Positive solutions of a fourth-order differential equation with integral boundary conditions

Abstract

We study the existence of positive solutions to the fourth-order two-point boundary value problem \begin{cases} u^{\prime\prime\prime\prime}(t) + f(t,u(t))=0, & 0 < t < 1, u^{\prime}(0) = u^\prime(1) = u^{\prime\prime}(0) =0, & u(0) = \alpha[u], \end{cases} where $\alpha[u]=\int^1_0u(t){\rm d}A(t)$ is a Riemann-Stieltjes integral with $A \geq0$ being a nondecreasing function of bounded variation and $f \in\mathcal{C}([0,1] \times\mathbb{R}_+, \mathbb{R}_+)$. The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii's fixed point theorem and the Avery-Peterson fixed point theorem.

Keywords

• boundary value problem
• fixed point
• positive solution
• cone
• existence theorem