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:

(1.1)

in a Banach space , with being the infinitesimal generator of a -semigroup defined on , and a scalar function ( and ), 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 , and the mild solution is then given by

(1.2)

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.

Suppose that generates a bounded -semigroup on a Banach space . If , then

(1.3)

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.

Let in (1.1). Then taking Laplace transform we can calculate

(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].

Let be the product Banach space with the norm

(2.2)

for each and . Then the operator matrix

(2.3)

generates a -semigroup on Here, is the vector-valued Sobolev space and the Dirac distribution, that is, for each ; the operator is given by

(2.4)

Denote by the -semigroup generated by . It follows that, for each , the first coordinate of

(2.5)

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 ;

   and

(2.6)

for some constant . Here,  : .

Then

  (1.1) is strongly stable if ;

  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.

Moreover, from [9, Proposition 2.8]) we know that if is a closed subspace of such that is -invariant and for all , then (the restriction of to ) generates the -semigroup

(2.7)

which is defined on the Banach space

(2.8)

Thus, by assumptions (i), (ii) and the well-known Hille-Yosida theorem for -semigroups, we know that is bounded. Hence, in view of Lemma 1.2, we get

(2.9)

Clearly

(2.10)

for each . So, combining (2.5) with (2.9), we have

(2.11)

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

(i)for each ,

(ii)there exists a constant satisfying

(2.12)

with

(2.13)

Then

  if and

(2.14)

for each , then (1.1) is strongly stable;

  if is reflexive and , then (1.1) is asymptotically almost periodic provided

(2.15)

is countable.

Proof.

As in [9, Section 3], we take

(2.16)

In view of the discussion in [8, Lemma ], we can infer that

(2.17)

Moreover, we have

(2.18)

Hence,

(2.19)

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

Next, for , we consider the eigenequation

(2.20)

Writing and , we see easily that (2.20) is equivalent to

(2.21)

Thus, if and

(2.22)

then by (2.21) we obtain

(2.23)

By the closed graph theorem, the operator

(2.24)

in the second equality of (2.23) is bounded. Hence, noting that

(2.25)

we have

(2.26)

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

(2.27)

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 all ,

(2.28)

is bounded on , where ,

   is analytic on and are bounded on , where

(2.29)

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

  for each and , the limit

(2.30)

exists uniformly for .

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.

Take . Then the solution to (1.1) is Lipschitz continuous and hence uniformly continuous. Actually, by assumption (i), we know that

(2.31)

and that

(2.32)

is analytic on . Thus, and

(2.33)

On the other hand, if (2.28) holds, then there exists such that

(2.34)

Hence, from [4, Chapter 1] (or [5]) and the uniqueness of the Laplace transform, it follows that

(2.35)

satisfies

(2.36)

and that

(2.37)

Moreover, by [3, Corollary ], the assumption (i) implies the boundedness of . Therefore,

(2.38)

is bounded and uniformly continuous on . In addition, the half-line spectrum set of is just the following set:

(2.39)

Write . Then

(2.40)
(2.41)

From assumption (ii) and [3, Corollary ], it follows that is bounded, which implies

(2.42)

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.

We reconsider Example 2.1. First, we note that

(3.1)

This implies that condition 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

(3.2)

and hence the estimate

(3.3)

Note and

(3.4)

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

(3.5)

where the constants satisfy

(3.6)

Let , and define

(3.7)

Then (3.5) can be formulated into the abstract form (1.1). It is well known that is self-adjoint (see, e.g., [12, page 280, (b) of Example 3]) and that generates an analytic -semigroup. The self-adjointness of implies

(3.8)

On the other hand, we can compute

(3.9)

It follows immediately that condition (i) in Corollary 2.3 holds. Moreover, corresponding to (2.13), we have

(3.10)

Combining this with (3.6), (3.8), and (3.9), we estimate

(3.11)

Note that (2.15) becomes

(3.12)

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