Eigenvalue criteria for existence of positive solutions of impulsive differential equations with non-separated boundary conditions

Ruixi Liang,Jianhua Shen

doi:10.1186/1687-2770-2013-3

Abstract

In this paper, we discuss the existence of positive solutions for second-order differential equations subject to nonlinear impulsive conditions and non-separated periodic boundary value conditions. Our criteria for the existence of positive solutions will be expressed in terms of the first eigenvalue of the corresponding nonimpulsive problem. The main tool of study is a fixed point theorem in a cone.

Highlights

We are concerned with the existence of positive solutions for the following boundary value problem (BVP) with impulses:
Motivated by the work above, in this paper we study the existence of positive solutions for the boundary value problem ( . )
Α ≤ u ≤ β and u(t) ≥ δα >, which means that u(t) is a positive solution of Eq ( . )

Summary

Introduction

We are concerned with the existence of positive solutions for the following boundary value problem (BVP) with impulses:. Based upon the properties of Green’s function obtained in [ ], the authors extended and improved the work of [ ] by using topological degree theory They derived new criteria for the existence of non-trivial solutions, positive solutions and negative solutions of the problem There is no result about nonlinear impulsive differential equations with non-separated periodic boundary conditions. Motivated by the work above, in this paper we study the existence of positive solutions for the boundary value problem Assume that there exist positive constants α, β such that fγα ≥ , fγβ ≥ , Iiα ≥ , Iiβ ≥ and φ(s) ds – δ δfγα φ(s).

When λ

Ii φ fγ