Existence of multiple solutions for nonlinear multi-point boundary value problems

Hongyu Li,Yang Chen,Junting Zhang

doi:10.1186/s13662-020-02604-1

Abstract

In this paper, we study some nonlinear second order multi-point boundary value problems. We first give a lemma about the characteristic values of the corresponding linear operator. Then, by fixed point theorems in the recent existing literature, we obtain the existence of multiple solutions for these nonlinear second order multi-point boundary value problems, including two positive solutions, two negative solutions, and one sign-changing solution.

Highlights

In this paper, we study the following nonlinear multi-point boundary value problem: ⎧⎨–u (t) = g(t, u(t)), 0 ≤ t ≤ 1,⎩u (0) = 0, u(1) = m–2 i=1 αiu(βi ), (1.1)where g : [0, 1] × (–∞, +∞) → (–∞, +∞) is continuous; 0 < β1 < β2 < · · · < βm–2 < 1; αi >0 (i = 1, 2, . . . , m – 2) with m–2 i=1 αi
Many authors have considered the existence of nontrivial solutions for nonlinear multi-point boundary value problems and obtained many great results
The authors proved the existence of the first eigenvalue of the relevant linear operator, and they considered the existence of positive solutions for BVP (1.3)

Summary

Introduction

We study the following nonlinear multi-point boundary value problem:. where g : [0, 1] × (–∞, +∞) → (–∞, +∞) is continuous; 0 < β1 < β2 < · · · < βm–2 < 1; αi >. Many authors have considered the existence of nontrivial solutions for nonlinear multi-point boundary value problems and obtained many great results. The authors proved the existence of the first eigenvalue of the relevant linear operator, and they considered the existence of positive solutions for BVP (1.3). The method they used is the fixed point index theory. 2. We give the main lemma about the characteristic values of the relevant linear operator, prove some auxiliary lemmas that we need, and obtain the main result of the existence of multiple solutions for BVP (1.1) in Sect.

It is obvious that we only need to prove that

By Lemma