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

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## 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## 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 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.

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].

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.

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 ,

with

Then

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

is countable.

Proof.

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.

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 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.

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.

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]).

## 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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