Existence of Multiple Positive Solutions for Third-Order Three-Point Boundary Value Problem

Qiufeng Chen,Jianli Li

doi:10.4236/jamp.2019.77098

Abstract

In this paper, we study the existence of positive solutions for a class of third-order three-point boundary value problem. By employing the fixed point theorem on cone, some new criteria to ensure the three-point boundary value problem has at least three positive solutions are obtained. An example illustrating our main result is given. Moreover, some previous results will be improved significantly in our paper.

Highlights

IntroductionThe earliest boundary value problem studied is Dirichlet problem. We need to find the solution of Laplace equation
As we all know, the earliest boundary value problem studied is Dirichlet problem
There have been many papers dealing with the positive solutions of boundary value problems for nonlinear differential equations with various boundary conditions

Summary

Introduction

The earliest boundary value problem studied is Dirichlet problem. We need to find the solution of Laplace equation. In [6] [7] [8] [9] [10], the authors have studied the third-order three-point boundary value problem and proved that the model has at least one positive solution. Considered the existence of a positive solution to the third-order three-point boundary value problem as follows u′′′(t ) + a (t ) f (u (t )) = 0, t ∈ (0,1), u= (0) u= ′(0) 0. Few papers can be found in the literature for three positive solutions of third-order three-point boundary value problems. By the properties of the Green’s function, existence results of at least three positive solution for the third-order three-point boundary value problem are established by a new method which is different from the method in [13].

Preliminaries

The Existence of Three Positive Solutions

Example

Conclusion