%0 Journal Article
%T Existence of positive solutions for singular fourth-order three-point boundary value problems
%A Yan Sun and Cun Zhu
%J Advances in Difference Equations
%D 2013
%I 
%R 10.1186/1687-1847-2013-51
%X In this paper, we consider the boundary value problem $u^{(4)}(t)+f(t,u(t))=0,\,0<t<1$, subject to the boundary conditions $u(0)=u'(0)=u''(0)=0$, and $u''(1)-\alpha u''(\eta)=\lambda$. In this setting $0 < \eta < 1$ and $\alpha\in [0, \frac{1}{\eta})$ are constants, and $\lambda \in[0, +\infty)$ is a parameter. By imposing sufficient structure on the nonlinearity $f(t, u)$, we deduce the existence of at least one positive solution to the problem. The novelty in our setting lies in the fact that $f(t, u)$ may be singular at $t=0$ and $t=1$. Our results here are achieved by making use of the Krasnosel'skii fixed point theorem. We conclude with an example illustrating our results and the improvements that they presents.
%U http://www.advancesindifferenceequations.com/content/2013/1/51/abstract