Eigenvalues of Fourth-Order Singular Sturm-Liouville Boundary Value Problems

Abstract

In this paper, by using Krasnoselskii's fixed-point theorem, some sufficient conditions of existence of positive solutions for the following fourth-order nonlinear Sturm-Liouville eigenvalue problem:\begin{equation*}\left\{\begin{array}{lll} \frac{1}{p(t)}(p(t)u''')'(t)+ \lambda f(t,u)=0, t\in(0,1), \\ u(0)=u(1)=0, \\ \alpha u''(0)- \beta \lim_{t \rightarrow 0^{+}} p(t)u'''(t)=0, \\ \gamma u''(1)+\delta\lim_{t \rightarrow 1^{-}} p(t)u'''(t)=0, \end{array}\right.\end{equation*} are established, where $\alpha,\beta,\gamma,\delta \geq 0,$ and $~\beta\gamma+\alpha\gamma+\alpha\delta >0$. The function $p$ may be singular at $t=0$ or $1$, and $f$ satisfies Carathéodory condition.