Positive solutions for a second-order p-Laplacian impulsive boundary value problem

Youzheng Ding,Donal O’Regan

doi:10.1186/1687-1847-2012-159

Abstract

In this work, we study the existence and multiplicity of positive solutions for a second-order p-Laplacian boundary value problem involving impulsive effects. We establish our main results via Jensen’s inequality, the first eigenvalue of a relevant linear operator and the Krasnoselskii-Zabreiko fixed point theorem. Some examples are presented to illustrate the main results.MSC:34B15, 34B18, 34B37, 45G15, 45M20.

Highlights

Second-order differential equations with the p-Laplacian operator arise in modeling some physical and natural phenomena and can occur, for example, in non-Newtonian mechanics, nonlinear elasticity, glaciology, population biology, combustion theory, and nonlinear flow laws, see [, ]
Many cases of the existence and multiplicity of positive solutions for boundary value problems of differential equations with the p-Laplacian operator have appeared in the literature
By virtue of Krasonsel’skii’s fixed point theorem, they obtained the existence of positive solutions and multiple positive solutions under suitable conditions imposed on the nonlinear term f ∈ C([, ] × [, +∞), [, +∞))

Summary

Introduction

Second-order differential equations with the p-Laplacian operator arise in modeling some physical and natural phenomena and can occur, for example, in non-Newtonian mechanics, nonlinear elasticity, glaciology, population biology, combustion theory, and nonlinear flow laws, see [ , ]. Many cases of the existence and multiplicity of positive solutions for boundary value problems of differential equations with the p-Laplacian operator have appeared in the literature. By virtue of Krasonsel’skii’s fixed point theorem, they obtained the existence of positive solutions and multiple positive solutions under suitable conditions imposed on the nonlinear term f ∈ C([ , ] × [ , +∞), [ , +∞)).

Results

Conclusion