Existence of positive solutions for nonlinear third-order three-point boundary value problems

Abstract

This paper is concerned with the following nonlinear third-order three-point boundary value problem:

$$u^{‴}\left(t\right)+a\left(t\right)f\left(u\left(t\right)\right)=0\text{,}t\in (0,1)\text{,}$$$$u\left(0\right)=u^{\prime }\left(0\right)=0\text{,}{u}^{\prime }\left(1\right)=\alpha u^{\prime }\left(\eta \right)\text{,}$$

where $0<\eta <1$ and $1<\alpha <\frac{1}{\eta }$. First, the Green’s function for the associated linear boundary value problem is constructed, and then, some useful properties of the Green’s function are obtained by a new method. Finally, existence results for at least one positive solution for the above problem are established when $f$ is superlinear or sublinear.

Introduction

Third-order differential equations arise in a variety of different areas of applied mathematics and physics, e.g., in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity driven flows and so on [5]. Recently, third-order boundary value problems (BVPs for short) have received much attention. For example, [3], [4], [7], [10], [14] discussed some third-order two-point BVPs, while [1], [2], [11], [12], [13] studied some third-order three-point BVPs. In particular, Anderson [1] obtained some existence results for positive solutions for the following BVP:

$$x^{‴}\left(t\right)=f(t,x\left(t\right))\text{,}{t}_1\le t\le t_3\text{,}$$$$x\left(t_1\right)=x^{\prime }\left(t_2\right)=0\text{,}\gamma x\left(t_3\right)+\delta x^{″}\left(t_3\right)=0$$

by using the well-known Guo–Krasnoselskii and Leggett–Williams fixed point theorems [6], [8], [9]. In 2005, Sun [12] established various results on the existence of single and multiple positive solutions to some third-order differential equations satisfying the following three-point boundary conditions:

$$x\left(0\right)=x^{\prime }\left(\eta \right)=x^{″}\left(1\right)=0\text{,}$$

where $\eta \in [\frac{1}{2},1)$. The main tool in [12] was the Guo–Krasnoselskii fixed point theorem [6], [8].

Motivated greatly by the above-mentioned excellent works, in this paper we will consider the existence of a positive solution to the third-order three-point BVP

$$u^{‴}\left(t\right)+a\left(t\right)f\left(u\left(t\right)\right)=0\text{,}t\in (0,1)\text{,}$$$$u\left(0\right)=u^{\prime }\left(0\right)=0\text{,}{u}^{\prime }\left(1\right)=\alpha u^{\prime }\left(\eta \right)\text{,}$$

where $0<\eta <1$ and $1<\alpha <\frac{1}{\eta }$. Throughout, we assume that the following conditions are satisfied:

(A1) $f\in C([0,\infty ),[0,\infty ))$;

(A2) $a\in C([0,1],[0,\infty ))$ and is not identically zero on $[\frac{\eta }{\alpha },\eta ]$.

First, the Green’s function for the associated linear BVP is constructed, and then, some useful properties of the Green’s function are obtained by a new method. Finally, existence results of at least one positive solution for the BVP (1.4), (1.5) are established when $f$ is superlinear or sublinear.

In order to obtain our main results, we need the following Guo–Krasnoselskii fixed point theorem [6], [8].

Theorem 1.1

Let $E$ be a Banach space, and let $K\subset E$ be a cone. Assume ${\Omega }_1,{\Omega }_2$ are bounded open subsets of $E$ with $0\in {\Omega }_1$ , ${\overline{\Omega }}_1\subset {\Omega }_2$ , and let

$$A:K\cap ({\overline{\Omega }}_2\setminus {\Omega }_1)\to K$$

be a completely continuous operator such that either

(i) $\Vert Au\Vert \le \Vert u\Vert$ for $u\in K\cap \partial {\Omega }_1$ and $\Vert Au\Vert \ge \Vert u\Vert$ for $u\in K\cap \partial {\Omega }_2$ ; or

(ii) $\Vert Au\Vert \ge \Vert u\Vert$ for $u\in K\cap \partial {\Omega }_1$ and $\Vert Au\Vert \le \Vert u\Vert$ for $u\in K\cap \partial {\Omega }_2$ .

Then $A$ has a fixed point in $K\cap ({\overline{\Omega }}_2\setminus {\Omega }_1)$ .