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Uniformly Almost Periodic Functions and Almost Periodic Solutions to Dynamic Equations on Time Scales

DOI: 10.1155/2011/341520

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Abstract:

Firstly, we propose a concept of uniformly almost periodic functions on almost periodic time scales and investigate some basic properties of them. When time scale or , our definition of the uniformly almost periodic functions is equivalent to the classical definitions of uniformly almost periodic functions and the uniformly almost periodic sequences, respectively. Then, based on these, we study the existence and uniqueness of almost periodic solutions and derive some fundamental conditions of admitting an exponential dichotomy to linear dynamic equations. Finally, as an application of our results, we study the existence of almost periodic solutions for an almost periodic nonlinear dynamic equations on time scales. 1. Introduction In recent years, researches in many fields on time scales have received much attention. The theory of calculus on time scales (see [1, 2] and references cited therein) was initiated by Hilger in his Ph.D. thesis in 1988 [3] in order to unify continuous and discrete analysis, and it has a tremendous potential for applications and has recently received much attention since his fundamental work. It has been created in order to unify the study of differential and difference equations. Many papers have been published on the theory of dynamic equations on time scales [4–10]. Also, the existence of almost periodic, asymptotically almost periodic, and pseudo-almost periodic solutions is among the most attractive topics in qualitative theory of differential equations and difference equations due to their applications, especially in biology, economics and physics [11–29]. However, there are no concepts of almost periodic functions on time scales so that it is impossible for us to study almost periodic solutions for dynamic equations on time scales. Motivated by the above, our main purpose of this paper is firstly to propose a concept of uniformly almost periodic functions on time scales and investigate some basic properties of them. Then we study the existence and uniqueness of almost periodic solutions to linear dynamic equations on almost time scales. Finally, as an application of our results, we study the existence of almost periodic solutions for almost periodic nonlinear dynamic equations on time scales. The organization of this paper is as follows. In Section 2, we introduce some notations and definitions and state some preliminary results needed in the later sections. In Section 3, we propose the concept of uniformly almost periodic functions on almost periodic time scales and investigate the basic properties of uniformly almost

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