n₀-order Δ-almost periodic functions and dynamic equations

Abstract

In this paper, using matched spaces for time scales, we introduce new types of almost periodic functions, including -almost periodic functions and -order -almost periodic functions (-almost periodic functions). Also we introduce the definition of hull equations for homogeneous dynamic equations and obtain some existence results. Under exponential dichotomy for the corresponding homogeneous equation, we obtain the form of a -almost periodic solution with the -almost periodic affiliated function for a type of nonhomogeneous dynamic equation and we use it to study the existence of -almost periodic solutions with the -almost periodic affiliated function for new delay dynamic equations.

Keywords:

• Time scales
• almost periodic functions
• matched spaces
• delay dynamic equations

AMS Subject Classifications:

• 26E70
• 33E30
• 34N05
• 34C27

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was done while Chao Wang was a visiting scholar at Texas A&M University-Kingsville. This work is supported by Youth Fund of NSFC [grant number 11601470], Tian Yuan Fund of NSFC [grant number 11526181] and Yunnan Province Science and Technology Department Applied Basic Research Project of China [grant number 2014FB102].