Positive Solutions to a Singular Third-Order Three-Point Boundary Value Problem

Abstract:

      In this paper, we study the existence of positive solutions for the nonlinear singular third-order three-point boundary value problem $$\left\{\begin{array}{l}u'''(t)=-\lambda a(t)f(t,u(t)),\\ u(0)=u'(1)=u''(\eta)=0,\end{array}\right.$$ where $\lambda $ is a positive parameter and $0\le\eta<\frac{1}{2}$. By using the classical Krasnosel'skii's fixed point theorem in cone, we obtain various new results on the existence of positive solution, and the solution is strictly increasing. Finally we give an example.