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Advances in Difference Equations

, 2010:310951 | Cite as

Asymptotically Almost Periodic Solutions for Abstract Partial Neutral Integro-Differential Equation

  • José Paulo C. dos SantosEmail author
  • Sandro M. Guzzo
  • Marcos N. Rabelo
Open Access
Research Article
Part of the following topical collections:
  1. Abstract Differential and Difference Equations

Abstract

The existence of asymptotically almost periodic mild solutions for a class of abstract partial neutral integro-differential equations with unbounded delay is studied.

Keywords

Periodic Solution Heat Flux Dense Subset Mild Solution Resolvent Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

In this paper, we study the existence of asymptotically almost periodic mild solutions for a class of abstract partial neutral integro-differential equations modelled in the form

where Open image in new window and Open image in new window , Open image in new window , are closed linear operators; Open image in new window is a Banach space; the history Open image in new window , Open image in new window , belongs to some abstract phase space Open image in new window defined axiomatically Open image in new window are appropriated functions.

The study of abstract neutral equations is motivated by different practical applications in different technical fields. The literature related to ordinary neutral functional differential equations is very extensive and we refer the reader to Chukwu [1], Hale and Lunel [2], Wu [3], and the references therein. As a practical application, we note that the equation

arises in the study of the dynamics of income, employment, value of capital stock, and cumulative balance of payment; see [1] for details. In the above system, Open image in new window is a real number, the state Open image in new window , Open image in new window , Open image in new window are Open image in new window continuous functions matrices, Open image in new window is a constant Open image in new window matrix, Open image in new window represents the government intervention, and Open image in new window the private initiative. We note that by assuming the solution Open image in new window is known on Open image in new window , we can transform this system into an abstract system with unbounded delay described as (1.1).

Abstract partial neutral differential equations also appear in the theory of heat conduction. In the classic theory of heat conduction, it is assumed that the internal energy and the heat flux depend linearly on the temperature Open image in new window and on its gradient Open image in new window . Under these conditions, the classic heat equation describes sufficiently well the evolution of the temperature in different types of materials. However, this description is not satisfactory in materials with fading memory. In the theory developed in [4, 5], the internal energy and the heat flux are described as functionals of Open image in new window and Open image in new window . The next system, see for instance [6, 7, 8, 9], has been frequently used to describe this phenomenon,

In this system, Open image in new window is open, bounded, and with smooth boundary; Open image in new window ; Open image in new window represents the temperature in Open image in new window at the time Open image in new window ; Open image in new window is a physical constant Open image in new window , Open image in new window are the internal energy and the heat flux relaxation, respectively. By assuming that the solution Open image in new window is known on Open image in new window and Open image in new window , we can transform this system into an abstract system with unbounded delay described in the form (1.1).

Recent contributions on the existence of solutions with some of the previously enumerated properties or another type of almost periodicity to neutral functional differential equations have been made in [10, 11], for the case of neutral ordinary differential equations, and in [12, 13, 14, 15] for partial functional differential systems.

The purpose of this work is to study the existence of asymptotically almost periodic mild solutions for the neutral system (1.1). To this end, we study the existence and qualitative properties of an exponentially stable resolvent operator for the integro-differential system

There exists an extensive literature related to the existence and qualitative properties of resolvent operator for integro-differential equations. We refer the reader to the book by Gripenberg et al. [16] which contains an overview of the theory for the case where the underlying space Open image in new window has finite dimension. For abstract integro-differential equations described on infinite dimensional spaces, we cite the Prüss book [17] and the papers [18, 19, 20], Da Prato et al. [21, 22], and Lunardi [9, 23]. To finish this short description of the related literature, we cite the papers [24, 25, 26] where some of the above topics for the case of abstract neutral integro-differential equations with unbounded delay are treated.

To the best of our knowledge, the study of the existence of asymptotically almost periodic solutions of neutral integro-differential equations with unbounded delay described in the abstract form (1.1) is an untreated topic in the literature and this is the main motivation of this article.

To finish this section, we emphasize some notations used in this paper. Let Open image in new window and Open image in new window be Banach spaces. We denote by Open image in new window the space of bounded linear operators from Open image in new window into Open image in new window endowed with norm of operators, and we write simply Open image in new window when Open image in new window . By Open image in new window , we denote the range of a map Open image in new window , and for a closed linear operator Open image in new window , the notation Open image in new window represents the domain of Open image in new window endowed with the graph norm, Open image in new window , Open image in new window . In the case Open image in new window , the notation Open image in new window stands for the resolvent set of Open image in new window and Open image in new window is the resolvent operator of Open image in new window . Furthermore, for appropriate functions Open image in new window and Open image in new window , the notation Open image in new window denotes the Laplace transform of Open image in new window and Open image in new window the convolution between Open image in new window and Open image in new window , which is defined by Open image in new window . The notation Open image in new window stands for the closed ball with center at Open image in new window and radius Open image in new window in Open image in new window . As usual, Open image in new window represents the subspace of Open image in new window formed by the functions which vanish at infinity.

2. Preliminaries

In this work, we will employ an axiomatic definition of the phase space Open image in new window similar to that in [27]. More precisely, Open image in new window will denote a vector space of functions defined from Open image in new window into Open image in new window endowed with a seminorm denoted by Open image in new window and such that the following axioms hold.
  1. (A)
    If Open image in new window with Open image in new window is continuous on Open image in new window and Open image in new window , then for each Open image in new window the following conditions hold:
    1. (i)
       
    2. (ii)
       
    3. (iii)

      Open image in new window where Open image in new window is a constant, and Open image in new window are functions such that Open image in new window and Open image in new window are respectively continuous and locally bounded, and Open image in new window are independent of Open image in new window .

       
     
(A1) If Open image in new window is a function as in Open image in new window , then Open image in new window is a Open image in new window -valued continuous function on Open image in new window .
  1. (B)

    The space Open image in new window is complete.

     

(C2) If Open image in new window is a sequence in Open image in new window formed by functions with compact support such that Open image in new window uniformly on compact, then Open image in new window and Open image in new window as Open image in new window

Remark 2.1.

In the remainder of this paper, Open image in new window is such that

for every Open image in new window continuous and bounded; see [27, Proposition Open image in new window ] for details.

Definition 2.2.

Let Open image in new window be the Open image in new window -semigroup defined by Open image in new window on Open image in new window and Open image in new window on Open image in new window . The phase space Open image in new window is called a fading memory if Open image in new window as Open image in new window for each Open image in new window with Open image in new window .

Remark 2.3.

In this work, we suppose that there exists a positive Open image in new window such that

for each Open image in new window . Observe that this condition is verified, for example, if Open image in new window is a fading memory, see [27, Proposition Open image in new window ].

Example 2.4 (The phase space Open image in new window ).

Open image in new window Let Open image in new window , and let Open image in new window be a nonnegative measurable function which satisfies the conditions (g-5) and (g-6) in the terminology of [27]. Briefly, this means that Open image in new window is locally integrable, and there exists a nonnegative, locally bounded function Open image in new window on Open image in new window such that Open image in new window for all Open image in new window and Open image in new window , where Open image in new window is a set with Lebesgue measure zero. The space Open image in new window consists of all classes of functions Open image in new window such that Open image in new window is continuous on Open image in new window , Lebesgue-measurable, and Open image in new window is Lebesgue integrable on Open image in new window . The seminorm in Open image in new window is defined by

The space Open image in new window satisfies axioms (A), (A-1), and (B).  Moreover, when Open image in new window and Open image in new window , we can take Open image in new window , Open image in new window , and Open image in new window , for Open image in new window ; see [27, Theorem Open image in new window ] for details.

Now, we need to introduce some concepts, definitions, and technicalities on almost periodical functions.

Definition 2.5.

A function Open image in new window is almost periodic (a.p.) if for every Open image in new window , there exists a relatively dense subset of Open image in new window , denoted by Open image in new window , such that

Definition 2.6.

A function Open image in new window is asymptotically almost periodic (a.a.p.) if there exists an almost periodic function Open image in new window and Open image in new window such that Open image in new window .

The next lemmas are useful characterizations of a.p and a.a.p functions.

Lemma 2.7 (see [28, Theorem Open image in new window ]).

A function Open image in new window is asymptotically almost periodic if and only if for every Open image in new window there exist Open image in new window and a relatively dense subset of Open image in new window , denoted by Open image in new window , such that

In this paper, Open image in new window and Open image in new window are the spaces

endowed with the norms Open image in new window and Open image in new window respectively. We know from the result in [28] that Open image in new window and Open image in new window are Banach spaces.

Next, Open image in new window and Open image in new window are abstract Banach spaces.

Definition 2.8.

Let Open image in new window be an open subset of Open image in new window
  1. (a)

    A continuous function Open image in new window (resp., Open image in new window ) is called pointwise almost periodic (p.a.p.), (resp., pointwise asymptotically almost periodic (p.a.a.p.) if Open image in new window (resp., Open image in new window ) for every Open image in new window .

     
  2. (b)
    A continuous function Open image in new window is called uniformly almost periodic (u.a.p.), if for every Open image in new window and every compact Open image in new window there exists a relatively dense subset of Open image in new window , denoted by Open image in new window , such that
     
  1. (c)
    A continuous function Open image in new window is called uniformly asymptotically almost periodic (u.a.a.p.), if for every Open image in new window and every compact Open image in new window there exists a relatively dense subset of Open image in new window , denoted by Open image in new window , and Open image in new window such that
     

The next lemma summarizes some properties which are fundamental to obtain our results.

Lemma 2.9 (see [29, Theorem Open image in new window ]).

Let Open image in new window be an open set. Then the following properties hold.
  1. (a)

    If Open image in new window is p.a.p. and satisfies a local Lipschitz condition at Open image in new window , uniformly at Open image in new window , then Open image in new window is u.a.p.

     
  2. (b)

    If Open image in new window is p.a.a.p. and satisfies a local Lipschitz condition at Open image in new window , uniformly at Open image in new window , then Open image in new window is u.a.a.p.

     
  3. (c)
     
  4. (d)
     
  5. (e)
     

3. Resolvent Operators

In this section, we study the existence and qualitative properties of an exponentially resolvent operator for the integro-differential abstract Cauchy problem

The results obtained for the resolvent operator in this section are similar to those that can be found, for instance, in the papers [21, 23, 30]. In this paper, we prove the necessary estimates for the proof of an existence theorem of asymptotically almost periodic solutions for (1.1). For the better comprehension of the subject, we will introduce the following definitions, hypothesis, and results.

We introduce the following concept of resolvent operator for integro-differential problem (3.1).

Definition 3.1.

A one-parameter family of bounded linear operators Open image in new window on Open image in new window is called a resolvent operator of (3.1) if the following conditions are verified.
for every Open image in new window ,
  1. (c)
     

Definition 3.2.

A resolvent operator Open image in new window of (3.1) is called exponentially stable if there exist positive constants Open image in new window such that Open image in new window

In this work, we always assume that the following conditions are verified.

(H1) The operator Open image in new window is the infinitesimal generator of an analytic semigroup Open image in new window on Open image in new window , and there are constants Open image in new window , and Open image in new window such that Open image in new window and Open image in new window for all Open image in new window .

(H2) For all Open image in new window Open image in new window is a closed linear operator, Open image in new window , and Open image in new window is strongly measurable on Open image in new window for each Open image in new window . There exists Open image in new window such that Open image in new window exists for Open image in new window and Open image in new window for all Open image in new window and Open image in new window . Moreover, the operator valued function Open image in new window has an analytical extension (still denoted by Open image in new window ) to Open image in new window such that Open image in new window for all Open image in new window , and Open image in new window as Open image in new window .

(H3) There exist a subspace Open image in new window dense in Open image in new window and positive constants Open image in new window , Open image in new window such that Open image in new window , Open image in new window , and Open image in new window for every Open image in new window and all Open image in new window .

In the sequel, for Open image in new window , Open image in new window , and Open image in new window , set

and for Open image in new window , Open image in new window , the paths

with Open image in new window are oriented counterclockwise. In addition, Open image in new window is the set

We next study some preliminary properties needed to establish the existence of a resolvent operator for the problem (3.1).

Lemma 3.3.

There exists Open image in new window such that Open image in new window and the function Open image in new window is analytic. Moreover,
and there exist constants Open image in new window for Open image in new window such that

for every Open image in new window .

Proof.

fixed Open image in new window there exists a positive number Open image in new window such that Open image in new window for Open image in new window . Consequently, the operator Open image in new window has a continuous inverse with Open image in new window . Moreover, for Open image in new window , we have
which shows (3.7) and that Open image in new window . Now, from (3.7) we obtain Open image in new window and
Consequently,
the functions Open image in new window are analytic, and estimates (3.8), and (3.10) are valid. In addition, for Open image in new window , we can write

for Open image in new window sufficiently large. This proves (3.9) and completes the proof.

Observation 1.

If Open image in new window is a resolvent operator for (3.1), it follows from (3.3) that Open image in new window for all Open image in new window . Applying Lemma 3.3 and the properties of the Laplace transform, we conclude that Open image in new window is the unique resolvent operator for (3.1).

In the remainder of this section, Open image in new window and Open image in new window are numbers such that Open image in new window and Open image in new window . Moreover, we denote by Open image in new window a generic constant that represents any of the constants involved in the statements of Lemma 3.3 as well as any other constant that arises in the estimate that follows. We now define the operator family Open image in new window by

We will next establish that Open image in new window is a resolvent operator for (3.1).

Lemma 3.4.

The function Open image in new window is exponentially bounded in Open image in new window .

Proof.

If Open image in new window , from (3.17) and estimate (3.8), we get
On the other hand, using that Open image in new window is analytic on Open image in new window , for Open image in new window , we obtain

This complete the proof.

Lemma 3.5.

The operator function Open image in new window is exponentially bounded in Open image in new window .

Proof.

It follows from (3.9) that the integral in
is absolutely convergent in Open image in new window and defines a linear operator Open image in new window . Using that Open image in new window is closed, we can affirm that Open image in new window . From Lemma 3.3, Open image in new window is analytic and Open image in new window . If Open image in new window and Open image in new window , we have

From before and Lemma 3.4, we infer that Open image in new window is exponentially bounded in Open image in new window . The proof is finished.

Lemma 3.6.

The function Open image in new window is strongly continuous.

Proof.

It is clear from (3.17) that Open image in new window is continuous at Open image in new window for every Open image in new window . We next establish the continuity at Open image in new window . Let Open image in new window and Open image in new window be sufficiently large, using that

where Open image in new window represent the curve Open image in new window for Open image in new window .

For Open image in new window and Open image in new window , we get

Furthermore, it follows from (3.8), and assumption (H2) that
where Open image in new window is integrable for Open image in new window . From the Lebesgue dominated convergence theorem, we infer that
Let now Open image in new window be the curve Open image in new window for Open image in new window . Turning to apply Cauchy's theorem combining with the estimate
we obtain

we can affirm that Open image in new window for all Open image in new window , which completes and the proof since Open image in new window is dense in Open image in new window and Open image in new window is bounded on Open image in new window .

Notice that Open image in new window the sectors Open image in new window from Lemma 3.3, Open image in new window is analytic. Consider the contours

and Open image in new window oriented counterclockwise. By Cauchy theorem for Open image in new window , we obtain
The following estimate:
is the one responsible for the fact that the integral Open image in new window tends to Open image in new window as Open image in new window tend to Open image in new window in a similar way the integral Open image in new window tend to Open image in new window as Open image in new window tend to Open image in new window so that

and proceeding as before, we obtain Open image in new window for all Open image in new window which ends the proof.

The following result can be proved with an argument similar to that used in the proof of the preceding lemma with changing Open image in new window by Open image in new window

Lemma 3.7.

The function Open image in new window is strongly continuous.

We next set Open image in new window .

Lemma 3.8.

The function Open image in new window has an analytic extension to Open image in new window , and

Proof.

This property allows us to define the extension Open image in new window by this integral.

Similarly, the integral on the right hand side of (3.34) is also absolutely convergent in Open image in new window and strong, continuous on Open image in new window for Open image in new window . For Open image in new window ,

where Open image in new window is integrable for Open image in new window . From the Lebesgue dominated convergence theorem, we obtain that Open image in new window verifies (3.34). The proof is ended.

Lemma 3.9.

For every Open image in new window with Open image in new window , Open image in new window .

Proof.

Using that Open image in new window is analytic on Open image in new window and that the integrals involved in the calculus are absolutely convergent, we have

Theorem 3.10.

The function Open image in new window is a resolvent operator for the system (3.1).

Proof.

which implies
Applying [31, Proposition Open image in new window , Corollary Open image in new window ], we get
which in turn implies that

Arguing as above but using the equality Open image in new window we obtain that (3.2) holds.

On the other hand, by Lemma 3.8 we infer that Open image in new window . Next, we analyze the differentiability on Open image in new window . Let Open image in new window and Open image in new window for all Open image in new window we can choose Open image in new window such that

Consequently, for Open image in new window we have that

which proves the existence of the right derivative of Open image in new window at zero and that Open image in new window This proves that resolvent equation (3.3) is valid for every Open image in new window and Open image in new window for every Open image in new window . This completes the proof.

Corollary 3.11.

If Open image in new window then the function Open image in new window is an exponentially stable resolvent operator for the system (3.1).

In the next result, we denote by Open image in new window the fractional power of the operator Open image in new window (see [32] for details).

Theorem 3.12.

Suppose that the conditions Open image in new window are satisfied. Then there exists a positive number Open image in new window such that

for all Open image in new window

Proof.

On the other hand, using that Open image in new window is analytic on Open image in new window , for Open image in new window , we get
From the previous facts, we conclude that

which ends the proof.

Corollary 3.13.

In the remainder of this section, we discuss the existence and regularity of solutions of

where Open image in new window . In the sequel, Open image in new window is the operator function defined by (3.17). We begin by introducing the following concept of classical solution.

Definition 3.14.

A function Open image in new window , Open image in new window , is called a classical solution of (3.53)-(3.54) on Open image in new window if Open image in new window , the condition (3.54) holds and (3.53) is verified on Open image in new window .

The next result has been established in [30].

Theorem 3.15 ([30, Theorem Open image in new window ]).

Let Open image in new window . Assume that Open image in new window and Open image in new window is a classical solution of (3.53)-(3.54) on Open image in new window . Then

An immediate consequence of the above theorem is the uniqueness of classical solutions.

Corollary 3.16.

If Open image in new window are classical solutions of (3.53)-(3.54) on Open image in new window , then Open image in new window on Open image in new window .

Motivated by (3.55), we introduce the following concept.

Definition 3.17.

A function Open image in new window is called a mild solution of (3.53)-(3.54) if

4. Existence Result of Asymptotically Almost Periodic Solutions

In this section, we study the existence of asymptotically almost periodic mild solutions for the abstract integro-differential system (1.1). To establish our existence result, motivated by the previous section we introduce the following assumptions.

(P1) There exists a Banach space Open image in new window continuously included in Open image in new window such that the following conditions are verified.

(P2) The continuous function Open image in new window is p.a.a.p, and there exists a continuous function Open image in new window such that

(P3) The continuous function Open image in new window is p.a.a.p, and there exists a continuous function Open image in new window such that

Motivated by the theory of resolvent operator, we introduce the following concept of mild solution for (1.1).

Definition 4.1.

Lemma 4.2.

Let condition Open image in new window hold and let Open image in new window be a function in Open image in new window . If Open image in new window is the function defined by Open image in new window then Open image in new window .

Proof.

which implies that

Now, from inequality (4.6) and Lemma 2.7, we conclude that Open image in new window is a.a.p. The proof is complete.

Lemma 4.3.

Assume that the condition Open image in new window is fulfilled. Let Open image in new window and let Open image in new window be the function defined by

Then Open image in new window .

Proof.

We obtain
which implies that

From inequality (4.11) and Lemma 2.7, we conclude that Open image in new window is a.a.p., which ends the proof.

Now, we can establish our existence result.

Theorem 4.4.

Assume that Open image in new window is a fading memory space and Open image in new window Open image in new window , and Open image in new window are held. If Open image in new window and Open image in new window for every Open image in new window , then there exists Open image in new window such that for each Open image in new window , there exists a mild solution, Open image in new window , of (1.1) on Open image in new window such that Open image in new window and Open image in new window .

Proof.

where Open image in new window is the constant introduced in Remark 2.3. We affirm that the assertion holds for Open image in new window Let Open image in new window On the space
endowed with the metric Open image in new window , we define the operator Open image in new window by

where Open image in new window is the function defined by the relation Open image in new window and Open image in new window on Open image in new window . From the hypothesis Open image in new window Open image in new window , and Open image in new window , we obtain that Open image in new window is well defined and that Open image in new window Moreover, from Lemmas 4.2 and 4.3 it follows that Open image in new window .

Next, we prove that Open image in new window is a contraction from Open image in new window into Open image in new window . If Open image in new window and Open image in new window , we get

where the inequality Open image in new window has been used and Open image in new window represent the continuous inclusion of Open image in new window on Open image in new window . Thus, Open image in new window . On the other hand, for Open image in new window we see that

which shows that Open image in new window is a contraction from Open image in new window into Open image in new window The assertion is now a consequence of the contraction mapping principle. The proof is complete.

5. Applications

In this section, we study the existence of asymptotically almost periodic solutions of the partial neutral integro-differential system

for Open image in new window , Open image in new window , and Open image in new window Moreover, we have identified Open image in new window .

To represent this system in the abstract form (1.1), we choose the spaces Open image in new window and Open image in new window ; see Example 2.4 for details. We also consider the operators Open image in new window , Open image in new window , given by Open image in new window , Open image in new window for Open image in new window Moreover, Open image in new window has discrete spectrum, the eigenvalues are Open image in new window , Open image in new window with corresponding eigenvectors Open image in new window , and the set of functions Open image in new window is an orthonormal basis of Open image in new window and Open image in new window for Open image in new window . For Open image in new window from [32] we can define the fractional power Open image in new window of Open image in new window is given by Open image in new window where Open image in new window In the next theorem, we consider Open image in new window . We observe that Open image in new window and Open image in new window for Open image in new window from [33, Proposition Open image in new window ], we obtain that Open image in new window is a sectorial operator satisfying Open image in new window Moreover, it is easy to see that conditions (H2)-(H3) in Section 3 are satisfied with Open image in new window , and Open image in new window is the space of infinitely differentiable functions that vanish at Open image in new window and Open image in new window . Under the above conditions, we can represent the system

in the abstract form

We define the functions Open image in new window by

Moreover, Open image in new window are bounded linear operators, Open image in new window , Open image in new window , and a straightforward estimation using (ii) shows that Open image in new window and

for all Open image in new window . This allows us to rewrite the system (5.1) in the abstract form (1.1) with Open image in new window

Theorem 5.1.

Assume that the previous conditions are verified. Let Open image in new window and Open image in new window such that Open image in new window then there exists a mild solution Open image in new window of (5.1) with Open image in new window .

Proof.

since Open image in new window . By using a similar procedure as in the proofs of Lemma 3.3 and Theorem 3.10, we obtain the existence of resolvent operator for (5.2). From the hypothesis, we obtain Open image in new window by the Lemma 3.3, Corollaries 3.11 and 3.13, the assumption Open image in new window is satisfied. From Theorem 4.4, the proof is complete.

Notes

Acknowledgment

José Paulo C. dos Santos is partially supported by FAPEMIG/Brazil under Grant CEX-APQ-00476-09.

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Copyright information

© José Paulo C. dos Santos et al. 2010

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  • José Paulo C. dos Santos
    • 1
    Email author
  • Sandro M. Guzzo
    • 2
  • Marcos N. Rabelo
    • 3
  1. 1.Departamento de Ciências Exatas, Universidade Federal de AlfenasRua Gabriel Monteiro da SilvaAlfenas-MGBrazil
  2. 2.Colegiado do curso de Matemática-UNIOESTERua UniversitáriaCascavel-PRBrazil
  3. 3.Departamento de MatemáticaUniversidade Federal de PernambucoRecife-PEBrazil

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