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Advances in Difference Equations

, 2011:414906 | Cite as

Strong Stability and Asymptotical Almost Periodicity of Volterra Equations in Banach Spaces

  • Jian-Hua Chen
  • Ti-Jun Xiao
Open Access
Research Article
  • 1.6k Downloads
Part of the following topical collections:
  1. Nonlinear Integrodifferential Difference Equations and Related Topics

Abstract

We study strong stability and asymptotical almost periodicity of solutions to abstract Volterra equations in Banach spaces. Relevant criteria are established, and examples are given to illustrate our results.

Keywords

Banach Space Mild Solution Strong Stability Infinitesimal Generator Volterra Equation 
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

Owing to the memory behavior (cf., e.g., [1, 2]) of materials, many practical problems in engineering related to viscoelasticity or thermoviscoelasticity can be reduced to the following Volterra equation:
in a Banach space Open image in new window , with Open image in new window being the infinitesimal generator of a Open image in new window -semigroup Open image in new window defined on Open image in new window , and Open image in new window a scalar function ( Open image in new window and Open image in new window ), which is often called kernel function or memory kernel (cf., e.g., [1]). It is known that the above equation is well-posed. This implies the existence of the resolvent operator Open image in new window , and the mild solution is then given by

which is actually a classical solution if Open image in new window . In the present paper, we investigate strong stability and asymptotical almost periodicity of the solutions. For more information and related topics about the two concepts, we refer to the monographs [3, 4]. In particular, their connections with the vector-valued Laplace transform and theorems of Widder type can be found in [4, 5, 6]. Recall the following.

Definition 1.1.

Let Open image in new window be a Banach space and Open image in new window  :  Open image in new window a bounded uniformly continuous function.

(i) Open image in new window is called almost periodic if it can be uniformly approximated by linear combinations of Open image in new window ( Open image in new window ). Denote by Open image in new window the space of all almost periodic functions on Open image in new window .

(ii) Open image in new window is called asymptotically almost periodic if Open image in new window with Open image in new window and Open image in new window . Denote by Open image in new window the space of all asymptotically almost periodic functions on Open image in new window .

(iii)We call (1.1) or Open image in new window   strongly stable if, for each Open image in new window . We call (1.1) or Open image in new window asymptotically almost periodic if for each Open image in new window .

The following two results on Open image in new window -semigroup will be used in our investigation, among which the first is due to Ingham (see, e.g., [7, Section 1] and the second is known as Countable Spectrum Theorem [3, Theorem Open image in new window ]. As usual, the letter Open image in new window denotes the imaginary unit and Open image in new window the imaginary axis.

Lemma 1.2.

Lemma 1.3.

Let Open image in new window be a bounded Open image in new window -semigroup on a reflexive Banach space Open image in new window with generator Open image in new window . If Open image in new window is countable, then Open image in new window is asymptotically almost periodic.

2. Results and Proofs

Asymptotic behaviors of solutions to the special case of Open image in new window have been studied systematically, see, for example, [3, Chapter 4] and [8, Chapter V]. The following example shows that asymptotic behaviors of solutions to (1.1) are more complicated even in the finite-dimensional case.

Example 2.1.

Let Open image in new window in (1.1). Then taking Laplace transform we can calculate

It is clear that the following assertions hold.

(a)The corresponding semigroup Open image in new window is exponentially stable.

(b)Each solution with initial value Open image in new window is not strongly stable and hence not exponentially stable.

(c)Each solution with Open image in new window is asymptotically almost periodic.

It is well known that the semigroup approach is useful in the study of (1.1). More information can be found in the book [8, Chapter VI.7] or the papers [9, 10, 11].

Let Open image in new window be the product Banach space with the norm
for each Open image in new window and Open image in new window . Then the operator matrix
generates a Open image in new window -semigroup on Open image in new window Here, Open image in new window is the vector-valued Sobolev space and Open image in new window the Dirac distribution, that is, Open image in new window for each Open image in new window ; the operator Open image in new window is given by
Denote by Open image in new window the Open image in new window -semigroup generated by Open image in new window . It follows that, for each Open image in new window , the first coordinate of

is the unique solution of (1.1).

Theorem 2.2.

Let Open image in new window be the generator of a Open image in new window -semigroup Open image in new window on the Banach space Open image in new window and Open image in new window with Open image in new window . Assume that

   Open image in new window is a left-shift invariant closed subspace of Open image in new window such that Open image in new window for all Open image in new window ;

for some constant Open image in new window . Here, Open image in new window  :  Open image in new window .

Then

  (1.1) is strongly stable if Open image in new window ;

  if Open image in new window is reflexive and Open image in new window , then every solution to (1.1) is asymptotically almost periodic provided Open image in new window is countable.

Proof.

Since the first coordinate of (2.5) is the unique solution of (1.1), it is easy to see that the strong stability and asymptotic almost periodicity of (1.1) follows from the strong stability and asymptotic almost periodicity of Open image in new window , respectively.

Moreover, from [9, Proposition 2.8]) we know that if Open image in new window is a closed subspace of Open image in new window such that Open image in new window is Open image in new window -invariant and Open image in new window for all Open image in new window , then Open image in new window (the restriction of Open image in new window to Open image in new window ) generates the Open image in new window -semigroup
which is defined on the Banach space
Thus, by assumptions (i), (ii) and the well-known Hille-Yosida theorem for Open image in new window -semigroups, we know that Open image in new window is bounded. Hence, in view of Lemma 1.2, we get
for each Open image in new window . So, combining (2.5) with (2.9), we have

This means that (a) holds.

On the other hand, we note that, to get (b), it is sufficient to show that Open image in new window is asymptotically almost periodic. Actually, if Open image in new window is reflexive and Open image in new window , then it is not hard to verify that Open image in new window is reflexive. Hence, Open image in new window is reflexive. By assumption (i), Open image in new window is a closed subspace of Open image in new window . Thus, Pettis's theorem shows that Open image in new window is also reflexive. Hence, in view of Lemma 1.3, we get (b). This completes the proof.

Corollary 2.3.

Let Open image in new window be the generator of a Open image in new window -semigroup Open image in new window on the Banach space Open image in new window and Open image in new window . Assume that

(i)for each Open image in new window ,

(ii)there exists a constant Open image in new window satisfying

with

Then

for each Open image in new window , then (1.1) is strongly stable;

  if Open image in new window is reflexive and Open image in new window , then (1.1) is asymptotically almost periodic provided

is countable.

Proof.

As in [9, Section 3], we take
In view of the discussion in [8, Lemma Open image in new window ], we can infer that
Moreover, we have

with Open image in new window being defined as in (2.13). Thus, it is clear that Open image in new window is bounded if (2.12) is satisfied.

Next, for Open image in new window , we consider the eigenequation
Writing Open image in new window and Open image in new window , we see easily that (2.20) is equivalent to
then by (2.21) we obtain
By the closed graph theorem, the operator
in the second equality of (2.23) is bounded. Hence, noting that

Consequently, in view of (a) of Theorem 2.2, we know that (1.1) is strongly stable if (2.14) holds.

Furthermore, by [9, Lemma 3.3], we have

Combining this with (b) of Theorem 2.2, we conclude that (1.1) is asymptotically almost periodic if Open image in new window is reflexive, Open image in new window , and the set in (2.15) is countable.

Theorem 2.4.

Let Open image in new window be the generator of a Open image in new window -semigroup Open image in new window on the Banach space Open image in new window and Open image in new window with Open image in new window . Assume that

Open image in new window is bounded on Open image in new window , where Open image in new window ,

for each Open image in new window is the set of half-line spectrum of Open image in new window and Open image in new window ,

exists uniformly for Open image in new window .

Then every solution to (1.1) is asymptotically almost periodic. Moreover, if for each Open image in new window and Open image in new window the limit in (2.30) equals 0 uniformly for Open image in new window , then (1.1) is strongly stable.

Proof.

Take Open image in new window . Then the solution Open image in new window to (1.1) is Lipschitz continuous and hence uniformly continuous. Actually, by assumption (i), we know that
and that
On the other hand, if (2.28) holds, then there exists Open image in new window such that
Hence, from [4, Chapter 1] (or [5]) and the uniqueness of the Laplace transform, it follows that
satisfies
and that
Moreover, by [3, Corollary Open image in new window ], the assumption (i) implies the boundedness of Open image in new window . Therefore,
is bounded and uniformly continuous on Open image in new window . In addition, the half-line spectrum set of Open image in new window is just the following set:
From assumption (ii) and [3, Corollary Open image in new window ], it follows that Open image in new window is bounded, which implies

uniformly for Open image in new window . Finally, combining (2.40) with Theorem [7, Theorem 4.1], we complete the proof.

3. Applications

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

First, we apply Corollary 2.3 to Example 2.1. As one will see, the previous result will be obtained by a different point of view.

Example 3.1.

We reconsider Example 2.1. First, we note that
This implies that condition Open image in new window is not satisfied. Therefore, part (a) of Corollary 2.3 is not applicable, and this explains partially why the corresponding Volterra equation is not strongly stable. However, it is easy to check that conditions (i) and (ii) in Corollary 2.3 are satisfied. In particular, we have accordingly
and hence the estimate

Applying part (b) of Corollary 2.3, we know that the corresponding Volterra equation is asymptotically almost periodic.

Example 3.2.

Consider the Volterra equation
where the constants satisfy
Then (3.5) can be formulated into the abstract form (1.1). It is well known that Open image in new window is self-adjoint (see, e.g., [12, page 280, (b) of Example 3]) and that Open image in new window generates an analytic Open image in new window -semigroup. The self-adjointness of Open image in new window implies
On the other hand, we can compute
It follows immediately that condition (i) in Corollary 2.3 holds. Moreover, corresponding to (2.13), we have
Combining this with (3.6), (3.8), and (3.9), we estimate
Note that (2.15) becomes

Applying part (b) of Corollary 2.3, by (3.11), we conclude that (3.5) is asymptotically almost periodic (cf. [9, Remark 3.6]).

Notes

Acknowledgments

This work was supported partially by the NSF of China (11071042) and the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900).

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Copyright information

© J.-H. Chen and T.-J. Xiao. 2011

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

  1. 1.School of Mathematical and Computational ScienceHunan University of Science and TechnologyXiangtanChina
  2. 2.School of Mathematical SciencesFudan UniversityShanghaiChina

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