Changing-periodic time scales and decomposition theorems of time scales with applications to functions with local almost periodicity and automorphy

Chao Wang,Ravi P Agarwal

doi:10.1186/s13662-015-0633-1

Chao Wang, Ravi P Agarwal

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https://doi.org/10.1186/s13662-015-0633-1

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In this work, we address the periodic coverage phenomenon on arbitrary unbounded time scales and initiate a new idea, namely, we introduce the concept of changing-periodic time scales. We discuss some properties of this new concept and illustrate several examples. Specially, we establish a basic decomposition theorem of time scales which provides bridges between periodic time scales and an arbitrary time scale with a bounded graininess function μ. Based on this result, we introduce local-almost periodic and local-almost automorphic functions on changing-periodic time scales and study some related properties. The concept of changing-periodic time scales introduced in this paper will help in understanding and removing the serious deficiency which arises in the study of classical functions on time scales.

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With the development and deepening of the real analysis on time scales, many works related to time scales have appeared recently
In Section, as applications of changing-periodic time scales, first we shall introduce two new concepts - local-almost periodic and local-almost automorphic functions, and we show that these concepts can be used as a powerful tool to investigate the local-almost periodicity and local-almost automorphy of solutions of dynamic equations on time scales
From Corollary . it follows that the concept of almost automorphic functions on periodic time scales is equivalent to the concept of globally combinable-almost automorphic functions on changing-periodic time scales

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Introduction

With the development and deepening of the real analysis on time scales, many works related to time scales have appeared recently (see [ – ]). If ( ) admits an exponential dichotomy on the local part Ti, ( ) has a unique local-almost automorphic solution on Ti as follows: In what follows, we will introduce the concept of combinable-almost automorphic functions on changing-periodic time scales. Let T be a changing-periodic time scale and f be a combinable-almost automorphic function on T, and I be a combinable index number set. Proof Since Ti is an ωi-periodic sub-timescale, from T = i∈I Ti it follows that T is an ω-periodic time scale, where ω is a lowest common multiple of {ωi}i∈I and I is a combinable index number set. From Corollary . it follows that the concept of almost automorphic functions on periodic time scales is equivalent to the concept of globally combinable-almost automorphic functions on changing-periodic time scales

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