Strong Stability and Asymptotical Almost Periodicity of Volterra Equations in Banach Spaces
© J.-H. Chen and T.-J. Xiao. 2011
Received: 1 January 2011
Accepted: 1 March 2011
Published: 14 March 2011
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.
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.
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.
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.
It is clear that the following assertions hold.
is the unique solution of (1.1).
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.
Consequently, in view of (a) of Theorem 2.2, we know that (1.1) is strongly stable if (2.14) holds.
uniformly for . Finally, combining (2.40) with Theorem [7, Theorem 4.1], we complete the proof.
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.
Applying part (b) of Corollary 2.3, we know that the corresponding Volterra equation is asymptotically almost periodic.
Applying part (b) of Corollary 2.3, by (3.11), we conclude that (3.5) is asymptotically almost periodic (cf. [9, Remark 3.6]).
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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