Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green’s function

Chenghua Gao,Tianmei Geng

doi:10.1186/s13662-016-0837-z

Abstract

In this paper, by using the Krasnosel’skii fixed point theorem in a cone, we discuss the existence of positive solutions to the discrete third-order three-point boundary value problem $$\left \{ \textstyle\begin{array}{l} \Delta^{3}u(t-1)=\lambda a(t)f(t,u(t)), \quad t\in[1,T-2]_{\mathbb{Z}}, \Delta u(0)=u(T)=\Delta^{2}u(\eta)=0, \end{array}\displaystyle \right . $$ where $T>4$ is an integer, $[1,T-2]_{\mathbb{Z}}=\{1,2,\ldots,T-2\}$ , $\lambda>0$ is a parameter, $\eta\in\{\frac{T-1}{2},\ldots,T-2\}$ for odd T, and $\eta\in\{\frac{T-2}{2},\ldots,T-2\}$ for even T. Despite the sign-changing Green’s function, we also give the explicit interval for λ to guarantee the existence of positive solutions of the problem when f satisfies different growth assumptions.

Highlights

Let [a, b]Z denote the integer set {a, a +, . . . , b} with b > a
Boundary value problems for third-order differential or difference equations arise in many problems of physics, control system and applied mathematics, such as the deflection of a curved beam having a constant or varying cross section, three-layer beam and the electromagnetic wave incident on a system of charges sets them into motion, etc. [ ]
In [ – ], by using different methods, such as the Krasnosel’skii’s fixed point theorem in a cone, the iterative technique, and the fixed point theory, the authors obtained the existence of positive solutions of the boundary value problems for third-order differential equations

Summary

Introduction

Let [a, b]Z denote the integer set {a, a + , . . . , b} with b > a. In [ – ], by using different methods, such as the Krasnosel’skii’s fixed point theorem in a cone, the iterative technique, and the fixed point theory, the authors obtained the existence of positive solutions of the boundary value problems for third-order differential equations. In , Wang and Gao [ ] discussed the existence of positive solutions of the following third-order difference equation boundary value problems: u(t – ) = a(t)f (t, u(t)), t ∈ [ , T – ]Z, u( ) = u(T) = u(η) =. The Green’s function we construct in Section changes its sign and is more complicated than in the continuous case This will takes lots of difficulties for us to obtain the existence of positive solutions.

These two equations imply that even number and η

Define the cone K by

This implies that

For any