Nontrivial solution of a nonlinear second-order three-point boundary value problem

Abstract

In this paper, for a second-order three-point boundary value problem $$\begin{gathered} u'' + f(t,u) = 0, 0 < t < 1, \hfill \\ au(0) - bu'(0) = 0, u(1) - \alpha u(\eta ) = 0, \hfill \\ \end{gathered} $$ where η ∈ (0, 1), a, b, α ∈ R with a2 + b2 > 0, the existence of its nontrivial solution is studied. The conditions on f which guarantee the existence of nontrivial solution are formulated. As an application, some examples to demonstrate the results are given.