Affine-periodic solutions for nonlinear dynamic equations on time scales

Chuanbiao Wang,Yong Li

doi:10.1186/s13662-015-0634-0

Abstract

The existence of affine-periodic solutions for dynamic equations on time scales is studied. Mainly, via the topological degree theory, a general existence theorem is proved, which provides an effective method in the qualitative theory for nonlinear dynamic equations on time scales.

Highlights

1 Introduction The periodicity problem is a very important topic in the study of differential equations, but not all the natural phenomena can be described by periodicity only
We found that some differential equations exhibit a certain symmetry rather than periodicity [ – ]
The aim of this paper is to touch on such a topic for APSs on time scales

Summary

Introduction

The periodicity problem is a very important topic in the study of differential equations, but not all the natural phenomena can be described by periodicity only. We found that some differential equations exhibit a certain symmetry rather than periodicity [ – ]. Dt where f : R × Rn → Rn is continuous, and for some Q ∈ GLn(R), the following affine symmetry holds:. In the sense of ( ), we have the concept of an affine-periodic system (APS for short). The system ( ) is said to be a (Q, T)-affine-periodic system, if there exists Q ∈ GLn(R) and T > such that f (t + T, x) = Qf t, Q– x holds for all (t, x) ∈ R × Rn

Objectives

Conclusion