Asymptotically Almost Periodic Solutions for Abstract Partial Neutral Integro Differential Equation

doi:10.1155/2010/310951

Research Article

Asymptotically Almost Periodic

Solutions for Abstract Partial Neutral

Integro-Differential Equation

Jos ´e Paulo C. dos Santos,

_{Sandro M. Guzzo,}

and Marcos N. Rabelo

1_{Departamento de Ciˆencias Exatas, Universidade Federal de Alfenas, Rua Gabriel Monteiro da Silva,}

700. 37130-000 Alfenas-MG, Brazil

2_{Colegiado do curso de Matem´atica-UNIOESTE, Rua Universit´aria, 2069, Caixa Postal 711,}

85819-110 Cascavel-PR, Brazil

3_{Departamento de Matem´atica, Universidade Federal de Pernambuco, Av. Prof. Luiz Freire,}

Cidade Universit´aria, 50740-540 Recife-PE, Brazil

Correspondence should be addressed to Jos´e Paulo C. dos Santos,[email protected]

Received 24 November 2009; Revised 23 February 2010; Accepted 24 February 2010

Academic Editor: Toka Diagana

Copyrightq2010 Jos´e Paulo C. dos Santos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The existence of asymptotically almost periodic mild solutions for a class of abstract partial neutral integro-differential equations with unbounded delay is studied.

1. Introduction

In this paper, we study the existence of asymptotically almost periodic mild solutions for a class of abstract partial neutral integro-differential equations modelled in the form

d dt

xt ft, xt

Axt

_t

0

Bt−sxsdsgt, xt, 1.1

whereA :DA ⊂X → XandBt:DBt⊂X → X,t≥ 0, are closed linear operators;

X, · is a Banach space; the historyxt :−∞,0 → X,xtθ xtθ, belongs to some abstract phase spaceBdefined axiomaticallyf, g:I× B → Xare appropriated functions.

equations is very extensive and we refer the reader to Chukwu1, Hale and Lunel2, Wu

3, and the references therein. As a practical application, we note that the equation

d dt

ut−λ

t

−∞Ct−susds

Aut λ

t

−∞Bt−susds−pt qt 1.2

arises in the study of the dynamics of income, employment, value of capital stock, and cumulative balance of payment; see1for details. In the above system,λis a real number, the stateut ∈Rn_{,C·,B·aren×ncontinuous functions matrices,Ais a constantn×n} matrix,p·represents the government intervention, andq·the private initiative. We note that by assuming the solutionuis known on−∞,0, we can transform this system into an abstract system with unbounded delay described as1.1.

Abstract partial neutral differential equations also appear in the theory of heat conduction. In the classic theory of heat conduction, it is assumed that the internal energy and the heat flux depend linearly on the temperature u and on its gradient ∇u. Under these conditions, the classic heat equation describes sufficiently well the evolution of the temperature in different types of materials. However, this description is not satisfactory in materials with fading memory. In the theory developed in4,5, the internal energy and the heat flux are described as functionals ofuandux. The next system, see for instance6–9, has been frequently used to describe this phenomenon,

d dt

ut, x

t

−∞k1t−sus, xds

cΔut, x

t

−∞k2t−sΔus, xds,

ut, x 0, x∈∂Ω.

1.3

In this system,Ω ⊂ Rn _{is open, bounded, and with smooth boundary; t, x ∈ 0,∞×Ω;}

ut, xrepresents the temperature in xat the time t;cis a physical constant ki : R → R,

i 1,2,are the internal energy and the heat flux relaxation, respectively. By assuming that the solutionuis known on−∞,0andk2≡0, we can transform this system into an abstract system with unbounded delay described in the form1.1.

Recent contributions on the existence of solutions with some of the previously enumerated properties or another type of almost periodicity to neutral functional differential equations have been made in10,11, for the case of neutral ordinary differential equations, and in12–15for partial functional differential systems.

The purpose of this work is to study the existence of asymptotically almost periodic mild solutions for the neutral system1.1. To this end, we study the existence and qualitative properties of an exponentially stable resolvent operator for the integro-differential system

dxt

dt Axt

_t

0

Bt−sxsds, t≥0,

x0 z∈X.

1.4

Gripenberg et al. 16 which contains an overview of the theory for the case where the underlying spaceXhas finite dimension. For abstract integro-differential equations described on infinite dimensional spaces, we cite the Pr ¨uss book17and the papers18–20, Da Prato et al.21,22, and Lunardi9,23. To finish this short description of the related literature, we cite the papers24–26 where some of the above topics for the case of abstract neutral integro-differential equations with unbounded delay are treated.

To the best of our knowledge, the study of the existence of asymptotically almost periodic solutions of neutral integro-differential equations with unbounded delay described in the abstract form1.1is an untreated topic in the literature and this is the main motivation of this article.

To finish this section, we emphasize some notations used in this paper. LetZ, · Z andW,·Wbe Banach spaces. We denote byLZ, Wthe space of bounded linear operators fromZintoWendowed with norm of operators, and we write simplyLZwhenZW. By

RQ, we denote the range of a mapQ, and for a closed linear operatorP :DP⊆Z → W, the notationDPrepresents the domain ofPendowed with the graph norm,z1zZ P zW,z∈DP. In the caseZW, the notationρPstands for the resolvent set ofP,and

Rλ, P λI −P−1 is the resolvent operator ofP. Furthermore, for appropriate functions

K:0,∞ → ZandS:0,∞ → LZ, W, the notationK denotes the Laplace transform of

KandS∗Kthe convolution betweenSandK, which is defined byS∗Kt t₀St−sKsds. The notationBrx, Zstands for the closed ball with center atxand radiusr > 0 in Z. As usual,C00,∞, Zrepresents the subspace ofCb0,∞, Zformed by the functions which vanish at infinity.

2. Preliminaries

In this work, we will employ an axiomatic definition of the phase spaceBsimilar to that in

27. More precisely,Bwill denote a vector space of functions defined from−∞,0intoX

endowed with a seminorm denoted by · Band such that the following axioms hold.

AIfx:−∞, σb → X,withb >0,is continuous onσ, σbandxσ ∈ B, then for eacht∈σ, σbthe following conditions hold:

ixtis inB,

iixt ≤HxtB,

iiixtB ≤ Kt−σsup{xs :σ ≤s ≤ t}Mt−σxσB,whereH > 0 is a constant, andK, M:0,∞→1,∞are functions such thatK·andM·are respectively continuous and locally bounded, andH, K, Mare independent of

x·.

A1Ifx·is a function as inA, thenxtis aB-valued continuous function onσ, σb.

BThe spaceBis complete.

C2If ϕn

n∈N is a sequence in Cb−∞,0, X formed by functions with compact support such that ϕn → ϕuniformly on compact, thenϕ ∈ Bandϕn−ϕB → 0

Remark 2.1. In the remainder of this paper,L>0 is such that

_ϕ

B≤Lsup

θ≤0

_ϕθ

2.1

for everyϕ:−∞,0 → Xcontinuous and bounded; see27, Proposition 7.1.1for details.

Definition 2.2. LetSt:B → Bbe theC0-semigroup defined byStϕθ ϕ0on−t,0and

Stϕθ ϕtθon−∞,−t. The phase spaceBis called a fading memory ifStϕ_B → 0 ast → ∞for eachϕ∈ Bwithϕ0 0.

Remark 2.3. In this work, we suppose that there exists a positiveKsuch that

max{Kt, Mt} ≤K 2.2

for eacht ≥ 0. Observe that this condition is verified, for example, ifBis a fading memory, see27, Proposition 7.1.5.

Example 2.4The phase spaceCr×Lpρ, X. Letr ≥0,1≤p < ∞, and let ρ :−∞,−r → R be a nonnegative measurable function which satisfies the conditions g-5 and g-6 in the terminology of 27. Briefly, this means that ρ is locally integrable, and there exists a nonnegative, locally bounded function γ on−∞,0 such that ρξθ ≤ γξρθ, for all

ξ ≤ 0 and θ ∈ −∞,−r\Nξ, whereNξ ⊆ −∞,−r is a set with Lebesgue measure zero. The spaceCr ×Lpρ, Xconsists of all classes of functions ϕ : −∞,0 → X such thatϕis continuous on−r,0, Lebesgue-measurable, and ρϕp_{is Lebesgue integrable on−∞,−r.} The seminorm in Cr×Lpρ, Xis defined by

ϕ_B: supϕθ:−r≤θ≤0

₋r

−∞ρθ

ϕθ pdθ

1/p

. 2.3

The space BCr×Lpρ;Xsatisfies axiomsA,A-1, andB. Moreover, whenr 0 and

p 2, we can takeH 1,Mt γ−t1/2, andKt 1 0_−tρθdθ1/2, fort ≥0; see27, Theorem 1.3.8for details.

Now, we need to introduce some concepts, definitions, and technicalities on almost periodical functions.

Definition 2.5. A functionf∈CR, Zis almost periodica.p.if for everyε >0, there exists a relatively dense subset ofR, denoted byHε, f, Z, such that

_ftξ−ft

Z< ε, t∈R, ξ∈ H

ε, f, Z. 2.4

Definition 2.6. A functionf ∈ C0,∞, Zis asymptotically almost periodica.a.p.if there exists an almost periodic functiong·andw∈C00,∞, Zsuch thatf· g· w·.

Lemma 2.7see28, Theorem 5.5. A functionf∈C0,∞, Zis asymptotically almost periodic if and only if for everyε >0there existLε, f, Z>0and a relatively dense subset of0,∞, denoted

byTε, f, Z, such that

_ftξ−ft

Z< ε, t≥L

ε, f, Z, ξ∈ Tε, f, Z. 2.5

In this paper,APZandAAPZare the spaces

APZ f∈CR, Z:f is a.p.,

AAPZ f∈C0,∞, Z:f is a.a.p. 2.6

endowed with the norms|u|Zsups∈RusZanduZsups≥0usZ,respectively. We know from the result in28thatAPZandAAPZare Banach spaces.

Next,Z, · ZandW, · Ware abstract Banach spaces.

Definition 2.8. LetΩbe an open subset ofW.

aA continuous function f ∈ CR× Ω, Z resp., f ∈ C0,∞×Ω;Z is called pointwise almost periodicp.a.p.,resp., pointwise asymptotically almost periodic

p.a.a.p.iff·, x∈APZ resp.,f·,x∈AAPZfor everyx∈Ω.

bA continuous functionF∈CR×Ω, Zis called uniformly almost periodicu.a.p., if for everyε >0 and every compactK⊂Ωthere exists a relatively dense subset of R, denoted byHε, f, K, Z, such that

ftξ, y−ft, y _Z≤ε, t, ξ, y∈R× Hε, f, K, Z×K. 2.7

cA continuous functionf:C0,∞×Ω, Zis called uniformly asymptotically almost periodic u.a.a.p., if for every ε > 0 and every compact K ⊂ Ω there exists a relatively dense subset of0,∞, denoted byTε, f, K, Z, andLε, f, K, Z>0 such that

_{ftξ, y−ft, y}

Z≤ε, t≥L

ε, f, K, Z, ξ, y∈ Tε, f, K, Z×K. 2.8

The next lemma summarizes some properties which are fundamental to obtain our results.

Lemma 2.9see29, Theorem 1.2.7. LetΩ ⊂ W be an open set. Then the following properties hold.

aIff∈CR×Ω, Zis p.a.p. and satisfies a local Lipschitz condition atx∈Ω, uniformly at t, thenfis u.a.p.

bIf f ∈ C0,∞×Ω, Zis p.a.a.p. and satisfies a local Lipschitz condition at x ∈ Ω,

uniformly att, thenfis u.a.a.p.

cIf x ∈ APX, thent → xt ∈ APB. Moreover, if B is a fading memory space and

dIff ∈CR×Ω, Zis u.a.p. andy∈APWis such that{yt:t∈R}W ⊂Ω, thent→ ft, yt∈APZ.

eIff ∈C0,∞×Ω, Zis u.a.a.p andy∈AAPWis such that{yt:t∈R}W ⊂ Ω,

thent→ft, yt∈AAPZ.

3. Resolvent Operators

In this section, we study the existence and qualitative properties of an exponentially resolvent operator for the integro-differential abstract Cauchy problem

dxt

dt Axt

_t

0

Bt−sxsds,

x0 x∈X.

3.1

The results obtained for the resolvent operator in this section are similar to those that can be found, for instance, in the papers21,23,30. In this paper, we prove the necessary estimates for the proof of an existence theorem of asymptotically almost periodic solutions for

1.1. For the better comprehension of the subject, we will introduce the following definitions, hypothesis, and results.

We introduce the following concept of resolvent operator for integro-differential problem3.1.

Definition 3.1. A one-parameter family of bounded linear operatorsRt_t≥0onXis called a resolvent operator of3.1if the following conditions are verified.

aFunctionR·:0,∞ → LXis strongly continuous andR0xxfor allx∈X.

bForx∈DA,R·x∈C0,∞,DAC1_{0,∞, X, and}

dRtx

dt ARtx

t

0

Bt−sRsxds, 3.2

dRtx

dt RtAx

t

0

Rt−sBsxds, 3.3

for everyt≥0,

cThere exist constantsM >0, βsuch thatRt ≤Meβtfor everyt≥0.

Definition 3.2. A resolvent operatorRt_t≥0of3.1is called exponentially stable if there exist positive constantsM, αsuch thatRt ≤Me−αt_.

In this work, we always assume that the following conditions are verified.

H2For allt ≥0, Bt:DBt ⊆X → X is a closed linear operator,DA⊆DBt, andB·xis strongly measurable on0,∞for eachx ∈ DA. There existsb· ∈ L1

locRsuch thatbλexists for Reλ>0 andBtx ≤btx1for allt >0 and

x∈DA. Moreover, the operator valued functionB:Λω,π/2 → LDA, Xhas an analytical extensionstill denoted byBtoΛω,ϑsuch thatBλx ≤ Bλ x1 for allx∈DA, andBλO1/|λ|as|λ| → ∞.

H3There exist a subspaceD⊆DAdense inDAand positive constantsCi,i1,2, such thatAD⊆DA,BλD⊆DA, andABλx ≤C1xfor everyx∈D and allλ∈Λω,ϑ.

In the sequel, forr >0,θ∈π/2, ϑ, andw∈R, set

Λr,ω,θ

λ∈C:λ /ω, |λ|> r,argλ−ω< θ, 3.4

and forω Γi

r,θ,i1,2,3, the paths

ω Γ1 r,θ

ωteiθ_:t≥r,

ω Γ2_r,θωreiξ_{:−θ≤ξ≤θ,}

ω Γ3_r,θωte−iθ:t≥r,

3.5

withω Γr,θ3i1ω Γir,θ are oriented counterclockwise. In addition,ΨGis the set

ΨG

λ∈C:Gλ:λI−A−Bλ−1∈ LX

. 3.6

We next study some preliminary properties needed to establish the existence of a resolvent operator for the problem3.1.

Lemma 3.3. There existsr1 >0such thatΛr1,ω,ϑ⊆ΨGand the functionG :Λr1,ω,ϑ → LXis

analytic. Moreover,

Gλ Rλ, AI−BλRλ, A−1, 3.7

and there exist constantsMifori1,2,3such that

Gλ ≤ M1

|λ−ω|, 3.8

AGλx ≤ M2

|λ−ω|x1, x∈DA, 3.9

AGλ ≤M3, 3.10

Proof. Since

BλRλ, A ≤ Bλ Rλ, A₁

≤

⎛

⎝M0

Bλ

|λ−ω|

M0|λ| Bλ |λ−ω|

Bλ

⎞

⎠_, 3.11

fixed ε < 1, there exists a positive number r1 such that BλRλ, A ≤ ε for λ ∈

Λr1,ω,ϑ. Consequently, the operator I − BλRλ, A has a continuous inverse with I −

BλRλ, A−1 ≤1/1−ε. Moreover, forx∈X, we have

λI−Bλ−ARλ, AI−BλRλ, A−1xx, 3.12

and forx∈DA,

Rλ, AI−BλRλ, A−1λI−Bλ−Axx, 3.13

which shows3.7and thatΛr1,ω,ϑ⊆ΨG. Now, from3.7we obtainRGλ⊆DAand

AGλ λRλ, A−II−BλRλ, A−1. 3.14

Consequently,

AGλ ≤ 1

1−ελRλ, A−I

≤ 1

1−ε

M0 M_| 0|ω|

λ−ω|1

≤M3,

3.15

the functionsG, AG:Λr1,ω,ϑ → LXare analytic, and estimates3.8, and3.10are valid.

In addition, forx∈DA, we can write

AGλx ≤ ARλ, AI−BλRλ, A−1x−ARλ, Ax Rλ, AAx

ARλ, AI−I−BλRλ, A I−BλRλ, A−1x Rλ, AAx

≤M3BλRλ, Ax1Rλ, AAx transform, we conclude thatR·is the unique resolvent operator for3.1.

Lemma 3.5. The operator functionR·is exponentially bounded inLDA.

Proof. It follows from3.9that the integral in

Furthermore, it follows from3.8, and assumptionH2that

_eλt_λ−1_{GλABλx ≤} eωrC

|λ||λ−ω| Φλ, 3.25

whereΦ·is integrable forλ∈ω Γr,θ. From the Lebesgue dominated convergence theorem, we infer that

lim

t→0Rtx−x 1 2πi

ωΓr,θ

λ−1GλABλxdλ. 3.26

Let nowωCL,θbe the curveωLeiξforθ≤ξ≤2π−θ. Turning to apply Cauchy’s theorem combining with the estimate

ωCL,θ

λ−1GλABλxdλ ≤ CθL

L−ω2, 3.27

we obtain

1 2πi

ωΓr,θ

λ−1GλABλxdλ

lim

L→ ∞

1 2πi

{ωΓr,θ:|r|≤L}∪ωCL,θ

λ−1GλABλxdλ0,

3.28

we can affirm that limt→0Rtx−x 0 for allx ∈DA, which completes and the proof sinceDAis dense inXandR·is bounded on0,1.

Notice thatω < 0,the sectorsΛr,0,ϑ ⊆ Λr,ω,ϑ,fromLemma 3.3,G :Λr,ω,ϑ → LXis analytic. Consider the contours

γ1 {λs−iNsinθ: cosθ≤s≤ωcosθ},

γ2 {λsiNsinθ:ωcosθ≤s≤cosθ},

ω ΓN r,θ

ωteiθ_{:r≤t≤N∪ωre}iξ_{:−θ≤ξ≤θ∪ωte}−iθ_{:r ≤t≤N,}

0 ΓN_r,θteiθ:r ≤t≤N∪reiξ:−θ≤ξ≤θ∪te−iθ:r ≤t≤N,

3.29

andRNωΓN_r,θ∪γ2∪0ΓN_r,θ∪γ1,oriented counterclockwise. By Cauchy theorem for 0< t≤1, we obtain

RN

The following estimate:

γ1

eλtGλdλ

≤

ωcosθ

cosθ

eRes−iNsinθt C

|s−iNsinθ−ω|ds

≤

_ωcosθ

cosθ

est C

N|sinθ|ds≤ C N|sinθ|

eωt₋₁

t

ecosθt

≤ C

N|sinθ| e

3.31

is the one responsible for the fact that the integral_γ

1e

λt_{Gλdλtends to 0 asNtend to∞,} in a similar way the integral_γ

2e

λt_{Gλdλ,tend to 0 asNtend to∞,so that}

1 2πi

ωΓr,θ

eλtGλdλ 1

2πi

0Γr,θ

eλtGλdλ. 3.32

Forx∈DA, we obtain

Rtx−x 1

2πi

0Γr,θ

eλtGλx−λ−1eλtxdλ

1

2πi

0Γr,θ

eλtλ−1GλABλxdλ,

3.33

and proceeding as before, we obtain limt→0Rtx−x 0 for allx ∈ X,which ends the proof.

The following result can be proved with an argument similar to that used in the proof of the preceding lemma with changingDAbyD.

Lemma 3.7. The functionR:0,∞ → LDAis strongly continuous.

We next setδmin{ϑ−π/2, π−ϑ}.

Lemma 3.8. The functionR:0,∞ → LXhas an analytic extension toΛδ,0, and

dRz dz

1 2πi

ωΓr,θ

Proof. Forλ ∈ ω Γr,θ and z ∈ Λδ,0, we can writeλz ω|z|eiargzs|z|eiargzξ,where

This property allows us to define the extensionRzby this integral.

Similarly, the integral on the right hand side of3.34is also absolutely convergent in LXand strong, continuous onXforz∈Λδ,0. Forλ∈ω Γr,θ,

Lemma 3.9. For everyλ∈CwithReλ>max{0, ωr},Rλ Gλ.

Proof. Using thatG·is analytic onΛr,ω,θ and that the integrals involved in the calculus are absolutely convergent, we have

Theorem 3.10. The functionR·is a resolvent operator for the system3.1.

Proof. Letx∈DA. FromLemma 3.9, for Reλ>max{0, ωr},

which in turn implies that

Forζ∈Xandt∈0, δ, there existscζ,t∈0, tsuch that

ζ◦ Rtx−ζ◦ R0x

t ζ

Rcζ,t

Ax

_cζ,t

0

Rcϕ,t−s

Bsxds

. 3.44

Consequently, fort∈0, δwe have that

Rtx− R_t 0x−Ax sup

ζ≤1

ζ◦ Rtx−_tζ◦ R0x−ζAx

≤ sup s∈0,δ

RsAx

_s

0

Rs−τBτxdτ−Ax ,

3.45

which proves the existence of the right derivative ofR·at zero and thatd/dtRt|_t0Ax.

This proves that resolvent equation3.3is valid for everyt≥0 andR·x∈C1_0,∞;Xfor everyx∈DA. This completes the proof.

Corollary 3.11. Ifωr <0,then the functionR·is an exponentially stable resolvent operator for the system3.1.

In the next result, we denote by−Aϑthe fractional power of the operator−A see

32for details.

Theorem 3.12. Suppose that the conditionsH1–H3are satisfied. Then there exists a positive numberCsuch that

_−Aϑ_{Rt ≤}

⎧ ⎨ ⎩

Cerωt_{, t≥1,}

Cerωt_t−ϑ_{, t∈0,1,} 3.46

for allϑ∈0,1.

Proof. Letϑ∈0,1.From32, Theorem 6.10, there existsCϑ>0 such that

_−Aϑ_{x ≤C}

ϑAxϑx1−ϑ, x∈DA. 3.47

SinceG·is aDAvalued function, for allx∈X

_−Aϑ_{Gλx ≤C}

ϑAGλxϑGλx1−ϑ

≤CϑMϑ3xϑ

M₁1−ϑ

|λ−ω|1−ϑx

1−ϑ

≤ C

|λ−ω|1−ϑx,

whereCis independent ofλ. From3.48, we get fort≥1

From the previous facts, we conclude that

_−Aϑ_{Rt ≤Ce}rωt_t−ϑ_{, t∈0,1, 3.51}

which ends the proof.

Corollary 3.13. Ifωr <0andϑ∈0,1, then there existsφ∈L1_{0,∞such that}

_−Aϑ_{Rt ≤φt. 3.52}

In the remainder of this section, we discuss the existence and regularity of solutions of

Definition 3.14. A functionx :0, b → X, 0 < b ≤ a, is called a classical solution of 3.53

3.54on0, bifx∈C0, b,DA∩C1_{0, b, X, the condition3.54holds and3.53is} verified on0, a.

The next result has been established in30.

Theorem 3.1530, Theorem 2. Letz∈X. Assume thatf ∈C0, a, Xandx·is a classical solution of 3.53-3.54on0, a. Then

xt Rtz

_t

0

Rt−sfsds, t∈0, a. 3.55

An immediate consequence of the above theorem is the uniqueness of classical solutions.

Corollary 3.16. Ifu, vare classical solutions of 3.53-3.54on0, b, thenuvon0, b.

Motivated by3.55, we introduce the following concept.

Definition 3.17. A functionu∈C0, a, Xis called a mild solution of3.53-3.54if

ut Rtz

_t

0

Rt−sfsds, t∈0, a. 3.56

4. Existence Result of Asymptotically Almost Periodic Solutions

In this section, we study the existence of asymptotically almost periodic mild solutions for the abstract integro-differential system1.1. To establish our existence result, motivated by the previous section we introduce the following assumptions.

P1There exists a Banach space Y, · Y continuously included in X such that the following conditions are verified.

aFor every t ∈ 0,∞, Rt ∈ LX∩ LY,DA and Bt ∈ LY, X. In addition,AR·x, B·x∈C0,∞, Xfor everyx∈Y.

bThere are positive constantsM, βsuch that

maxRs,BsLY,X

≤Me−βt, s≥0. 4.1

cThere existsφ∈L1_{0,∞such that ARt}

LY,X≤φt, t≥0.

P2The continuous functionf : R× B → Y is p.a.a.p, and there exists a continuous functionLf :0,∞ → 0,∞,such that

ft, ψ1

−ft, ψ2

Y ≤Lfrψ1−ψ2B, t, ψj

P3The continuous functiong : R× B → X is p.a.a.p, and there exists a continuous

which implies that

be the function defined by

which implies that

wtξ−wt ≤ε, t≥T

ε

3η

−1_{, v, Y}

T1, ξ∈ T

ε

3η

−1_{, v, Y}

. 4.11

From inequality4.11andLemma 2.7, we conclude thatw·is a.a.p., which ends the proof.

Now, we can establish our existence result.

Theorem 4.4. Assume thatBis a fading memory space andP1,P2, andP3are held. IfLf0

Lg0 0 andft,0 gt,0 0 for every t ∈ R, then there existsε > 0 such that for each

ϕ∈Bε0,B, there exists a mild solution,u·, ϕ, of1.1on0,∞such thatu·, ϕ∈AAPXand

u0·, ϕ ϕ.

Proof. Letr >0 and 0< λ <1 be such that

Θ MHλMLfλrλLfλ1Krλ1K

!

ic_LY,X φ L1

M2 β2

"

Lgλ1Krλ1K

M β <1,

4.12

whereKis the constant introduced inRemark 2.3. We affirm that the assertion holds forε λr.Letϕ∈Bε0,B.On the space

Dx∈AAPX:x0 ϕ0, xt ≤r, t≥0 4.13

endowed with the metricdu, v u−v, we define the operatorΓ:D → C0,∞;Xby

Γut Rtϕ0 f0, ϕ−ft,u#t−

_t

0

ARt−sfs,u#sds

−

_t

0

Rt−s

_s

0

Bs−ξfξ,u#ξ

dξds

_t

0

Rt−sgs,u#sds, t≥0,

4.14

Next, we prove thatΓ·is a contraction fromDintoD. Ifu∈Dandt≥0, we get

5. Applications

In this section, we study the existence of asymptotically almost periodic solutions of the partial neutral integro-differential system

∂ space of infinitely differentiable functions that vanish atξ 0 andξ π. Under the above conditions, we can represent the system

We define the functionsf, g:B → Xby

≤ 1

s−γμ

!

1

s−γμ1

γ

s−γμ

"

M

λ−μ

≤ 1

K

_1Kγ

K

M

_λ−μ

≤ 1

K

1

K,

5.7

since s ≥ r max{MK 1 γ, K |γ μ|}. By using a similar procedure as in the proofs ofLemma 3.3andTheorem 3.10, we obtain the existence of resolvent operator for5.2. From the hypothesis, we obtainμr < 0; by theLemma 3.3, Corollaries3.11and3.13, the assumptionP1is satisfied. FromTheorem 4.4, the proof is complete.

Acknowledgment

Jos´e Paulo C. dos Santos is partially supported by FAPEMIG/Brazil under Grant CEX-APQ-00476-09.

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