Variational approach to a class of p-Laplacian systems on time scales

Jianwen Zhou,Yongkun Li

doi:10.1186/1687-1847-2013-297

Abstract

In this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of nontrivial periodic solutions for a class of p-Laplacian systems on time scales. By establishing a proper variational setting, three existence results are obtained. Finally, two examples are presented to illustrate the feasibility and effectiveness of our results.MSC:34N05, 37J45, 34C25.

Highlights

Many authors have been devoted to the investigation concerning the existence of periodic solutions of ( . ) by using critical point theory
In [ ], the authors studied Sobolev’s spaces on time scales and their properties. They presented a recent approach via variational methods and the critical point theory to obtain the existence of solutions of ( . )
In [ ], authors used the fixed point theorem of strictset-contraction to study the existence of positive periodic solutions for functional dynamic equations with impulse effects on time scales

Summary

Introduction

In [ ], the authors studied Sobolev’s spaces on time scales and their properties As applications, they presented a recent approach via variational methods and the critical point theory to obtain the existence of solutions of In [ ], authors used the fixed point theorem of strictset-contraction to study the existence of positive periodic solutions for functional dynamic equations with impulse effects on time scales. As is well known, it is very difficult to use Hilger’s integral to deal with the existence of solutions to a dynamic equation on time scales since it is only concerned with antiderivatives It is still worth making an attempt to extend variational methods to study of periodic solutions for various Hamiltonian systems because there are tremendous applications on periodic solutions to discrete or continuous Hamiltonian systems theoretically and practically [ – ]. As shown in [ ], a deformation lemma can be proved with the weaker condition (C) replacing the usual P.S. condition, and it turns out that the saddle point theorem (Theorem . ) holds true under condition (C)

Variational setting

Main results