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

Abstract

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.

Highlights

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

Summary

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

Conclusion

Methods