On third-order three-point right focal boundary value problems

Abstract

In this paper, a fixed point theorem in a cone, some inequalities of the associated Green's function and the concavity of solutions are applied to obtain the existence of positive solutions of third-order three-point boundary value problem with dependence on the first-order derivative

$\begin{cases}& x'''(t) = f(t, x(t), x'(t)), \quad 0 < t < 1, \\ & x(0) = x'(\eta) = x''(1) = 0, \end{cases}$

where $f:[0, 1]\times[0, \infty)\times R\to [0,\infty)$ is a nonnegative continuous function, $\eta\in(1/2, 1).$