Strong Stability and Asymptotical Almost Periodicity of Volterra Equations in Banach Spaces
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Research Article Strong Stability and Asymptotical Almost Periodicity of Volterra Equations in Banach Spaces Jian-Hua Chen1 and Ti-Jun Xiao2 1
School of Mathematical and Computational Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China 2 School of Mathematical Sciences, Fudan University, Shanghai 200433, China Correspondence should be addressed to Ti-Jun Xiao, [email protected] Received 1 January 2011; Accepted 1 March 2011 Academic Editor: Toka Diagana Copyright q 2011 J.-H. Chen and T.-J. Xiao. 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. 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 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: u t Aut
t
at − sAusds,
0
t ≥ 0,
1.1
u0 x in a Banach space X, with A being the infinitesimal generator of a C0 -semigroup T t defined on X, and a· ∈ Lp R , C a scalar function R : 0, ∞ and 1 ≤ p < ∞, 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 St, and the mild solution is then given by ut Stx,
t ≥ 0,
1.2
2
Advances in Difference Equations
which is actually a classical solution if x ∈ DA. 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 X be a Banach space and f : R → X a bounded uniformly continuous function. i f is called almost periodic if it can be uniformly approximated by linear combinations of eibt x b ∈ R, x ∈ X. Denote by AP R , X the space of all almost periodic functions on R . ii f is called asymptotically almost periodic if f f1 f2 with limt → ∞ f1 t 0 and f2 ∈ AP R , X. Denote by AP P R , X the space of all asymptotically almost periodic functions on R . iii We call 1.1 or St strongly stable if, for each x ∈ DA, limt → ∞ Stx 0. We call 1.1 or St asymptotically almost periodic if for each x ∈ DA, S·x ∈ AP P R , X. The following two results on C0 -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 5.5.6. As usual, the letter i denotes the imaginary unit and iR the imagina
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