# Pseudo Asymptotic Behavior of Mild Solution for Nonautonomous Integrodifferential Equations with Nondense Domain

Authors: Zhinan Xia
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

By the weighted ergodic function based on the measure theory, we study pseudo asymptotic behavior of mild solution for nonautonomous integrodifferential equations with nondense domain. The existence and uniqueness ofμ-pseudo antiperiodic (μ-pseudo periodic, μ-pseudo almost periodic, and μ-pseudo automorphic) solution are investigated. Some interesting examples are presented to illustrate the main findings.


---

## Body

## 1. Introduction

The study of pseudo asymptotic behavior of solution is one of the most interesting and important topics in the qualitative theory of differential equations. Much work has been done to investigate the existence of pseudo antiperiodic, pseudo periodic, pseudo almost periodic, and pseudo almost automorphic solution for differential equations [1–5]. Recently, Blot et al. [6, 7] used the results of the measure theory to establish μ-ergodic and introduce the new concepts of μ-pseudo almost periodic and μ-pseudo almost automorphic function, which are more general than pseudo almost periodic and pseudo almost automorphic function, respectively. They developed some results like completeness and composition theorems to investigate differential equations in Banach space.Integrodifferential equations play a crucial role in qualitative theory of differential equations due to their application to physics, engineering, biology, and other subjects. This type of equations has received much attention in recent years and the general asymptotic behavior of solution is at present an active source of research.In this paper, we study pseudo asymptotic behavior of solution to the following nonautonomous integrodifferential equations with nondense domain:(1)u′(t)=A(t)u(t)+f(t,u(t),(Ku)(t)),t∈R,(Ku)(t)=∫-∞tk(t-s)g(s,u(s))ds,
where the linear operators A(t):D(A(t))⊂X→X have a domain D(A(t)) not necessarily dense in Banach space X and satisfy “Acquistapace-Terreni” conditions and f, g, and k are continuous functions.Some recent contributions on almost periodic, almost automorphic, pseudo almost periodic, and pseudo almost automorphic solution to integrodifferential equations of the form (1) in the case A(t)=A are constant [4, 8–13]. However, for the nonautonomous case, that is, (1), the study of pseudo asymptotic behavior of solution is rare [14]. In this paper, we will make use of the so-called “Acquistapace-Terreni” conditions associated with exponential dichotomy, fixed point theorem to derive some sufficient conditions to the existence and uniqueness of μ-pseudo antiperiodic (μ-pseudo periodic, μ-pseudo almost periodic, and μ-pseudo almost automorphic) mild solution to (1).The paper is organized as follows. In Section2, we recall some fundamental results about the notion of P-class of functions including composition theorem. Section 3 is devoted to pseudo asymptotic behavior of mild solution to nonautonomous integrodifferential equations with nondense domain. In Section 4, we provide some examples to illustrate our main results.


## 2. Preliminaries and Basic Results

Let(X,∥·∥) and (Y,∥·∥) be two Banach spaces and N, Z, R, and C stand for the set of natural numbers, integers, real numbers, and complex numbers, respectively. For A being a linear operator on X, D(A), ρ(A), R(λ,A), and σ(A) stand for the domain, the resolvent set, the resolvent, and spectrum of A. In order to facilitate the discussion below, we further introduce the following notations:(i)
BC(R,X) (resp., BC(R×Y,X)): the Banach space of bounded continuous functions from R to X (resp., from R×Y to X) with the supremum norm;(ii)
C(R,X) (resp. C(R×Y,X)): the set of continuous functions from R to X (resp., from R×Y to X);(iii)
L(X,Y): the Banach space of bounded linear operators from X to Y endowed with the operator topology; in particular, we write L(X) when X=Y;(iv)
Lp(R,X): the space of all classes of equivalence (with respect to the equality almost everywhere on R) of measurable functions f:R→X such that ∥f∥∈Lp(R,R).


### 2.1. Evolution Family and Exponential Dichotomy

Definition 1.
A family of bounded linear operators(U(t,s))t≥s on a Banach space X is called a strong continuous evolution family if(i)
U(t,r)U(r,s)=U(t,s) and U(s,s)=I for all t≥r≥s and t,r,s∈R;(ii)
the map(t,s)→U(t,s)x is continuous for all x∈X,t≥s and t,s∈R.Definition 2.
An evolution family(U(t,s))t≥s on a Banach space X is called hyperbolic (or has an exponential dichotomy) if there exist projections P(t), t∈R, uniformly bounded and strongly continuous in t and constants c>0, δ>0 such that(i)
U(t,s)P(s)=P(t)U(t,s) for t≥s and t,s∈R,(ii)
the restrictionUQ(t,s):Q(s)X→Q(t)X of U(t,s) is invertible for t≥s (and set UQ(s,t)∶=U(t,s)-1),(iii)
(2)∥U(t,s)P(s)∥≤Me-δ(t-s),∥UQ(s,t)Q(t)∥≤Me-δ(t-s),for t≥s and t,s∈R. Here and next we set Q∶=I-P.Remark 3.
Exponential dichotomy is a classical concept in the study of long-time behaviour of evolution equations. IfP(t)=I for t∈R, then (U(t,s))t≥s is exponentially stable. One can see [15–17] for more details.
If(U(t,s))t≥s is hyperbolic, then
(3)Γ(t,s)∶={U(t,s)P(s),t≥s,t,s∈R,-UQ(t,s)Q(s),t<s,t,s∈R,
is called Green’s function corresponding to (U(t,s))t≥s, P(·) and
(4)∥Γ(t,s)∥≤{Me-δ(t-s),t≥s,t,s∈R,Me-δ(s-t),t<s,t,s∈R.


### 2.2.μ-Ergodic and Functions by Measure Theory

B denotes the Lebesgue σ-field of R and M stands for the set of all positive measures μ on B satisfying μ(R)=∞ and μ([a,b])<∞ for all a,b∈R(a≤b). We formulate the following hypothesis.(H0)
For allτ∈R, there exist β>0 and a bounded interval I such that
(5)μ({a+τ,a∈A})≤βμ(A)ifA∈BsatisfiesA∩I=∅.Definition 4 (see [6]).
Letμ∈M. A function f∈BC(R,X) is said to be μ-ergodic if
(6)lim⁡T→+∞1μ([-T,T])∫[-T,T]‍∥f(t)∥dμ(t)=0.
Denote by E(R,X,μ) the set of such functions.Lemma 5 (see [6]).
Letμ∈M and satisfy (H0); then E(R,X,μ) is a translation invariant.Definition 6.
A functionf∈C(R,X) is said to be antiperiodic if there exists a ω∈R∖{0} with the property that f(t+ω)=-f(t) for all t∈R. If there exists a less positive ω with this property, it is called the antiperiodic of f. The collection of those functions is denoted by Pωap(R,X).Definition 7.
A functionf∈C(R,X) is said to be periodic if there exists a ω∈R∖{0} with the property that f(t+ω)=f(t) for all t∈R. If there exists a less positive ω with this property, it is called the periodic of f. The collection of those ω-periodic functions is denoted by Pω(R,X).Definition 8 (see [18]).
A functionf∈C(R,X) is said to be almost periodic if for each ε>0, there exists an l(ε)>0, such that every interval I of length l(ε) contains a number τ with the property that ∥f(t+τ)-f(t)∥<ε for all t∈R. Denote by AP(R,X) the set of such functions.Definition 9 (see [19]).
A functionf∈C(R,X) is said to be almost automorphic if for every sequence of real numbers (sn′)n∈N, there exists a subsequence (sn)n∈N such that
(7)g(t)∶=lim⁡n→∞f(t+sn)
is well defined for each t∈R, and
(8)lim⁡n→∞g(t-sn)=f(t)
for each t∈R. Denote by AA(R,X) the set of such functions.Next, we recall theμ-class of functions by the measure theory.Definition 10.
Letμ∈M. A function f∈C(R,X) is said to be μ-pseudo antiperiodic if it can be decomposed as f=g+φ, where g∈Pωap(R,X) and φ∈E(R,X,μ). Denote by PPωap(R,X,μ) the collection of such functions.Definition 11.
Letμ∈M. A function f∈C(R,X) is said to be a μ-pseudo periodic if it can be decomposed as f=g+φ, where g∈Pω(R,X) and φ∈E(R,X,μ). Denote by PPω(R,X,μ) the collection of such functions.Definition 12 (see [6]).
Letμ∈M. A function f∈C(R,X) is said to be μ-pseudo almost periodic if it can be decomposed as f=g+φ, where g∈AP(R,X) and φ∈E(R,X,μ). Denote by PAP(R,X,μ) the collection of such functions.Definition 13 (see [7]).
Letμ∈M. A function f∈C(R,X) is said to be μ-pseudo almost automorphic if it can be decomposed as f=g+φ, where g∈AA(R,X) and φ∈E(R,X,μ). Denote by PAA(R,X,μ) the collection of such functions.Remark 14.
(i) If the measureμ is the Lebesgue measure, then PPωap(R,X,μ), PPω(R,X,μ), PAP(R,X,μ), and PAA(R,X,μ) are the following functions: pseudo antiperiodic (PPωap(R,X) [5]), pseudo periodic (PPω(R,X) [4]), pseudo almost periodic (PAP(R,X) [20]), and pseudo almost automorphic (PAA(R,X) [21]), respectively. One can see [6, 7, 22] for more details.
(ii) Letρ(t)>0 a.e. on R for the Lebesgue measure. μ denotes the positive measure defined by
(9)μ(A)=∫A‍ρ(t)dtforA∈B,
where dt denotes the Lebesgue measure on R; then PPωap(R,X,μ), PPω(R,X,μ), PAP(R,X,μ), and PAA(R,X,μ) are the weighted class of functions: weighted pseudo antiperiodic (WPPωap(R,X) [1]), weighted pseudo periodic (WPPω(R,X) [1]), weighted pseudo almost periodic (WPAP(R,X) [3]), and weighted pseudo almost automorphic (WPAA(R,X) [2]), respectively.
Let(10)A(R,X)={Pωap(R,X),Pω(R,X),AP(R,X),AA(R,X)}P(R,X,μ)={PPωap(R,X,μ),PPω(R,X,μ),kPAP(R,X,μ),PAA(R,X,μ)PPωap}.
It is not difficult to see that f∈P(R,X,μ) if and only if it can be decomposed as f=g+φ, where g∈A(R,X) and φ∈E(R,X,μ).Definition 15.
Letμ1,μ2∈M; μ1 is said to be equivalent to μ2(μ1~μ2) if there exist constants α,β>0 and a bounded interval I (eventually I=∅) such that
(11)αμ1(A)≤μ2(A)≤βμ1(A)forA∈BsatisfiesA∩I=∅.Similarly as the proof of [6, 7], we have the following results for the class of functions P(R,X,μ).Lemma 16.
Letμ∈M; then the following properties hold:(i)
f±g∈P(R,X,μ) if f,g∈P(R,X,μ);(ii)
λf∈P(R,X,μ) if λ∈R, f∈P(R,X,μ);(iii)
P(R,X,μ) is a Banach space with the supremum norm ∥·∥;(iv)
PPωap(R,X,μ)⊂PPω(R,X,μ)⊂PAP(R,X,μ)⊂PAA(R,X,μ)⊂BC(R,X).Lemma 17.
Letμ1,μ1∈M. If μ1~μ2, then E(R,X,μ1)=E(R,X,μ2),P(R,X,μ1)=P(R,X,μ2).Theorem 18.
Letμ∈M, f∈P(R×X×X,X,μ) and satisfy the following:(i)
f is uniformly continuous on each compact set K1×K2 in X×X with respect to the second and third variables u, v;(ii)
for all bounded subsetsE1,E2 of X, f is bounded on R×E1×E2.Then f(·,h1(·),h2(·))∈P(R,X,μ) if h1(·),h2(·)∈P(R,X,μ).Corollary 19.
Letμ∈M and f∈P(R×X×X,X,μ), and there exists a constant Lf>0 such that
(12)∥f(t,u1,u2)-f(t,v1,v2)∥≤Lf(∥u1-v1∥+∥u2-v2∥),hhhhhhhhhhhhhhhhhhhhhhhht∈R,u1,u2,v1,v2∈X;
then f(·,h1(·),h2(·))∈P(R,X,μ) if h1(·),h2(·)∈P(R,X,μ).


## 2.1. Evolution Family and Exponential Dichotomy

Definition 1.
A family of bounded linear operators(U(t,s))t≥s on a Banach space X is called a strong continuous evolution family if(i)
U(t,r)U(r,s)=U(t,s) and U(s,s)=I for all t≥r≥s and t,r,s∈R;(ii)
the map(t,s)→U(t,s)x is continuous for all x∈X,t≥s and t,s∈R.Definition 2.
An evolution family(U(t,s))t≥s on a Banach space X is called hyperbolic (or has an exponential dichotomy) if there exist projections P(t), t∈R, uniformly bounded and strongly continuous in t and constants c>0, δ>0 such that(i)
U(t,s)P(s)=P(t)U(t,s) for t≥s and t,s∈R,(ii)
the restrictionUQ(t,s):Q(s)X→Q(t)X of U(t,s) is invertible for t≥s (and set UQ(s,t)∶=U(t,s)-1),(iii)
(2)∥U(t,s)P(s)∥≤Me-δ(t-s),∥UQ(s,t)Q(t)∥≤Me-δ(t-s),for t≥s and t,s∈R. Here and next we set Q∶=I-P.Remark 3.
Exponential dichotomy is a classical concept in the study of long-time behaviour of evolution equations. IfP(t)=I for t∈R, then (U(t,s))t≥s is exponentially stable. One can see [15–17] for more details.
If(U(t,s))t≥s is hyperbolic, then
(3)Γ(t,s)∶={U(t,s)P(s),t≥s,t,s∈R,-UQ(t,s)Q(s),t<s,t,s∈R,
is called Green’s function corresponding to (U(t,s))t≥s, P(·) and
(4)∥Γ(t,s)∥≤{Me-δ(t-s),t≥s,t,s∈R,Me-δ(s-t),t<s,t,s∈R.


## 2.2.μ-Ergodic and Functions by Measure Theory

B denotes the Lebesgue σ-field of R and M stands for the set of all positive measures μ on B satisfying μ(R)=∞ and μ([a,b])<∞ for all a,b∈R(a≤b). We formulate the following hypothesis.(H0)
For allτ∈R, there exist β>0 and a bounded interval I such that
(5)μ({a+τ,a∈A})≤βμ(A)ifA∈BsatisfiesA∩I=∅.Definition 4 (see [6]).
Letμ∈M. A function f∈BC(R,X) is said to be μ-ergodic if
(6)lim⁡T→+∞1μ([-T,T])∫[-T,T]‍∥f(t)∥dμ(t)=0.
Denote by E(R,X,μ) the set of such functions.Lemma 5 (see [6]).
Letμ∈M and satisfy (H0); then E(R,X,μ) is a translation invariant.Definition 6.
A functionf∈C(R,X) is said to be antiperiodic if there exists a ω∈R∖{0} with the property that f(t+ω)=-f(t) for all t∈R. If there exists a less positive ω with this property, it is called the antiperiodic of f. The collection of those functions is denoted by Pωap(R,X).Definition 7.
A functionf∈C(R,X) is said to be periodic if there exists a ω∈R∖{0} with the property that f(t+ω)=f(t) for all t∈R. If there exists a less positive ω with this property, it is called the periodic of f. The collection of those ω-periodic functions is denoted by Pω(R,X).Definition 8 (see [18]).
A functionf∈C(R,X) is said to be almost periodic if for each ε>0, there exists an l(ε)>0, such that every interval I of length l(ε) contains a number τ with the property that ∥f(t+τ)-f(t)∥<ε for all t∈R. Denote by AP(R,X) the set of such functions.Definition 9 (see [19]).
A functionf∈C(R,X) is said to be almost automorphic if for every sequence of real numbers (sn′)n∈N, there exists a subsequence (sn)n∈N such that
(7)g(t)∶=lim⁡n→∞f(t+sn)
is well defined for each t∈R, and
(8)lim⁡n→∞g(t-sn)=f(t)
for each t∈R. Denote by AA(R,X) the set of such functions.Next, we recall theμ-class of functions by the measure theory.Definition 10.
Letμ∈M. A function f∈C(R,X) is said to be μ-pseudo antiperiodic if it can be decomposed as f=g+φ, where g∈Pωap(R,X) and φ∈E(R,X,μ). Denote by PPωap(R,X,μ) the collection of such functions.Definition 11.
Letμ∈M. A function f∈C(R,X) is said to be a μ-pseudo periodic if it can be decomposed as f=g+φ, where g∈Pω(R,X) and φ∈E(R,X,μ). Denote by PPω(R,X,μ) the collection of such functions.Definition 12 (see [6]).
Letμ∈M. A function f∈C(R,X) is said to be μ-pseudo almost periodic if it can be decomposed as f=g+φ, where g∈AP(R,X) and φ∈E(R,X,μ). Denote by PAP(R,X,μ) the collection of such functions.Definition 13 (see [7]).
Letμ∈M. A function f∈C(R,X) is said to be μ-pseudo almost automorphic if it can be decomposed as f=g+φ, where g∈AA(R,X) and φ∈E(R,X,μ). Denote by PAA(R,X,μ) the collection of such functions.Remark 14.
(i) If the measureμ is the Lebesgue measure, then PPωap(R,X,μ), PPω(R,X,μ), PAP(R,X,μ), and PAA(R,X,μ) are the following functions: pseudo antiperiodic (PPωap(R,X) [5]), pseudo periodic (PPω(R,X) [4]), pseudo almost periodic (PAP(R,X) [20]), and pseudo almost automorphic (PAA(R,X) [21]), respectively. One can see [6, 7, 22] for more details.
(ii) Letρ(t)>0 a.e. on R for the Lebesgue measure. μ denotes the positive measure defined by
(9)μ(A)=∫A‍ρ(t)dtforA∈B,
where dt denotes the Lebesgue measure on R; then PPωap(R,X,μ), PPω(R,X,μ), PAP(R,X,μ), and PAA(R,X,μ) are the weighted class of functions: weighted pseudo antiperiodic (WPPωap(R,X) [1]), weighted pseudo periodic (WPPω(R,X) [1]), weighted pseudo almost periodic (WPAP(R,X) [3]), and weighted pseudo almost automorphic (WPAA(R,X) [2]), respectively.
Let(10)A(R,X)={Pωap(R,X),Pω(R,X),AP(R,X),AA(R,X)}P(R,X,μ)={PPωap(R,X,μ),PPω(R,X,μ),kPAP(R,X,μ),PAA(R,X,μ)PPωap}.
It is not difficult to see that f∈P(R,X,μ) if and only if it can be decomposed as f=g+φ, where g∈A(R,X) and φ∈E(R,X,μ).Definition 15.
Letμ1,μ2∈M; μ1 is said to be equivalent to μ2(μ1~μ2) if there exist constants α,β>0 and a bounded interval I (eventually I=∅) such that
(11)αμ1(A)≤μ2(A)≤βμ1(A)forA∈BsatisfiesA∩I=∅.Similarly as the proof of [6, 7], we have the following results for the class of functions P(R,X,μ).Lemma 16.
Letμ∈M; then the following properties hold:(i)
f±g∈P(R,X,μ) if f,g∈P(R,X,μ);(ii)
λf∈P(R,X,μ) if λ∈R, f∈P(R,X,μ);(iii)
P(R,X,μ) is a Banach space with the supremum norm ∥·∥;(iv)
PPωap(R,X,μ)⊂PPω(R,X,μ)⊂PAP(R,X,μ)⊂PAA(R,X,μ)⊂BC(R,X).Lemma 17.
Letμ1,μ1∈M. If μ1~μ2, then E(R,X,μ1)=E(R,X,μ2),P(R,X,μ1)=P(R,X,μ2).Theorem 18.
Letμ∈M, f∈P(R×X×X,X,μ) and satisfy the following:(i)
f is uniformly continuous on each compact set K1×K2 in X×X with respect to the second and third variables u, v;(ii)
for all bounded subsetsE1,E2 of X, f is bounded on R×E1×E2.Then f(·,h1(·),h2(·))∈P(R,X,μ) if h1(·),h2(·)∈P(R,X,μ).Corollary 19.
Letμ∈M and f∈P(R×X×X,X,μ), and there exists a constant Lf>0 such that
(12)∥f(t,u1,u2)-f(t,v1,v2)∥≤Lf(∥u1-v1∥+∥u2-v2∥),hhhhhhhhhhhhhhhhhhhhhhhht∈R,u1,u2,v1,v2∈X;
then f(·,h1(·),h2(·))∈P(R,X,μ) if h1(·),h2(·)∈P(R,X,μ).


## 3. Nonautonomous Integrodifferential Equations

This section is devoted to pseudo asymptotic behavior of mild solution to (1). In this section, we make the following assumptions.(H1)
There exist constantsλ0≥0, θ∈(π/2,π), L,M~≥0, and β,γ∈(0,1) with β+γ>1 such that
(13)Σθ∪{0}⊂ρ(A(t)-λ0),∥R(λ,A(t)-λ0)∥≤M~1+|λ|,∥(A(t)-λ0)R(λ,A(t)-λ0)×[R(λ0,A(t))-R(λ0,A(s))]∥≤L|t-s|β|λ|-γfort,s∈R, Σθ={λ∈C∖{0}:|argλ|≤θ}.(H2)
The evolution family(U(t,s))t≥s generated by A(t) has an exponential dichotomy with constants M>0 and δ>0; dichotomy projections P(t), t∈R; and Green’s function Γ(t,s).(H3)
Considerk∈C(R+,R) and |k(t)|≤Cke-bt for some positive constants Ck,b.(H4)
There exists a constantLg>0 such that
(14)∥g(t,u)-g(t,v)∥≤Lg∥u-v∥,t∈R,u,v∈X.(H5)
There exists a constantLf>0 such that
(15)∥f(t,u1,u2)-f(t,v1,v2)∥≤Lf(∥u1-v1∥+∥u2-v2∥),hhhhhhhhhhhhhhhhhhhhhhhht∈R,u1,u2,v1,v2∈X.(H6)
μ∈M and satisfies (H0).Remark 20.
(
H
1
) is usually called “Acquistapace-Terreni” conditions, which was first introduced in [23] and widely used to study nonautonomous differential equations in [16, 17, 23–25]. If (H1) holds, there exists a unique evolution family (U(t,s))t≥s on X, which governs the homogeneous version of (1) [24].Before starting our main results, we recall the definition of the mild solution to (1).Definition 21 (see [26]).
A mild solution of (1) is a continuous function u:R→X satisfying
(16)u(t)=U(t,s)u(s)+∫st‍U(t,σ)f(σ,u(σ),(Ku)(σ))dσ
for all t≥s, t,s∈R.Lemma 22.
Assume thath∈E(R,X,μ) and (H1), (H2), and (H6) hold; then
(17)(Πh)(t)∶=∫-∞t‍U(t,σ)P(σ)h(σ)dσ-∫t∞‍UQ(t,σ)Q(σ)h(σ)dσ∈E(R,X,μ).Proof.
First, we show thatΠh is well defined. In fact, if h∈E(R,X,μ), so ∥h∥<∞. By (2),
(18)∥U(t,σ)P(σ)h(σ)∥≤Me-δ(t-σ)∥h(σ)∥≤Me-δ(t-σ)∥h∥,∥UQ(t,σ)Q(σ)h(σ)∥≤Me-δ(σ-t)∥h(σ)∥≤Me-δ(σ-t)∥h∥;
it follows that U(t,σ)P(σ)h(σ) is integrable over (-∞,t) and UQ(t,σ)Q(σ)h(σ) is integrable over (t,∞) for t∈R.
Note thatΠh∈BC(R,X). Next, we show that
(19)lim⁡T→+∞1μ([-T,T])∫[-T,T]‍∥(Πh)(t)∥dμ(t)=0.
In fact, for T>0, by using Fubini’s theorem, one has
(20)1μ([-T,T])∫[-T,T]‍∥(Πh)(t)∥dμ(t)≤1μ([-T,T])∫[-T,T]‍∫-∞t‍∥U(t,σ)P(σ)h(σ)∥dσdμ(t)+1μ([-T,T])×∫[-T,T]‍∫t∞‍∥UQ(t,σ)Q(σ)h(σ)∥dσdμ(t)≤1μ([-T,T])∫[-T,T]‍∫-∞t‍Me-δ(t-σ)∥h(σ)∥dσdμ(t)+1μ([-T,T])∫[-T,T]‍∫t∞‍Me-δ(σ-t)∥h(σ)∥dσdμ(t)=1μ([-T,T])∫[-T,T]‍∫0∞‍Me-δσ∥h(t-σ)∥dσdμ(t)+1μ([-T,T])∫[-T,T]‍∫0∞‍Me-δσ∥h(t+σ)∥dσdμ(t)=∫0∞‍Me-δσΦT(σ)dσ+∫0∞‍Me-δσΨT(σ)dσ,
where
(21)ΦT(σ)∶=1μ([-T,T])∫[-T,T]‍∥h(t-σ)∥dμ(t),ΨT(σ)∶=1μ([-T,T])∫[-T,T]‍∥h(t+σ)∥dμ(t).
Since μ∈M and satisfies (H0), by Lemma 5, it follows that h(·-σ), h(·+σ)∈E(R,X,μ) for each σ∈R; hence, lim⁡T→∞ΦT(σ)=0 and lim⁡T→∞ΨT(σ)=0 for all σ∈R. Since ∥ΦT∥<∞ and ∥ΨT∥<∞, then Πh∈E(R,X,μ) by using the Lebesgue dominated convergence theorem.Lemma 23.
Assume thath∈P(R,X,μ) and (H3) and (H6) hold; then
(22)(Λh)(t)=∫-∞t‍k(t-s)h(s)ds∈P(R,X,μ).Proof.
Similarly as the proof of Lemma22, it is not difficult to see that Λh is well defined and Λh∈BC(R,X).
(i) Note thath∈PPωap(R,X,μ), μ∈M.
Leth=h1+h2, where h1∈Pωap(R,X) and h2∈E(R,X,μ); then
(23)(Λh)(t)=∫-∞t‍k(t-s)h(s)ds∶=(Λ1h)(t)+(Λ2h)(t),
where
(24)(Λ1h)(t)=∫-∞tk(t-s)h1(s)ds,(Λ2h)(t)=∫-∞t‍k(t-s)h2(s)ds.
Note thath1∈Pωap(R,X); then
(25)(Λ1h)(t+ω)=∫-∞t+ω‍k(t+ω-s)h1(s)ds=∫-∞t‍k(t-s)h1(s+ω)ds=-(Λ1h)(t);
hence, Λ1h∈Pωp(R,X).
By using Fubini’s theorem, one has(26)1μ([-T,T])∫[-T,T]‍∫-∞t‍∥k(t-s)h2(s)∥dsdμ(t)≤1μ([-T,T])∫[-T,T]‍∫-∞t‍Cke-b(t-s)∥h2(s)∥dsdμ(t)≤∫0∞‍Ckμ([-T,T])∫[-T,T]‍∥h2(t-s)∥dμ(t)ds.
Since h2∈E(R,X,μ), it follows that h2(·-s)∈E(R,X,μ) for each s∈R by Lemma 5; hence, Λ2h∈E(R,X,μ) by using the Lebesgue dominated convergence theorem. Hence, Λh∈PPωap(R,X,μ).
(ii) Note thath∈PPω(R,X,μ), μ∈M.
Leth=h1+h2, where h1∈Pω(R,X) and h2∈E(R,X,μ); then
(27)(Λ1h)(t+ω)=∫-∞t+ω‍k(t+ω-s)h1(s)ds=∫-∞t‍k(t-s)h1(s+ω)ds=(Λ1h)(t);
hence, Λ1h∈Pω(R,X). Since Λ2h∈E(R,X,μ) by (i), hence, Λh∈PPω(R,X,μ).
(iii) Note thath∈PAP(R,X,μ), μ∈M.
By [27], Λ1h∈AP(R,X). Since Λ2h∈E(R,X,μ) by (i), hence, Λh∈PAP(R,X,μ).
(iv) Note thath∈PAA(R,X,μ), μ∈M.
Similarly as the proof of [28], Λ1h∈AA(R,X). Since Λ2h∈E(R,X,μ) by (i); hence, Λh∈PAA(R,X,μ).


### 3.1. Pseudo Almost Automorphic Perturbation

In this subsection, we investigated the existence and uniqueness of pseudo almost automorphic mild solution of (1).First, we introduce the concept of bi-almost automorphic function.Definition 24 (see [29]).
A functionf∈C(R×R,X) is called bi-almost automorphic if for every sequence of real numbers (sn′)n∈N, there exists a subsequence (sn)n∈N such that
(28)g(t,s)∶=lim⁡n→∞f(t+sn,s+sn)
is well defined for each t,s∈R, and
(29)lim⁡n→∞g(t-sn,s-sn)=f(t,s)
for each t,s∈R. The collection of all such functions will be denoted by bAA(R×R,X).
Now, we make the following assumptions:(H71)
Γ(t,s)x∈bAA(R×R,X) for each x∈X;(H81)
g∈PAA(R×X,X,μ) and f∈PAA(R×X×X,X,μ).Lemma 25 (see [25]).
Assume thath∈AA(R,X), (H1), (H2), and (H71) hold; then
(30)(Πh)(t)=∫-∞t‍U(t,σ)P(σ)h(σ)dσ-∫t∞‍UQ(t,σ)Q(σ)h(σ)dσ∈AA(R,X).Theorem 26.
Suppose(H1)–(H6), (H71), and (H81) hold; if 2MLf(b+LgCk)<bδ, then (1) has a unique mild solution u∈PAA(R,X,μ) such that
(31)u(t)=∫-∞t‍U(t,σ)P(σ)f(σ,u(σ),(Ku)(σ))dσ-∫t∞‍UQ(t,σ)Q(σ)f(σ,u(σ),(Ku)(σ))dσ.Proof.
First, we show that (1) admits a unique bounded solution given by (31), which is similar to the proof of [26, Theorem 3.3]. For u∈PAA(R,X,μ), it is clear that h(·)∶=f(·,u(·),(Ku)(·))∈PAA(R,X,μ) by Lemma 23 and Corollary 19; then ∥h∥<∞. By the definition of exponential dichotomy of (U(t,s))t≥s, it is not difficult to see that (31) is well defined for each t∈R.
For allt≥s, t,s∈R,
(32)u(s)=∫-∞s‍U(s,σ)P(σ)h(σ)dσ-∫s∞‍UQ(s,σ)Q(σ)h(σ)dσ,s∈R;
then
(33)U(t,s)u(s)=∫-∞s‍U(t,s)U(s,σ)P(σ)h(σ)dσ-∫s∞‍U(t,s)UQ(s,σ)Q(σ)h(σ)dσ=∫-∞s‍U(t,σ)P(σ)h(σ)dσ-∫s∞‍UQ(t,σ)Q(σ)h(σ)dσ=∫-∞t‍U(t,σ)P(σ)h(σ)dσ-∫st‍U(t,σ)P(σ)h(σ)dσ-∫t∞‍UQ(t,σ)Q(σ)h(σ)dσ-∫st‍UQ(t,σ)Q(σ)h(σ)dσ=u(t)-∫st‍U(t,σ)h(σ)dσ;
that is, u is a mild solution of (1). To prove the uniqueness, let v be another mild solution of (1); then
(34)v(t)=U(t,s)u(s)+∫st‍U(t,σ)h(σ)dσ,
by the exponential dichotomy of (U(t,s))t≥s,
(35)P(t)v(t)=∫-∞t‍U(t,σ)P(σ)h(σ)dσ,t∈R.
Similarly,
(36)Q(t)v(t)=∫∞t‍UQ(t,σ)Q(σ)h(σ)dσ,t∈R.
So,
(37)v(t)=P(t)v(t)+Q(t)v(t)=∫-∞t‍U(t,σ)P(σ)h(σ)dσ-∫t∞‍UQ(t,σ)Q(σ)h(σ)dσ=u(t).
Next, define the operatorF:PAA(R,X,μ)→PAA(R,X,μ) by
(38)(Fu)(t)=∫-∞t‍U(t,σ)P(σ)f(σ,u(σ),(Ku)(σ))dσ-∫t∞‍UQ(t,σ)Q(σ)f(σ,u(σ),(Ku)(σ))dσ.
By Lemma 22, Lemma 23, Lemma 25, and Corollary 19, F maps PAA(R,X,μ) into itself.
For anyu,v∈PAA(R,X,μ),
(39)∥(Fu)(t)-(Fv)(t)∥≤∫-∞t‍∥U(t,σ)P(σ)[f(σ,u(σ),(Ku)(σ))hhhhhhhhhhhhhhhhhi-f(σ,v(σ),(Kv)(σ))]∥dσ+∫t∞‍∥UQ(t,σ)Q(σ)[f(σ,u(σ),(Ku)(σ))hhhhhhhhhhhhhhhhhhhi-f(σ,v(σ),(Kv)(σ))]∥dσ≤Lf∫-∞t‍Me-δ(t-σ)×(∥u(σ)-v(σ)∥+∥(Ku)(σ)-(Kv)(σ)∥)dσ+Lf∫t∞‍Me-δ(σ-t)k×(∥u(σ)-v(σ)∥+∥(Ku)(σ)-(Kv)(σ)∥)dσ≤Lf(1+LgCkb)∫-∞t‍Me-δ(t-σ)∥u(σ)-v(σ)∥dσ+Lf(1+LgCkb)∫t∞‍Me-δ(σ-t)∥u(σ)-v(σ)∥dσ≤2MLf(b+LgCk)bδ∥u-v∥.
By the Banach contraction mapping principle, F has a unique fixed point in PAA(R,X,μ), which is the unique PAA mild solution to (1).Next, consider the following nonautonomous Volterra integrodifferential equations:(40)u′(t)=A(t)u(t)+∫-∞t‍k(t-s)u(s)ds+f(t,u(t)),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhht∈R,
where the linear operators A(t):D(A(t))⊂X→X have a domain D(A(t)) not necessarily dense in X and satisfy “Acquistapace-Terreni” conditions and f is a continuous function.For the pseudo almost automorphy of (40), one has the following.Theorem 27.
Suppose(H1), (H2), (H3), (H6), and (H71) hold, and f∈PAA(R×X,X,μ) satisfies(H9)
∥f(t,u)-f(t,v)∥≤Lf∥u-v∥,t∈R and u,v∈X, where Lf>0 is a constant.Then (40) has a unique mild solution u∈PAA(R,X,μ) if 2M(bLf+Ck)<bδ.Proof.
Let(41)(Ku)(t)=∫-∞t‍k(t-s)u(s)ds,
and define
(42)Γu=∫-∞t‍U(t,σ)P(σ)((Ku)(σ)+f(σ,u(σ)))dσ-∫t∞‍UQ(t,σ)Q(σ)((Ku)(σ)+f(σ,u(σ)))dσ.
Similarly as the proof of Theorem 26, Γ is well defined and (42) is a mild solution of (40).
For anyu,v∈PAA(R,X,μ),
(43)∥(Γu)(t)-(Γv)(t)∥≤∫-∞t‍Me-δ(t-σ)(∥(Ku)(σ)-(Kv)(σ)∥∥f(σ,u(σ))-f(σ,v(σ))∥hhhhhhhhhhhhhhh+∥f(σ,u(σ))-f(σ,v(σ))∥)dσ+∫t∞‍Me-δ(σ-t)(∥(Ku)(σ)-(Kv)(σ)∥∥f(σ,u(σ))-f(σ,v(σ))∥hhhhhhhhhhhhhhhh+∥f(σ,u(σ))-f(σ,v(σ))∥)dσ≤(Ckb+Lf)∫-∞t‍Me-δ(t-σ)∥u(σ)-v(σ)∥dσ+(Ckb+Lf)∫t∞‍Me-δ(σ-t)∥u(σ)-v(σ)∥dσ≤2M(bLf+Ck)bδ∥u-v∥.
By the Banach contraction mapping principle, Γ has a unique fixed point in PAA(R,X,μ), which is the unique PAA mild solution to (40).


### 3.2. Pseudo Almost Periodic Perturbation

In this subsection, we investigated the existence and uniqueness of pseudo almost periodic mild solution of (1) and (40). We make the following assumptions:(H72)
R(λ0,A(·))∈AP(R,L(X)) for λ0 in (H1);(H82)
g∈PAP(R×X,X,μ) and f∈PAP(R×X×X,X,μ).Similarly as the proof of [16], we have the following results.Lemma 28.
Assume thath∈AP(R,X), (H1), (H2), and (H72) hold; then
(44)(Πh)(t)=∫-∞t‍U(t,σ)P(σ)h(σ)dσ-∫t∞‍UQ(t,σ)Q(σ)h(σ)dσ∈AP(R,X).By Lemma22, Lemma 23, and Lemma 28, similarly as the proof of Theorem 26, Theorem 27, the following results hold.Theorem 29.
Suppose(H1)–(H6), (H72), and (H82) hold; then (1) has a unique mild solution u∈PAP(R,X,μ) if 2MLf(b+LgCk)<bδ.Theorem 30.
Suppose(H1), (H2), (H3), (H6), and (H72) hold, and f∈PAP(R×X,X,μ) satisfies (H9), then (40) has a unique mild solution u∈PAP(R,X,μ) if 2M(bLf+Ck)<bδ.


### 3.3. Pseudo Periodic (Antiperiodic) Perturbation

In this subsection, we investigated the existence and uniqueness of pseudo periodic (antiperiodic) mild solution of (1), (40). We make the following assumptions:(H73)
there existsω∈R∖{0} such that Γ(t+ω,s+ω)=Γ(t,s);(H83)
g∈PPω(R×X,X,μ), f∈PPω(R×X×X,X,μ).Lemma 31.
Assume thath∈Pω(R,X), (H1), (H2), and (H73) hold; then
(45)(Πh)(t)=∫-∞t‍U(t,σ)P(σ)h(σ)dσ-∫t∞‍UQ(t,σ)Q(σ)h(σ)dσ∈Pω(R,X).Proof.
Sinceh∈Pω(R,X), then h(t+ω)=h(t) for t∈R. By (H73), one has
(46)(Πh)(t+ω)=∫-∞t+ω‍U(t+ω,σ)P(σ)h(σ)dσ-∫t+ω∞‍UQ(t+ω,σ)Q(σ)h(σ)dσ=∫-∞t‍U(t+ω,σ+ω)P(σ+ω)h(σ+ω)dσ-∫t∞‍UQ(t+ω,σ+ω)Q(σ+ω)h(σ+ω)dσ=(Πh)(t);
hence, Πh∈Pω(R,X).By Lemma22, Lemma 23, and Lemma 28, similarly as the proof of Theorem 26 and Theorem 27, the following results hold.Theorem 32.
Suppose(H1)–(H6), (H73), and (H83) hold; if 2MLf(b+LgCk)<bδ, then (1) has a unique mild solution u∈PPω(R,X,μ).Theorem 33.
Suppose(H1), (H2), (H3), (H6), and (H73) hold, and f∈Pω(R×X,X,μ) satisfies (H9); then (40) has a unique mild solution u∈PPω(R,X,μ) if 2M(bLf+Ck)<bδ.Remark 34.
(i) Ifg∈PPωap(R×X,X,μ) and f∈PPωap(R×X×X,X,μ) in (H83), then Theorem 32 holds for PPωap(R,X,μ); that is, (1) has a unique mild solution u∈PPωap(R,X,μ).
(ii) Iff∈PPωap(R×X,X,μ), then Theorem 33 holds for PPωap(R,X,μ)); that is, (40) has a unique mild solution u∈PPωap(R,X,μ).


## 3.1. Pseudo Almost Automorphic Perturbation

In this subsection, we investigated the existence and uniqueness of pseudo almost automorphic mild solution of (1).First, we introduce the concept of bi-almost automorphic function.Definition 24 (see [29]).
A functionf∈C(R×R,X) is called bi-almost automorphic if for every sequence of real numbers (sn′)n∈N, there exists a subsequence (sn)n∈N such that
(28)g(t,s)∶=lim⁡n→∞f(t+sn,s+sn)
is well defined for each t,s∈R, and
(29)lim⁡n→∞g(t-sn,s-sn)=f(t,s)
for each t,s∈R. The collection of all such functions will be denoted by bAA(R×R,X).
Now, we make the following assumptions:(H71)
Γ(t,s)x∈bAA(R×R,X) for each x∈X;(H81)
g∈PAA(R×X,X,μ) and f∈PAA(R×X×X,X,μ).Lemma 25 (see [25]).
Assume thath∈AA(R,X), (H1), (H2), and (H71) hold; then
(30)(Πh)(t)=∫-∞t‍U(t,σ)P(σ)h(σ)dσ-∫t∞‍UQ(t,σ)Q(σ)h(σ)dσ∈AA(R,X).Theorem 26.
Suppose(H1)–(H6), (H71), and (H81) hold; if 2MLf(b+LgCk)<bδ, then (1) has a unique mild solution u∈PAA(R,X,μ) such that
(31)u(t)=∫-∞t‍U(t,σ)P(σ)f(σ,u(σ),(Ku)(σ))dσ-∫t∞‍UQ(t,σ)Q(σ)f(σ,u(σ),(Ku)(σ))dσ.Proof.
First, we show that (1) admits a unique bounded solution given by (31), which is similar to the proof of [26, Theorem 3.3]. For u∈PAA(R,X,μ), it is clear that h(·)∶=f(·,u(·),(Ku)(·))∈PAA(R,X,μ) by Lemma 23 and Corollary 19; then ∥h∥<∞. By the definition of exponential dichotomy of (U(t,s))t≥s, it is not difficult to see that (31) is well defined for each t∈R.
For allt≥s, t,s∈R,
(32)u(s)=∫-∞s‍U(s,σ)P(σ)h(σ)dσ-∫s∞‍UQ(s,σ)Q(σ)h(σ)dσ,s∈R;
then
(33)U(t,s)u(s)=∫-∞s‍U(t,s)U(s,σ)P(σ)h(σ)dσ-∫s∞‍U(t,s)UQ(s,σ)Q(σ)h(σ)dσ=∫-∞s‍U(t,σ)P(σ)h(σ)dσ-∫s∞‍UQ(t,σ)Q(σ)h(σ)dσ=∫-∞t‍U(t,σ)P(σ)h(σ)dσ-∫st‍U(t,σ)P(σ)h(σ)dσ-∫t∞‍UQ(t,σ)Q(σ)h(σ)dσ-∫st‍UQ(t,σ)Q(σ)h(σ)dσ=u(t)-∫st‍U(t,σ)h(σ)dσ;
that is, u is a mild solution of (1). To prove the uniqueness, let v be another mild solution of (1); then
(34)v(t)=U(t,s)u(s)+∫st‍U(t,σ)h(σ)dσ,
by the exponential dichotomy of (U(t,s))t≥s,
(35)P(t)v(t)=∫-∞t‍U(t,σ)P(σ)h(σ)dσ,t∈R.
Similarly,
(36)Q(t)v(t)=∫∞t‍UQ(t,σ)Q(σ)h(σ)dσ,t∈R.
So,
(37)v(t)=P(t)v(t)+Q(t)v(t)=∫-∞t‍U(t,σ)P(σ)h(σ)dσ-∫t∞‍UQ(t,σ)Q(σ)h(σ)dσ=u(t).
Next, define the operatorF:PAA(R,X,μ)→PAA(R,X,μ) by
(38)(Fu)(t)=∫-∞t‍U(t,σ)P(σ)f(σ,u(σ),(Ku)(σ))dσ-∫t∞‍UQ(t,σ)Q(σ)f(σ,u(σ),(Ku)(σ))dσ.
By Lemma 22, Lemma 23, Lemma 25, and Corollary 19, F maps PAA(R,X,μ) into itself.
For anyu,v∈PAA(R,X,μ),
(39)∥(Fu)(t)-(Fv)(t)∥≤∫-∞t‍∥U(t,σ)P(σ)[f(σ,u(σ),(Ku)(σ))hhhhhhhhhhhhhhhhhi-f(σ,v(σ),(Kv)(σ))]∥dσ+∫t∞‍∥UQ(t,σ)Q(σ)[f(σ,u(σ),(Ku)(σ))hhhhhhhhhhhhhhhhhhhi-f(σ,v(σ),(Kv)(σ))]∥dσ≤Lf∫-∞t‍Me-δ(t-σ)×(∥u(σ)-v(σ)∥+∥(Ku)(σ)-(Kv)(σ)∥)dσ+Lf∫t∞‍Me-δ(σ-t)k×(∥u(σ)-v(σ)∥+∥(Ku)(σ)-(Kv)(σ)∥)dσ≤Lf(1+LgCkb)∫-∞t‍Me-δ(t-σ)∥u(σ)-v(σ)∥dσ+Lf(1+LgCkb)∫t∞‍Me-δ(σ-t)∥u(σ)-v(σ)∥dσ≤2MLf(b+LgCk)bδ∥u-v∥.
By the Banach contraction mapping principle, F has a unique fixed point in PAA(R,X,μ), which is the unique PAA mild solution to (1).Next, consider the following nonautonomous Volterra integrodifferential equations:(40)u′(t)=A(t)u(t)+∫-∞t‍k(t-s)u(s)ds+f(t,u(t)),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhht∈R,
where the linear operators A(t):D(A(t))⊂X→X have a domain D(A(t)) not necessarily dense in X and satisfy “Acquistapace-Terreni” conditions and f is a continuous function.For the pseudo almost automorphy of (40), one has the following.Theorem 27.
Suppose(H1), (H2), (H3), (H6), and (H71) hold, and f∈PAA(R×X,X,μ) satisfies(H9)
∥f(t,u)-f(t,v)∥≤Lf∥u-v∥,t∈R and u,v∈X, where Lf>0 is a constant.Then (40) has a unique mild solution u∈PAA(R,X,μ) if 2M(bLf+Ck)<bδ.Proof.
Let(41)(Ku)(t)=∫-∞t‍k(t-s)u(s)ds,
and define
(42)Γu=∫-∞t‍U(t,σ)P(σ)((Ku)(σ)+f(σ,u(σ)))dσ-∫t∞‍UQ(t,σ)Q(σ)((Ku)(σ)+f(σ,u(σ)))dσ.
Similarly as the proof of Theorem 26, Γ is well defined and (42) is a mild solution of (40).
For anyu,v∈PAA(R,X,μ),
(43)∥(Γu)(t)-(Γv)(t)∥≤∫-∞t‍Me-δ(t-σ)(∥(Ku)(σ)-(Kv)(σ)∥∥f(σ,u(σ))-f(σ,v(σ))∥hhhhhhhhhhhhhhh+∥f(σ,u(σ))-f(σ,v(σ))∥)dσ+∫t∞‍Me-δ(σ-t)(∥(Ku)(σ)-(Kv)(σ)∥∥f(σ,u(σ))-f(σ,v(σ))∥hhhhhhhhhhhhhhhh+∥f(σ,u(σ))-f(σ,v(σ))∥)dσ≤(Ckb+Lf)∫-∞t‍Me-δ(t-σ)∥u(σ)-v(σ)∥dσ+(Ckb+Lf)∫t∞‍Me-δ(σ-t)∥u(σ)-v(σ)∥dσ≤2M(bLf+Ck)bδ∥u-v∥.
By the Banach contraction mapping principle, Γ has a unique fixed point in PAA(R,X,μ), which is the unique PAA mild solution to (40).


## 3.2. Pseudo Almost Periodic Perturbation

In this subsection, we investigated the existence and uniqueness of pseudo almost periodic mild solution of (1) and (40). We make the following assumptions:(H72)
R(λ0,A(·))∈AP(R,L(X)) for λ0 in (H1);(H82)
g∈PAP(R×X,X,μ) and f∈PAP(R×X×X,X,μ).Similarly as the proof of [16], we have the following results.Lemma 28.
Assume thath∈AP(R,X), (H1), (H2), and (H72) hold; then
(44)(Πh)(t)=∫-∞t‍U(t,σ)P(σ)h(σ)dσ-∫t∞‍UQ(t,σ)Q(σ)h(σ)dσ∈AP(R,X).By Lemma22, Lemma 23, and Lemma 28, similarly as the proof of Theorem 26, Theorem 27, the following results hold.Theorem 29.
Suppose(H1)–(H6), (H72), and (H82) hold; then (1) has a unique mild solution u∈PAP(R,X,μ) if 2MLf(b+LgCk)<bδ.Theorem 30.
Suppose(H1), (H2), (H3), (H6), and (H72) hold, and f∈PAP(R×X,X,μ) satisfies (H9), then (40) has a unique mild solution u∈PAP(R,X,μ) if 2M(bLf+Ck)<bδ.


## 3.3. Pseudo Periodic (Antiperiodic) Perturbation

In this subsection, we investigated the existence and uniqueness of pseudo periodic (antiperiodic) mild solution of (1), (40). We make the following assumptions:(H73)
there existsω∈R∖{0} such that Γ(t+ω,s+ω)=Γ(t,s);(H83)
g∈PPω(R×X,X,μ), f∈PPω(R×X×X,X,μ).Lemma 31.
Assume thath∈Pω(R,X), (H1), (H2), and (H73) hold; then
(45)(Πh)(t)=∫-∞t‍U(t,σ)P(σ)h(σ)dσ-∫t∞‍UQ(t,σ)Q(σ)h(σ)dσ∈Pω(R,X).Proof.
Sinceh∈Pω(R,X), then h(t+ω)=h(t) for t∈R. By (H73), one has
(46)(Πh)(t+ω)=∫-∞t+ω‍U(t+ω,σ)P(σ)h(σ)dσ-∫t+ω∞‍UQ(t+ω,σ)Q(σ)h(σ)dσ=∫-∞t‍U(t+ω,σ+ω)P(σ+ω)h(σ+ω)dσ-∫t∞‍UQ(t+ω,σ+ω)Q(σ+ω)h(σ+ω)dσ=(Πh)(t);
hence, Πh∈Pω(R,X).By Lemma22, Lemma 23, and Lemma 28, similarly as the proof of Theorem 26 and Theorem 27, the following results hold.Theorem 32.
Suppose(H1)–(H6), (H73), and (H83) hold; if 2MLf(b+LgCk)<bδ, then (1) has a unique mild solution u∈PPω(R,X,μ).Theorem 33.
Suppose(H1), (H2), (H3), (H6), and (H73) hold, and f∈Pω(R×X,X,μ) satisfies (H9); then (40) has a unique mild solution u∈PPω(R,X,μ) if 2M(bLf+Ck)<bδ.Remark 34.
(i) Ifg∈PPωap(R×X,X,μ) and f∈PPωap(R×X×X,X,μ) in (H83), then Theorem 32 holds for PPωap(R,X,μ); that is, (1) has a unique mild solution u∈PPωap(R,X,μ).
(ii) Iff∈PPωap(R×X,X,μ), then Theorem 33 holds for PPωap(R,X,μ)); that is, (40) has a unique mild solution u∈PPωap(R,X,μ).


## 4. Example

In this section, we provide some examples to illustrate our main results.Example 1.
Consider the heat equations with Dirichlet conditions(47)∂u(t,x)∂t=∂2u(t,x)∂x2+(-2+sin1cos⁡t+cos⁡πt)u(t,x)+f(t,u(t,x),(Ku)(t,x)),(Ku)(t,x)=∫-∞tk(t-s)g(s,u(s,x))ds,hhhhhhhhhhhhhhhhht∈R,x∈[0,π],u(t,0)=u(t,π)=0,t∈R,
where f∈PAA(R×L2[0,π]×L2[0,π],L2[0,π],μ), g∈PAA(R×L2[0,π],L2[0,π],μ), and μ∈M.
Note thatX=L2[0,π] is equipped with its natural topology and define
(48)D(A)={u∈L2[0,π]:u′′∈L2[0,π],u(0)=u(π)=0},Au=u′′-2u,∀u∈D(A).
Letφn(t)=2/πsin(nt) for all n∈N. It is well known that A is the infinitesimal generator of an analytic semigroup (T(t))t≥0 on L2[0,π] with ∥T(t)∥≤e-3t for t≥0 [30]. Moreover,
(49)T(t)φ=∑n=1∞‍e-(n2+2)t〈φ,φn〉φn,
for each φ∈L2[0,π].
Define a family of linear operatorsA(t) by
(50)D(A(t))=D(A),A(t)φ(x)=(A+sin1cos⁡t+cos⁡πt)φ(x),hhhhhhhhhhhhhi∀x∈[0,π],φ∈D(A).
Then the system
(51)u′(t)=A(t)u(t),t≥s,u(s)=φ∈L2[0,π],
has an associated evolution family (U(t,s))t≥s on L2[0,π], which can be explicitly expressed by
(52)U(t,s)φ=T(t-s)e∫st‍sin(1/(cos⁡τ+cos⁡πτ))dτφ.
Moreover,
(53)∥U(t,s)∥≤e-2(t-s)foreveryt≥s.
Note that sin(1/(cos⁡t+cos⁡πt))∈AA(R,R), and it is not difficult to verify that A(t) satisfies (H1), (H2), (H71), and (H81) with M=1 and δ=2. One can see [26, 31] for more details.
Now, the following theorem is an immediate consequence of Theorem26.Theorem 35.
Under the assumptions(H3)–(H6), (47) admits a unique PAA mild solution u(t)∈PAA(R,L2[0,π],μ) if Lf(b+LgCk)<b.Example 2.
For (40), let Ω⊂Rn be a bounded domain with C2-boundary and X=L2(Ω). For t∈R, define
(54)A(t)=a(t)Δu,u∈D(A(t))∶=H01(Ω)∩H2(Ω),
where a(·)∈AA(R,X) is Hölder continuous and inf⁡t∈Ra(t)≥a>0. It is not difficult to see that (H1),(H71) hold and the evolution family U(t,s) generated by A(t) has an exponential dichotomy provided that the Hölder constant of a(·) is sufficiently small; hence, (H2) holds. One can see [32, 33] for more details. Suppose further
(55)k(t-s)=Cke-b(t-s),∀t≥s,f(t,u(x))=γ(sin12+cos⁡t+cos⁡2t+e-|t|(1+|t|)2)cos⁡(u(x)),hhhhhhhhhhhhhhhhhhhhhh∀t∈R,u∈X,x∈Ω,
where Ck, b, and γ are positive constants; then f∈PAA(R,X,μ); μ is the Lebesgue measure, so (H3), (H6) hold. Moreover,
(56)∥f(t,u)-f(t,v)∥≤2γ∥u-v∥,∀t∈R,u∈X,x∈Ω,
which implies that (H9) holds. Therefore, (40) has a unique mild solution u∈PAA(R,X,μ) if γ is sufficiently small by Theorem 27.



---

*Source: 419103-2014-08-11.xml*