Research Article | Open Access

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

**Academic Editor:**Bogdan Sasu

#### 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 of -pseudo antiperiodic (-pseudo periodic, -pseudo almost periodic, and -pseudo automorphic) solution are investigated. Some interesting examples are presented to illustrate the main findings.

#### 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 -ergodic and introduce the new concepts of -pseudo almost periodic and -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: where the linear operators have a domain not necessarily dense in Banach space and satisfy “Acquistapace-Terreni” conditions and , , and 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 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 -pseudo antiperiodic (-pseudo periodic, -pseudo almost periodic, and -pseudo almost automorphic) mild solution to (1).

The paper is organized as follows. In Section 2, we recall some fundamental results about the notion of -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 and be two Banach spaces and , , , and stand for the set of natural numbers, integers, real numbers, and complex numbers, respectively. For being a linear operator on , , , , and stand for the domain, the resolvent set, the resolvent, and spectrum of . In order to facilitate the discussion below, we further introduce the following notations:(i) (resp., ): the Banach space of bounded continuous functions from to (resp., from to ) with the supremum norm;(ii) (resp. ): the set of continuous functions from to (resp., from to );(iii): the Banach space of bounded linear operators from to endowed with the operator topology; in particular, we write when ;(iv): the space of all classes of equivalence (with respect to the equality almost everywhere on ) of measurable functions such that .

##### 2.1. Evolution Family and Exponential Dichotomy

*Definition 1. *A family of bounded linear operators on a Banach space is called a strong continuous evolution family if(i) and for all and ;(ii)the map is continuous for all and .

*Definition 2. *An evolution family on a Banach space is called hyperbolic (or has an exponential dichotomy) if there exist projections , , uniformly bounded and strongly continuous in and constants , such that(i) for and ,(ii)the restriction of is invertible for (and set ),(iii)for and . Here and next we set .

*Remark 3. *Exponential dichotomy is a classical concept in the study of long-time behaviour of evolution equations. If for , then is exponentially stable. One can see [15–17] for more details.

If is hyperbolic, then
is called Green’s function corresponding to , and

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

denotes the Lebesgue -field of and stands for the set of all positive measures on satisfying and for all . We formulate the following hypothesis.For all , there exist and a bounded interval such that

*Definition 4 (see [6]). *Let . A function is said to be -ergodic if
Denote by the set of such functions.

Lemma 5 (see [6]). *Let and satisfy ; then is a translation invariant.*

*Definition 6. *A function is said to be antiperiodic if there exists a with the property that for all . If there exists a less positive with this property, it is called the antiperiodic of . The collection of those functions is denoted by .

*Definition 7. *A function is said to be periodic if there exists a with the property that for all . If there exists a less positive with this property, it is called the periodic of . The collection of those -periodic functions is denoted by .

*Definition 8 (see [18]). *A function is said to be almost periodic if for each , there exists an , such that every interval of length contains a number with the property that for all . Denote by the set of such functions.

*Definition 9 (see [19]). *A function is said to be almost automorphic if for every sequence of real numbers , there exists a subsequence such that
is well defined for each , and
for each . Denote by the set of such functions.

Next, we recall the -class of functions by the measure theory.

*Definition 10. *Let . A function is said to be -pseudo antiperiodic if it can be decomposed as , where and . Denote by the collection of such functions.

*Definition 11. *Let . A function is said to be a -pseudo periodic if it can be decomposed as , where and . Denote by the collection of such functions.

*Definition 12 (see [6]). *Let . A function is said to be -pseudo almost periodic if it can be decomposed as , where and . Denote by the collection of such functions.

*Definition 13 (see [7]). *Let . A function is said to be -pseudo almost automorphic if it can be decomposed as , where and . Denote by the collection of such functions.

*Remark 14. *(i) If the measure is the Lebesgue measure, then , , , and are the following functions: pseudo antiperiodic ( [5]), pseudo periodic ( [4]), pseudo almost periodic ( [20]), and pseudo almost automorphic ( [21]), respectively. One can see [6, 7, 22] for more details.

(ii) Let a.e. on for the Lebesgue measure. denotes the positive measure defined by
where denotes the Lebesgue measure on ; then , , , and are the weighted class of functions: weighted pseudo antiperiodic ( [1]), weighted pseudo periodic ( [1]), weighted pseudo almost periodic ( [3]), and weighted pseudo almost automorphic ( [2]), respectively.

Let
It is not difficult to see that if and only if it can be decomposed as , where and .

*Definition 15. *Let ; is said to be equivalent to if there exist constants and a bounded interval (eventually ) such that

Similarly as the proof of [6, 7], we have the following results for the class of functions .

Lemma 16. *Let ; then the following properties hold:*(i)* if ;*(ii)* if , ;*(iii)* is a Banach space with the supremum norm ;*(iv)*.*

Lemma 17. *Let . If , then .*

Theorem 18. *Let , and satisfy the following:*(i)* is uniformly continuous on each compact set in with respect to the second and third variables , ;*(ii)*for all bounded subsets of , is bounded on .**Then if .*

Corollary 19. *Let and , and there exists a constant such that
**
then if .*

#### 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.()There exist constants , , , and with such that for , .()The evolution family generated by has an exponential dichotomy with constants and ; dichotomy projections , ; and Green’s function .()Consider and for some positive constants .()There exists a constant such that ()There exists a constant such that () and satisfies .

*Remark 20. * 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 holds, there exists a unique evolution family on , 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 satisfying
for all , .

Lemma 22. *Assume that and , , and hold; then
*

*Proof. *First, we show that is well defined. In fact, if , so . By (2),
it follows that is integrable over and is integrable over for .

Note that . Next, we show that
In fact, for , by using Fubini’s theorem, one has
where
Since and satisfies , by Lemma 5, it follows that , for each ; hence, and for all . Since and , then by using the Lebesgue dominated convergence theorem.

Lemma 23. *Assume that and and hold; then
*

*Proof. *Similarly as the proof of Lemma 22, it is not difficult to see that is well defined and .

(i) Note that , .

Let , where and ; then
where

Note that ; then
hence, .

By using Fubini’s theorem, one has
Since , it follows that for each by Lemma 5; hence, by using the Lebesgue dominated convergence theorem. Hence, .

(ii) Note that , .

Let , where and ; then
hence, . Since by (i), hence, .

(iii) Note that , .

By [27], . Since by (i), hence, .

(iv) Note that , .

Similarly as the proof of [28], . Since by (i); hence, .

##### 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 function is called bi-almost automorphic if for every sequence of real numbers , there exists a subsequence such that
is well defined for each , and
for each . The collection of all such functions will be denoted by .

Now, we make the following assumptions:() for each ;() and .

Lemma 25 (see [25]). *Assume that , , , and hold; then
*

Theorem 26. *Suppose , , and hold; if , then (1) has a unique mild solution such that
*

*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 , it is clear that by Lemma 23 and Corollary 19; then . By the definition of exponential dichotomy of , it is not difficult to see that (31) is well defined for each .

For all , ,
then
that is, is a mild solution of (1). To prove the uniqueness, let be another mild solution of (1); then
by the exponential dichotomy of ,
Similarly,
So,

Next, define the operator by
By Lemma 22, Lemma 23, Lemma 25, and Corollary 19, maps into itself.

For any ,
By the Banach contraction mapping principle, has a unique fixed point in , which is the unique mild solution to (1).

Next, consider the following nonautonomous Volterra integrodifferential equations: where the linear operators have a domain not necessarily dense in and satisfy “Acquistapace-Terreni” conditions and is a continuous function.

For the pseudo almost automorphy of (40), one has the following.

Theorem 27. *Suppose , , , , and hold, and satisfies*()* and , where is a constant.**Then (40) has a unique mild solution if .*

*Proof. *Let
and define
Similarly as the proof of Theorem 26, is well defined and (42) is a mild solution of (40).

For any ,