Existence of positive solutions for nonlinear three-point boundary value problem

Abstract:

In this paper,~we study the existence of positive solutions for second-order three-point boundary value problem \[ \begin{cases} u''-k^{2}u+a(t)f(u)=0,~~t\in(0,1),\u(0)=0,~~u(1)=\alpha u(\eta), \end{cases} \] where~$a\in C([0,1],[0,\infty)),~\eta\in(0,1),~\alpha\in\Big(0,\frac{\sinh(k)}{\sinh(k\eta)}\Big),~f\in C([0,\infty),[0,\infty))$.~The proof of the main result is based on fixed point theorem.