On almost periodicity of solutions of second-order differential equations involving reflection of the argument

Peiguang Wang,Dhaou Lassoued,Syed Abbas,Akbar Zada,Tongxing Li

doi:10.1186/s13662-018-1938-7

Abstract

We study almost periodic solutions for a class of nonlinear second-order differential equations involving reflection of the argument. We establish existence results of almost periodic solutions as critical points by a variational approach. We also prove structure results on the set of strong almost periodic solutions, existence results of weak almost periodic solutions, and a density result on the almost periodic forcing term for which the equation possesses usual almost periodic solutions.

Highlights

The study of existence, uniqueness, and stability of periodic and almost periodic solutions has become one of the most attractive topics in the qualitative theory of ordinary and functional differential equations for its significance in the physical sciences, mathematical biology, control theory, and other fields; see, for instance, [3, 8, 11, 19, 20, 28] and the references cited therein
The almost periodic functions are closely connected with harmonic analysis, differential equations, and dynamical systems; cf
Wiener and Aftabizadeh [29] initiated the analysis of boundary value problems involving reflection of the argument

Summary

Introduction

The study of existence, uniqueness, and stability of periodic and almost periodic solutions has become one of the most attractive topics in the qualitative theory of ordinary and functional differential equations for its significance in the physical sciences, mathematical biology, control theory, and other fields; see, for instance, [3, 8, 11, 19, 20, 28] and the references cited therein. The almost periodic functions are closely connected with harmonic analysis, differential equations, and dynamical systems; cf Corduneanu [12] and Fink [14]. These functions are basically generalizations of continuous periodic and quasi-periodic functions. Differential equations involving reflection of the argument have numerous applications in the study of stability of differential-difference equations. Such equations show very interesting properties by themselves, and so many authors have worked on this category of equations.

Wang et al Advances in Difference Equations

Every almost periodic function u possesses a mean time

Proof The relation

The mapping q

We consider the operator