Weighted Pseudo Almost Automorphic Mild Solutions to Abstract Functional Differential Equations

LEI Guo-liang,

(1.School of Science, Hubei University of Automotive Technology, Shiyan 442002, China；2.College of Science, China University of Mining and Technology, Xuzhou 442002, China)

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WeightedPseudoAlmostAutomorphicMildSolutionstoAbstractFunctionalDifferentialEquations

LEI Guo-liang1,YUETian1

(1.SchoolofScience,HubeiUniversityofAutomotiveTechnology,Shiyan442002,China；2.CollegeofScience,ChinaUniversityofMiningandTechnology,Xuzhou442002,China)

AbstractIn this paper, by using the theory of semigroups of operators and Banach fixed point theorem, the existence and uniqueness of weighted pseudo almost automorphic mild solutions to a class of abstract functional differential equation in Banach space are obtained under some suitable hypotheses, which extend some known results.

Key wordsweighted pseudo almost automorphic; abstract functional differential equation; exponentially stable; existence and uniqueness

ThetheoryofalmostautomorphywasfirstintroducedintheliteraturebyBochnerintheearliersixties,whichisanaturalgeneralizationofalmostperiodicity[1],formoredetailsaboutthistopicswerefertotherecentbook[2]wheretheauthorgaveanimportantoverviewonthetheoryofalmostautomorphicfunctionsandtheirapplicationstodifferentialequations.Inthelastdecade,severalauthorsincludingEzzinbi,Goldstein,N’Guérékataandothers,haveextendedthetheoryofalmostautomorphyanditsapplicationstodifferentialequations[1-7].

Xiao,LiangandZhang[8]postulatedanewconceptofafunctioncalledapseudo-almostautomorphicfunction,establishedexistenceanduniquenesstheoremsofpseudo-almostautomorphicsolutionstosomesemilinearabstractdifferentialequationsandstudiedtwocompositiontheoremsaboutpseudo-almostautomorphicfunctionsaswellasasymptoticallyalmostautomorphicfunctions(Theorems2.3and2.4,[8]).

Weightedpseudo-almostautomorphicfunctionsaremoregeneralthanweightedpseudo-almostperiodicfunctionswhichwereintroducedbyDiagana[9-11]andrecentlystudiedbyHacene,Ezzinbi[12-13],Ding[14].Blot,Mophou,N’Guérékata,Pennequin[15]andLiu[16-17]havestudiedbasicpropertiesofweightedpseudo-almostautomorphicfunctionsandthenusedtheseresultstostudytheexistenceanduniquenessofweightedpseudo-almostautomorphicmildsolutionstosomeabstractdifferentialequations.

Motivatedbyworks[13,16,18],weconsidertheexistenceanduniquenessoftheweightedpseudoalmostautomorphicmildsolutionofthefollowingsemilinearevolutionequationinaBanachspaceX

(1)

where WPAA(R,ρ) is the set of all weighted pseudo almost automorphic functions fromRtoXandthefamily{A(t),t∈R} of operators inXgenerates an exponentially stable evolution family {U(t,s),t.s}.

1Preliminaries

ItisclearthatUB?U∞?Uwithstrictinclusions. {U (t, s),t≥s}isanexponentiallystableevolutionfamily,ifthereexistsM≥1andδ>0suchthat‖U(t,s)‖≤Me-δ(t-s)fort≥s.

ThecollectionofallsuchfunctionswillbedenotedbyAA(X).

Definition1.2[19]Acontinuousfunctionf:R×X→Xis said to be almost automorphic iff(t,x) is almost automorphic for eacht∈R uniformly for allx∈B, whereBis any bounded subset ofX.

The collection of all such functions will be denoted by AA(R×X,X).

Lemma1.3[1]Assumethatf:R→Xis almost automorphic, thenfis bounded.

Lemma1.5[20]Letf:R×X→Xbe almost automorphic int∈R,x∈Xand assume thatf(t,x) satisfies a Lipschitz condition inxuniformly int∈R Thenx(t)∈AA(X) impliesf(t,x(t))∈AA(X).

The notation PAA0,PAA0(R×X,X)respectively,standforthespaceoffunctions

Definition 1.6[8]A continuous functionf:R→X(R×X→X)issaidtobepseudoalmostautomorphicifitcanbedecomposedasf=g+φ,whereg∈AA(X)(AA(R×X,X))andφ∈PAA0(X)(PAA0(R×X,X)).

DenotebyPAA(X)(PAA(R×X, X))thesetofallsuchfunctions.

Nowforρ∈U∞,wedefine

Definition 1.7[15]A bounded continuous functionf:R→X(R×X→X)issaidtobeweightedpseudoalmostautomorphicifitcanbedecomposedasf=g+φ,where

g∈AA(X)(AA(R×X,X))andφ∈PAA0(R,ρ)(PAA0(R×X,ρ)).

DenotebyWPAA(R,ρ)(WPAA(R×X,ρ))thesetofallsuchfunctions.

Lemma 1.8[15]The decomposition of a weighted pseudo almost automorphic function is unique for anyρ∈UB.

Lemma 1.10[15]Letf=g+φ∈WPAA(R,ρ)whereρ∈U∞,g∈AA(R×X,X)andφ∈PAA0(R×X,ρ).AssumebothfandgareLipschitzianinx∈Xuniformlyint∈RThenx(t)∈WPAA(R,ρ)impliesf(t, x(t))∈WPAA(R,ρ).

Lemma 1.11[16]Let ∑θ={z∈C:|argz|≤θ}∪{0}?ρ(A(t)),θ∈(-π/2,π),ifthereexistaconstantK0andasetofrealnumbersα1,α2,…,αk,β1,…,βkwith0≤βi<αi≤2,(i=1,2,…,k)suchthat

fort,s∈R,λ∈∑θ{0}andthereexistsaconstantM≥0suchthat

Thenthereexistsauniqueevolutionfamily{U(t,s),t≥s>-∞}.

Definition 1.12A continuous functionx(t):R→Xiscalledweightedpseudoalmostautomorphicmildsolutiontoequation(1)ifitsatisfies

(2)

fort≥sands∈R.

2The Main Results

To show our main results, we assume that the following conditions are satisfied.

(H1)F1(t,·)∈WPAA(R×X,X),(i=1,2)andthereexisttwopositiveconstantsLi(i=1,2)suchthat‖Fi(t,x)-Fi(t,y)‖≤Li‖x-y‖WPAA(R×X,ρ)forallt∈Randx,y∈WPAA(R,ρ), ρ∈U∞.

(H2) a,b∈C(R,R),a(R)=R,b(R)=RandthereexisttwopositiveconstantsKi(i=1,2)suchthat‖x(a)-y(a)‖≤K1‖x-y‖WPAA(R,ρ),‖x(b)-y(b)‖≤K2‖x-y‖WPAA(R,ρ),witha,b∈WPAA(R,ρ)wheneverx,y∈WPAA(R,ρ).

(H3) {A(t),t∈R}satisfiesLemma1.11and{U (t,s),t≥s}isexponentiallystable.

(H4)Foreverysequenceofrealnumbers{sn}n∈N,thereexistsasubsequence{τn}n∈Nandforanyfixeds∈R,ε>0,thereexistsanN∈Nsuchthat,foralln>N,itfollowsthat

‖U(t+τn,s+τn)-U(t,s)‖≤εe-δ(t-s)/2,

forallt≥s∈R.Moreover

‖U(t-τn,s-τn)-U(t,s)‖≤εe-δ(t-s)/2,forallt≥s∈R.

ProofFirst we observe thatv(t) is bounded. By Lemma 1.3,h(s) is bounded, we assume that there existsM1>0, such that ‖h(·)‖AA(X)≤M1. So

Hencev(t) is bounded. Now we show thatv(t) is almost automorphic with respect tot∈R.Let{sn}n∈Nbeanarbitrarysequenceofrealnumbers.Sinceh(t)∈AA(X),thereexistasubsequence{τn}n∈Nsuchthat

Nowweconsider

Obviously, v(t+τn)isboundedforalln=1,2,….

For(A1),foranyfixeds∈Randε>0,thereexistsanN0∈Nsuchthat,foralln>N0,whichfollowsthat‖h(s+τn)-g(s)‖<ε.Inaddition,bycondition(H4),forsandεabove,thereexistsanN1∈Nsuchthat,foralln>N1,itfollowsthat‖U(t+τn,s+τn)-U(t,s)‖≤εe-δ(t-s)/2.

NowtakingN=max{N0,N1},foralln>N,

‖U(t+τn,s+τn)h(s+τn)-U(t,s)g(s)‖≤

‖U(t,s)‖‖h(s+τn)-g(s)‖+‖U(t+τn,s+τn)-U(t,s)‖‖h(s+τn)‖≤

Mεe-δ(t-s)+M1εe-δ(t-s)/2

Asn→∞,foreachs∈Rfixedandanyt≥s,wehave

U(t+τn,s+τn)h(s+τn)→U(t,s)g(s).

WedefineamappingTby

ProofLetx∈WPAA(R,ρ).Obviously,thefunctionx(a(t)), x(b(t))areweightedpseudoalmostautomorphic.Bythecompositiontheoremofweightedalmostautomorphicfunctionsin[14]orLemma1.10,itfollowsthatH(t)=F1(t,x(a(t))),F2(t,x(b(t)))∈WPAA(R×X,X).

Let

F2(t,x(b(t)))=h(t)+φ(t),whereh∈AA(X), φ∈PAA0(R,ρ).

Then

ByLemma2.1,weknowthatG1∈AA(X),soG1(t)isalmostautomorphic.

ByusingtheFubinitheorem,wehave

Furthermore, Txisweightedpseudoalmostautomorphic.Theproofiscompleted.

ProofBy Theorem 2.2, we can seeTmaps WPAA(R,ρ)intoWPAA(R,ρ) .

Letx,y∈WPAA(R,ρ),andnoticethat

‖(Tx)(t)-(Ty)(t)‖≤‖F1(t,x(a(t)))-F1(t,y(a(t)))‖+

Sowehave

Therefore,bytheBanachfixedpointtheorem, Thasauniquefixedpointx∈WPAA(R,ρ)suchthatTx=x.

Fixings∈Rwehave

SinceU(t,s)=U(t,r)U(r,s),fort≥r≥s (see[21,Chapter5,Theorem5.2]),

Fort≥τ,

x(t)-F1(t,x(a(t)))-U(t,τ)x(τ)+U(t,τ)F1(τ,x(a(τ))).

Sothat

Itfollowsthatx(t)satisfiesequation(2).Hencex(t)isamildsolutiontoequation(1).

Inconclusion, x(t)istheuniquemildsolutiontoequation(1),whichcompletestheproof.

Remark 2.4WhenU(t,s)=T(t-s), we can deal with the existence and uniqueness of a weighted pseudo almost automorphic solution for

whereAistheinfinitesimalgeneratorofaC0-semigroup{T(t)}t≥0.Inthiscasewehavethemildsolutiongivenby

Remark 2.5Whenρ=1, we can deal with the existence and uniqueness of a pseudo almost automorphic solution for

Inthiscasewehavethemildsolutiongivenby

Remark 2.6WhenU(t,s)=T(t-s),ρ=1, we can also deal with the existence and uniqueness of a weighted pseudo almost automorphic solution for

Inthiscasewehavetheweightedpseudoalmostautomorphicmildsolutiongivenby

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(編輯胡文杰)

抽象泛函微分方程的權偽概自守溫和解

雷國梁1，岳田1*，宋曉秋2

(1.湖北汽車工業學院理學院，中國 十堰442002；2. 中國礦業大學理學院,中國 徐州221116)

摘要利用算子半群理論和Banach 不動點定理研究了一類抽象泛函微分方程權偽概自守溫和解的存在唯一性，所得結論拓展了已有結果．

關鍵詞權偽概自守；抽象泛函微分方程；指數穩定；存在唯一性

中圖分類號O175.1

文獻標識碼A

文章編號1000-2537(2015)05-0084-07

通訊作者*,E-mail：yuetian@cumt.edu.cn, SONG Xiao-qiu2

基金項目：國家自然科學基金資助項目(51374199); 中央高?；究蒲袠I務費專項資金項目(2012LWB53); 湖北省自然科學基金資助項目(2014CFB629)

收稿日期：2014-07-14

DOI:10.7612/j.issn.1000-2537.2015.05.014