Pseudo almost periodic functions and pseudo almost periodic solutions to dynamic equations on time scales

Abstract

In this paper, we first introduce a concept of the mean-value of uniformly almost periodic functions on time scales and give some of its basic properties. Then, we propose a concept of pseudo almost periodic functions on time scales and study some basic properties of pseudo almost periodic functions on time scales. Finally, we establish some results about the existence of pseudo almost periodic solutions to dynamic equations on time scales.

Uniformly rd-piecewise almost periodic functions with applications to the analysis of impulsive Δ-dynamic system on time scales

“…In Ref. [9], the properties of mean-value were obtained under assuming that f ðt; xÞ is almost periodic in t uniformly for x 2 R n . For more details, the reader can consult Ref.…”

“…For more details, the reader can consult Ref. [9]. The following two definitions were introduced and studied in Ref.…”

“…Using the concept and relative properties of mean-value of almost periodic functions on time scales in the work [9], we establish some new theorems for admitting an exponential dichotomy of homogenous dynamic systems with impulsive effects, which are completely new even the time scale T ¼ R or T ¼ Z. Furthermore, the expressions of almost periodic solutions for nonlinear impulsive dynamic systems are also obtained under the sense of the associate homogeneous dynamic equations of the system admitting an exponential dichotomy on time scales.…”

“…Because of this fact, it has been attracting the attention of many mathematicians (see Refs. [3][4][5][6][7][8][9][10][11][12][13][14][15]). …”

Smooth stable invariant manifolds on time scales

“…The concept of uniform or nonuniform exponential dichotomies on time scales introduced in [3,4,5,6] is a very important method and tool to explore the dynamic behavior of nonautonomous dynamical systems such as the existence and roughness [4,5,7], the Hartman-Grobman theorems [6,8,9], periodic solutions [4,11], (pseudo) almost-periodic solutions [10,12,13,14,15,16] and impulsive dynamic systems [17].…”