Multiple solutions for three-point boundary value problem with nonlinear terms depending on the first order derivative

Abstract.

In this paper, we consider the following second-order three-point boundary value problem

$$ \begin{array}{*{20}l} {u''(t) + f(t,u(t),u'(t)) = 0,\quad t \in (0,1),} \\ {u(0) = 0,u(1) = \xi u(n),} \\ \end{array} $$

where f : [0, 1] × R² → R is continuous, ξ > 0, 0 < η < 1 such that ξη < 1. We give conditions on f and two pairs of lower and upper solutions to ensure the existence of at least three solutions of the given problem. Our method is based upon Leray-Schauder degree theory. The emphasis here is that f depends on the first derivative. Our results extend some results in the references.