Journal of Applied Mathematics

Volume 2014 (2014), Article ID 419103, 10 pages

http://dx.doi.org/10.1155/2014/419103

Research Article

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

Zhinan Xia

Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, Zhejiang 310023, China

Received 6 April 2014; Accepted 15 July 2014; Published 11 August 2014

Academic Editor: Bogdan Sasu

Copyright © 2014 Zhinan Xia. 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.

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 ,

By the Banach contraction mapping principle, has a unique fixed point in , which is the unique 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:

() for in ;() and .

Similarly as the proof of [16], we have the following results.

Lemma 28. Assume that , , , and hold; then

By Lemma 22, Lemma 23, and Lemma 28, similarly as the proof of Theorem 26, Theorem 27, the following results hold.

Theorem 29. Suppose , , and hold; then (1) has a unique mild solution if .

Theorem 30. Suppose , , , , and hold, and satisfies , then (40) has a unique mild solution if .

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:

()there exists such that ;(), .

Lemma 31. Assume that , , , and hold; then

Proof. Since , then for . By , one has

hence, .

By Lemma 22, Lemma 23, and Lemma 28, similarly as the proof of Theorem 26 and Theorem 27, the following results hold.

Theorem 32. Suppose , , and hold; if , then (1) has a unique mild solution .

Theorem 33. Suppose , , , , and hold, and satisfies ; then (40) has a unique mild solution if .

Remark 34. (i) If and in , then Theorem 32 holds for ; that is, (1) has a unique mild solution .

(ii) If , then Theorem 33 holds for ; that is, (40) has a unique mild solution .

4. Example

In this section, we provide some examples to illustrate our main results.

Example 1. Consider the heat equations with Dirichlet conditions

where , , and .

Note that is equipped with its natural topology and define

Let for all . It is well known that is the infinitesimal generator of an analytic semigroup on with for [30]. Moreover,

for each .

Define a family of linear operators by

Then the system

has an associated evolution family on , which can be explicitly expressed by

Moreover,

Note that , and it is not difficult to verify that satisfies , , , and with and . One can see [26, 31] for more details.

Now, the following theorem is an immediate consequence of Theorem 26.

Theorem 35. Under the assumptions , (47) admits a unique mild solution if .

Example 2. For (40), let be a bounded domain with -boundary and . For , define

where is Hölder continuous and . It is not difficult to see that hold and the evolution family generated by has an exponential dichotomy provided that the Hölder constant of is sufficiently small; hence, holds. One can see [32, 33] for more details. Suppose further

where , , and are positive constants; then ; is the Lebesgue measure, so , hold. Moreover,

which implies that holds. Therefore, (40) has a unique mild solution if is sufficiently small by Theorem 27.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was supported by Zhejiang Provincial Natural Science Foundation of China under Grant no. LQ13A010015.

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