抽象泛函微分方程的权伪概自守温和解（英文）

雷国梁　岳田　宋晓秋

摘要利用算子半群理论和Banach 不动点定理研究了一类抽象泛函微分方程权伪概自守温和解的存在唯一性，所得结论拓展了已有结果.

关键词权伪概自守；抽象泛函微分方程；指数稳定；存在唯一性

The theory of almost automorphy was first introduced in the literature by Bochner in the earlier sixties， which is a natural generalization of almost periodicity[1]， for more details about this topics we refer to the recent book[2] where the author gave an important overview on the theory of almost automorphic functions and their applications to differential equations. In the last decade， several authors including Ezzinbi， Goldstein， NGuérékata and others， have extended the theory of almost automorphy and its applications to differential equations[17].

Xiao， Liang and Zhang[8] postulated a new concept of a function called a pseudoalmost automorphic function， established existence and uniqueness theorems of pseudoalmost automorphic solutions to some semilinear abstract differential equations and studied two composition theorems about pseudoalmost automorphic functions as well as asymptotically almost automorphic functions （Theorems 2.3 and 2.4， [8]）.

Weighted pseudoalmost automorphic functions are more general than weighted pseudoalmost periodic functions which were introduced by Diagana[911] and recently studied by Hacene， Ezzinbi[1213]， Ding[14]. Blot， Mophou， NGuérékata， Pennequin[15] and Liu[1617] have studied basic properties of weighted pseudoalmost automorphic functions and then used these results to study the existence and uniqueness of weighted pseudoalmost automorphic mild solutions to some abstract differential equations.

Motivated by works [13，16，18]， we consider the existence and uniqueness of the weighted pseudo almost automorphic mild solution of the following semilinear evolution equation in a Banach space X

dx（t）dt=A（t）x（t）+ddtF1（t，x（a（t）））+F2（t，x（b（t））），t∈R，x∈WPPA（R，ρ），（1）

where WPAA（R，ρ ） is the set of all weighted pseudo almost automorphic functions from R to X and the family {A（t），t∈R} of operators in X generates an exponentially stable evolution family {U （t， s），t. s}.

湖南师范大学自然科学学报第38卷第5期雷国梁等：抽象泛函微分方程的权伪概自守温和解1Preliminaries

In this section， we introduce definitions， notations， lemmas and preliminary facts which are used throughout this work. We assume that X is a Banach space endowed with the norm ||·||.N， R and C stand for the sets of positive integer， real and complex numbers. We denote by B（X） the Banach space of all bounded linear operators from X to itself. BC（R， X ）（BC（R×X， X）） is the space of all bounded continuous functions from R to X（R× X to X）. L1loc（R） denote the space of locally integrable functions on R. Let U be the collection of functions （weights） ρ：R→ （0，+∞）， which are locally integrable over R with.ρ>0（a.e.）. From now， if ρ∈U and for r>0， we then set m（r，ρ）=∫r-rρ（t）dt， U∞：={ρ∈U：limr→∞ m（r，ρ）=∞}， UB：={ρ∈U∞： ρ is bounded and infx∈R ρ（x）>0}.

It is clear that UBU∞U with strict inclusions. {U （t， s），t≥s} is an exponentially stable evolution family， if there exists M≥1 and δ>0 such that ‖U（t，s）‖≤Me-δ（t-s） for t≥s.

Definition 1.1[19]A continuous function f：R→X is said to be almost automorphic if for every sequence of real numbers {sn}n∈N， there exists a subsequence {τn}n∈N such that g（t）=limn→∞ f（t+τn） is well defined for each t∈R and limn→∞ g（t-τn）=f（t） for each t∈R.

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

Definition 1.2[19]A continuous function f：R×X→X is said to be almost automorphic if f （t， x） is almost automorphic for each t∈R uniformly for all x∈B， where B is any bounded subset of X.

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

Lemma 1.3[1]Assume that f：R→X is almost automorphic， then f is bounded.

Lemma 1.4[1]（AA（X），‖·‖AA（X）） is a Banach space endowed with the supremum norm given by ‖f‖AA（X）=supt∈R‖f（t）‖.

Lemma 1.5[20]Let f：R×X→X be almost automorphic in t∈R，x∈X and assume that f（t，x） satisfies a Lipschitz condition in x uniformly in t∈R Then x（t）∈AA（X） implies f（t， x（t））∈AA（X）.

The notation PAA0，PAA0（R×X，X） respectively， stand for the space of functions

PAA0（X）={φ（t）∈BC（R，X）：limr→∞∫r-r‖φ（t）‖dt=0}，

PAA0（R×X，X）=φ（t）∈BC（R×X，X）：limr→∞∫r-r‖φ（t，x）‖dt=0

uniformly for x in any bounded subset of X.

Definition 1.6[8]A continuous function f：R→X（R×X→X） is said to be pseudo almost automorphic if it can be decomposed as f=g+φ， where g∈AA（X）（AA（R×X，X）） and φ∈PAA0（X）（PAA0（R×X，X））.

Denote by PAA（X）（PAA（R×X， X）） the set of all such functions.

Now for ρ∈U∞， we define

PAA0（R，ρ）：={φ（t）∈BC（R，X）：limr→∞1m（r，ρ）∫r-r‖φ（t）‖ρ（t）dt=0}，

PAA0（R×X，ρ）=φ（t，x）∈BC（R×X，X）：limr→∞1m（r，ρ）∫r-r‖φ（t，x）‖ρ（t）dt=0

uniformly for x∈X.

Definition 1.7[15]A bounded continuous function f：R→X（R×X→X） is said to be weighted pseudo almost automorphic if it can be decomposed as f=g+φ， where

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

Denote by WPAA（R，ρ）（WPAA（R×X，ρ）） the set of all such functions.

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

Lemma 1.9[15]If ρ∈UB，（WPAA（R，ρ），‖·‖WPAA（R，ρ）） is a Banach space endowed with the supremum norm given by ‖f‖WPAA（R，ρ）=supt∈R‖f（t）‖.

Lemma 1.10[15]Let f=g+φ∈WPAA（R，ρ） where ρ∈U∞，g∈AA（R×X，X） and φ∈PAA0（R×X，ρ）. Assume both f and g are Lipschitzian in x∈X uniformly in t∈R Then x（t）∈WPAA（R，ρ） implies f（t， x（t））∈WPAA（R，ρ）.

Lemma 1.11[16]Let ∑θ={z∈C：|arg z|≤θ}∪{0}ρ（A（t）），θ∈（-π/2，π）， if there exist a constant K0 and a set of real numbers α1，α2，…，αk，β1，…，βk with 0≤βi<αi≤2，（i=1，2，…，k） such that

‖A（t）（λ-A（t））-1（A（t）-1-A（s）-1）‖≤K0∑ki=1（t-s）αi|λ|βi-1，

for t，s∈R，λ∈∑θ＼{0} and there exists a constant M≥0 such that

‖（λ-A（t））-1‖≤M1+λ，λ∈∑θ.

Then there exists a unique evolution family {U（t，s），t≥s>-∞}.

Definition 1.12A continuous function x（t）：R→X is called weighted pseudo almost automorphic mild solution to equation （1） if it satisfies

x（t）=F1（t，x（a（t）））+U（t，s）[x（s）-F1（s，x（a（s）））]+∫tsU（t，r）F2（r，x（b（r）））dr，（2）

for t≥s and s∈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） and there exist two positive constants Li（i=1，2） such that ‖Fi（t，x）-Fi（t，y）‖≤Li‖x-y‖WPAA（R×X，ρ） for all t∈R and x，y∈WPAA（R，ρ）， ρ∈U∞.

（H2） a，b∈C（R，R），a（R）=R，b（R）=R and there exist two positive constants Ki（i=1，2） such that ‖x（a）-y（a）‖≤K1‖x-y‖WPAA（R，ρ），‖x（b）-y（b）‖≤K2‖x-y‖WPAA（R，ρ）， with a，b∈WPAA（R，ρ） whenever x，y∈WPAA（R，ρ）.

（H3） {A（t），t∈R} satisfies Lemma 1.11 and {U （t，s），t≥s} is exponentially stable.

（H4） For every sequence of real numbers {sn}n∈N， there exists a subsequence {τn}n∈N and for any fixed s∈R，ε>0， there exists an N∈N such that， for all n>N， it follows that

‖U（t+τn，s+τn）-U（t，s）‖≤εe-δ（t-s）/2，

for all t≥s∈R. Moreover

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

Lemma 2.1If h（s） is almost automorphic， the function v（t）=∫t-∞U（t，s）h（s）ds is almost automorphic.

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

‖v（t）‖≤∫t-∞‖U（t，s）‖·‖h（s）‖ds≤MM1∫t-∞e-δ（t-s）ds≤MM1δ<∞.

Hence v（t） is bounded. Now we show that v（t） is almost automorphic with respect to t∈R. Let {sn}n∈N be an arbitrary sequence of real numbers. Since h（t）∈AA（X）， there exist a subsequence {τn}n∈N such that

（A1） g（t）=limn→∞ h（t+τn） is well defined for each t∈R；

（A2） limn→∞ g（t-τn）=h（t） for each t∈R.

Now we consider

v（t+τn）=∫t+τn-∞U（t+τn，s）h（s）ds=∫t-∞U（t+τn，s+τn）h（s+τn）ds.

Obviously， v（t+τn） is bounded for all n=1，2，….

For （A1）， for any fixed s∈R and ε>0， there exists an N0∈N such that， for all n>N0， which follows that ‖h（s+τn）-g（s）‖<ε. In addition， by condition （H4）， for s and ε above， there exists an N1∈N such that， for all n>N1， it follows that ‖U（t+τn，s+τn）-U（t，s）‖≤εe-δ（t-s）/2.

Now taking N=max{N0，N1}， for all n>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

As n→∞， for each s∈R fixed and any t≥s， we have

U（t+τn，s+τn）h（s+τn）→U（t，s）g（s）.

If we let u（t）=∫t-∞U（t，s）g（s）ds， we observe that the integral is absolutely convergent for each t. So， by Lebesgues dominated convergent theorem， v（t+τn）→u（t） as n→∞ for each t∈R. We can show a similar way that u（t-τn）→v（t） as n→∞ for each t∈R. Hence v（t） is almost automorphic with respect to t∈R. This completes the proof.

We define a mapping T by

（Tx）（t）=F1（t，x（a（t）））+∫t-∞U（t，s）F2（s，x（b（s）））ds，t∈R.

Theorem 2.2F1（t，·），F2（t，·）∈WPAA（R×X，ρ） and U（t，s） satisfies the conditions （H1）（H4）. If x（t） is weighted pseudo almost automorphic， then G（t）=∫t-∞U（t，s）F2（s，x（b（s）））ds and H（t）=F1（t，x（a（t））） are weighted pseudo almost automorphic. Moreover， the function Tx is weighted pseudo almost automorphic.

ProofLet x∈WPAA（R，ρ）. Obviously， the function x（a（t））， x（b（t）） are weighted pseudo almost automorphic. By the composition theorem of weighted almost automorphic functions in [14] or Lemma 1.10， it follows that H（t）=F1（t，x（a（t））），F2（t，x（b（t）））∈WPAA（R×X，X）.

Let

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

Then

G（t）=∫t-∞U（t，s）h（s）ds+∫t-∞U（t，s）（s）ds：=G1（t）+G2（t）.

By Lemma 2.1， we know that G1∈AA（X）， so G1（t） is almost automorphic.

Next， in order to show that G（t） is weighted pseudo almost automorphic， we need to show G2（t）∈PAA0（R，ρ）. Let Γ（ρ）=1m（r，ρ）∫r-r‖G2（t）‖ρ（t）dt， we will show limr→∞ Γ（ρ）=0. It is obvious that Γ（ρ）≥0.

By using the Fubini theorem， we have

Γ（ρ）≤limr→∞1m（r，ρ）∫r-rdt∫-r-∞Me-δ（t-s）‖（s）‖ρ（s）ds+limr→∞1m（r，ρ）∫r-rdt∫t-rMe-δ（t-s）‖（s）‖ρ（s）ds=

limr→∞1m（r，ρ）∫-r-∞eδs‖（s）‖ds∫r-rMe-δtρ（t）dt+limr→∞1m（r，ρ）∫r-r‖（s）‖ρ（t）dt∫t-rMe-δ（t-s）ds≤

limr→∞1m（r，ρ）supt∈R‖（t）‖‖ρ‖L1loc（R）Mδ2[1-e-2δr]+

limr→∞1m（r，ρ）Mδ2[1-e-δ（t+r）]∫r-r‖（t）‖ρ（t）dt：=Γ1（ρ）+Γ2（ρ）.

Since （t） is bounded and limr→∞ m（r，ρ）=∞， then Γ1（ρ）=0. In addition， we know the fact that δ>0，-r≤t≤r and ∈PAA0（R，ρ）， so Γ2（ρ）=0.

Furthermore， Tx is weighted pseudo almost automorphic. The proof is completed.

Theorem 2.3Let F1（T，·），F2（t，·） and U （t，s） satisfy all the conditions of （H1）， （H2）， （H3）， （H4）， and 0

ProofBy Theorem 2.2， we can see T maps WPAA（R，ρ） into WPAA（R，ρ） .

Let x，y∈WPAA（R，ρ）， and notice that

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

‖∫t-∞U（t，s）[F2（s，x（b（s）））-F2（s，y（b（s）））]ds‖≤

L1‖x（a（t））-y（a（t））‖+L2∫t-∞‖U（t，s）‖·‖x（b（s））-y（b（s））‖ds≤

L1K1‖x-y‖WPAA（R，ρ）+ML2K2‖x-y‖WPAA（R，ρ）∫t-∞e-δ（t-s）ds≤

（L1K1+ML2K2δ）‖x-y‖WPAA（R，ρ）.

So we have

‖（Tx）（t）-（Ty）（t）‖WPAA（R，ρ）≤（L1K1+ML2K2δ）‖x-y‖WPAA（R，ρ）.

For 0

Therefore， by the Banach fixed point theorem， T has a unique fixed point x∈WPAA（R，ρ） such that Tx=x.

Fixing s∈R we have

x（t）=F1（t，x（a（t）））+∫t-∞U（t，r）F2（r，x（b（r）））dr.

Since U（t，s）=U（t，r）U（r，s）， for t≥r≥s （see[21， Chapter 5， Theorem 5.2]），

let x（τ）=F1（τ，x（a（τ）））+∫τ-∞U（τ，s）F2（s，x（b（s）））ds. So

U（t，τ）x（τ）=U（t，τ）F1（τ，x（a（τ）））+∫τ-∞U（t，s）F2（s，x（b（s）））ds.

For t≥τ，

∫τU（t，s）F2（s，x（b（s）））ds=∫t-∞U（t，s）F2（s，x（b（s）））ds-∫τ-∞U（t，s）F2（s，x（b（s）））ds=

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

So that

x（t）=F1（t，x（a（t）））+U（t，τ）[x（τ）-F1（τ，x（a（τ）））]+∫tτU（t，s）F2（s，x（b（s）））ds.

It follows that x（t） satisfies equation （2）. Hence x（t） is a mild solution to equation （1）.

In conclusion， x（t） is the unique mild solution to equation （1）， which completes the proof.

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

dx（t）dt=Ax（t）+ddtF1（t，x（a（t）））+F2（t，x（b（t）））， t∈R，x∈WPAA（X），

where A is the infinitesimal generator of a C0semigroup {T（t）}t≥0. In this case we have the mild solution given by

x（t）=F1（t，x（a（t）））+∫t-∞T（t-s）F2（s，x（b（s）））ds， for t∈R.

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

dx（t）dt=A（t）x（t）+ddtF1（t，x（a（t）））+F2（t，x（b（t）））， t∈R，x∈PAA（X）.

In this case we have the mild solution given by

x（t）=F1（t，x（a（t）））+∫t-∞U（t，s）F2（s，x（b（s）））ds， for t∈R.

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

dx（t）dt=Ax（t）+ddtF1（t，x（a（t）））+F2（t，x（b（t）））， t∈R，x∈PAA（X）.

In this case we have the weighted pseudo almost automorphic mild solution given by

x（t）=F1（t，x（a（t）））+∫t-∞T（t-s）F2（s，x（b（s）））ds， for t∈R.

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（编辑胡文杰）