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

- Jian-Hua Chen
^{1}and - Ti-Jun Xiao
^{2}Email author

**2011**:414906

https://doi.org/10.1155/2011/414906

© J.-H. Chen and T.-J. Xiao. 2011

**Received: **1 January 2011

**Accepted: **1 March 2011

**Published: **14 March 2011

## 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.

## 1. Introduction

*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:

*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 , and the mild solution is then given by

which is actually a classical solution if . 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 [4–6]. Recall the following.

Definition 1.1.

Let be a Banach space and : a bounded uniformly continuous function.

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

(ii)
is called *asymptotically almost periodic* if
with
and
. Denote by
the space of all asymptotically almost periodic functions on
.

(iii)We call (1.1) or
*strongly stable* if, for each
. We call (1.1) or
asymptotically almost periodic if for each
.

The following two results on
-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
]. As usual, the letter
denotes the imaginary unit and
the imaginary axis.

Lemma 1.2.

Lemma 1.3.

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

## 2. Results and Proofs

Asymptotic behaviors of solutions to the special case of 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.

It is clear that the following assertions hold.

(a)The corresponding semigroup is exponentially stable.

(b)Each solution with initial value
is *not* strongly stable and hence not exponentially stable.

(c)Each solution with 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 [9–11].

is the unique solution of (1.1).

Theorem 2.2.

Let be the generator of a -semigroup on the Banach space and with . Assume that

is a left-shift invariant closed subspace of such that for all ;

Then

if is reflexive and , then every solution to (1.1) is asymptotically almost periodic provided 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 , respectively.

This means that (a) holds.

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

Corollary 2.3.

Let be the generator of a -semigroup on the Banach space and . Assume that

with

Then

for each , then (1.1) is strongly stable;

is countable.

Proof.

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

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

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

Theorem 2.4.

Let be the generator of a -semigroup on the Banach space and with . Assume that

for each is the set of half-line spectrum of and ,

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

Proof.

uniformly for . 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.

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

Example 3.2.

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

## Declarations

### 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).

## Authors’ Affiliations

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