909) ⩽ 3 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 Vinfin 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 1003817100381710038171003817119866119905

1003817100381710038171003817119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

+ 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩽ 119888radic119873 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 V1198671 1003817100381710038171003817119885119909100381710038171003817100381721198671

+ (10038171003817100381710038171198661199051003817100381710038171003817119867minus1 + 119866119867minus1) 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩽ ( 1198880radic119873 + 12057216) 1003817100381710038171003817119885119909

100381710038171003817100381721198671+ 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1)

(118)

Letting11987310158401 = max([256119888091205722] + 11198730) then we have

Λ 2 (119885119909) ⩽ 1205724 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1) for 119873 ⩾ 1198731015840

1(119)

On the other hand we have the lower bound of Λ 1(119885119909)Λ 1 (119885119909) ⩾ 1003817100381710038171003817119885119909

100381710038171003817100381721198671 minus 3 V2infin 1003817100381710038171003817119885119909100381710038171003817100381721198712

minus 2 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩾ 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 1198881198732

1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 2 119866119867minus1 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩾ (34 minus 11988811198732

) 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1

(120)

Letting1198731 = max([radic41198881] + 111987310158401) we see that for119873 ⩾ 1198731

Λ 1 (119885119909) ⩾ 12 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1 for 119905 ⩾ 1199051 (121)

Therefore we substitute (119)-(121) into (115) and use (114) tofind

119889119889119905Λ 1 (119885119909) + 120572Λ 1 (119885119909) ⩽ 119888 (100381710038171003817100381711986611990510038171003817100381710038172119867minus1 + 1198662119867minus1)

⩽ 119888 (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172) (122)

Applying the Gronwall inequality on [1199051 119905] and observingΛ 1(119885119909(1199051)) = Λ 1(0) = 0 we getΛ 1 (119885119909 (119905)) ⩽ int119905

1199051

119888119890120572(119903minus119905) (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172) 119889119903⩽ 119888 (1 + 119873 + 1198652 (119905))

(123)

which along withA21015840 and (121) imply the needed result (105)


54 Exponential Decay for Large Times Let 119911 = 119911(119905) =119911(119905 119904 1199060) = 119876119873119906(119904 + 119905 119904 1199060) which is a solution of (75) Let119885 = 119885(119905) = 119885(119905 119904 1199060) be the corresponding solution of (76)We need to prove the exponential decay of 119911 minus 119885Proposition 13 Suppose A11015840 A21015840 are satisfied Then thereexists1198732 ⩾ 1198731 such that for119873 ⩾ 1198732

sup119904⩽0

119911 (119905 119904) minus 119885 (119905 119904)1198671 ⩽ 119888119890minus120572(119905minus1199051) for 119905 ⩾ 1199051 (124)

Proof Recall that V = 119910 + 119885 and 119908 = 119911 minus 119885 = 119906 minus V Then itfollows from (75) and (76) that 119908 is the solution of

119908119905 + 120572119908 + 119894119908119909119909 minus 119894119908= 119894119876119873 ((|119906|2 + |V|2)119908 + 119906V119908) 119905 ⩾ 1199051 (125)

Multiplying (125) by minus119908119905 minus 120572119908 taking the imaginary part ofthe resulting equation and then integrating overΩ we obtainan energy equation

12 119889119889119905Υ1 (119908) + 120572Υ1 (119908) = Υ2 (119908) (126)

with

Υ1 (119908) = 11990821198671 minus intΩ(|119906|2 + |V|2) |119908|2 119889119909

minus ReintΩ119906V1199082119889119909

Υ2 (119908) = minus12 intΩ(|119906|2 + |V|2)

119905|119908|2 119889119909

minus 12ReintΩ (119906V)119905 1199082119889119909

(127)

By the boundedness of 119911 119885 in 1198761198731198671 and 119906119905 V119905 in 119867minus1 wehave the upper bound of Υ2(119908)

Υ2 (119908) ⩽ 119888 (1003817100381710038171003817119906119905 + V1199051003817100381710038171003817119867minus1) (119906 + V1198671) 1199081198671 119908119871infin

⩽ 119888radic119873 11990821198671 (128)

and also the lower bound of Υ1(119908)Υ1 (119908) ⩾ (1 minus 1198881198732

) 11990821198671 ⩾ 12 11990821198671 (129)

if119873 is large enough By (126)ndash(129) we have

119889119889119905Υ1 (119908) + 120572Υ1 (119908) ⩽ ( 1198882radic119873 minus 120572) 11990821198671 ⩽ 0 (130)

if 119873 ⩾ 1198732 fl max([411988822 1205722] + 11198731) and 119905 ⩾ 1199051 By theGronwall inequality on [1199051 119905] we obtain

Υ1 (119908 (119905)) ⩽ 119890minus120572(119905minus1199051)Υ1 (119908 (1199051)) (131)

By Lemma 10 119911(1199051)1198671 = 119876119873119906(119904 + 1199051 119904 1199060)1198671 ⩽ 119862 Notethat 119885(1199051) = 0 and so 119908(1199051)1198671 ⩽ 119862 which easily impliesthat Υ1(119908(1199051)) is bounded Therefore by (129) we have

119908 (119905)21198671 ⩽ 119862119890minus120572(119905minus1199051) forall119905 ⩾ 1199051 (132)

which completes the proof

6 Backward Compact Attractors

Finally we state and prove the main result We need to spiltthe evolution process 119878 Let119873 ⩾ 1198732 be fixed where1198732 is thelevel given in Proposition 13 Then for all 119905 ⩾ 0 119904 isin R and1199060 isin 1198671 we write 119878 = 1198781 + 1198782 with

1198781 (119904 + 119905 119904) 1199060=

119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) if 0 ⩽ 119905 ⩽ 1199051 or 119904 gt 0V (119905 119904) = 119910 (119905 119904) + 119885 (119905 119904) for 119905 ⩾ 1199051 119904 ⩽ 0

(133)

1198782 (119904 + 119905 119904) 1199060=

119911 (119905 119904) = 119876119873119906 (119904 + 119905 119904 1199060) for 0 ⩽ 119905 ⩽ 1199051 or 119904 gt 0119908 (119905 119904) = 119911 (119905 119904) minus 119885 (119905 119904) for 119905 ⩾ 1199051 119904 ⩽ 0

(134)

where 1199051 = 1199051(11990601198671) is the forward entering time

Theorem 14 (a) Suppose A1 A2 Then the nonautonomousSchrodinger equation has an increasing bounded and pullbackabsorbing setK = K(119904)119904isinR

(b) Suppose A11015840 A21015840 A3 Then the nonautonomousSchrodinger equation has a backward compact pullback attrac-torA = A(119904)119904isinR given by

A (119903) = 120596 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

119878 (119903 119903 minus 119905)K (119903)

= 1205961 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

1198781 (119903 119903 minus 119905)K (119903)119903 isin R

(135)

Proof The assertion (a) follows from Theorem 8 To prove(b) by the abstract result (ie Theorem 4) it suffices to provethat the evolution process 119878 has a backward compact-decaydecomposition

We need to prove that 1198781 is backward asymptoticallycompact Indeed let 119903 isin R and take some sequences 119903119899 ⩽ 119903120591119899 rarr +infin and 1199060119899 isin 119861119877 sub 1198671 Without loss of generalitywe assume 120591119899 ⩾ max1199051 119903 where 1199051 is the forward enteringtime Then by (133) we have

1198781 (119903119899 119903119899 minus 120591119899) 1199060119899 = 1198781 ((119903119899 minus 120591119899) + 120591119899 119903119899 minus 120591119899) 1199060119899= 119910 (120591119899 119903119899 minus 120591119899 1199060119899)

+ 119885 (120591119899 119903119899 minus 120591119899 1199060119899) (136)

By Lemma 10 we know that

1003817100381710038171003817119910 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198671⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119910 (119905 119904 1199060)10038171003817100381710038171198671 ⩽ 119862 (137)


which means that 119910(120591119899 119903119899 minus 120591119899 1199060119899) is a bounded sequencein the finite-dimensional space 1198751198731198671 and so passing to asubsequence it is convergent By Proposition 12 we know

1003817100381710038171003817119885 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198672⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119885 (119905 119904 1199060)10038171003817100381710038171198672 ⩽ 119862 (119873 + 1) (138)

which means that 119885(120591119899 119903119899 minus 120591119899 1199060119899) is bounded in 1198672 Bythe Sobolev compact embedding it is precompact in1198671 andso is 1198781(119903119899 119903119899 minus 120591119899)1199060119899

On the other hand by Proposition 13 we know for 119903 isin R120591 ⩾ max1199051 119903 and 1199060 isin 119861119877sup119903⩽119903

10038171003817100381710038171198782 (119903 119903 minus 120591) 119906010038171003817100381710038171198671⩽ sup

119904⩽0

1003817100381710038171003817119911 (120591 119904 1199060) minus 119885 (120591 119904 1199060)10038171003817100381710038171198671 ⩽ 119862119890minus120572(120591minus1199051) (139)

which implies 1198782 is backward decay

Remark 15 (I) In fact under Assumptions A11015840-A21015840 it ispossible to prove the existence of a forward attractor (see[32 33]) which may be different from the pullback attractorIn the present paper we only impose the forward absorptionto deduce a backward compact-decay composition whichseems not to be deduced from the pullback absorption for thenonautonomous Schrodinger equation

(II) Suppose 119891 119891119905 isin 119862(R 1198712) and they are time-periodicthen 119891 satisfies both conditions A11015840 and A21015840 In this case byusing the theoretical result as given in [17] one can obtaina periodic pullback attractor and maybe a periodic forwardattractor

Conflicts of Interest

There are no conflicts of interest to declare

Acknowledgments

The authors were supported by National Natural ScienceFoundation of China Grant 11571283 and L She was sup-ported by Natural Science Foundation of Education ofGuizhou Province KY[2016]103References

[1] G P Agrawal Nonlinear Fiber Optics vol 55 Academic PressSan Diego CA USA 2002

[2] M Hayashi and T Ozawa ldquoWell-posedness for a generalizedderivative nonlinear Schrodinger equationrdquo Journal of Differen-tial Equations vol 261 no 10 pp 5424ndash5445 2016

[3] N Akroune ldquoRegularity of the attractor for a weakly dampednonlinear Schrodinger equation on Rrdquo Applied MathematicsLetters vol 12 no 3 pp 45ndash48 1999

[4] E Emna K Wided and Z Ezzeddine ldquoFinite dimensionalglobal attractor for a semi-discrete nonlinear Schrodingerequation with a point defectrdquo Applied Mathematics and Com-putation vol 217 no 19 pp 7818ndash7830 2011

[5] J-M Ghidaglia ldquoFinite-dimensional behavior for weaklydamped driven Schrodinger equationsrdquo Annales de lrsquoInstitutHenri Poincarre Analyse Non Lineaire vol 5 no 4 pp 365ndash4051988

[6] O Goubet and L Molinet ldquoGlobal attractor for weakly dampednonlinear Schrodinger equations in L2(R)rdquo Nonlinear AnalysisTheory Methods amp Applications An International Multidisci-plinary Journal vol 71 no 1 pp 317ndash320 2009

[7] O Goubet ldquoRegularity of the attractor for a weakly dampednonlinear Schrodinger equationrdquo Applicable Analysis An Inter-national Journal vol 60 no 1-2 pp 99ndash119 1996

[8] R Temam Infinite Dimensional Dynamical Systems in Mechan-ics and Physics vol 68 Springer New York NY USA 1988

[9] X Wang ldquoAn energy equation for the weakly damped drivennonlinear Schrodinger equations and its application to theirattractorsrdquo Physica D Nonlinear Phenomena vol 88 no 3-4pp 167ndash175 1995

[10] X Yang C Zhao and J Cao ldquoDynamics of the discretecoupled nonlinear Schrodinger-Boussinesq equationsrdquo AppliedMathematics and Computation vol 219 no 16 pp 8508ndash85242013

[11] C Zhu C Mu and Z Pu ldquoAttractor for the nonlinearSchrodinger equation with a nonlocal nonlinear termrdquo Journalof Dynamical and Control Systems vol 16 no 4 pp 585ndash6032010

[12] A N Carvalho J A Langa and J Robinson Attractors forinfinite-dimensional non-autonomous dynamical systems vol182 of Applied Mathematical Sciences Springer New York 2013

[13] P E Kloeden and M Rasmussen Nonautonomous DynamicalSystems vol 176 of Mathematical Surveys and MonographsAmerican Mathematical Society Providence RI USA 2011

[14] P E Kloeden J Real and C Sun ldquoPullback attractors for asemilinear heat equation on time-varying domainsrdquo Journal ofDifferential Equations vol 246 no 12 pp 4702ndash4730 2009

[15] Y Guo S Cheng and Y Tang ldquoApproximate Kelvin-Voigtfluid driven by an external force depending on velocity withdistributed delayrdquoDiscrete Dynamics in Nature and Society vol2015 Article ID 721673 2015

[16] S Zhou H Chen and Z Wang ldquoPullback exponential attrac-tor for second order nonautonomous lattice systemrdquo DiscreteDynamics in Nature and Society vol 2014 Article ID 2370272014

[17] B Wang ldquoSufficient and necessary criteria for existence of pull-back attractors for non-compact random dynamical systemsrdquoJournal of Differential Equations vol 253 no 5 pp 1544ndash15832012

[18] Z Wang and S Zhou ldquoRandom attractor for non-autonomousstochastic strongly damped wave equation on unboundeddomainsrdquo Journal of Applied Analysis and Computation vol 5no 3 pp 363ndash387 2015

[19] J Yin Y Li and A Gu ldquoRegularity of pullback attractorsfor non-autonomous stochastic coupled reaction-diffusion sys-temsrdquo Journal of Applied Analysis and Computation vol 7 no3 pp 884ndash898 2017

[20] Y You ldquoRobustness of random attractors for a stochasticreaction-diffusion systemrdquo Journal of Applied Analysis andComputation vol 6 no 4 pp 1000ndash1022 2016

[21] H Cui and Y Li ldquoExistence and upper semicontinuity ofrandom attractors for stochastic degenerate parabolic equationswith multiplicative noisesrdquo Applied Mathematics and Computa-tion vol 271 pp 777ndash789 2015


[22] H Cui Y Li and J Yin ldquoLong time behavior of stochasticMHDequations perturbed bymultiplicative noisesrdquo Journal of AppliedAnalysis and Computation vol 6 no 4 pp 1081ndash1104 2016

[23] A Krause M Lewis and B Wang ldquoDynamics of the non-autonomous stochastic p-Laplace equation driven by multi-plicative noiserdquo Applied Mathematics and Computation vol246 pp 365ndash376 2014

[24] A Krause and B Wang ldquoPullback attractors of non-autonomous stochastic degenerate parabolic equations onunbounded domainsrdquo Journal of Mathematical Analysis andApplications vol 417 no 2 pp 1018ndash1038 2014

[25] Y Li A Gu and J Li ldquoExistence and continuity of bi-spatialrandom attractors and application to stochastic semilinearLaplacian equationsrdquo Journal of Differential Equations vol 258no 2 pp 504ndash534 2015

[26] Y Li and J Yin ldquoA modified proof of pullback attractors ina sobolev space for stochastic fitzhugh-nagumo equationsrdquoDiscrete and Continuous Dynamical Systems - Series B vol 21no 4 pp 1203ndash1223 2016

[27] W-Q Zhao ldquoRegularity of random attractors for a degenerateparabolic equations driven by additive noisesrdquo Applied Mathe-matics and Computation vol 239 pp 358ndash374 2014

[28] W Zhao and Y Zhang ldquoCompactness and attracting of randomattractors for non-autonomous stochastic lattice dynamicalsystems in weighted spacerdquo Applied Mathematics and Compu-tation vol 291 pp 226ndash243 2016

[29] H Cui J A Langa and Y Li ldquoRegularity and structure ofpullback attractors for reaction-diffusion type systems withoutuniquenessrdquo Nonlinear Analysis Theory Methods amp Applica-tions An International Multidisciplinary Journal vol 140 pp208ndash235 2016

[30] Y Li R Wang and J Yin ldquoBackward compact attractorsfor non-autonomous Benjamin-BONa-Mahony equations onunbounded channelsrdquo Discrete and Continuous DynamicalSystems - Series B vol 22 no 7 pp 2569ndash2586 2017

[31] J Yin A Gu and Y Li ldquoBackwards compact attractors for non-autonomous damped 3DNavier-Stokes equationsrdquoDynamics ofPartial Differential Equations vol 14 no 2 pp 201ndash218 2017

[32] P E Kloeden and T Lorenz ldquoConstruction of nonautonomousforward attractorsrdquo Proceedings of the American MathematicalSociety vol 144 no 1 pp 259ndash268 2016

[33] P E Kloeden and M Yang ldquoForward attraction in nonau-tonomous difference equationsrdquo Journal of Difference Equationsand Applications vol 22 no 4 pp 513ndash525 2016

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evolution process that is it has a backward compact-decaydecomposition For such a decomposable process we showthat the existence of a backward compact attractor is equiva-lent to the existence of an increasing bounded and pullbackabsorbing set (see Theorem 4)

We then apply the abstract result to the nonautonomousSchrodinger equation In Section 3 the increasingly pullbackabsorption is verified if119891 is assumed to be backward boundedand the time-derivative 119891119905 is backward tempered

The difficulty arises from verifying the compact-decaydecomposition according to the high and low frequency ofthe Fourier series In this case the pullback absorption maynot be suitable for verifying such a decomposable propertySo in Section 4 we have to give an auxiliary result on theforward absorption It may be possible to deduce a forwardattractor (cf [32 33]) but we do not pursue this forwardattractor in the present paper

In Section 5 we present some techniques of splittingthe solutions of (1) into high and low frequency parts andestablish a new equation with initial value zero in the high-frequency part Then the forward absorption obtained inSection 4 can be applied to prove that the new equation hasa forward uniformly bounded solution in 1198672 which furtherproves that the component system is backward asymptoticallycompact in1198671 Also we prove that the difference of solutionsfrom both equations in the high-frequency part is backwardexponential decay and so obtain the compact-decay decom-position as required

The final existence result of a backward compact attractoris given in Theorem 14 It is worth pointing out that thepullback-forward method (involving the high-low frequencydecomposition) may be special for the nonautonomousSchrodinger equation which is different from treating thepullback dynamics for other nonautonomous dissipativeequations

2 Backward Compact Dynamics forDecomposable Systems

Let (119883 sdot 119883) be a Banach space equipped with the class Bof all bounded subset in 119883 We consider a nonautonomousprocess 119878 on 119883 which means 119878(119905 119904) 119883 rarr 119883 is a continuousnonlinear mapping such that 119878(119904 119904) = id119883 and 119878(119905 119904) =119878(119905 119903)119878(119903 119904) for all 119905 ⩾ 119903 ⩾ 119904 with 119904 isin R

We assume that the process is decomposable in thefollowing sense

Definition 1 A nonautonomous process 119878 is said to have abackward compact-decay decomposition 119878(119905 119904) = 1198781(119905 119904) +1198782(119905 119904) (119905 ⩾ 119904) if

(i) 1198781 is backward asymptotically compact that is thesequence 1198781(119904119899 119904119899minus120591119899)119909119899infin119899=1 is precompact whenever 119904119899 ⩽ 119905with fixed 119905 isin R 120591119899 rarr +infin and 119909119899infin119899=1 is bounded in119883

(ii) 1198782 is backward exponential decay that is for each 119905 isin R

and 119861 isin B there exist two positive constants 1198881 = 1198881(119905 119861) and1198882 = 1198882(119905 119861) such that for some 1205910 ⩾ 0sup119904⩽119905

sup119909isin119861

10038171003817100381710038171198782 (119904 119904 minus 120591) 1199091003817100381710038171003817119883 ⩽ 1198881119890minus1198882(120591minus1205910) forall120591 ⩾ 1205910 (3)

A nonautonomous set D = D(119905)119905isinR in 119883 means aset-valued mapping D(sdot) R rarr 2119883 0 which is saidto have some topological properties (such as compactnessboundedness and closedness) if the component setD(119905) hasthe corresponding properties

Definition 2 A nonautonomous setD = D(119905)119905isinR is called(i) backward (resp forward or globally) compact if D(119905)

and cup119904⩽119905D(119904) (resp cup119903⩾119905D(119903) or cup119903isinRD(119903)) are compact foreach 119905 isin R

(ii) increasing (resp decreasing) if D(119904) sub D(119905) (respD(119904) sup D(119905)) for all 119904 ⩽ 119905Definition 3 A backward compact set A = A(119905)119905isinR in 119883is called a backward compact attractor for a nonautonomousprocess 119878 ifA is invariant that is 119878(119905 119904)A(119904) = A(119905) for 119905 ⩾ 119904and pullback attracting that is for any 119905 isin R and 119861 isin B

lim120591rarr+infin

dist119883 (119878 (119905 119905 minus 120591) 119861A (119905)) = 0 (4)

where the Hausdorff semi-distance dist119883(119860 119861) =sup119886isin119860inf119887isin119861119886 minus 119887119883

A backward compact attractor must be unique andminimal where the minimality meansA sub K for any closedattracting setK (see [30])

For the purpose of applying to the Schrodinger equationwe need to establish a new existence theorem of a backwardcompact attractor for a backward compact-decay processalthough other existence criteria were established in [29 30]

Recall that a nonautonomous set K is called pullbackabsorbing for 119878 if for each 119905 isin R and 119861 isin B there is a1205910 = 1205910(119905 119861) such that

119878 (119905 119905 minus 120591) 119861 sub K (119905) forall120591 ⩾ 1205910 (5)

Theorem 4 Suppose a nonautonomous process 119878 has a back-ward compact-decay decomposition 119878(119905 119904) = 1198781(119905 119904) + 1198782(119905 119904)for 119905 ⩾ 119904 Then the following statements are equivalent

(i) There exists an increasing bounded and pullbackabsorbing setK = K(119905)119905isinR

(ii) There is a backward compact attractor A = A(119905)119905isinRgiven by

A (119905) = 120596 (K (119905) 119905) fl ⋂1205910gt0

⋃120591⩾1205910

119878 (119905 119905 minus 120591)K (119905)119905 isin R

(6)

Proof The necessity is easily proved by setting K(119905) =1198731(cup119904⩽119905A(119904)) the 1-neighborhood of cup119904⩽119905A(119904) for each 119905 isinR Since cup119904⩽119905A(119904) is obviously an increasing family in 119905 isin Rit follows thatK is increasing Since cup119904⩽119905A(119904) is precompactit follows thatK(119905) is bounded (not necessarily precompact)Finally it is easy to deduce the pullback absorption ofK fromthe pullback attraction ofA

Conversely suppose (i) is true we show that A(119905) fl120596(K(119905) 119905) is a backward compact attractor in three steps

Step 1 We show the invariance As usual (cf [12]) it is easyto prove the positive invariance 119878(119905 119904)A(119904) sub A(119905) On the



lim119899rarrinfin

119878 (119905 119905 minus 120591119899) 119910119899 = 119910 (7)








119878 (119904 119904 minus 119894) 119911119894 = 119878 (119904 119904 minus 120591119896119894) 119910119896119894 997888rarr 119911 (11)




119878 (119905 119904) 119878 (119904 119904 minus 120591119896119894) 119910119896119894= 119878 (119905 119904) 119911 isin 119878 (119905 119904)A (119904) (12)





lim119896rarrinfin


1198782 (119905 119905 minus 120591119899119896) 119909119899119896 = 0 (14)


lim119896rarrinfin



1198782 (119905 119905 minus 120591119899119896) 119909119899119896 = 119909 (15)



and by (15)

lim119896rarrinfin


119878 (119905 119905 minus 120591119896119894) 119909119896119894 = 119909 (17)



⋃120591⩾1205910

119878 (119904 119904 minus 120591)K (119904) sub K (119904) sub K (119905) (18)


119861119905 = ⋃119904⩽119905

A (119904) = ⋃119904⩽119905

120596 (K (119904) 119904) sub K (119905) (19)




1198781 (119904119899119896 119904119899119896 minus 120591119899119896) 119909119899119896 997888rarr 119909 (21)




(23)





119906119905 + 119894119906119909119909 + 119894 |119906|2 119906 + 120572119906 minus 119894119906 = 119891 (119905 119909) 119906 (119905 0) = 119906 (119905 1) 119905 ⩾ 119904119906 (119904 119909) = 1199060 (119909)

119904 isin R 119909 isin Ω = (0 1) (24)



1198651 (119905) fl sup119904⩽119905



1198652 (119905) fl sup119904⩽119905

int119904





119904⩽119905int119904



1198653 (119905) = sup119904⩽119905

infinsum119898=0


119904minus(119898+1)119890120572(119903minus119904) 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 119889119903

⩽ sup119904⩽119905

infinsum119898=0

119890minus119898120572 int119904minus119898

119904minus(119898+1)

1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 119889119903


119890minus119898120572sup119903⩽119905

1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 = 1198651 (119905)1 minus 119890minus120572

(28)




119889119889119905 1199062 + 120572 1199062 ⩽ 1120572 1003817100381710038171003817119891 (119905 sdot)10038171003817100381710038172 (29)

119889119889119905Φ1 (119906) + 120572Φ1 (119906) ⩽ 9120572 1199066 + 3120572 1199064+ 1120572 1003817100381710038171003817119891119905 (119905 sdot)10038171003817100381710038172

(30)

where

Φ1 (119906) = 100381710038171003817100381711990611990910038171003817100381710038172 + 1199062 minus 12 11990644 + 2 ImintΩ119891119906119889119909 (31)



119889119889119905Φ1 (119906) + 2120572Φ2 (119906) minus 2ReintΩ119891119905119906 119889119909 = 0 (32)


Φ2 (119906) = 100381710038171003817100381711990611990910038171003817100381710038172 + 1199062 minus 11990644 + ImintΩ119891119906119889119909 (33)


⩽ minus 100381710038171003817100381711990611990910038171003817100381710038172 minus 1199062+ 3 1199063 (10038171003817100381710038171199061199091003817100381710038171003817 + 119906)

⩽ 9 1199066 + 3 1199064 minus 1199062

(34)


119889119889119905Φ1 (119906) + 120572Φ1 (119906) = 120572 (Φ1 (119906) minus 2Φ2 (119906))+ 2 Imint

Ω119891119905119906 119889119909

⩽ 9120572 1199066 + 3120572 1199064 + 1120572 100381710038171003817100381711989111990510038171003817100381710038172

(35)



by

119878 (119905 119904) 1199060 = 119906 (119905 119904 1199060) for 119905 ⩾ 119904 1199060 isin 1198671 (36)





sup119904⩽119905

int119904


sup119904⩽119905

int119904





119904minus120591119890120572(119903minus119903) 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 119889119903

(40)






which yields

int119904


⩽ 8 1003817100381710038171003817119906010038171003817100381710038176 int119904


+ 11988811986531 (119905) int119904

119904minus120591119890120572(119903minus119904)119889119903

⩽ 8119890minus3120572120591 1003817100381710038171003817119906010038171003817100381710038176 int0

minus120591119890minus2120572119903119889119903 + 11988811986531 (119905) int119904


⩽ 4120572119890minus120572120591 1003817100381710038171003817119906010038171003817100381710038176 + 11988811986531 (119905)

(44)



as follows



sup119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)100381710038171003817100381721198671 ⩽ 1198721 (119905)fl 1198883 (1 + 11986531 (119905) + 1198652 (119905))

(45)





+ 3120572int119904


+ 1120572 int119904

119904minus120591119890120572(119903minus119904) 1003817100381710038171003817119891119903 (119903 sdot)10038171003817100381710038172 119889119903

(46)


Φ1 (119906 (119904 119904 minus 120591 1199060))⩽ 119890minus120572120591Φ1 (1199060) + 119888 (1 + 11986531 (119905) + 11986521 (119905) + 1198652 (119905))

(47)



(48)


119890minus120572120591Φ1 (1199060) ⩽ 2119890minus120572120591 (10038171003817100381710038171199060100381710038171003817100381721198671 + 1198651 (119905)) ⩽ 1 (49)


sup119904⩽119905

Φ1 (119906 (119904 119904 minus 120591 1199060)) ⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905)) (50)


Φ1 (119906) ⩾ 100381710038171003817100381711990611990910038171003817100381710038172 + 1199062 minus 1199063 (10038171003817100381710038171199061199091003817100381710038171003817 + 119906)+ 2 Imint

Ω119891119906119889119909




1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)100381710038171003817100381721198671⩽ 2 sup

119904⩽119905Φ1 (119906 (119904 119904 minus 120591 1199060))

+ 4 sup119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)10038171003817100381710038176+ 2 sup

119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)10038171003817100381710038174 + 4 sup119904⩽119905

1003817100381710038171003817119891 (119904 sdot)10038171003817100381710038172⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905)) + 4 (1 + 1198881198651 (119905))3

+ 2 (1 + 1198881198651 (119905))2 + 41198651 (119905)⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905))

(52)




K (119905) = 119908 isin 1198671 11990821198671 ⩽ 1198721 (119905)fl 1198883 (1 + 11986531 (119905) + 1198652 (119905)) 119905 isin R (53)











sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038171198671 ⩽ 1205883 (54)

sup119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119906119905 (119904 + 119905 119904 1199060)1003817100381710038171003817119867minus1 ⩽ 1205884 (55)


1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038172 ⩽ 119890minus120572(119903minus119904) 1003817100381710038171003817119906010038171003817100381710038172 + 11205721198653 (119903) (56)

Taking 119903 = 119904 + 119905 in (56) we have

1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038172 ⩽ 119890minus120572119905 10038171003817100381710038171199060100381710038171003817100381721198671 + 11205721198653 (119904 + 119905)⩽ 119890minus120572119905 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198881198651 (119905)

(57)



1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 ⩽ 8119890minus3120572(119903minus119904) 1003817100381710038171003817119906010038171003817100381710038176 + 11988812058831 119904 ⩽ 119903 ⩽ 119904 + 119905 (59)

which yields

int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 119889119903

⩽ 8119890minus120572119905 1003817100381710038171003817119906010038171003817100381710038176 int119904+119905

119904119890minus2120572(119903minus119904)119889119903

+ 11988812058831 int119904+119905

119904119890120572(119903minus119904minus119905)119889119903


(60)


int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 119889119903 ⩽ 1 + 11988812058831 (61)


int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038174 119889119903 ⩽ 1 + 11988812058821 (62)



+ 9120572int119904+119905

119904119890120572(119903minus119904minus119905) 119906 (119903 119904)6 119889119903

+ 3120572int119904+119905

119904119890120572(119903minus119904minus119905) 119906 (119903 119904)4

+ 1120572 int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119891119903 (119903 sdot)10038171003817100381710038172 119889119903

(63)


+ 1198881198652 (119904 + 119905) (64)


Φ1 (119906 (119903 119904 1199060)) ⩽ 2 11990621198671 + 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172⩽ 2 11990621198671 + 1198651 (119903) 119903 ⩾ 119904 (65)


⩽ 119890minus120572119905 (2 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198651 (0)) ⩽ 1 (66)




(68)


+ 2 1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038174+ 4 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172

⩽ 119888 (1 + 12058831 + 12058821 + 1198651 (119905))⩽ 119888 (1 + 12058831 + 12058821 + 1205881)

(69)




⩽ 100381710038171003817100381711990611990910038171003817100381710038172 + 100381710038171003817100381712059311990910038171003817100381710038172 + 11990644 + 1003817100381710038171003817120593100381710038171003817100381744+ 119888 (1199062 + 100381710038171003817100381712059310038171003817100381710038172) + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172

⩽ 119888 100381710038171003817100381711990611990910038171003817100381710038172 + 119862 + 1198651 (119905) ⩽ 119888 100381710038171003817100381711990611990910038171003817100381710038172 + 119862

(70)




119906 (119905) = sum119896isinZ

119906119896 (119905) 1198902119896120587119894119909 (71)


119875119873119906 (119905) = sum|119896|⩽119873

119906119896 (119905) 1198902119896120587119894119909119876119873119906 (119905) = sum

|119896|gt119873

119906119896 (119905) 1198902119896120587119894119909(72)


119910 = 119910 (119905) = 119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) 119911 = 119911 (119905) = 119911 (119905 119904) = 119876119873119906 (119904 + 119905 119904 1199060) (73)


sup119905⩾1199051

sup119904⩽0

(1003817100381710038171003817119910 (119905 119904)100381710038171003817100381721198671 + 119911 (119905 119904)21198671) ⩽ 119862 (74)


= 119876119873119891 (119904 + 119905) 119905 ⩾ 1199051119911 (1199051) = 119876119873119906 (119904 + 1199051 119904 1199060)

(75)


119885119905 + 120572119885 + 119894119885119909119909 minus 119894119885 + 119894119876119873 (1003816100381610038161003816119910 + 11988510038161003816100381610038162 (119910 + 119885))= 119876119873119891 (119904 + 119905) 119905 ⩾ 1199051

119885 (1199051) = 119885 (1199051 119904) = 0(76)






119875119898119876N1198671 = 119885119898 (119905) 119885119898 (119905) = sum119873lt|119896|⩽119898

119906119896 (119905) 1198902119896120587119894119909 119898 gt 119873

(78)




sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198671 ⩽ 119862 (79)



12 119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) = Ψ2 (119885) 119905 ⩾ 1199051 (80)

with

Ψ1 (119885) = 11988521198671 + 2 ImintΩ119891119885119889119909 minus 12 11988544

minus 2ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus 2Reint

Ω|119885|2 119885119910119889119909 minus 2int

Ω

100381610038161003816100381611991010038161003816100381610038162 |119885|2 119889119909minus 2int

Ω(Re (119910119885))2 119889119909

Ψ2 (119885) = 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909 + 1205722 11988544

minus ReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909

minus 120572ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus int

ΩRe (119910119905119885) |119885|2 119889119909

+ 120572intΩRe (119910119885) |119885|2 119889119909

minus ReintΩ119910119910119905 |119885|2 119889119909

minus 2intΩRe (119910119905119885)Re (119910119885) 119889119909

(81)


1198681 fl 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909

⩽ 12057216 1198852 + 119888 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172⩽ 12057216 11988521198671 + 1198881205881 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(82)


1198854 ⩽ 119888 11988534 119885141198671

119885 ⩽ 119888119873 1198851198671 (83)


1198682 fl 1205722 11988544 ⩽ 119888 1198853 1198851198671 ⩽ 119888119873311988541198671 (84)


119906V1199081198671 ⩽ 119888 1199061198671 V1198671 1199081198671 (85)


1198683 fl minusReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909


(86)




100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909 ⩽ 12057232 1198852 + 119888 1003817100381710038171003817119910100381710038171003817100381766⩽ 12057232 1198852 + 119862

(87)


119885infin ⩽ 119888 11988512 119885121198671

⩽ 119888radic119873 1198851198671 (88)



⩽ 119888119873 11988531198671 ⩽ 12057232 11988521198671 + 119888119873211988541198671

(89)


1198686 fl 120572intΩRe (119910119885) |119885|2 119889119909 ⩽ 120572 100381710038171003817100381711991010038171003817100381710038174 1003817100381710038171003817100381711988531003817100381710038171003817100381743

= 120572 100381710038171003817100381711991010038171003817100381710038174 11988534 ⩽ 119888 100381710038171003817100381711991010038171003817100381710038171198671 11988594 119885341198671

⩽ 1198881198739411988531198671 ⩽ 12057232 11988521198671 + 11988811987392

11988541198671 (90)



ΩRe (119910119905119885)Re (119910119885) 119889119909



(91)


Ψ2 (119885) = 7sum119894=1

119868119894⩽ (312057216 + 1198881radic119873)11988521198671 + 1198881198732

11988541198671 + 1198881205881+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(92)



Ψ2 (119885) ⩽ 1205724 11988521198671 + 119888119873211988541198671 + 1198881205881

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (93)



11988541198671 minus 119862⩾ 12 11988521198671 minus 1198881198732

11988541198671 minus 119862

(94)


2Ψ2 (119885) minus 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (95)


119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (96)


Ψ1 (119885 (119905)) ⩽ 1198881198732int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903+ 119862int119905

1199051

119890120572(119903minus119905)119889119903+ 119888int119905

1199051

119890120572(119903minus119905) 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172 119889119903⩽ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862120572+ 1198881198652 (119904 + 119905)

(97)


119885 (119905)41198671+ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862 (98)



120585 (119905) ⩽ 119888211987321205852 (119905) + 1198622

ie 11988821205852 (119905) minus 1198732120585 (119905) + 11986221198732 ⩾ 0(99)




Δ = 1198734 minus 4119888211986221198732 = 1198732 (1198732 minus 411988821198622) gt 0 (100)



21198882

1205722 = 1198732 + 119873radic1198732 minus 411988821198622

21198882 (101)



radic1198732 minus 411988821198622 ⩽ 119873(102)


120585 (119905) ⩽ 1205721or 120585 (119905) ⩾ 1205722

119905 ⩾ 1199051(103)





sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198672 ⩽ 119862 (119873 + 1) (105)





119885119909119905 + 120572119885119909 + 119876119873 (119860 minus 1198651015840 (V)) 119885119909 = 119866 119905 ⩾ 1199051 (106)



119866 = 119876119873 (119891119909 (119904 + 119905 sdot) + 1198651015840 (V) 119910119909) (107)



⟨120593 119891119909 (119904 + 119905)⟩ = int120593119891119909 (119904 + 119905)⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172)⩽ 119888 (1003817100381710038171003817120593100381710038171003817100381721198671 + 1198651 (119904 + 119905)) ⩽ 119888 (1 + 1205881)

(108)





⩽ 119888 V21198671 100381710038171003817100381711991011990910038171003817100381710038171198671 ⩽ 119888119873 100381710038171003817100381711991010038171003817100381710038171198671 ⩽ 1198621198731003817100381710038171003817minus2119894119876119873 Re (V (119905) 119910119909 (119905)) V (119905)10038171003817100381710038171198671 ⩽ 119888119873 V21198671 100381710038171003817100381711991010038171003817100381710038171198671

⩽ 119862119873(110)





⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891119905 (119904 + 119905)10038171003817100381710038172) (112)




(113)


119905 ⩾ 1199051 119904 ⩽ 0 (114)


119889119889119905Λ 1 (119885119909) + 2120572Λ 1 (119885119909) = 2Λ 2 (119885119909) (115)

where

Λ 1 (119885119909) = 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus int

Ω|V|2 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119885119909)2 119889119909

+ 2intΩIm (119866119885119909) 119889119909

(116)

Λ 2 (119885119909) = minusintΩRe (VV119905) 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119905119885119909) sdot Re (V119885119909) 119889119909

+ intΩIm (119866119905119885119909) 119889119909

+ 120572intΩIm (119866119885119909) 119889119909

(117)


Λ 2 (119885119909) ⩽ 3 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 Vinfin 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 1003817100381710038171003817119866119905

1003817100381710038171003817119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

+ 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩽ 119888radic119873 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 V1198671 1003817100381710038171003817119885119909100381710038171003817100381721198671

+ (10038171003817100381710038171198661199051003817100381710038171003817119867minus1 + 119866119867minus1) 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩽ ( 1198880radic119873 + 12057216) 1003817100381710038171003817119885119909

100381710038171003817100381721198671+ 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1)

(118)


Λ 2 (119885119909) ⩽ 1205724 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1) for 119873 ⩾ 1198731015840

1(119)


100381710038171003817100381721198671 minus 3 V2infin 1003817100381710038171003817119885119909100381710038171003817100381721198712

minus 2 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩾ 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 1198881198732

1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 2 119866119867minus1 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩾ (34 minus 11988811198732

) 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1

(120)


Λ 1 (119885119909) ⩾ 12 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1 for 119905 ⩾ 1199051 (121)


119889119889119905Λ 1 (119885119909) + 120572Λ 1 (119885119909) ⩽ 119888 (100381710038171003817100381711986611990510038171003817100381710038172119867minus1 + 1198662119867minus1)

⩽ 119888 (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172) (122)


1199051

119888119890120572(119903minus119905) (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172) 119889119903⩽ 119888 (1 + 119873 + 1198652 (119905))

(123)




sup119904⩽0



119908119905 + 120572119908 + 119894119908119909119909 minus 119894119908= 119894119876119873 ((|119906|2 + |V|2)119908 + 119906V119908) 119905 ⩾ 1199051 (125)


12 119889119889119905Υ1 (119908) + 120572Υ1 (119908) = Υ2 (119908) (126)

with

Υ1 (119908) = 11990821198671 minus intΩ(|119906|2 + |V|2) |119908|2 119889119909

minus ReintΩ119906V1199082119889119909

Υ2 (119908) = minus12 intΩ(|119906|2 + |V|2)

119905|119908|2 119889119909

minus 12ReintΩ (119906V)119905 1199082119889119909

(127)


Υ2 (119908) ⩽ 119888 (1003817100381710038171003817119906119905 + V1199051003817100381710038171003817119867minus1) (119906 + V1198671) 1199081198671 119908119871infin

⩽ 119888radic119873 11990821198671 (128)


) 11990821198671 ⩾ 12 11990821198671 (129)


119889119889119905Υ1 (119908) + 120572Υ1 (119908) ⩽ ( 1198882radic119873 minus 120572) 11990821198671 ⩽ 0 (130)


Υ1 (119908 (119905)) ⩽ 119890minus120572(119905minus1199051)Υ1 (119908 (1199051)) (131)






1198781 (119904 + 119905 119904) 1199060=

119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) if 0 ⩽ 119905 ⩽ 1199051 or 119904 gt 0V (119905 119904) = 119910 (119905 119904) + 119885 (119905 119904) for 119905 ⩾ 1199051 119904 ⩽ 0

(133)

1198782 (119904 + 119905 119904) 1199060=


(134)




A (119903) = 120596 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

119878 (119903 119903 minus 119905)K (119903)

= 1205961 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

1198781 (119903 119903 minus 119905)K (119903)119903 isin R

(135)




+ 119885 (120591119899 119903119899 minus 120591119899 1199060119899) (136)


1003817100381710038171003817119910 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198671⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119910 (119905 119904 1199060)10038171003817100381710038171198671 ⩽ 119862 (137)



1003817100381710038171003817119885 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198672⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119885 (119905 119904 1199060)10038171003817100381710038171198672 ⩽ 119862 (119873 + 1) (138)



10038171003817100381710038171198782 (119903 119903 minus 120591) 119906010038171003817100381710038171198671⩽ sup

119904⩽0







Acknowledgments











































Journal of












Journal of







Volume 2018






Volume 2018










lim119899rarrinfin

119878 (119905 119905 minus 120591119899) 119910119899 = 119910 (7)








119878 (119904 119904 minus 119894) 119911119894 = 119878 (119904 119904 minus 120591119896119894) 119910119896119894 997888rarr 119911 (11)




119878 (119905 119904) 119878 (119904 119904 minus 120591119896119894) 119910119896119894= 119878 (119905 119904) 119911 isin 119878 (119905 119904)A (119904) (12)





lim119896rarrinfin


1198782 (119905 119905 minus 120591119899119896) 119909119899119896 = 0 (14)


lim119896rarrinfin



1198782 (119905 119905 minus 120591119899119896) 119909119899119896 = 119909 (15)



and by (15)

lim119896rarrinfin


119878 (119905 119905 minus 120591119896119894) 119909119896119894 = 119909 (17)



⋃120591⩾1205910

119878 (119904 119904 minus 120591)K (119904) sub K (119904) sub K (119905) (18)


119861119905 = ⋃119904⩽119905

A (119904) = ⋃119904⩽119905

120596 (K (119904) 119904) sub K (119905) (19)




1198781 (119904119899119896 119904119899119896 minus 120591119899119896) 119909119899119896 997888rarr 119909 (21)




(23)





119906119905 + 119894119906119909119909 + 119894 |119906|2 119906 + 120572119906 minus 119894119906 = 119891 (119905 119909) 119906 (119905 0) = 119906 (119905 1) 119905 ⩾ 119904119906 (119904 119909) = 1199060 (119909)

119904 isin R 119909 isin Ω = (0 1) (24)



1198651 (119905) fl sup119904⩽119905



1198652 (119905) fl sup119904⩽119905

int119904





119904⩽119905int119904



1198653 (119905) = sup119904⩽119905

infinsum119898=0


119904minus(119898+1)119890120572(119903minus119904) 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 119889119903

⩽ sup119904⩽119905

infinsum119898=0

119890minus119898120572 int119904minus119898

119904minus(119898+1)

1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 119889119903


119890minus119898120572sup119903⩽119905

1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 = 1198651 (119905)1 minus 119890minus120572

(28)




119889119889119905 1199062 + 120572 1199062 ⩽ 1120572 1003817100381710038171003817119891 (119905 sdot)10038171003817100381710038172 (29)

119889119889119905Φ1 (119906) + 120572Φ1 (119906) ⩽ 9120572 1199066 + 3120572 1199064+ 1120572 1003817100381710038171003817119891119905 (119905 sdot)10038171003817100381710038172

(30)

where

Φ1 (119906) = 100381710038171003817100381711990611990910038171003817100381710038172 + 1199062 minus 12 11990644 + 2 ImintΩ119891119906119889119909 (31)



119889119889119905Φ1 (119906) + 2120572Φ2 (119906) minus 2ReintΩ119891119905119906 119889119909 = 0 (32)


Φ2 (119906) = 100381710038171003817100381711990611990910038171003817100381710038172 + 1199062 minus 11990644 + ImintΩ119891119906119889119909 (33)


⩽ minus 100381710038171003817100381711990611990910038171003817100381710038172 minus 1199062+ 3 1199063 (10038171003817100381710038171199061199091003817100381710038171003817 + 119906)

⩽ 9 1199066 + 3 1199064 minus 1199062

(34)


119889119889119905Φ1 (119906) + 120572Φ1 (119906) = 120572 (Φ1 (119906) minus 2Φ2 (119906))+ 2 Imint

Ω119891119905119906 119889119909

⩽ 9120572 1199066 + 3120572 1199064 + 1120572 100381710038171003817100381711989111990510038171003817100381710038172

(35)



by

119878 (119905 119904) 1199060 = 119906 (119905 119904 1199060) for 119905 ⩾ 119904 1199060 isin 1198671 (36)





sup119904⩽119905

int119904


sup119904⩽119905

int119904





119904minus120591119890120572(119903minus119903) 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 119889119903

(40)






which yields

int119904


⩽ 8 1003817100381710038171003817119906010038171003817100381710038176 int119904


+ 11988811986531 (119905) int119904

119904minus120591119890120572(119903minus119904)119889119903

⩽ 8119890minus3120572120591 1003817100381710038171003817119906010038171003817100381710038176 int0

minus120591119890minus2120572119903119889119903 + 11988811986531 (119905) int119904


⩽ 4120572119890minus120572120591 1003817100381710038171003817119906010038171003817100381710038176 + 11988811986531 (119905)

(44)



as follows



sup119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)100381710038171003817100381721198671 ⩽ 1198721 (119905)fl 1198883 (1 + 11986531 (119905) + 1198652 (119905))

(45)





+ 3120572int119904


+ 1120572 int119904

119904minus120591119890120572(119903minus119904) 1003817100381710038171003817119891119903 (119903 sdot)10038171003817100381710038172 119889119903

(46)


Φ1 (119906 (119904 119904 minus 120591 1199060))⩽ 119890minus120572120591Φ1 (1199060) + 119888 (1 + 11986531 (119905) + 11986521 (119905) + 1198652 (119905))

(47)



(48)


119890minus120572120591Φ1 (1199060) ⩽ 2119890minus120572120591 (10038171003817100381710038171199060100381710038171003817100381721198671 + 1198651 (119905)) ⩽ 1 (49)


sup119904⩽119905

Φ1 (119906 (119904 119904 minus 120591 1199060)) ⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905)) (50)


Φ1 (119906) ⩾ 100381710038171003817100381711990611990910038171003817100381710038172 + 1199062 minus 1199063 (10038171003817100381710038171199061199091003817100381710038171003817 + 119906)+ 2 Imint

Ω119891119906119889119909




1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)100381710038171003817100381721198671⩽ 2 sup

119904⩽119905Φ1 (119906 (119904 119904 minus 120591 1199060))

+ 4 sup119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)10038171003817100381710038176+ 2 sup

119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)10038171003817100381710038174 + 4 sup119904⩽119905

1003817100381710038171003817119891 (119904 sdot)10038171003817100381710038172⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905)) + 4 (1 + 1198881198651 (119905))3

+ 2 (1 + 1198881198651 (119905))2 + 41198651 (119905)⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905))

(52)




K (119905) = 119908 isin 1198671 11990821198671 ⩽ 1198721 (119905)fl 1198883 (1 + 11986531 (119905) + 1198652 (119905)) 119905 isin R (53)











sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038171198671 ⩽ 1205883 (54)

sup119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119906119905 (119904 + 119905 119904 1199060)1003817100381710038171003817119867minus1 ⩽ 1205884 (55)


1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038172 ⩽ 119890minus120572(119903minus119904) 1003817100381710038171003817119906010038171003817100381710038172 + 11205721198653 (119903) (56)

Taking 119903 = 119904 + 119905 in (56) we have

1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038172 ⩽ 119890minus120572119905 10038171003817100381710038171199060100381710038171003817100381721198671 + 11205721198653 (119904 + 119905)⩽ 119890minus120572119905 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198881198651 (119905)

(57)



1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 ⩽ 8119890minus3120572(119903minus119904) 1003817100381710038171003817119906010038171003817100381710038176 + 11988812058831 119904 ⩽ 119903 ⩽ 119904 + 119905 (59)

which yields

int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 119889119903

⩽ 8119890minus120572119905 1003817100381710038171003817119906010038171003817100381710038176 int119904+119905

119904119890minus2120572(119903minus119904)119889119903

+ 11988812058831 int119904+119905

119904119890120572(119903minus119904minus119905)119889119903


(60)


int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 119889119903 ⩽ 1 + 11988812058831 (61)


int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038174 119889119903 ⩽ 1 + 11988812058821 (62)



+ 9120572int119904+119905

119904119890120572(119903minus119904minus119905) 119906 (119903 119904)6 119889119903

+ 3120572int119904+119905

119904119890120572(119903minus119904minus119905) 119906 (119903 119904)4

+ 1120572 int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119891119903 (119903 sdot)10038171003817100381710038172 119889119903

(63)


+ 1198881198652 (119904 + 119905) (64)


Φ1 (119906 (119903 119904 1199060)) ⩽ 2 11990621198671 + 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172⩽ 2 11990621198671 + 1198651 (119903) 119903 ⩾ 119904 (65)


⩽ 119890minus120572119905 (2 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198651 (0)) ⩽ 1 (66)




(68)


+ 2 1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038174+ 4 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172

⩽ 119888 (1 + 12058831 + 12058821 + 1198651 (119905))⩽ 119888 (1 + 12058831 + 12058821 + 1205881)

(69)




⩽ 100381710038171003817100381711990611990910038171003817100381710038172 + 100381710038171003817100381712059311990910038171003817100381710038172 + 11990644 + 1003817100381710038171003817120593100381710038171003817100381744+ 119888 (1199062 + 100381710038171003817100381712059310038171003817100381710038172) + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172

⩽ 119888 100381710038171003817100381711990611990910038171003817100381710038172 + 119862 + 1198651 (119905) ⩽ 119888 100381710038171003817100381711990611990910038171003817100381710038172 + 119862

(70)




119906 (119905) = sum119896isinZ

119906119896 (119905) 1198902119896120587119894119909 (71)


119875119873119906 (119905) = sum|119896|⩽119873

119906119896 (119905) 1198902119896120587119894119909119876119873119906 (119905) = sum

|119896|gt119873

119906119896 (119905) 1198902119896120587119894119909(72)


119910 = 119910 (119905) = 119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) 119911 = 119911 (119905) = 119911 (119905 119904) = 119876119873119906 (119904 + 119905 119904 1199060) (73)


sup119905⩾1199051

sup119904⩽0

(1003817100381710038171003817119910 (119905 119904)100381710038171003817100381721198671 + 119911 (119905 119904)21198671) ⩽ 119862 (74)


= 119876119873119891 (119904 + 119905) 119905 ⩾ 1199051119911 (1199051) = 119876119873119906 (119904 + 1199051 119904 1199060)

(75)


119885119905 + 120572119885 + 119894119885119909119909 minus 119894119885 + 119894119876119873 (1003816100381610038161003816119910 + 11988510038161003816100381610038162 (119910 + 119885))= 119876119873119891 (119904 + 119905) 119905 ⩾ 1199051

119885 (1199051) = 119885 (1199051 119904) = 0(76)






119875119898119876N1198671 = 119885119898 (119905) 119885119898 (119905) = sum119873lt|119896|⩽119898

119906119896 (119905) 1198902119896120587119894119909 119898 gt 119873

(78)




sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198671 ⩽ 119862 (79)



12 119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) = Ψ2 (119885) 119905 ⩾ 1199051 (80)

with

Ψ1 (119885) = 11988521198671 + 2 ImintΩ119891119885119889119909 minus 12 11988544

minus 2ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus 2Reint

Ω|119885|2 119885119910119889119909 minus 2int

Ω

100381610038161003816100381611991010038161003816100381610038162 |119885|2 119889119909minus 2int

Ω(Re (119910119885))2 119889119909

Ψ2 (119885) = 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909 + 1205722 11988544

minus ReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909

minus 120572ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus int

ΩRe (119910119905119885) |119885|2 119889119909

+ 120572intΩRe (119910119885) |119885|2 119889119909

minus ReintΩ119910119910119905 |119885|2 119889119909

minus 2intΩRe (119910119905119885)Re (119910119885) 119889119909

(81)


1198681 fl 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909

⩽ 12057216 1198852 + 119888 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172⩽ 12057216 11988521198671 + 1198881205881 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(82)


1198854 ⩽ 119888 11988534 119885141198671

119885 ⩽ 119888119873 1198851198671 (83)


1198682 fl 1205722 11988544 ⩽ 119888 1198853 1198851198671 ⩽ 119888119873311988541198671 (84)


119906V1199081198671 ⩽ 119888 1199061198671 V1198671 1199081198671 (85)


1198683 fl minusReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909


(86)




100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909 ⩽ 12057232 1198852 + 119888 1003817100381710038171003817119910100381710038171003817100381766⩽ 12057232 1198852 + 119862

(87)


119885infin ⩽ 119888 11988512 119885121198671

⩽ 119888radic119873 1198851198671 (88)



⩽ 119888119873 11988531198671 ⩽ 12057232 11988521198671 + 119888119873211988541198671

(89)


1198686 fl 120572intΩRe (119910119885) |119885|2 119889119909 ⩽ 120572 100381710038171003817100381711991010038171003817100381710038174 1003817100381710038171003817100381711988531003817100381710038171003817100381743

= 120572 100381710038171003817100381711991010038171003817100381710038174 11988534 ⩽ 119888 100381710038171003817100381711991010038171003817100381710038171198671 11988594 119885341198671

⩽ 1198881198739411988531198671 ⩽ 12057232 11988521198671 + 11988811987392

11988541198671 (90)



ΩRe (119910119905119885)Re (119910119885) 119889119909



(91)


Ψ2 (119885) = 7sum119894=1

119868119894⩽ (312057216 + 1198881radic119873)11988521198671 + 1198881198732

11988541198671 + 1198881205881+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(92)



Ψ2 (119885) ⩽ 1205724 11988521198671 + 119888119873211988541198671 + 1198881205881

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (93)



11988541198671 minus 119862⩾ 12 11988521198671 minus 1198881198732

11988541198671 minus 119862

(94)


2Ψ2 (119885) minus 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (95)


119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (96)


Ψ1 (119885 (119905)) ⩽ 1198881198732int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903+ 119862int119905

1199051

119890120572(119903minus119905)119889119903+ 119888int119905

1199051

119890120572(119903minus119905) 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172 119889119903⩽ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862120572+ 1198881198652 (119904 + 119905)

(97)


119885 (119905)41198671+ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862 (98)



120585 (119905) ⩽ 119888211987321205852 (119905) + 1198622

ie 11988821205852 (119905) minus 1198732120585 (119905) + 11986221198732 ⩾ 0(99)




Δ = 1198734 minus 4119888211986221198732 = 1198732 (1198732 minus 411988821198622) gt 0 (100)



21198882

1205722 = 1198732 + 119873radic1198732 minus 411988821198622

21198882 (101)



radic1198732 minus 411988821198622 ⩽ 119873(102)


120585 (119905) ⩽ 1205721or 120585 (119905) ⩾ 1205722

119905 ⩾ 1199051(103)





sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198672 ⩽ 119862 (119873 + 1) (105)





119885119909119905 + 120572119885119909 + 119876119873 (119860 minus 1198651015840 (V)) 119885119909 = 119866 119905 ⩾ 1199051 (106)



119866 = 119876119873 (119891119909 (119904 + 119905 sdot) + 1198651015840 (V) 119910119909) (107)



⟨120593 119891119909 (119904 + 119905)⟩ = int120593119891119909 (119904 + 119905)⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172)⩽ 119888 (1003817100381710038171003817120593100381710038171003817100381721198671 + 1198651 (119904 + 119905)) ⩽ 119888 (1 + 1205881)

(108)





⩽ 119888 V21198671 100381710038171003817100381711991011990910038171003817100381710038171198671 ⩽ 119888119873 100381710038171003817100381711991010038171003817100381710038171198671 ⩽ 1198621198731003817100381710038171003817minus2119894119876119873 Re (V (119905) 119910119909 (119905)) V (119905)10038171003817100381710038171198671 ⩽ 119888119873 V21198671 100381710038171003817100381711991010038171003817100381710038171198671

⩽ 119862119873(110)





⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891119905 (119904 + 119905)10038171003817100381710038172) (112)




(113)


119905 ⩾ 1199051 119904 ⩽ 0 (114)


119889119889119905Λ 1 (119885119909) + 2120572Λ 1 (119885119909) = 2Λ 2 (119885119909) (115)

where

Λ 1 (119885119909) = 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus int

Ω|V|2 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119885119909)2 119889119909

+ 2intΩIm (119866119885119909) 119889119909

(116)

Λ 2 (119885119909) = minusintΩRe (VV119905) 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119905119885119909) sdot Re (V119885119909) 119889119909

+ intΩIm (119866119905119885119909) 119889119909

+ 120572intΩIm (119866119885119909) 119889119909

(117)


Λ 2 (119885119909) ⩽ 3 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 Vinfin 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 1003817100381710038171003817119866119905

1003817100381710038171003817119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

+ 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩽ 119888radic119873 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 V1198671 1003817100381710038171003817119885119909100381710038171003817100381721198671

+ (10038171003817100381710038171198661199051003817100381710038171003817119867minus1 + 119866119867minus1) 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩽ ( 1198880radic119873 + 12057216) 1003817100381710038171003817119885119909

100381710038171003817100381721198671+ 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1)

(118)


Λ 2 (119885119909) ⩽ 1205724 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1) for 119873 ⩾ 1198731015840

1(119)


100381710038171003817100381721198671 minus 3 V2infin 1003817100381710038171003817119885119909100381710038171003817100381721198712

minus 2 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩾ 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 1198881198732

1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 2 119866119867minus1 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩾ (34 minus 11988811198732

) 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1

(120)


Λ 1 (119885119909) ⩾ 12 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1 for 119905 ⩾ 1199051 (121)


119889119889119905Λ 1 (119885119909) + 120572Λ 1 (119885119909) ⩽ 119888 (100381710038171003817100381711986611990510038171003817100381710038172119867minus1 + 1198662119867minus1)

⩽ 119888 (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172) (122)


1199051

119888119890120572(119903minus119905) (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172) 119889119903⩽ 119888 (1 + 119873 + 1198652 (119905))

(123)




sup119904⩽0



119908119905 + 120572119908 + 119894119908119909119909 minus 119894119908= 119894119876119873 ((|119906|2 + |V|2)119908 + 119906V119908) 119905 ⩾ 1199051 (125)


12 119889119889119905Υ1 (119908) + 120572Υ1 (119908) = Υ2 (119908) (126)

with

Υ1 (119908) = 11990821198671 minus intΩ(|119906|2 + |V|2) |119908|2 119889119909

minus ReintΩ119906V1199082119889119909

Υ2 (119908) = minus12 intΩ(|119906|2 + |V|2)

119905|119908|2 119889119909

minus 12ReintΩ (119906V)119905 1199082119889119909

(127)


Υ2 (119908) ⩽ 119888 (1003817100381710038171003817119906119905 + V1199051003817100381710038171003817119867minus1) (119906 + V1198671) 1199081198671 119908119871infin

⩽ 119888radic119873 11990821198671 (128)


) 11990821198671 ⩾ 12 11990821198671 (129)


119889119889119905Υ1 (119908) + 120572Υ1 (119908) ⩽ ( 1198882radic119873 minus 120572) 11990821198671 ⩽ 0 (130)


Υ1 (119908 (119905)) ⩽ 119890minus120572(119905minus1199051)Υ1 (119908 (1199051)) (131)






1198781 (119904 + 119905 119904) 1199060=

119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) if 0 ⩽ 119905 ⩽ 1199051 or 119904 gt 0V (119905 119904) = 119910 (119905 119904) + 119885 (119905 119904) for 119905 ⩾ 1199051 119904 ⩽ 0

(133)

1198782 (119904 + 119905 119904) 1199060=


(134)




A (119903) = 120596 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

119878 (119903 119903 minus 119905)K (119903)

= 1205961 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

1198781 (119903 119903 minus 119905)K (119903)119903 isin R

(135)




+ 119885 (120591119899 119903119899 minus 120591119899 1199060119899) (136)


1003817100381710038171003817119910 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198671⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119910 (119905 119904 1199060)10038171003817100381710038171198671 ⩽ 119862 (137)



1003817100381710038171003817119885 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198672⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119885 (119905 119904 1199060)10038171003817100381710038171198672 ⩽ 119862 (119873 + 1) (138)



10038171003817100381710038171198782 (119903 119903 minus 120591) 119906010038171003817100381710038171198671⩽ sup

119904⩽0







Acknowledgments











































Journal of












Journal of







Volume 2018






Volume 2018











119906119905 + 119894119906119909119909 + 119894 |119906|2 119906 + 120572119906 minus 119894119906 = 119891 (119905 119909) 119906 (119905 0) = 119906 (119905 1) 119905 ⩾ 119904119906 (119904 119909) = 1199060 (119909)

119904 isin R 119909 isin Ω = (0 1) (24)



1198651 (119905) fl sup119904⩽119905



1198652 (119905) fl sup119904⩽119905

int119904





119904⩽119905int119904



1198653 (119905) = sup119904⩽119905

infinsum119898=0


119904minus(119898+1)119890120572(119903minus119904) 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 119889119903

⩽ sup119904⩽119905

infinsum119898=0

119890minus119898120572 int119904minus119898

119904minus(119898+1)

1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 119889119903


119890minus119898120572sup119903⩽119905

1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 = 1198651 (119905)1 minus 119890minus120572

(28)




119889119889119905 1199062 + 120572 1199062 ⩽ 1120572 1003817100381710038171003817119891 (119905 sdot)10038171003817100381710038172 (29)

119889119889119905Φ1 (119906) + 120572Φ1 (119906) ⩽ 9120572 1199066 + 3120572 1199064+ 1120572 1003817100381710038171003817119891119905 (119905 sdot)10038171003817100381710038172

(30)

where

Φ1 (119906) = 100381710038171003817100381711990611990910038171003817100381710038172 + 1199062 minus 12 11990644 + 2 ImintΩ119891119906119889119909 (31)



119889119889119905Φ1 (119906) + 2120572Φ2 (119906) minus 2ReintΩ119891119905119906 119889119909 = 0 (32)


Φ2 (119906) = 100381710038171003817100381711990611990910038171003817100381710038172 + 1199062 minus 11990644 + ImintΩ119891119906119889119909 (33)


⩽ minus 100381710038171003817100381711990611990910038171003817100381710038172 minus 1199062+ 3 1199063 (10038171003817100381710038171199061199091003817100381710038171003817 + 119906)

⩽ 9 1199066 + 3 1199064 minus 1199062

(34)


119889119889119905Φ1 (119906) + 120572Φ1 (119906) = 120572 (Φ1 (119906) minus 2Φ2 (119906))+ 2 Imint

Ω119891119905119906 119889119909

⩽ 9120572 1199066 + 3120572 1199064 + 1120572 100381710038171003817100381711989111990510038171003817100381710038172

(35)



by

119878 (119905 119904) 1199060 = 119906 (119905 119904 1199060) for 119905 ⩾ 119904 1199060 isin 1198671 (36)





sup119904⩽119905

int119904


sup119904⩽119905

int119904





119904minus120591119890120572(119903minus119903) 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 119889119903

(40)






which yields

int119904


⩽ 8 1003817100381710038171003817119906010038171003817100381710038176 int119904


+ 11988811986531 (119905) int119904

119904minus120591119890120572(119903minus119904)119889119903

⩽ 8119890minus3120572120591 1003817100381710038171003817119906010038171003817100381710038176 int0

minus120591119890minus2120572119903119889119903 + 11988811986531 (119905) int119904


⩽ 4120572119890minus120572120591 1003817100381710038171003817119906010038171003817100381710038176 + 11988811986531 (119905)

(44)



as follows



sup119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)100381710038171003817100381721198671 ⩽ 1198721 (119905)fl 1198883 (1 + 11986531 (119905) + 1198652 (119905))

(45)





+ 3120572int119904


+ 1120572 int119904

119904minus120591119890120572(119903minus119904) 1003817100381710038171003817119891119903 (119903 sdot)10038171003817100381710038172 119889119903

(46)


Φ1 (119906 (119904 119904 minus 120591 1199060))⩽ 119890minus120572120591Φ1 (1199060) + 119888 (1 + 11986531 (119905) + 11986521 (119905) + 1198652 (119905))

(47)



(48)


119890minus120572120591Φ1 (1199060) ⩽ 2119890minus120572120591 (10038171003817100381710038171199060100381710038171003817100381721198671 + 1198651 (119905)) ⩽ 1 (49)


sup119904⩽119905

Φ1 (119906 (119904 119904 minus 120591 1199060)) ⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905)) (50)


Φ1 (119906) ⩾ 100381710038171003817100381711990611990910038171003817100381710038172 + 1199062 minus 1199063 (10038171003817100381710038171199061199091003817100381710038171003817 + 119906)+ 2 Imint

Ω119891119906119889119909




1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)100381710038171003817100381721198671⩽ 2 sup

119904⩽119905Φ1 (119906 (119904 119904 minus 120591 1199060))

+ 4 sup119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)10038171003817100381710038176+ 2 sup

119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)10038171003817100381710038174 + 4 sup119904⩽119905

1003817100381710038171003817119891 (119904 sdot)10038171003817100381710038172⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905)) + 4 (1 + 1198881198651 (119905))3

+ 2 (1 + 1198881198651 (119905))2 + 41198651 (119905)⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905))

(52)




K (119905) = 119908 isin 1198671 11990821198671 ⩽ 1198721 (119905)fl 1198883 (1 + 11986531 (119905) + 1198652 (119905)) 119905 isin R (53)











sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038171198671 ⩽ 1205883 (54)

sup119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119906119905 (119904 + 119905 119904 1199060)1003817100381710038171003817119867minus1 ⩽ 1205884 (55)


1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038172 ⩽ 119890minus120572(119903minus119904) 1003817100381710038171003817119906010038171003817100381710038172 + 11205721198653 (119903) (56)

Taking 119903 = 119904 + 119905 in (56) we have

1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038172 ⩽ 119890minus120572119905 10038171003817100381710038171199060100381710038171003817100381721198671 + 11205721198653 (119904 + 119905)⩽ 119890minus120572119905 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198881198651 (119905)

(57)



1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 ⩽ 8119890minus3120572(119903minus119904) 1003817100381710038171003817119906010038171003817100381710038176 + 11988812058831 119904 ⩽ 119903 ⩽ 119904 + 119905 (59)

which yields

int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 119889119903

⩽ 8119890minus120572119905 1003817100381710038171003817119906010038171003817100381710038176 int119904+119905

119904119890minus2120572(119903minus119904)119889119903

+ 11988812058831 int119904+119905

119904119890120572(119903minus119904minus119905)119889119903


(60)


int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 119889119903 ⩽ 1 + 11988812058831 (61)


int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038174 119889119903 ⩽ 1 + 11988812058821 (62)



+ 9120572int119904+119905

119904119890120572(119903minus119904minus119905) 119906 (119903 119904)6 119889119903

+ 3120572int119904+119905

119904119890120572(119903minus119904minus119905) 119906 (119903 119904)4

+ 1120572 int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119891119903 (119903 sdot)10038171003817100381710038172 119889119903

(63)


+ 1198881198652 (119904 + 119905) (64)


Φ1 (119906 (119903 119904 1199060)) ⩽ 2 11990621198671 + 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172⩽ 2 11990621198671 + 1198651 (119903) 119903 ⩾ 119904 (65)


⩽ 119890minus120572119905 (2 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198651 (0)) ⩽ 1 (66)




(68)


+ 2 1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038174+ 4 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172

⩽ 119888 (1 + 12058831 + 12058821 + 1198651 (119905))⩽ 119888 (1 + 12058831 + 12058821 + 1205881)

(69)




⩽ 100381710038171003817100381711990611990910038171003817100381710038172 + 100381710038171003817100381712059311990910038171003817100381710038172 + 11990644 + 1003817100381710038171003817120593100381710038171003817100381744+ 119888 (1199062 + 100381710038171003817100381712059310038171003817100381710038172) + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172

⩽ 119888 100381710038171003817100381711990611990910038171003817100381710038172 + 119862 + 1198651 (119905) ⩽ 119888 100381710038171003817100381711990611990910038171003817100381710038172 + 119862

(70)




119906 (119905) = sum119896isinZ

119906119896 (119905) 1198902119896120587119894119909 (71)


119875119873119906 (119905) = sum|119896|⩽119873

119906119896 (119905) 1198902119896120587119894119909119876119873119906 (119905) = sum

|119896|gt119873

119906119896 (119905) 1198902119896120587119894119909(72)


119910 = 119910 (119905) = 119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) 119911 = 119911 (119905) = 119911 (119905 119904) = 119876119873119906 (119904 + 119905 119904 1199060) (73)


sup119905⩾1199051

sup119904⩽0

(1003817100381710038171003817119910 (119905 119904)100381710038171003817100381721198671 + 119911 (119905 119904)21198671) ⩽ 119862 (74)


= 119876119873119891 (119904 + 119905) 119905 ⩾ 1199051119911 (1199051) = 119876119873119906 (119904 + 1199051 119904 1199060)

(75)


119885119905 + 120572119885 + 119894119885119909119909 minus 119894119885 + 119894119876119873 (1003816100381610038161003816119910 + 11988510038161003816100381610038162 (119910 + 119885))= 119876119873119891 (119904 + 119905) 119905 ⩾ 1199051

119885 (1199051) = 119885 (1199051 119904) = 0(76)






119875119898119876N1198671 = 119885119898 (119905) 119885119898 (119905) = sum119873lt|119896|⩽119898

119906119896 (119905) 1198902119896120587119894119909 119898 gt 119873

(78)




sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198671 ⩽ 119862 (79)



12 119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) = Ψ2 (119885) 119905 ⩾ 1199051 (80)

with

Ψ1 (119885) = 11988521198671 + 2 ImintΩ119891119885119889119909 minus 12 11988544

minus 2ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus 2Reint

Ω|119885|2 119885119910119889119909 minus 2int

Ω

100381610038161003816100381611991010038161003816100381610038162 |119885|2 119889119909minus 2int

Ω(Re (119910119885))2 119889119909

Ψ2 (119885) = 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909 + 1205722 11988544

minus ReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909

minus 120572ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus int

ΩRe (119910119905119885) |119885|2 119889119909

+ 120572intΩRe (119910119885) |119885|2 119889119909

minus ReintΩ119910119910119905 |119885|2 119889119909

minus 2intΩRe (119910119905119885)Re (119910119885) 119889119909

(81)


1198681 fl 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909

⩽ 12057216 1198852 + 119888 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172⩽ 12057216 11988521198671 + 1198881205881 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(82)


1198854 ⩽ 119888 11988534 119885141198671

119885 ⩽ 119888119873 1198851198671 (83)


1198682 fl 1205722 11988544 ⩽ 119888 1198853 1198851198671 ⩽ 119888119873311988541198671 (84)


119906V1199081198671 ⩽ 119888 1199061198671 V1198671 1199081198671 (85)


1198683 fl minusReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909


(86)




100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909 ⩽ 12057232 1198852 + 119888 1003817100381710038171003817119910100381710038171003817100381766⩽ 12057232 1198852 + 119862

(87)


119885infin ⩽ 119888 11988512 119885121198671

⩽ 119888radic119873 1198851198671 (88)



⩽ 119888119873 11988531198671 ⩽ 12057232 11988521198671 + 119888119873211988541198671

(89)


1198686 fl 120572intΩRe (119910119885) |119885|2 119889119909 ⩽ 120572 100381710038171003817100381711991010038171003817100381710038174 1003817100381710038171003817100381711988531003817100381710038171003817100381743

= 120572 100381710038171003817100381711991010038171003817100381710038174 11988534 ⩽ 119888 100381710038171003817100381711991010038171003817100381710038171198671 11988594 119885341198671

⩽ 1198881198739411988531198671 ⩽ 12057232 11988521198671 + 11988811987392

11988541198671 (90)



ΩRe (119910119905119885)Re (119910119885) 119889119909



(91)


Ψ2 (119885) = 7sum119894=1

119868119894⩽ (312057216 + 1198881radic119873)11988521198671 + 1198881198732

11988541198671 + 1198881205881+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(92)



Ψ2 (119885) ⩽ 1205724 11988521198671 + 119888119873211988541198671 + 1198881205881

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (93)



11988541198671 minus 119862⩾ 12 11988521198671 minus 1198881198732

11988541198671 minus 119862

(94)


2Ψ2 (119885) minus 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (95)


119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (96)


Ψ1 (119885 (119905)) ⩽ 1198881198732int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903+ 119862int119905

1199051

119890120572(119903minus119905)119889119903+ 119888int119905

1199051

119890120572(119903minus119905) 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172 119889119903⩽ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862120572+ 1198881198652 (119904 + 119905)

(97)


119885 (119905)41198671+ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862 (98)



120585 (119905) ⩽ 119888211987321205852 (119905) + 1198622

ie 11988821205852 (119905) minus 1198732120585 (119905) + 11986221198732 ⩾ 0(99)




Δ = 1198734 minus 4119888211986221198732 = 1198732 (1198732 minus 411988821198622) gt 0 (100)



21198882

1205722 = 1198732 + 119873radic1198732 minus 411988821198622

21198882 (101)



radic1198732 minus 411988821198622 ⩽ 119873(102)


120585 (119905) ⩽ 1205721or 120585 (119905) ⩾ 1205722

119905 ⩾ 1199051(103)





sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198672 ⩽ 119862 (119873 + 1) (105)





119885119909119905 + 120572119885119909 + 119876119873 (119860 minus 1198651015840 (V)) 119885119909 = 119866 119905 ⩾ 1199051 (106)



119866 = 119876119873 (119891119909 (119904 + 119905 sdot) + 1198651015840 (V) 119910119909) (107)



⟨120593 119891119909 (119904 + 119905)⟩ = int120593119891119909 (119904 + 119905)⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172)⩽ 119888 (1003817100381710038171003817120593100381710038171003817100381721198671 + 1198651 (119904 + 119905)) ⩽ 119888 (1 + 1205881)

(108)





⩽ 119888 V21198671 100381710038171003817100381711991011990910038171003817100381710038171198671 ⩽ 119888119873 100381710038171003817100381711991010038171003817100381710038171198671 ⩽ 1198621198731003817100381710038171003817minus2119894119876119873 Re (V (119905) 119910119909 (119905)) V (119905)10038171003817100381710038171198671 ⩽ 119888119873 V21198671 100381710038171003817100381711991010038171003817100381710038171198671

⩽ 119862119873(110)





⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891119905 (119904 + 119905)10038171003817100381710038172) (112)




(113)


119905 ⩾ 1199051 119904 ⩽ 0 (114)


119889119889119905Λ 1 (119885119909) + 2120572Λ 1 (119885119909) = 2Λ 2 (119885119909) (115)

where

Λ 1 (119885119909) = 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus int

Ω|V|2 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119885119909)2 119889119909

+ 2intΩIm (119866119885119909) 119889119909

(116)

Λ 2 (119885119909) = minusintΩRe (VV119905) 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119905119885119909) sdot Re (V119885119909) 119889119909

+ intΩIm (119866119905119885119909) 119889119909

+ 120572intΩIm (119866119885119909) 119889119909

(117)


Λ 2 (119885119909) ⩽ 3 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 Vinfin 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 1003817100381710038171003817119866119905

1003817100381710038171003817119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

+ 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩽ 119888radic119873 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 V1198671 1003817100381710038171003817119885119909100381710038171003817100381721198671

+ (10038171003817100381710038171198661199051003817100381710038171003817119867minus1 + 119866119867minus1) 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩽ ( 1198880radic119873 + 12057216) 1003817100381710038171003817119885119909

100381710038171003817100381721198671+ 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1)

(118)


Λ 2 (119885119909) ⩽ 1205724 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1) for 119873 ⩾ 1198731015840

1(119)


100381710038171003817100381721198671 minus 3 V2infin 1003817100381710038171003817119885119909100381710038171003817100381721198712

minus 2 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩾ 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 1198881198732

1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 2 119866119867minus1 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩾ (34 minus 11988811198732

) 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1

(120)


Λ 1 (119885119909) ⩾ 12 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1 for 119905 ⩾ 1199051 (121)


119889119889119905Λ 1 (119885119909) + 120572Λ 1 (119885119909) ⩽ 119888 (100381710038171003817100381711986611990510038171003817100381710038172119867minus1 + 1198662119867minus1)

⩽ 119888 (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172) (122)


1199051

119888119890120572(119903minus119905) (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172) 119889119903⩽ 119888 (1 + 119873 + 1198652 (119905))

(123)




sup119904⩽0



119908119905 + 120572119908 + 119894119908119909119909 minus 119894119908= 119894119876119873 ((|119906|2 + |V|2)119908 + 119906V119908) 119905 ⩾ 1199051 (125)


12 119889119889119905Υ1 (119908) + 120572Υ1 (119908) = Υ2 (119908) (126)

with

Υ1 (119908) = 11990821198671 minus intΩ(|119906|2 + |V|2) |119908|2 119889119909

minus ReintΩ119906V1199082119889119909

Υ2 (119908) = minus12 intΩ(|119906|2 + |V|2)

119905|119908|2 119889119909

minus 12ReintΩ (119906V)119905 1199082119889119909

(127)


Υ2 (119908) ⩽ 119888 (1003817100381710038171003817119906119905 + V1199051003817100381710038171003817119867minus1) (119906 + V1198671) 1199081198671 119908119871infin

⩽ 119888radic119873 11990821198671 (128)


) 11990821198671 ⩾ 12 11990821198671 (129)


119889119889119905Υ1 (119908) + 120572Υ1 (119908) ⩽ ( 1198882radic119873 minus 120572) 11990821198671 ⩽ 0 (130)


Υ1 (119908 (119905)) ⩽ 119890minus120572(119905minus1199051)Υ1 (119908 (1199051)) (131)






1198781 (119904 + 119905 119904) 1199060=

119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) if 0 ⩽ 119905 ⩽ 1199051 or 119904 gt 0V (119905 119904) = 119910 (119905 119904) + 119885 (119905 119904) for 119905 ⩾ 1199051 119904 ⩽ 0

(133)

1198782 (119904 + 119905 119904) 1199060=


(134)




A (119903) = 120596 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

119878 (119903 119903 minus 119905)K (119903)

= 1205961 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

1198781 (119903 119903 minus 119905)K (119903)119903 isin R

(135)




+ 119885 (120591119899 119903119899 minus 120591119899 1199060119899) (136)


1003817100381710038171003817119910 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198671⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119910 (119905 119904 1199060)10038171003817100381710038171198671 ⩽ 119862 (137)



1003817100381710038171003817119885 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198672⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119885 (119905 119904 1199060)10038171003817100381710038171198672 ⩽ 119862 (119873 + 1) (138)



10038171003817100381710038171198782 (119903 119903 minus 120591) 119906010038171003817100381710038171198671⩽ sup

119904⩽0







Acknowledgments











































Journal of












Journal of







Volume 2018






Volume 2018












sup119904⩽119905

int119904


sup119904⩽119905

int119904





119904minus120591119890120572(119903minus119903) 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172 119889119903

(40)






which yields

int119904


⩽ 8 1003817100381710038171003817119906010038171003817100381710038176 int119904


+ 11988811986531 (119905) int119904

119904minus120591119890120572(119903minus119904)119889119903

⩽ 8119890minus3120572120591 1003817100381710038171003817119906010038171003817100381710038176 int0

minus120591119890minus2120572119903119889119903 + 11988811986531 (119905) int119904


⩽ 4120572119890minus120572120591 1003817100381710038171003817119906010038171003817100381710038176 + 11988811986531 (119905)

(44)



as follows



sup119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)100381710038171003817100381721198671 ⩽ 1198721 (119905)fl 1198883 (1 + 11986531 (119905) + 1198652 (119905))

(45)





+ 3120572int119904


+ 1120572 int119904

119904minus120591119890120572(119903minus119904) 1003817100381710038171003817119891119903 (119903 sdot)10038171003817100381710038172 119889119903

(46)


Φ1 (119906 (119904 119904 minus 120591 1199060))⩽ 119890minus120572120591Φ1 (1199060) + 119888 (1 + 11986531 (119905) + 11986521 (119905) + 1198652 (119905))

(47)



(48)


119890minus120572120591Φ1 (1199060) ⩽ 2119890minus120572120591 (10038171003817100381710038171199060100381710038171003817100381721198671 + 1198651 (119905)) ⩽ 1 (49)


sup119904⩽119905

Φ1 (119906 (119904 119904 minus 120591 1199060)) ⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905)) (50)


Φ1 (119906) ⩾ 100381710038171003817100381711990611990910038171003817100381710038172 + 1199062 minus 1199063 (10038171003817100381710038171199061199091003817100381710038171003817 + 119906)+ 2 Imint

Ω119891119906119889119909




1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)100381710038171003817100381721198671⩽ 2 sup

119904⩽119905Φ1 (119906 (119904 119904 minus 120591 1199060))

+ 4 sup119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)10038171003817100381710038176+ 2 sup

119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)10038171003817100381710038174 + 4 sup119904⩽119905

1003817100381710038171003817119891 (119904 sdot)10038171003817100381710038172⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905)) + 4 (1 + 1198881198651 (119905))3

+ 2 (1 + 1198881198651 (119905))2 + 41198651 (119905)⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905))

(52)




K (119905) = 119908 isin 1198671 11990821198671 ⩽ 1198721 (119905)fl 1198883 (1 + 11986531 (119905) + 1198652 (119905)) 119905 isin R (53)











sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038171198671 ⩽ 1205883 (54)

sup119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119906119905 (119904 + 119905 119904 1199060)1003817100381710038171003817119867minus1 ⩽ 1205884 (55)


1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038172 ⩽ 119890minus120572(119903minus119904) 1003817100381710038171003817119906010038171003817100381710038172 + 11205721198653 (119903) (56)

Taking 119903 = 119904 + 119905 in (56) we have

1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038172 ⩽ 119890minus120572119905 10038171003817100381710038171199060100381710038171003817100381721198671 + 11205721198653 (119904 + 119905)⩽ 119890minus120572119905 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198881198651 (119905)

(57)



1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 ⩽ 8119890minus3120572(119903minus119904) 1003817100381710038171003817119906010038171003817100381710038176 + 11988812058831 119904 ⩽ 119903 ⩽ 119904 + 119905 (59)

which yields

int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 119889119903

⩽ 8119890minus120572119905 1003817100381710038171003817119906010038171003817100381710038176 int119904+119905

119904119890minus2120572(119903minus119904)119889119903

+ 11988812058831 int119904+119905

119904119890120572(119903minus119904minus119905)119889119903


(60)


int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 119889119903 ⩽ 1 + 11988812058831 (61)


int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038174 119889119903 ⩽ 1 + 11988812058821 (62)



+ 9120572int119904+119905

119904119890120572(119903minus119904minus119905) 119906 (119903 119904)6 119889119903

+ 3120572int119904+119905

119904119890120572(119903minus119904minus119905) 119906 (119903 119904)4

+ 1120572 int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119891119903 (119903 sdot)10038171003817100381710038172 119889119903

(63)


+ 1198881198652 (119904 + 119905) (64)


Φ1 (119906 (119903 119904 1199060)) ⩽ 2 11990621198671 + 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172⩽ 2 11990621198671 + 1198651 (119903) 119903 ⩾ 119904 (65)


⩽ 119890minus120572119905 (2 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198651 (0)) ⩽ 1 (66)




(68)


+ 2 1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038174+ 4 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172

⩽ 119888 (1 + 12058831 + 12058821 + 1198651 (119905))⩽ 119888 (1 + 12058831 + 12058821 + 1205881)

(69)




⩽ 100381710038171003817100381711990611990910038171003817100381710038172 + 100381710038171003817100381712059311990910038171003817100381710038172 + 11990644 + 1003817100381710038171003817120593100381710038171003817100381744+ 119888 (1199062 + 100381710038171003817100381712059310038171003817100381710038172) + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172

⩽ 119888 100381710038171003817100381711990611990910038171003817100381710038172 + 119862 + 1198651 (119905) ⩽ 119888 100381710038171003817100381711990611990910038171003817100381710038172 + 119862

(70)




119906 (119905) = sum119896isinZ

119906119896 (119905) 1198902119896120587119894119909 (71)


119875119873119906 (119905) = sum|119896|⩽119873

119906119896 (119905) 1198902119896120587119894119909119876119873119906 (119905) = sum

|119896|gt119873

119906119896 (119905) 1198902119896120587119894119909(72)


119910 = 119910 (119905) = 119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) 119911 = 119911 (119905) = 119911 (119905 119904) = 119876119873119906 (119904 + 119905 119904 1199060) (73)


sup119905⩾1199051

sup119904⩽0

(1003817100381710038171003817119910 (119905 119904)100381710038171003817100381721198671 + 119911 (119905 119904)21198671) ⩽ 119862 (74)


= 119876119873119891 (119904 + 119905) 119905 ⩾ 1199051119911 (1199051) = 119876119873119906 (119904 + 1199051 119904 1199060)

(75)


119885119905 + 120572119885 + 119894119885119909119909 minus 119894119885 + 119894119876119873 (1003816100381610038161003816119910 + 11988510038161003816100381610038162 (119910 + 119885))= 119876119873119891 (119904 + 119905) 119905 ⩾ 1199051

119885 (1199051) = 119885 (1199051 119904) = 0(76)






119875119898119876N1198671 = 119885119898 (119905) 119885119898 (119905) = sum119873lt|119896|⩽119898

119906119896 (119905) 1198902119896120587119894119909 119898 gt 119873

(78)




sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198671 ⩽ 119862 (79)



12 119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) = Ψ2 (119885) 119905 ⩾ 1199051 (80)

with

Ψ1 (119885) = 11988521198671 + 2 ImintΩ119891119885119889119909 minus 12 11988544

minus 2ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus 2Reint

Ω|119885|2 119885119910119889119909 minus 2int

Ω

100381610038161003816100381611991010038161003816100381610038162 |119885|2 119889119909minus 2int

Ω(Re (119910119885))2 119889119909

Ψ2 (119885) = 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909 + 1205722 11988544

minus ReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909

minus 120572ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus int

ΩRe (119910119905119885) |119885|2 119889119909

+ 120572intΩRe (119910119885) |119885|2 119889119909

minus ReintΩ119910119910119905 |119885|2 119889119909

minus 2intΩRe (119910119905119885)Re (119910119885) 119889119909

(81)


1198681 fl 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909

⩽ 12057216 1198852 + 119888 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172⩽ 12057216 11988521198671 + 1198881205881 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(82)


1198854 ⩽ 119888 11988534 119885141198671

119885 ⩽ 119888119873 1198851198671 (83)


1198682 fl 1205722 11988544 ⩽ 119888 1198853 1198851198671 ⩽ 119888119873311988541198671 (84)


119906V1199081198671 ⩽ 119888 1199061198671 V1198671 1199081198671 (85)


1198683 fl minusReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909


(86)




100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909 ⩽ 12057232 1198852 + 119888 1003817100381710038171003817119910100381710038171003817100381766⩽ 12057232 1198852 + 119862

(87)


119885infin ⩽ 119888 11988512 119885121198671

⩽ 119888radic119873 1198851198671 (88)



⩽ 119888119873 11988531198671 ⩽ 12057232 11988521198671 + 119888119873211988541198671

(89)


1198686 fl 120572intΩRe (119910119885) |119885|2 119889119909 ⩽ 120572 100381710038171003817100381711991010038171003817100381710038174 1003817100381710038171003817100381711988531003817100381710038171003817100381743

= 120572 100381710038171003817100381711991010038171003817100381710038174 11988534 ⩽ 119888 100381710038171003817100381711991010038171003817100381710038171198671 11988594 119885341198671

⩽ 1198881198739411988531198671 ⩽ 12057232 11988521198671 + 11988811987392

11988541198671 (90)



ΩRe (119910119905119885)Re (119910119885) 119889119909



(91)


Ψ2 (119885) = 7sum119894=1

119868119894⩽ (312057216 + 1198881radic119873)11988521198671 + 1198881198732

11988541198671 + 1198881205881+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(92)



Ψ2 (119885) ⩽ 1205724 11988521198671 + 119888119873211988541198671 + 1198881205881

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (93)



11988541198671 minus 119862⩾ 12 11988521198671 minus 1198881198732

11988541198671 minus 119862

(94)


2Ψ2 (119885) minus 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (95)


119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (96)


Ψ1 (119885 (119905)) ⩽ 1198881198732int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903+ 119862int119905

1199051

119890120572(119903minus119905)119889119903+ 119888int119905

1199051

119890120572(119903minus119905) 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172 119889119903⩽ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862120572+ 1198881198652 (119904 + 119905)

(97)


119885 (119905)41198671+ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862 (98)



120585 (119905) ⩽ 119888211987321205852 (119905) + 1198622

ie 11988821205852 (119905) minus 1198732120585 (119905) + 11986221198732 ⩾ 0(99)




Δ = 1198734 minus 4119888211986221198732 = 1198732 (1198732 minus 411988821198622) gt 0 (100)



21198882

1205722 = 1198732 + 119873radic1198732 minus 411988821198622

21198882 (101)



radic1198732 minus 411988821198622 ⩽ 119873(102)


120585 (119905) ⩽ 1205721or 120585 (119905) ⩾ 1205722

119905 ⩾ 1199051(103)





sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198672 ⩽ 119862 (119873 + 1) (105)





119885119909119905 + 120572119885119909 + 119876119873 (119860 minus 1198651015840 (V)) 119885119909 = 119866 119905 ⩾ 1199051 (106)



119866 = 119876119873 (119891119909 (119904 + 119905 sdot) + 1198651015840 (V) 119910119909) (107)



⟨120593 119891119909 (119904 + 119905)⟩ = int120593119891119909 (119904 + 119905)⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172)⩽ 119888 (1003817100381710038171003817120593100381710038171003817100381721198671 + 1198651 (119904 + 119905)) ⩽ 119888 (1 + 1205881)

(108)





⩽ 119888 V21198671 100381710038171003817100381711991011990910038171003817100381710038171198671 ⩽ 119888119873 100381710038171003817100381711991010038171003817100381710038171198671 ⩽ 1198621198731003817100381710038171003817minus2119894119876119873 Re (V (119905) 119910119909 (119905)) V (119905)10038171003817100381710038171198671 ⩽ 119888119873 V21198671 100381710038171003817100381711991010038171003817100381710038171198671

⩽ 119862119873(110)





⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891119905 (119904 + 119905)10038171003817100381710038172) (112)




(113)


119905 ⩾ 1199051 119904 ⩽ 0 (114)


119889119889119905Λ 1 (119885119909) + 2120572Λ 1 (119885119909) = 2Λ 2 (119885119909) (115)

where

Λ 1 (119885119909) = 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus int

Ω|V|2 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119885119909)2 119889119909

+ 2intΩIm (119866119885119909) 119889119909

(116)

Λ 2 (119885119909) = minusintΩRe (VV119905) 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119905119885119909) sdot Re (V119885119909) 119889119909

+ intΩIm (119866119905119885119909) 119889119909

+ 120572intΩIm (119866119885119909) 119889119909

(117)


Λ 2 (119885119909) ⩽ 3 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 Vinfin 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 1003817100381710038171003817119866119905

1003817100381710038171003817119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

+ 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩽ 119888radic119873 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 V1198671 1003817100381710038171003817119885119909100381710038171003817100381721198671

+ (10038171003817100381710038171198661199051003817100381710038171003817119867minus1 + 119866119867minus1) 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩽ ( 1198880radic119873 + 12057216) 1003817100381710038171003817119885119909

100381710038171003817100381721198671+ 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1)

(118)


Λ 2 (119885119909) ⩽ 1205724 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1) for 119873 ⩾ 1198731015840

1(119)


100381710038171003817100381721198671 minus 3 V2infin 1003817100381710038171003817119885119909100381710038171003817100381721198712

minus 2 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩾ 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 1198881198732

1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 2 119866119867minus1 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩾ (34 minus 11988811198732

) 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1

(120)


Λ 1 (119885119909) ⩾ 12 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1 for 119905 ⩾ 1199051 (121)


119889119889119905Λ 1 (119885119909) + 120572Λ 1 (119885119909) ⩽ 119888 (100381710038171003817100381711986611990510038171003817100381710038172119867minus1 + 1198662119867minus1)

⩽ 119888 (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172) (122)


1199051

119888119890120572(119903minus119905) (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172) 119889119903⩽ 119888 (1 + 119873 + 1198652 (119905))

(123)




sup119904⩽0



119908119905 + 120572119908 + 119894119908119909119909 minus 119894119908= 119894119876119873 ((|119906|2 + |V|2)119908 + 119906V119908) 119905 ⩾ 1199051 (125)


12 119889119889119905Υ1 (119908) + 120572Υ1 (119908) = Υ2 (119908) (126)

with

Υ1 (119908) = 11990821198671 minus intΩ(|119906|2 + |V|2) |119908|2 119889119909

minus ReintΩ119906V1199082119889119909

Υ2 (119908) = minus12 intΩ(|119906|2 + |V|2)

119905|119908|2 119889119909

minus 12ReintΩ (119906V)119905 1199082119889119909

(127)


Υ2 (119908) ⩽ 119888 (1003817100381710038171003817119906119905 + V1199051003817100381710038171003817119867minus1) (119906 + V1198671) 1199081198671 119908119871infin

⩽ 119888radic119873 11990821198671 (128)


) 11990821198671 ⩾ 12 11990821198671 (129)


119889119889119905Υ1 (119908) + 120572Υ1 (119908) ⩽ ( 1198882radic119873 minus 120572) 11990821198671 ⩽ 0 (130)


Υ1 (119908 (119905)) ⩽ 119890minus120572(119905minus1199051)Υ1 (119908 (1199051)) (131)






1198781 (119904 + 119905 119904) 1199060=

119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) if 0 ⩽ 119905 ⩽ 1199051 or 119904 gt 0V (119905 119904) = 119910 (119905 119904) + 119885 (119905 119904) for 119905 ⩾ 1199051 119904 ⩽ 0

(133)

1198782 (119904 + 119905 119904) 1199060=


(134)




A (119903) = 120596 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

119878 (119903 119903 minus 119905)K (119903)

= 1205961 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

1198781 (119903 119903 minus 119905)K (119903)119903 isin R

(135)




+ 119885 (120591119899 119903119899 minus 120591119899 1199060119899) (136)


1003817100381710038171003817119910 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198671⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119910 (119905 119904 1199060)10038171003817100381710038171198671 ⩽ 119862 (137)



1003817100381710038171003817119885 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198672⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119885 (119905 119904 1199060)10038171003817100381710038171198672 ⩽ 119862 (119873 + 1) (138)



10038171003817100381710038171198782 (119903 119903 minus 120591) 119906010038171003817100381710038171198671⩽ sup

119904⩽0







Acknowledgments











































Journal of












Journal of







Volume 2018






Volume 2018










1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)100381710038171003817100381721198671⩽ 2 sup

119904⩽119905Φ1 (119906 (119904 119904 minus 120591 1199060))

+ 4 sup119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)10038171003817100381710038176+ 2 sup

119904⩽119905

1003817100381710038171003817119906 (119904 119904 minus 120591 1199060)10038171003817100381710038174 + 4 sup119904⩽119905

1003817100381710038171003817119891 (119904 sdot)10038171003817100381710038172⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905)) + 4 (1 + 1198881198651 (119905))3

+ 2 (1 + 1198881198651 (119905))2 + 41198651 (119905)⩽ 119888 (1 + 11986531 (119905) + 1198652 (119905))

(52)




K (119905) = 119908 isin 1198671 11990821198671 ⩽ 1198721 (119905)fl 1198883 (1 + 11986531 (119905) + 1198652 (119905)) 119905 isin R (53)











sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038171198671 ⩽ 1205883 (54)

sup119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119906119905 (119904 + 119905 119904 1199060)1003817100381710038171003817119867minus1 ⩽ 1205884 (55)


1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038172 ⩽ 119890minus120572(119903minus119904) 1003817100381710038171003817119906010038171003817100381710038172 + 11205721198653 (119903) (56)

Taking 119903 = 119904 + 119905 in (56) we have

1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038172 ⩽ 119890minus120572119905 10038171003817100381710038171199060100381710038171003817100381721198671 + 11205721198653 (119904 + 119905)⩽ 119890minus120572119905 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198881198651 (119905)

(57)



1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 ⩽ 8119890minus3120572(119903minus119904) 1003817100381710038171003817119906010038171003817100381710038176 + 11988812058831 119904 ⩽ 119903 ⩽ 119904 + 119905 (59)

which yields

int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 119889119903

⩽ 8119890minus120572119905 1003817100381710038171003817119906010038171003817100381710038176 int119904+119905

119904119890minus2120572(119903minus119904)119889119903

+ 11988812058831 int119904+119905

119904119890120572(119903minus119904minus119905)119889119903


(60)


int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038176 119889119903 ⩽ 1 + 11988812058831 (61)


int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119906 (119903 119904 1199060)10038171003817100381710038174 119889119903 ⩽ 1 + 11988812058821 (62)



+ 9120572int119904+119905

119904119890120572(119903minus119904minus119905) 119906 (119903 119904)6 119889119903

+ 3120572int119904+119905

119904119890120572(119903minus119904minus119905) 119906 (119903 119904)4

+ 1120572 int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119891119903 (119903 sdot)10038171003817100381710038172 119889119903

(63)


+ 1198881198652 (119904 + 119905) (64)


Φ1 (119906 (119903 119904 1199060)) ⩽ 2 11990621198671 + 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172⩽ 2 11990621198671 + 1198651 (119903) 119903 ⩾ 119904 (65)


⩽ 119890minus120572119905 (2 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198651 (0)) ⩽ 1 (66)




(68)


+ 2 1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038174+ 4 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172

⩽ 119888 (1 + 12058831 + 12058821 + 1198651 (119905))⩽ 119888 (1 + 12058831 + 12058821 + 1205881)

(69)




⩽ 100381710038171003817100381711990611990910038171003817100381710038172 + 100381710038171003817100381712059311990910038171003817100381710038172 + 11990644 + 1003817100381710038171003817120593100381710038171003817100381744+ 119888 (1199062 + 100381710038171003817100381712059310038171003817100381710038172) + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172

⩽ 119888 100381710038171003817100381711990611990910038171003817100381710038172 + 119862 + 1198651 (119905) ⩽ 119888 100381710038171003817100381711990611990910038171003817100381710038172 + 119862

(70)




119906 (119905) = sum119896isinZ

119906119896 (119905) 1198902119896120587119894119909 (71)


119875119873119906 (119905) = sum|119896|⩽119873

119906119896 (119905) 1198902119896120587119894119909119876119873119906 (119905) = sum

|119896|gt119873

119906119896 (119905) 1198902119896120587119894119909(72)


119910 = 119910 (119905) = 119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) 119911 = 119911 (119905) = 119911 (119905 119904) = 119876119873119906 (119904 + 119905 119904 1199060) (73)


sup119905⩾1199051

sup119904⩽0

(1003817100381710038171003817119910 (119905 119904)100381710038171003817100381721198671 + 119911 (119905 119904)21198671) ⩽ 119862 (74)


= 119876119873119891 (119904 + 119905) 119905 ⩾ 1199051119911 (1199051) = 119876119873119906 (119904 + 1199051 119904 1199060)

(75)


119885119905 + 120572119885 + 119894119885119909119909 minus 119894119885 + 119894119876119873 (1003816100381610038161003816119910 + 11988510038161003816100381610038162 (119910 + 119885))= 119876119873119891 (119904 + 119905) 119905 ⩾ 1199051

119885 (1199051) = 119885 (1199051 119904) = 0(76)






119875119898119876N1198671 = 119885119898 (119905) 119885119898 (119905) = sum119873lt|119896|⩽119898

119906119896 (119905) 1198902119896120587119894119909 119898 gt 119873

(78)




sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198671 ⩽ 119862 (79)



12 119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) = Ψ2 (119885) 119905 ⩾ 1199051 (80)

with

Ψ1 (119885) = 11988521198671 + 2 ImintΩ119891119885119889119909 minus 12 11988544

minus 2ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus 2Reint

Ω|119885|2 119885119910119889119909 minus 2int

Ω

100381610038161003816100381611991010038161003816100381610038162 |119885|2 119889119909minus 2int

Ω(Re (119910119885))2 119889119909

Ψ2 (119885) = 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909 + 1205722 11988544

minus ReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909

minus 120572ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus int

ΩRe (119910119905119885) |119885|2 119889119909

+ 120572intΩRe (119910119885) |119885|2 119889119909

minus ReintΩ119910119910119905 |119885|2 119889119909

minus 2intΩRe (119910119905119885)Re (119910119885) 119889119909

(81)


1198681 fl 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909

⩽ 12057216 1198852 + 119888 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172⩽ 12057216 11988521198671 + 1198881205881 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(82)


1198854 ⩽ 119888 11988534 119885141198671

119885 ⩽ 119888119873 1198851198671 (83)


1198682 fl 1205722 11988544 ⩽ 119888 1198853 1198851198671 ⩽ 119888119873311988541198671 (84)


119906V1199081198671 ⩽ 119888 1199061198671 V1198671 1199081198671 (85)


1198683 fl minusReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909


(86)




100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909 ⩽ 12057232 1198852 + 119888 1003817100381710038171003817119910100381710038171003817100381766⩽ 12057232 1198852 + 119862

(87)


119885infin ⩽ 119888 11988512 119885121198671

⩽ 119888radic119873 1198851198671 (88)



⩽ 119888119873 11988531198671 ⩽ 12057232 11988521198671 + 119888119873211988541198671

(89)


1198686 fl 120572intΩRe (119910119885) |119885|2 119889119909 ⩽ 120572 100381710038171003817100381711991010038171003817100381710038174 1003817100381710038171003817100381711988531003817100381710038171003817100381743

= 120572 100381710038171003817100381711991010038171003817100381710038174 11988534 ⩽ 119888 100381710038171003817100381711991010038171003817100381710038171198671 11988594 119885341198671

⩽ 1198881198739411988531198671 ⩽ 12057232 11988521198671 + 11988811987392

11988541198671 (90)



ΩRe (119910119905119885)Re (119910119885) 119889119909



(91)


Ψ2 (119885) = 7sum119894=1

119868119894⩽ (312057216 + 1198881radic119873)11988521198671 + 1198881198732

11988541198671 + 1198881205881+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(92)



Ψ2 (119885) ⩽ 1205724 11988521198671 + 119888119873211988541198671 + 1198881205881

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (93)



11988541198671 minus 119862⩾ 12 11988521198671 minus 1198881198732

11988541198671 minus 119862

(94)


2Ψ2 (119885) minus 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (95)


119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (96)


Ψ1 (119885 (119905)) ⩽ 1198881198732int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903+ 119862int119905

1199051

119890120572(119903minus119905)119889119903+ 119888int119905

1199051

119890120572(119903minus119905) 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172 119889119903⩽ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862120572+ 1198881198652 (119904 + 119905)

(97)


119885 (119905)41198671+ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862 (98)



120585 (119905) ⩽ 119888211987321205852 (119905) + 1198622

ie 11988821205852 (119905) minus 1198732120585 (119905) + 11986221198732 ⩾ 0(99)




Δ = 1198734 minus 4119888211986221198732 = 1198732 (1198732 minus 411988821198622) gt 0 (100)



21198882

1205722 = 1198732 + 119873radic1198732 minus 411988821198622

21198882 (101)



radic1198732 minus 411988821198622 ⩽ 119873(102)


120585 (119905) ⩽ 1205721or 120585 (119905) ⩾ 1205722

119905 ⩾ 1199051(103)





sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198672 ⩽ 119862 (119873 + 1) (105)





119885119909119905 + 120572119885119909 + 119876119873 (119860 minus 1198651015840 (V)) 119885119909 = 119866 119905 ⩾ 1199051 (106)



119866 = 119876119873 (119891119909 (119904 + 119905 sdot) + 1198651015840 (V) 119910119909) (107)



⟨120593 119891119909 (119904 + 119905)⟩ = int120593119891119909 (119904 + 119905)⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172)⩽ 119888 (1003817100381710038171003817120593100381710038171003817100381721198671 + 1198651 (119904 + 119905)) ⩽ 119888 (1 + 1205881)

(108)





⩽ 119888 V21198671 100381710038171003817100381711991011990910038171003817100381710038171198671 ⩽ 119888119873 100381710038171003817100381711991010038171003817100381710038171198671 ⩽ 1198621198731003817100381710038171003817minus2119894119876119873 Re (V (119905) 119910119909 (119905)) V (119905)10038171003817100381710038171198671 ⩽ 119888119873 V21198671 100381710038171003817100381711991010038171003817100381710038171198671

⩽ 119862119873(110)





⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891119905 (119904 + 119905)10038171003817100381710038172) (112)




(113)


119905 ⩾ 1199051 119904 ⩽ 0 (114)


119889119889119905Λ 1 (119885119909) + 2120572Λ 1 (119885119909) = 2Λ 2 (119885119909) (115)

where

Λ 1 (119885119909) = 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus int

Ω|V|2 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119885119909)2 119889119909

+ 2intΩIm (119866119885119909) 119889119909

(116)

Λ 2 (119885119909) = minusintΩRe (VV119905) 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119905119885119909) sdot Re (V119885119909) 119889119909

+ intΩIm (119866119905119885119909) 119889119909

+ 120572intΩIm (119866119885119909) 119889119909

(117)


Λ 2 (119885119909) ⩽ 3 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 Vinfin 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 1003817100381710038171003817119866119905

1003817100381710038171003817119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

+ 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩽ 119888radic119873 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 V1198671 1003817100381710038171003817119885119909100381710038171003817100381721198671

+ (10038171003817100381710038171198661199051003817100381710038171003817119867minus1 + 119866119867minus1) 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩽ ( 1198880radic119873 + 12057216) 1003817100381710038171003817119885119909

100381710038171003817100381721198671+ 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1)

(118)


Λ 2 (119885119909) ⩽ 1205724 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1) for 119873 ⩾ 1198731015840

1(119)


100381710038171003817100381721198671 minus 3 V2infin 1003817100381710038171003817119885119909100381710038171003817100381721198712

minus 2 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩾ 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 1198881198732

1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 2 119866119867minus1 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩾ (34 minus 11988811198732

) 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1

(120)


Λ 1 (119885119909) ⩾ 12 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1 for 119905 ⩾ 1199051 (121)


119889119889119905Λ 1 (119885119909) + 120572Λ 1 (119885119909) ⩽ 119888 (100381710038171003817100381711986611990510038171003817100381710038172119867minus1 + 1198662119867minus1)

⩽ 119888 (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172) (122)


1199051

119888119890120572(119903minus119905) (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172) 119889119903⩽ 119888 (1 + 119873 + 1198652 (119905))

(123)




sup119904⩽0



119908119905 + 120572119908 + 119894119908119909119909 minus 119894119908= 119894119876119873 ((|119906|2 + |V|2)119908 + 119906V119908) 119905 ⩾ 1199051 (125)


12 119889119889119905Υ1 (119908) + 120572Υ1 (119908) = Υ2 (119908) (126)

with

Υ1 (119908) = 11990821198671 minus intΩ(|119906|2 + |V|2) |119908|2 119889119909

minus ReintΩ119906V1199082119889119909

Υ2 (119908) = minus12 intΩ(|119906|2 + |V|2)

119905|119908|2 119889119909

minus 12ReintΩ (119906V)119905 1199082119889119909

(127)


Υ2 (119908) ⩽ 119888 (1003817100381710038171003817119906119905 + V1199051003817100381710038171003817119867minus1) (119906 + V1198671) 1199081198671 119908119871infin

⩽ 119888radic119873 11990821198671 (128)


) 11990821198671 ⩾ 12 11990821198671 (129)


119889119889119905Υ1 (119908) + 120572Υ1 (119908) ⩽ ( 1198882radic119873 minus 120572) 11990821198671 ⩽ 0 (130)


Υ1 (119908 (119905)) ⩽ 119890minus120572(119905minus1199051)Υ1 (119908 (1199051)) (131)






1198781 (119904 + 119905 119904) 1199060=

119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) if 0 ⩽ 119905 ⩽ 1199051 or 119904 gt 0V (119905 119904) = 119910 (119905 119904) + 119885 (119905 119904) for 119905 ⩾ 1199051 119904 ⩽ 0

(133)

1198782 (119904 + 119905 119904) 1199060=


(134)




A (119903) = 120596 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

119878 (119903 119903 minus 119905)K (119903)

= 1205961 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

1198781 (119903 119903 minus 119905)K (119903)119903 isin R

(135)




+ 119885 (120591119899 119903119899 minus 120591119899 1199060119899) (136)


1003817100381710038171003817119910 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198671⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119910 (119905 119904 1199060)10038171003817100381710038171198671 ⩽ 119862 (137)



1003817100381710038171003817119885 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198672⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119885 (119905 119904 1199060)10038171003817100381710038171198672 ⩽ 119862 (119873 + 1) (138)



10038171003817100381710038171198782 (119903 119903 minus 120591) 119906010038171003817100381710038171198671⩽ sup

119904⩽0







Acknowledgments











































Journal of












Journal of







Volume 2018






Volume 2018










+ 9120572int119904+119905

119904119890120572(119903minus119904minus119905) 119906 (119903 119904)6 119889119903

+ 3120572int119904+119905

119904119890120572(119903minus119904minus119905) 119906 (119903 119904)4

+ 1120572 int119904+119905

119904119890120572(119903minus119904minus119905) 1003817100381710038171003817119891119903 (119903 sdot)10038171003817100381710038172 119889119903

(63)


+ 1198881198652 (119904 + 119905) (64)


Φ1 (119906 (119903 119904 1199060)) ⩽ 2 11990621198671 + 1003817100381710038171003817119891 (119903 sdot)10038171003817100381710038172⩽ 2 11990621198671 + 1198651 (119903) 119903 ⩾ 119904 (65)


⩽ 119890minus120572119905 (2 10038171003817100381710038171199060100381710038171003817100381721198671 + 1198651 (0)) ⩽ 1 (66)




(68)


+ 2 1003817100381710038171003817119906 (119904 + 119905 119904 1199060)10038171003817100381710038174+ 4 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172

⩽ 119888 (1 + 12058831 + 12058821 + 1198651 (119905))⩽ 119888 (1 + 12058831 + 12058821 + 1205881)

(69)




⩽ 100381710038171003817100381711990611990910038171003817100381710038172 + 100381710038171003817100381712059311990910038171003817100381710038172 + 11990644 + 1003817100381710038171003817120593100381710038171003817100381744+ 119888 (1199062 + 100381710038171003817100381712059310038171003817100381710038172) + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172

⩽ 119888 100381710038171003817100381711990611990910038171003817100381710038172 + 119862 + 1198651 (119905) ⩽ 119888 100381710038171003817100381711990611990910038171003817100381710038172 + 119862

(70)




119906 (119905) = sum119896isinZ

119906119896 (119905) 1198902119896120587119894119909 (71)


119875119873119906 (119905) = sum|119896|⩽119873

119906119896 (119905) 1198902119896120587119894119909119876119873119906 (119905) = sum

|119896|gt119873

119906119896 (119905) 1198902119896120587119894119909(72)


119910 = 119910 (119905) = 119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) 119911 = 119911 (119905) = 119911 (119905 119904) = 119876119873119906 (119904 + 119905 119904 1199060) (73)


sup119905⩾1199051

sup119904⩽0

(1003817100381710038171003817119910 (119905 119904)100381710038171003817100381721198671 + 119911 (119905 119904)21198671) ⩽ 119862 (74)


= 119876119873119891 (119904 + 119905) 119905 ⩾ 1199051119911 (1199051) = 119876119873119906 (119904 + 1199051 119904 1199060)

(75)


119885119905 + 120572119885 + 119894119885119909119909 minus 119894119885 + 119894119876119873 (1003816100381610038161003816119910 + 11988510038161003816100381610038162 (119910 + 119885))= 119876119873119891 (119904 + 119905) 119905 ⩾ 1199051

119885 (1199051) = 119885 (1199051 119904) = 0(76)






119875119898119876N1198671 = 119885119898 (119905) 119885119898 (119905) = sum119873lt|119896|⩽119898

119906119896 (119905) 1198902119896120587119894119909 119898 gt 119873

(78)




sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198671 ⩽ 119862 (79)



12 119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) = Ψ2 (119885) 119905 ⩾ 1199051 (80)

with

Ψ1 (119885) = 11988521198671 + 2 ImintΩ119891119885119889119909 minus 12 11988544

minus 2ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus 2Reint

Ω|119885|2 119885119910119889119909 minus 2int

Ω

100381610038161003816100381611991010038161003816100381610038162 |119885|2 119889119909minus 2int

Ω(Re (119910119885))2 119889119909

Ψ2 (119885) = 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909 + 1205722 11988544

minus ReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909

minus 120572ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus int

ΩRe (119910119905119885) |119885|2 119889119909

+ 120572intΩRe (119910119885) |119885|2 119889119909

minus ReintΩ119910119910119905 |119885|2 119889119909

minus 2intΩRe (119910119905119885)Re (119910119885) 119889119909

(81)


1198681 fl 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909

⩽ 12057216 1198852 + 119888 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172⩽ 12057216 11988521198671 + 1198881205881 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(82)


1198854 ⩽ 119888 11988534 119885141198671

119885 ⩽ 119888119873 1198851198671 (83)


1198682 fl 1205722 11988544 ⩽ 119888 1198853 1198851198671 ⩽ 119888119873311988541198671 (84)


119906V1199081198671 ⩽ 119888 1199061198671 V1198671 1199081198671 (85)


1198683 fl minusReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909


(86)




100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909 ⩽ 12057232 1198852 + 119888 1003817100381710038171003817119910100381710038171003817100381766⩽ 12057232 1198852 + 119862

(87)


119885infin ⩽ 119888 11988512 119885121198671

⩽ 119888radic119873 1198851198671 (88)



⩽ 119888119873 11988531198671 ⩽ 12057232 11988521198671 + 119888119873211988541198671

(89)


1198686 fl 120572intΩRe (119910119885) |119885|2 119889119909 ⩽ 120572 100381710038171003817100381711991010038171003817100381710038174 1003817100381710038171003817100381711988531003817100381710038171003817100381743

= 120572 100381710038171003817100381711991010038171003817100381710038174 11988534 ⩽ 119888 100381710038171003817100381711991010038171003817100381710038171198671 11988594 119885341198671

⩽ 1198881198739411988531198671 ⩽ 12057232 11988521198671 + 11988811987392

11988541198671 (90)



ΩRe (119910119905119885)Re (119910119885) 119889119909



(91)


Ψ2 (119885) = 7sum119894=1

119868119894⩽ (312057216 + 1198881radic119873)11988521198671 + 1198881198732

11988541198671 + 1198881205881+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(92)



Ψ2 (119885) ⩽ 1205724 11988521198671 + 119888119873211988541198671 + 1198881205881

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (93)



11988541198671 minus 119862⩾ 12 11988521198671 minus 1198881198732

11988541198671 minus 119862

(94)


2Ψ2 (119885) minus 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (95)


119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (96)


Ψ1 (119885 (119905)) ⩽ 1198881198732int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903+ 119862int119905

1199051

119890120572(119903minus119905)119889119903+ 119888int119905

1199051

119890120572(119903minus119905) 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172 119889119903⩽ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862120572+ 1198881198652 (119904 + 119905)

(97)


119885 (119905)41198671+ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862 (98)



120585 (119905) ⩽ 119888211987321205852 (119905) + 1198622

ie 11988821205852 (119905) minus 1198732120585 (119905) + 11986221198732 ⩾ 0(99)




Δ = 1198734 minus 4119888211986221198732 = 1198732 (1198732 minus 411988821198622) gt 0 (100)



21198882

1205722 = 1198732 + 119873radic1198732 minus 411988821198622

21198882 (101)



radic1198732 minus 411988821198622 ⩽ 119873(102)


120585 (119905) ⩽ 1205721or 120585 (119905) ⩾ 1205722

119905 ⩾ 1199051(103)





sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198672 ⩽ 119862 (119873 + 1) (105)





119885119909119905 + 120572119885119909 + 119876119873 (119860 minus 1198651015840 (V)) 119885119909 = 119866 119905 ⩾ 1199051 (106)



119866 = 119876119873 (119891119909 (119904 + 119905 sdot) + 1198651015840 (V) 119910119909) (107)



⟨120593 119891119909 (119904 + 119905)⟩ = int120593119891119909 (119904 + 119905)⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172)⩽ 119888 (1003817100381710038171003817120593100381710038171003817100381721198671 + 1198651 (119904 + 119905)) ⩽ 119888 (1 + 1205881)

(108)





⩽ 119888 V21198671 100381710038171003817100381711991011990910038171003817100381710038171198671 ⩽ 119888119873 100381710038171003817100381711991010038171003817100381710038171198671 ⩽ 1198621198731003817100381710038171003817minus2119894119876119873 Re (V (119905) 119910119909 (119905)) V (119905)10038171003817100381710038171198671 ⩽ 119888119873 V21198671 100381710038171003817100381711991010038171003817100381710038171198671

⩽ 119862119873(110)





⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891119905 (119904 + 119905)10038171003817100381710038172) (112)




(113)


119905 ⩾ 1199051 119904 ⩽ 0 (114)


119889119889119905Λ 1 (119885119909) + 2120572Λ 1 (119885119909) = 2Λ 2 (119885119909) (115)

where

Λ 1 (119885119909) = 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus int

Ω|V|2 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119885119909)2 119889119909

+ 2intΩIm (119866119885119909) 119889119909

(116)

Λ 2 (119885119909) = minusintΩRe (VV119905) 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119905119885119909) sdot Re (V119885119909) 119889119909

+ intΩIm (119866119905119885119909) 119889119909

+ 120572intΩIm (119866119885119909) 119889119909

(117)


Λ 2 (119885119909) ⩽ 3 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 Vinfin 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 1003817100381710038171003817119866119905

1003817100381710038171003817119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

+ 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩽ 119888radic119873 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 V1198671 1003817100381710038171003817119885119909100381710038171003817100381721198671

+ (10038171003817100381710038171198661199051003817100381710038171003817119867minus1 + 119866119867minus1) 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩽ ( 1198880radic119873 + 12057216) 1003817100381710038171003817119885119909

100381710038171003817100381721198671+ 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1)

(118)


Λ 2 (119885119909) ⩽ 1205724 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1) for 119873 ⩾ 1198731015840

1(119)


100381710038171003817100381721198671 minus 3 V2infin 1003817100381710038171003817119885119909100381710038171003817100381721198712

minus 2 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩾ 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 1198881198732

1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 2 119866119867minus1 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩾ (34 minus 11988811198732

) 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1

(120)


Λ 1 (119885119909) ⩾ 12 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1 for 119905 ⩾ 1199051 (121)


119889119889119905Λ 1 (119885119909) + 120572Λ 1 (119885119909) ⩽ 119888 (100381710038171003817100381711986611990510038171003817100381710038172119867minus1 + 1198662119867minus1)

⩽ 119888 (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172) (122)


1199051

119888119890120572(119903minus119905) (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172) 119889119903⩽ 119888 (1 + 119873 + 1198652 (119905))

(123)




sup119904⩽0



119908119905 + 120572119908 + 119894119908119909119909 minus 119894119908= 119894119876119873 ((|119906|2 + |V|2)119908 + 119906V119908) 119905 ⩾ 1199051 (125)


12 119889119889119905Υ1 (119908) + 120572Υ1 (119908) = Υ2 (119908) (126)

with

Υ1 (119908) = 11990821198671 minus intΩ(|119906|2 + |V|2) |119908|2 119889119909

minus ReintΩ119906V1199082119889119909

Υ2 (119908) = minus12 intΩ(|119906|2 + |V|2)

119905|119908|2 119889119909

minus 12ReintΩ (119906V)119905 1199082119889119909

(127)


Υ2 (119908) ⩽ 119888 (1003817100381710038171003817119906119905 + V1199051003817100381710038171003817119867minus1) (119906 + V1198671) 1199081198671 119908119871infin

⩽ 119888radic119873 11990821198671 (128)


) 11990821198671 ⩾ 12 11990821198671 (129)


119889119889119905Υ1 (119908) + 120572Υ1 (119908) ⩽ ( 1198882radic119873 minus 120572) 11990821198671 ⩽ 0 (130)


Υ1 (119908 (119905)) ⩽ 119890minus120572(119905minus1199051)Υ1 (119908 (1199051)) (131)






1198781 (119904 + 119905 119904) 1199060=

119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) if 0 ⩽ 119905 ⩽ 1199051 or 119904 gt 0V (119905 119904) = 119910 (119905 119904) + 119885 (119905 119904) for 119905 ⩾ 1199051 119904 ⩽ 0

(133)

1198782 (119904 + 119905 119904) 1199060=


(134)




A (119903) = 120596 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

119878 (119903 119903 minus 119905)K (119903)

= 1205961 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

1198781 (119903 119903 minus 119905)K (119903)119903 isin R

(135)




+ 119885 (120591119899 119903119899 minus 120591119899 1199060119899) (136)


1003817100381710038171003817119910 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198671⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119910 (119905 119904 1199060)10038171003817100381710038171198671 ⩽ 119862 (137)



1003817100381710038171003817119885 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198672⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119885 (119905 119904 1199060)10038171003817100381710038171198672 ⩽ 119862 (119873 + 1) (138)



10038171003817100381710038171198782 (119903 119903 minus 120591) 119906010038171003817100381710038171198671⩽ sup

119904⩽0







Acknowledgments











































Journal of












Journal of







Volume 2018






Volume 2018













119875119898119876N1198671 = 119885119898 (119905) 119885119898 (119905) = sum119873lt|119896|⩽119898

119906119896 (119905) 1198902119896120587119894119909 119898 gt 119873

(78)




sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198671 ⩽ 119862 (79)



12 119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) = Ψ2 (119885) 119905 ⩾ 1199051 (80)

with

Ψ1 (119885) = 11988521198671 + 2 ImintΩ119891119885119889119909 minus 12 11988544

minus 2ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus 2Reint

Ω|119885|2 119885119910119889119909 minus 2int

Ω

100381610038161003816100381611991010038161003816100381610038162 |119885|2 119889119909minus 2int

Ω(Re (119910119885))2 119889119909

Ψ2 (119885) = 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909 + 1205722 11988544

minus ReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909

minus 120572ReintΩ

100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909minus int

ΩRe (119910119905119885) |119885|2 119889119909

+ 120572intΩRe (119910119885) |119885|2 119889119909

minus ReintΩ119910119910119905 |119885|2 119889119909

minus 2intΩRe (119910119905119885)Re (119910119885) 119889119909

(81)


1198681 fl 120572 ImintΩ119891119885119889119909 + Imint

Ω119891119905119885119889119909

⩽ 12057216 1198852 + 119888 1003817100381710038171003817119891 (119904 + 119905 sdot)10038171003817100381710038172 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172⩽ 12057216 11988521198671 + 1198881205881 + 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(82)


1198854 ⩽ 119888 11988534 119885141198671

119885 ⩽ 119888119873 1198851198671 (83)


1198682 fl 1205722 11988544 ⩽ 119888 1198853 1198851198671 ⩽ 119888119873311988541198671 (84)


119906V1199081198671 ⩽ 119888 1199061198671 V1198671 1199081198671 (85)


1198683 fl minusReintΩ(100381610038161003816100381611991010038161003816100381610038162 119910)119905 119885119889119909


(86)




100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909 ⩽ 12057232 1198852 + 119888 1003817100381710038171003817119910100381710038171003817100381766⩽ 12057232 1198852 + 119862

(87)


119885infin ⩽ 119888 11988512 119885121198671

⩽ 119888radic119873 1198851198671 (88)



⩽ 119888119873 11988531198671 ⩽ 12057232 11988521198671 + 119888119873211988541198671

(89)


1198686 fl 120572intΩRe (119910119885) |119885|2 119889119909 ⩽ 120572 100381710038171003817100381711991010038171003817100381710038174 1003817100381710038171003817100381711988531003817100381710038171003817100381743

= 120572 100381710038171003817100381711991010038171003817100381710038174 11988534 ⩽ 119888 100381710038171003817100381711991010038171003817100381710038171198671 11988594 119885341198671

⩽ 1198881198739411988531198671 ⩽ 12057232 11988521198671 + 11988811987392

11988541198671 (90)



ΩRe (119910119905119885)Re (119910119885) 119889119909



(91)


Ψ2 (119885) = 7sum119894=1

119868119894⩽ (312057216 + 1198881radic119873)11988521198671 + 1198881198732

11988541198671 + 1198881205881+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(92)



Ψ2 (119885) ⩽ 1205724 11988521198671 + 119888119873211988541198671 + 1198881205881

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (93)



11988541198671 minus 119862⩾ 12 11988521198671 minus 1198881198732

11988541198671 minus 119862

(94)


2Ψ2 (119885) minus 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (95)


119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (96)


Ψ1 (119885 (119905)) ⩽ 1198881198732int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903+ 119862int119905

1199051

119890120572(119903minus119905)119889119903+ 119888int119905

1199051

119890120572(119903minus119905) 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172 119889119903⩽ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862120572+ 1198881198652 (119904 + 119905)

(97)


119885 (119905)41198671+ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862 (98)



120585 (119905) ⩽ 119888211987321205852 (119905) + 1198622

ie 11988821205852 (119905) minus 1198732120585 (119905) + 11986221198732 ⩾ 0(99)




Δ = 1198734 minus 4119888211986221198732 = 1198732 (1198732 minus 411988821198622) gt 0 (100)



21198882

1205722 = 1198732 + 119873radic1198732 minus 411988821198622

21198882 (101)



radic1198732 minus 411988821198622 ⩽ 119873(102)


120585 (119905) ⩽ 1205721or 120585 (119905) ⩾ 1205722

119905 ⩾ 1199051(103)





sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198672 ⩽ 119862 (119873 + 1) (105)





119885119909119905 + 120572119885119909 + 119876119873 (119860 minus 1198651015840 (V)) 119885119909 = 119866 119905 ⩾ 1199051 (106)



119866 = 119876119873 (119891119909 (119904 + 119905 sdot) + 1198651015840 (V) 119910119909) (107)



⟨120593 119891119909 (119904 + 119905)⟩ = int120593119891119909 (119904 + 119905)⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172)⩽ 119888 (1003817100381710038171003817120593100381710038171003817100381721198671 + 1198651 (119904 + 119905)) ⩽ 119888 (1 + 1205881)

(108)





⩽ 119888 V21198671 100381710038171003817100381711991011990910038171003817100381710038171198671 ⩽ 119888119873 100381710038171003817100381711991010038171003817100381710038171198671 ⩽ 1198621198731003817100381710038171003817minus2119894119876119873 Re (V (119905) 119910119909 (119905)) V (119905)10038171003817100381710038171198671 ⩽ 119888119873 V21198671 100381710038171003817100381711991010038171003817100381710038171198671

⩽ 119862119873(110)





⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891119905 (119904 + 119905)10038171003817100381710038172) (112)




(113)


119905 ⩾ 1199051 119904 ⩽ 0 (114)


119889119889119905Λ 1 (119885119909) + 2120572Λ 1 (119885119909) = 2Λ 2 (119885119909) (115)

where

Λ 1 (119885119909) = 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus int

Ω|V|2 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119885119909)2 119889119909

+ 2intΩIm (119866119885119909) 119889119909

(116)

Λ 2 (119885119909) = minusintΩRe (VV119905) 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119905119885119909) sdot Re (V119885119909) 119889119909

+ intΩIm (119866119905119885119909) 119889119909

+ 120572intΩIm (119866119885119909) 119889119909

(117)


Λ 2 (119885119909) ⩽ 3 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 Vinfin 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 1003817100381710038171003817119866119905

1003817100381710038171003817119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

+ 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩽ 119888radic119873 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 V1198671 1003817100381710038171003817119885119909100381710038171003817100381721198671

+ (10038171003817100381710038171198661199051003817100381710038171003817119867minus1 + 119866119867minus1) 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩽ ( 1198880radic119873 + 12057216) 1003817100381710038171003817119885119909

100381710038171003817100381721198671+ 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1)

(118)


Λ 2 (119885119909) ⩽ 1205724 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1) for 119873 ⩾ 1198731015840

1(119)


100381710038171003817100381721198671 minus 3 V2infin 1003817100381710038171003817119885119909100381710038171003817100381721198712

minus 2 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩾ 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 1198881198732

1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 2 119866119867minus1 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩾ (34 minus 11988811198732

) 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1

(120)


Λ 1 (119885119909) ⩾ 12 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1 for 119905 ⩾ 1199051 (121)


119889119889119905Λ 1 (119885119909) + 120572Λ 1 (119885119909) ⩽ 119888 (100381710038171003817100381711986611990510038171003817100381710038172119867minus1 + 1198662119867minus1)

⩽ 119888 (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172) (122)


1199051

119888119890120572(119903minus119905) (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172) 119889119903⩽ 119888 (1 + 119873 + 1198652 (119905))

(123)




sup119904⩽0



119908119905 + 120572119908 + 119894119908119909119909 minus 119894119908= 119894119876119873 ((|119906|2 + |V|2)119908 + 119906V119908) 119905 ⩾ 1199051 (125)


12 119889119889119905Υ1 (119908) + 120572Υ1 (119908) = Υ2 (119908) (126)

with

Υ1 (119908) = 11990821198671 minus intΩ(|119906|2 + |V|2) |119908|2 119889119909

minus ReintΩ119906V1199082119889119909

Υ2 (119908) = minus12 intΩ(|119906|2 + |V|2)

119905|119908|2 119889119909

minus 12ReintΩ (119906V)119905 1199082119889119909

(127)


Υ2 (119908) ⩽ 119888 (1003817100381710038171003817119906119905 + V1199051003817100381710038171003817119867minus1) (119906 + V1198671) 1199081198671 119908119871infin

⩽ 119888radic119873 11990821198671 (128)


) 11990821198671 ⩾ 12 11990821198671 (129)


119889119889119905Υ1 (119908) + 120572Υ1 (119908) ⩽ ( 1198882radic119873 minus 120572) 11990821198671 ⩽ 0 (130)


Υ1 (119908 (119905)) ⩽ 119890minus120572(119905minus1199051)Υ1 (119908 (1199051)) (131)






1198781 (119904 + 119905 119904) 1199060=

119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) if 0 ⩽ 119905 ⩽ 1199051 or 119904 gt 0V (119905 119904) = 119910 (119905 119904) + 119885 (119905 119904) for 119905 ⩾ 1199051 119904 ⩽ 0

(133)

1198782 (119904 + 119905 119904) 1199060=


(134)




A (119903) = 120596 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

119878 (119903 119903 minus 119905)K (119903)

= 1205961 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

1198781 (119903 119903 minus 119905)K (119903)119903 isin R

(135)




+ 119885 (120591119899 119903119899 minus 120591119899 1199060119899) (136)


1003817100381710038171003817119910 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198671⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119910 (119905 119904 1199060)10038171003817100381710038171198671 ⩽ 119862 (137)



1003817100381710038171003817119885 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198672⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119885 (119905 119904 1199060)10038171003817100381710038171198672 ⩽ 119862 (119873 + 1) (138)



10038171003817100381710038171198782 (119903 119903 minus 120591) 119906010038171003817100381710038171198671⩽ sup

119904⩽0







Acknowledgments











































Journal of












Journal of







Volume 2018






Volume 2018











100381610038161003816100381611991010038161003816100381610038162 119910119885119889119909 ⩽ 12057232 1198852 + 119888 1003817100381710038171003817119910100381710038171003817100381766⩽ 12057232 1198852 + 119862

(87)


119885infin ⩽ 119888 11988512 119885121198671

⩽ 119888radic119873 1198851198671 (88)



⩽ 119888119873 11988531198671 ⩽ 12057232 11988521198671 + 119888119873211988541198671

(89)


1198686 fl 120572intΩRe (119910119885) |119885|2 119889119909 ⩽ 120572 100381710038171003817100381711991010038171003817100381710038174 1003817100381710038171003817100381711988531003817100381710038171003817100381743

= 120572 100381710038171003817100381711991010038171003817100381710038174 11988534 ⩽ 119888 100381710038171003817100381711991010038171003817100381710038171198671 11988594 119885341198671

⩽ 1198881198739411988531198671 ⩽ 12057232 11988521198671 + 11988811987392

11988541198671 (90)



ΩRe (119910119905119885)Re (119910119885) 119889119909



(91)


Ψ2 (119885) = 7sum119894=1

119868119894⩽ (312057216 + 1198881radic119873)11988521198671 + 1198881198732

11988541198671 + 1198881205881+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172

(92)



Ψ2 (119885) ⩽ 1205724 11988521198671 + 119888119873211988541198671 + 1198881205881

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (93)



11988541198671 minus 119862⩾ 12 11988521198671 minus 1198881198732

11988541198671 minus 119862

(94)


2Ψ2 (119885) minus 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (95)


119889119889119905Ψ1 (119885) + 120572Ψ1 (119885) ⩽ 119888119873211988541198671 + 119862

+ 119888 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172 (96)


Ψ1 (119885 (119905)) ⩽ 1198881198732int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903+ 119862int119905

1199051

119890120572(119903minus119905)119889119903+ 119888int119905

1199051

119890120572(119903minus119905) 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172 119889119903⩽ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862120572+ 1198881198652 (119904 + 119905)

(97)


119885 (119905)41198671+ 1198881198732

int119905

1199051

119885 (119903)41198671 119890120572(119903minus119905)119889119903 + 119862 (98)



120585 (119905) ⩽ 119888211987321205852 (119905) + 1198622

ie 11988821205852 (119905) minus 1198732120585 (119905) + 11986221198732 ⩾ 0(99)




Δ = 1198734 minus 4119888211986221198732 = 1198732 (1198732 minus 411988821198622) gt 0 (100)



21198882

1205722 = 1198732 + 119873radic1198732 minus 411988821198622

21198882 (101)



radic1198732 minus 411988821198622 ⩽ 119873(102)


120585 (119905) ⩽ 1205721or 120585 (119905) ⩾ 1205722

119905 ⩾ 1199051(103)





sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198672 ⩽ 119862 (119873 + 1) (105)





119885119909119905 + 120572119885119909 + 119876119873 (119860 minus 1198651015840 (V)) 119885119909 = 119866 119905 ⩾ 1199051 (106)



119866 = 119876119873 (119891119909 (119904 + 119905 sdot) + 1198651015840 (V) 119910119909) (107)



⟨120593 119891119909 (119904 + 119905)⟩ = int120593119891119909 (119904 + 119905)⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172)⩽ 119888 (1003817100381710038171003817120593100381710038171003817100381721198671 + 1198651 (119904 + 119905)) ⩽ 119888 (1 + 1205881)

(108)





⩽ 119888 V21198671 100381710038171003817100381711991011990910038171003817100381710038171198671 ⩽ 119888119873 100381710038171003817100381711991010038171003817100381710038171198671 ⩽ 1198621198731003817100381710038171003817minus2119894119876119873 Re (V (119905) 119910119909 (119905)) V (119905)10038171003817100381710038171198671 ⩽ 119888119873 V21198671 100381710038171003817100381711991010038171003817100381710038171198671

⩽ 119862119873(110)





⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891119905 (119904 + 119905)10038171003817100381710038172) (112)




(113)


119905 ⩾ 1199051 119904 ⩽ 0 (114)


119889119889119905Λ 1 (119885119909) + 2120572Λ 1 (119885119909) = 2Λ 2 (119885119909) (115)

where

Λ 1 (119885119909) = 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus int

Ω|V|2 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119885119909)2 119889119909

+ 2intΩIm (119866119885119909) 119889119909

(116)

Λ 2 (119885119909) = minusintΩRe (VV119905) 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119905119885119909) sdot Re (V119885119909) 119889119909

+ intΩIm (119866119905119885119909) 119889119909

+ 120572intΩIm (119866119885119909) 119889119909

(117)


Λ 2 (119885119909) ⩽ 3 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 Vinfin 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 1003817100381710038171003817119866119905

1003817100381710038171003817119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

+ 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩽ 119888radic119873 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 V1198671 1003817100381710038171003817119885119909100381710038171003817100381721198671

+ (10038171003817100381710038171198661199051003817100381710038171003817119867minus1 + 119866119867minus1) 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩽ ( 1198880radic119873 + 12057216) 1003817100381710038171003817119885119909

100381710038171003817100381721198671+ 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1)

(118)


Λ 2 (119885119909) ⩽ 1205724 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1) for 119873 ⩾ 1198731015840

1(119)


100381710038171003817100381721198671 minus 3 V2infin 1003817100381710038171003817119885119909100381710038171003817100381721198712

minus 2 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩾ 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 1198881198732

1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 2 119866119867minus1 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩾ (34 minus 11988811198732

) 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1

(120)


Λ 1 (119885119909) ⩾ 12 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1 for 119905 ⩾ 1199051 (121)


119889119889119905Λ 1 (119885119909) + 120572Λ 1 (119885119909) ⩽ 119888 (100381710038171003817100381711986611990510038171003817100381710038172119867minus1 + 1198662119867minus1)

⩽ 119888 (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172) (122)


1199051

119888119890120572(119903minus119905) (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172) 119889119903⩽ 119888 (1 + 119873 + 1198652 (119905))

(123)




sup119904⩽0



119908119905 + 120572119908 + 119894119908119909119909 minus 119894119908= 119894119876119873 ((|119906|2 + |V|2)119908 + 119906V119908) 119905 ⩾ 1199051 (125)


12 119889119889119905Υ1 (119908) + 120572Υ1 (119908) = Υ2 (119908) (126)

with

Υ1 (119908) = 11990821198671 minus intΩ(|119906|2 + |V|2) |119908|2 119889119909

minus ReintΩ119906V1199082119889119909

Υ2 (119908) = minus12 intΩ(|119906|2 + |V|2)

119905|119908|2 119889119909

minus 12ReintΩ (119906V)119905 1199082119889119909

(127)


Υ2 (119908) ⩽ 119888 (1003817100381710038171003817119906119905 + V1199051003817100381710038171003817119867minus1) (119906 + V1198671) 1199081198671 119908119871infin

⩽ 119888radic119873 11990821198671 (128)


) 11990821198671 ⩾ 12 11990821198671 (129)


119889119889119905Υ1 (119908) + 120572Υ1 (119908) ⩽ ( 1198882radic119873 minus 120572) 11990821198671 ⩽ 0 (130)


Υ1 (119908 (119905)) ⩽ 119890minus120572(119905minus1199051)Υ1 (119908 (1199051)) (131)






1198781 (119904 + 119905 119904) 1199060=

119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) if 0 ⩽ 119905 ⩽ 1199051 or 119904 gt 0V (119905 119904) = 119910 (119905 119904) + 119885 (119905 119904) for 119905 ⩾ 1199051 119904 ⩽ 0

(133)

1198782 (119904 + 119905 119904) 1199060=


(134)




A (119903) = 120596 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

119878 (119903 119903 minus 119905)K (119903)

= 1205961 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

1198781 (119903 119903 minus 119905)K (119903)119903 isin R

(135)




+ 119885 (120591119899 119903119899 minus 120591119899 1199060119899) (136)


1003817100381710038171003817119910 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198671⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119910 (119905 119904 1199060)10038171003817100381710038171198671 ⩽ 119862 (137)



1003817100381710038171003817119885 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198672⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119885 (119905 119904 1199060)10038171003817100381710038171198672 ⩽ 119862 (119873 + 1) (138)



10038171003817100381710038171198782 (119903 119903 minus 120591) 119906010038171003817100381710038171198671⩽ sup

119904⩽0







Acknowledgments











































Journal of












Journal of







Volume 2018






Volume 2018










120585 (119905) ⩽ 119888211987321205852 (119905) + 1198622

ie 11988821205852 (119905) minus 1198732120585 (119905) + 11986221198732 ⩾ 0(99)




Δ = 1198734 minus 4119888211986221198732 = 1198732 (1198732 minus 411988821198622) gt 0 (100)



21198882

1205722 = 1198732 + 119873radic1198732 minus 411988821198622

21198882 (101)



radic1198732 minus 411988821198622 ⩽ 119873(102)


120585 (119905) ⩽ 1205721or 120585 (119905) ⩾ 1205722

119905 ⩾ 1199051(103)





sup119905⩾1199051

sup119904⩽0

119885 (119905 119904)21198672 ⩽ 119862 (119873 + 1) (105)





119885119909119905 + 120572119885119909 + 119876119873 (119860 minus 1198651015840 (V)) 119885119909 = 119866 119905 ⩾ 1199051 (106)



119866 = 119876119873 (119891119909 (119904 + 119905 sdot) + 1198651015840 (V) 119910119909) (107)



⟨120593 119891119909 (119904 + 119905)⟩ = int120593119891119909 (119904 + 119905)⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891 (119904 + 119905)10038171003817100381710038172)⩽ 119888 (1003817100381710038171003817120593100381710038171003817100381721198671 + 1198651 (119904 + 119905)) ⩽ 119888 (1 + 1205881)

(108)





⩽ 119888 V21198671 100381710038171003817100381711991011990910038171003817100381710038171198671 ⩽ 119888119873 100381710038171003817100381711991010038171003817100381710038171198671 ⩽ 1198621198731003817100381710038171003817minus2119894119876119873 Re (V (119905) 119910119909 (119905)) V (119905)10038171003817100381710038171198671 ⩽ 119888119873 V21198671 100381710038171003817100381711991010038171003817100381710038171198671

⩽ 119862119873(110)





⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891119905 (119904 + 119905)10038171003817100381710038172) (112)




(113)


119905 ⩾ 1199051 119904 ⩽ 0 (114)


119889119889119905Λ 1 (119885119909) + 2120572Λ 1 (119885119909) = 2Λ 2 (119885119909) (115)

where

Λ 1 (119885119909) = 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus int

Ω|V|2 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119885119909)2 119889119909

+ 2intΩIm (119866119885119909) 119889119909

(116)

Λ 2 (119885119909) = minusintΩRe (VV119905) 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119905119885119909) sdot Re (V119885119909) 119889119909

+ intΩIm (119866119905119885119909) 119889119909

+ 120572intΩIm (119866119885119909) 119889119909

(117)


Λ 2 (119885119909) ⩽ 3 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 Vinfin 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 1003817100381710038171003817119866119905

1003817100381710038171003817119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

+ 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩽ 119888radic119873 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 V1198671 1003817100381710038171003817119885119909100381710038171003817100381721198671

+ (10038171003817100381710038171198661199051003817100381710038171003817119867minus1 + 119866119867minus1) 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩽ ( 1198880radic119873 + 12057216) 1003817100381710038171003817119885119909

100381710038171003817100381721198671+ 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1)

(118)


Λ 2 (119885119909) ⩽ 1205724 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1) for 119873 ⩾ 1198731015840

1(119)


100381710038171003817100381721198671 minus 3 V2infin 1003817100381710038171003817119885119909100381710038171003817100381721198712

minus 2 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩾ 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 1198881198732

1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 2 119866119867minus1 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩾ (34 minus 11988811198732

) 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1

(120)


Λ 1 (119885119909) ⩾ 12 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1 for 119905 ⩾ 1199051 (121)


119889119889119905Λ 1 (119885119909) + 120572Λ 1 (119885119909) ⩽ 119888 (100381710038171003817100381711986611990510038171003817100381710038172119867minus1 + 1198662119867minus1)

⩽ 119888 (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172) (122)


1199051

119888119890120572(119903minus119905) (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172) 119889119903⩽ 119888 (1 + 119873 + 1198652 (119905))

(123)




sup119904⩽0



119908119905 + 120572119908 + 119894119908119909119909 minus 119894119908= 119894119876119873 ((|119906|2 + |V|2)119908 + 119906V119908) 119905 ⩾ 1199051 (125)


12 119889119889119905Υ1 (119908) + 120572Υ1 (119908) = Υ2 (119908) (126)

with

Υ1 (119908) = 11990821198671 minus intΩ(|119906|2 + |V|2) |119908|2 119889119909

minus ReintΩ119906V1199082119889119909

Υ2 (119908) = minus12 intΩ(|119906|2 + |V|2)

119905|119908|2 119889119909

minus 12ReintΩ (119906V)119905 1199082119889119909

(127)


Υ2 (119908) ⩽ 119888 (1003817100381710038171003817119906119905 + V1199051003817100381710038171003817119867minus1) (119906 + V1198671) 1199081198671 119908119871infin

⩽ 119888radic119873 11990821198671 (128)


) 11990821198671 ⩾ 12 11990821198671 (129)


119889119889119905Υ1 (119908) + 120572Υ1 (119908) ⩽ ( 1198882radic119873 minus 120572) 11990821198671 ⩽ 0 (130)


Υ1 (119908 (119905)) ⩽ 119890minus120572(119905minus1199051)Υ1 (119908 (1199051)) (131)






1198781 (119904 + 119905 119904) 1199060=

119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) if 0 ⩽ 119905 ⩽ 1199051 or 119904 gt 0V (119905 119904) = 119910 (119905 119904) + 119885 (119905 119904) for 119905 ⩾ 1199051 119904 ⩽ 0

(133)

1198782 (119904 + 119905 119904) 1199060=


(134)




A (119903) = 120596 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

119878 (119903 119903 minus 119905)K (119903)

= 1205961 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

1198781 (119903 119903 minus 119905)K (119903)119903 isin R

(135)




+ 119885 (120591119899 119903119899 minus 120591119899 1199060119899) (136)


1003817100381710038171003817119910 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198671⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119910 (119905 119904 1199060)10038171003817100381710038171198671 ⩽ 119862 (137)



1003817100381710038171003817119885 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198672⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119885 (119905 119904 1199060)10038171003817100381710038171198672 ⩽ 119862 (119873 + 1) (138)



10038171003817100381710038171198782 (119903 119903 minus 120591) 119906010038171003817100381710038171198671⩽ sup

119904⩽0







Acknowledgments











































Journal of












Journal of







Volume 2018






Volume 2018










⩽ 119888 (100381710038171003817100381712059311990910038171003817100381710038172 + 1003817100381710038171003817119891119905 (119904 + 119905)10038171003817100381710038172) (112)




(113)


119905 ⩾ 1199051 119904 ⩽ 0 (114)


119889119889119905Λ 1 (119885119909) + 2120572Λ 1 (119885119909) = 2Λ 2 (119885119909) (115)

where

Λ 1 (119885119909) = 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus int

Ω|V|2 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119885119909)2 119889119909

+ 2intΩIm (119866119885119909) 119889119909

(116)

Λ 2 (119885119909) = minusintΩRe (VV119905) 1003816100381610038161003816119885119909

10038161003816100381610038162 119889119909minus 2int

ΩRe (V119905119885119909) sdot Re (V119885119909) 119889119909

+ intΩIm (119866119905119885119909) 119889119909

+ 120572intΩIm (119866119885119909) 119889119909

(117)


Λ 2 (119885119909) ⩽ 3 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 Vinfin 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 1003817100381710038171003817119866119905

1003817100381710038171003817119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

+ 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩽ 119888radic119873 1003817100381710038171003817V1199051003817100381710038171003817119867minus1 V1198671 1003817100381710038171003817119885119909100381710038171003817100381721198671

+ (10038171003817100381710038171198661199051003817100381710038171003817119867minus1 + 119866119867minus1) 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩽ ( 1198880radic119873 + 12057216) 1003817100381710038171003817119885119909

100381710038171003817100381721198671+ 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1)

(118)


Λ 2 (119885119909) ⩽ 1205724 1003817100381710038171003817119885119909100381710038171003817100381721198671 + 119888 (1003817100381710038171003817119866119905

10038171003817100381710038172119867minus1 + 1198662119867minus1) for 119873 ⩾ 1198731015840

1(119)


100381710038171003817100381721198671 minus 3 V2infin 1003817100381710038171003817119885119909100381710038171003817100381721198712

minus 2 119866119867minus1 100381710038171003817100381711988511990910038171003817100381710038171198671

⩾ 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 1198881198732

1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 2 119866119867minus1 1003817100381710038171003817119885119909

10038171003817100381710038171198671⩾ (34 minus 11988811198732

) 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1

(120)


Λ 1 (119885119909) ⩾ 12 1003817100381710038171003817119885119909100381710038171003817100381721198671 minus 119888 1198662119867minus1 for 119905 ⩾ 1199051 (121)


119889119889119905Λ 1 (119885119909) + 120572Λ 1 (119885119909) ⩽ 119888 (100381710038171003817100381711986611990510038171003817100381710038172119867minus1 + 1198662119867minus1)

⩽ 119888 (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119905 sdot)10038171003817100381710038172) (122)


1199051

119888119890120572(119903minus119905) (1 + 119873 + 1003817100381710038171003817119891119905 (119904 + 119903 sdot)10038171003817100381710038172) 119889119903⩽ 119888 (1 + 119873 + 1198652 (119905))

(123)




sup119904⩽0



119908119905 + 120572119908 + 119894119908119909119909 minus 119894119908= 119894119876119873 ((|119906|2 + |V|2)119908 + 119906V119908) 119905 ⩾ 1199051 (125)


12 119889119889119905Υ1 (119908) + 120572Υ1 (119908) = Υ2 (119908) (126)

with

Υ1 (119908) = 11990821198671 minus intΩ(|119906|2 + |V|2) |119908|2 119889119909

minus ReintΩ119906V1199082119889119909

Υ2 (119908) = minus12 intΩ(|119906|2 + |V|2)

119905|119908|2 119889119909

minus 12ReintΩ (119906V)119905 1199082119889119909

(127)


Υ2 (119908) ⩽ 119888 (1003817100381710038171003817119906119905 + V1199051003817100381710038171003817119867minus1) (119906 + V1198671) 1199081198671 119908119871infin

⩽ 119888radic119873 11990821198671 (128)


) 11990821198671 ⩾ 12 11990821198671 (129)


119889119889119905Υ1 (119908) + 120572Υ1 (119908) ⩽ ( 1198882radic119873 minus 120572) 11990821198671 ⩽ 0 (130)


Υ1 (119908 (119905)) ⩽ 119890minus120572(119905minus1199051)Υ1 (119908 (1199051)) (131)






1198781 (119904 + 119905 119904) 1199060=

119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) if 0 ⩽ 119905 ⩽ 1199051 or 119904 gt 0V (119905 119904) = 119910 (119905 119904) + 119885 (119905 119904) for 119905 ⩾ 1199051 119904 ⩽ 0

(133)

1198782 (119904 + 119905 119904) 1199060=


(134)




A (119903) = 120596 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

119878 (119903 119903 minus 119905)K (119903)

= 1205961 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

1198781 (119903 119903 minus 119905)K (119903)119903 isin R

(135)




+ 119885 (120591119899 119903119899 minus 120591119899 1199060119899) (136)


1003817100381710038171003817119910 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198671⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119910 (119905 119904 1199060)10038171003817100381710038171198671 ⩽ 119862 (137)



1003817100381710038171003817119885 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198672⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119885 (119905 119904 1199060)10038171003817100381710038171198672 ⩽ 119862 (119873 + 1) (138)



10038171003817100381710038171198782 (119903 119903 minus 120591) 119906010038171003817100381710038171198671⩽ sup

119904⩽0







Acknowledgments











































Journal of












Journal of







Volume 2018






Volume 2018










sup119904⩽0



119908119905 + 120572119908 + 119894119908119909119909 minus 119894119908= 119894119876119873 ((|119906|2 + |V|2)119908 + 119906V119908) 119905 ⩾ 1199051 (125)


12 119889119889119905Υ1 (119908) + 120572Υ1 (119908) = Υ2 (119908) (126)

with

Υ1 (119908) = 11990821198671 minus intΩ(|119906|2 + |V|2) |119908|2 119889119909

minus ReintΩ119906V1199082119889119909

Υ2 (119908) = minus12 intΩ(|119906|2 + |V|2)

119905|119908|2 119889119909

minus 12ReintΩ (119906V)119905 1199082119889119909

(127)


Υ2 (119908) ⩽ 119888 (1003817100381710038171003817119906119905 + V1199051003817100381710038171003817119867minus1) (119906 + V1198671) 1199081198671 119908119871infin

⩽ 119888radic119873 11990821198671 (128)


) 11990821198671 ⩾ 12 11990821198671 (129)


119889119889119905Υ1 (119908) + 120572Υ1 (119908) ⩽ ( 1198882radic119873 minus 120572) 11990821198671 ⩽ 0 (130)


Υ1 (119908 (119905)) ⩽ 119890minus120572(119905minus1199051)Υ1 (119908 (1199051)) (131)






1198781 (119904 + 119905 119904) 1199060=

119910 (119905 119904) = 119875119873119906 (119904 + 119905 119904 1199060) if 0 ⩽ 119905 ⩽ 1199051 or 119904 gt 0V (119905 119904) = 119910 (119905 119904) + 119885 (119905 119904) for 119905 ⩾ 1199051 119904 ⩽ 0

(133)

1198782 (119904 + 119905 119904) 1199060=


(134)




A (119903) = 120596 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

119878 (119903 119903 minus 119905)K (119903)

= 1205961 (K (119903) 119903) = ⋂1199050gt0

⋃119905⩾1199050

1198781 (119903 119903 minus 119905)K (119903)119903 isin R

(135)




+ 119885 (120591119899 119903119899 minus 120591119899 1199060119899) (136)


1003817100381710038171003817119910 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198671⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119910 (119905 119904 1199060)10038171003817100381710038171198671 ⩽ 119862 (137)



1003817100381710038171003817119885 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198672⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119885 (119905 119904 1199060)10038171003817100381710038171198672 ⩽ 119862 (119873 + 1) (138)



10038171003817100381710038171198782 (119903 119903 minus 120591) 119906010038171003817100381710038171198671⩽ sup

119904⩽0







Acknowledgments











































Journal of












Journal of







Volume 2018






Volume 2018










1003817100381710038171003817119885 (120591119899 119903119899 minus 120591119899 1199060119899)10038171003817100381710038171198672⩽ sup

119905⩾1199051

sup119904⩽0

sup1199060isin119861119877

1003817100381710038171003817119885 (119905 119904 1199060)10038171003817100381710038171198672 ⩽ 119862 (119873 + 1) (138)



10038171003817100381710038171198782 (119903 119903 minus 120591) 119906010038171003817100381710038171198671⩽ sup

119904⩽0







Acknowledgments











































Journal of












Journal of







Volume 2018






Volume 2018




























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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Pullback-Forward Dynamics for Damped Schrödinger Equations ...downloads.hindawi.com/journals/ddns/2018/2139792.pdf · ResearchArticle Pullback-Forward Dynamics for Damped Schrödinger