Multiplicity of solutions to a four-point boundary value problem of a differential system via variational approach

Weigao Ge,Zhihong Zhao

doi:10.1186/s13661-016-0559-x

Abstract

By constructing an adequate real functional and choosing an appropriate admissible function space, the existence of multiple solutions to a four-point boundary value problem, which may be taken as an extension of Sturm-Liouville boundary value problems, is proved via a variational approach for a second-order differential system with a p-Laplacian.

Highlights

The variational approach, together with the critical point theory, is one of the important methods in the study of two-point boundary value problems of ordinary differential equation [ – ], as well as impulsive differential equations [ – ]
U( ) – u(T) = u ( ) – u (T) =, where φp(x) = |x|p– x for x ∈ Rn, and they obtained an existence theorem of periodic solutions under the condition p
By use of the existence theorem of three critical points given by Ricceri [ ], they obtained sufficient conditions for the existence of three solutions to the discussed system, when the parameter λ is defined in a certain interval [, δ]

Summary

Introduction

The variational approach, together with the critical point theory, is one of the important methods in the study of two-point boundary value problems of ordinary differential equation [ – ], as well as impulsive differential equations [ – ]. U( ) – u(T) = u ( ) – u (T) = , where φp(x) = |x|p– x for x ∈ Rn, and they obtained an existence theorem of periodic solutions under the condition p– By use of the existence theorem of three critical points given by Ricceri [ ], they obtained sufficient conditions for the existence of three solutions to the discussed system, when the parameter λ is defined in a certain interval [ , δ].

Results

Conclusion