Existence of two positive solutions for a class of second order impulsive singular integro-differential equations on the half line

Dajun Guo

doi:10.1186/s13661-015-0337-1

Abstract

In this paper, the author discusses the existence of two positive solutions for an infinite boundary value problem of second order impulsive singular integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.

Highlights

1 Introduction The theory of impulsive differential equations has been emerging as an important area of investigation in recent years
Many problems have been investigated for impulsive differential equations, impulsive functional differential equations and impulsive differential inclusions
There is a vast literature on existence of solutions: by using upper and lower solutions together with the monotone iterative technique to obtain the extremal solutions [ – ]; by using fixed point theorems to obtain the existence of solution and multiple solutions [ – ]; by using the Leray-Schauder degree theory or fixed point index theory to obtain multiple solutions [ – ]; by using the variational method to obtain the existence of solution and existence of infinite many solutions [ – ]

Summary

Introduction

The theory of impulsive differential equations has been emerging as an important area of investigation in recent years (see [ – ]). In recent article [ ], the author discussed the existence of two positive solutions for an infinite boundary value problem of first order impulsive singular integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type, which was established by the author in [ ] (see [ – ]). It is clear: if condition (H ) is satisfied, ( ) implies ( ).

Results

Conclusion