Infinitely many singularities and denumerably many positive solutions for a second-order impulsive Neumann boundary value problem

Minmin Wang,Meiqiang Feng

doi:10.1186/s13661-017-0784-y

Abstract

Using a fixed point theorem of cone expansion and compression of norm type and a new method to deal with the impulsive term, we prove that the second-order singular impulsive Neumann boundary value problem has denumerably many positive solutions. Noticing that M>0, our main results improve many previous results.

Highlights

1 Introduction We are concerned with the existence of denumerably many positive solutions of the second-order singular impulsive Neumann boundary value problem
Wang and Feng Boundary Value Problems (2017) 2017:50 and Hölder’s inequality, the authors showed the existence of countably many positive solutions
The Arzelà-Ascoli theorem implies that T is completely continuous, and the lemma is proved

Summary

Introduction

Wang and Feng Boundary Value Problems (2017) 2017:50 and Hölder’s inequality, the authors showed the existence of countably many positive solutions. Under the case M = and ω(t) with infinitely many singularities, the properties of the corresponding Green’s function for problem We present some new properties of Green’s function under the case M = and ω(t) with infinitely many singularities. In Section , we obtain some new sufficient conditions for the existence of denumerably many positive solutions for problem Suppose that G(t, s) is the Green’s function of the boundary value problem. By the definition of G(t, s) and the properties of sinhx and coshx, we have the following results. To establish the existence of positive solutions to problem To establish the existence of positive solutions to problem ( . ), for a fixed τ ∈ ( , δ), we construct the cone Kτ in PC [ , ] by

Dik sinh γ

From π