Positive solutions for nonlinear singular third order boundary value problem

Abstract

In this paper, we investigate the problem of existence of positive solutions for the nonlinear third order boundary value problem:

$$\begin{array}{cc}\hfill & u^{‴}(t)+\lambda a(t)f(u(t))=0\text{,}0<t<1\text{,}\hfill \\ \hfill & u(0)=u^{\prime }(0)=0\text{,}\alpha u^{\prime }(1)+\beta u^{″}(1)=0\text{,}\hfill \end{array}$$

where $\lambda$ is a positive parameter. By using Krasnoselskii’s fixed-point theorem of cone, we establish various results on the existence of positive solutions of the boundary value problem.

Under various assumptions on $a(t)$ and $f(u(t))$, we give the intervals of the parameter $\lambda$ which yield the existence of the positive solutions. An example is also given to illustrate the main results.

Introduction

One of the most frequently used tools for proving the existence of positive solutions to the integral equations and boundary value problems is Krasnoselskii’s theorem on cone expansion and compression and its norm-type version due to Guo and Lakshmikantham [3]. To the best of our knowledge, the first paper taking this approach is by Wang in [7]. Since this pioneering work a lot research has been done in this area. Third order equations arise in a variety of different areas of applied mathematics and physics, as the deflection of a curved beam having a constant or varying cross section, three layer beam, electromagnetic waves or gravity driven flows and so on. Different type of techniques has been used to study such problems [4].

Anderson [1] proved that there exist at least three positive solutions to the boundary value problem by using the famous Leggett–Williams fixed-point theorem. Anderson and Avery [2] applied a generalization of the Leggett–Williams fixed-point theorem to obtain the existence of at least three solutions for to a third order discrete focal boundary value problem. In [8], Yao and Feng used the upper and lower solutions method to prove some existence results for the following third order two-point boundary value problem:

$$\begin{array}{cc}\hfill & u^{‴}(t)+f(t\text{,}(u(t))=0\text{,}0⩽t⩽1\text{,}\hfill \\ \hfill & u(0)=u^{\prime }(0)=u^{\prime }(1)=0\text{.}\hfill \end{array}$$

Li [4] used Krasnoselskii’s fixed-point theorem to prove some existence results to the nonlinear third order singular boundary value problem

$$\begin{array}{cc}\hfill & u^{‴}(t)+\lambda a(t)f(u(t))=0\text{,}0<t<1\text{,}\hfill \\ \hfill & u(0)=u^{\prime }(0)=u^{‴}(1)=0\text{.}\hfill \end{array}$$

Sun and Wen [6] considered the nonlinear third order singular boundary value problem

$$\begin{array}{cc}\hfill & u^{‴}(t)+\lambda a(t)f(u(t))=0\text{,}0<t<1\text{,}\hfill \\ \hfill & \alpha u^{\prime }(0)-\beta u^{″}(0)=0\text{,}u(1)=u^{\prime }(1)=0\text{.}\hfill \end{array}$$

Very recently, Liu et al. [5] used Krasnoselskii’s fixed-point theorem to prove some existence results to the nonlinear third order singular boundary value problem

$$\begin{array}{cc}\hfill & u^{‴}(t)-\alpha (t)f(t\text{,}u(t))=0\text{,}a<t<b\text{,}\hfill \\ \hfill & u(a)=u(b)=u^{″}(b)=0\text{.}\hfill \end{array}$$

The purpose of this paper is to establish the existence of positive solutions to nonlinear third order boundary value problem:

$$u^{‴}(t)+\lambda a(t)f(u(t))=0\text{,}0<t<1\text{,}$$

$$u(0)=u^{\prime }(0)=0\text{,}\alpha u^{\prime }(1)+\beta u^{″}(1)=0\text{,}$$

where $\lambda >0$ is a positive parameter and $a:(0\text{,}1)\to [0\text{,}\infty )$ is continuous and ${\int }_0^{1}a(t)\mathrm{d}t>0$, $f:[0\text{,}\infty )\to [0\text{,}\infty )$ is continuous and $\alpha \text{,}\beta ⩾0\text{,}\alpha +\beta >0$. Here, by a positive solution of the boundary value problem we mean a function which is positive on (0, 1) and satisfies differential equation (1.1) and the boundary condition (1.2).