New Green\u2019s function and two infinite families of positive solutions for a second order impulsive singular parametric equation

Minmin Wang,Meiqiang Feng

doi:10.1186/s13662-017-1211-5

Minmin Wang, Meiqiang Feng

Open Access

https://doi.org/10.1186/s13662-017-1211-5

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Abstract

Using the theorem and properties of the fixed point index in a Banach space and applying a new method to dispose of the impulsive term, we prove that there exists a solvable interval of positive parameter λ in which the second order impulsive singular equation has two infinite families of positive solutions. Moreover, we also establish the new expression of Green’s function for the above equation. Noticing that lambda>0 and c_{k}neq0 (k=1,2,ldots,n), our main results improve many previous results. This is probably the first time that the existence of two infinite families of positive solutions for second order impulsive singular parametric equations has been studied.

Highlights

1 Introduction In this paper, we consider the existence of two infinite families of positive solutions for the second impulsive singular parametric differential equation
By using the transformation technique to deal with impulsive term of second impulsive differential equations, the authors obtained the existence results of positive solutions by using fixed point theorems in a cone
For each natural number i, letf satisfy (H ) and (H ), there exists τ > such that, for < λ < τ , problem

Summary

Introduction

We consider the existence of two infinite families of positive solutions for the second impulsive singular parametric differential equation. By using the transformation technique to deal with impulsive term of second impulsive differential equations, the authors obtained the existence results of positive solutions by using fixed point theorems in a cone. They only gave the sufficient conditions for the existence of finite positive solutions. Positive solutions for second order singular impulsive parametric equations; for details, see [ – ]. There are almost no papers considering a second order impulsive parametric equation with infinitely many singularities.

Let θ

Define a cone K in E by