Multiplicity Results for Kirchhoff-Type Three-Point Boundary Value Problems

This article is concerned with the multiplicity results of the solutions for a Kirchhoff-type three-point boundary value problem. The method observed here is according to variational methods and critical point theory. In fact, through a consequence of the local minimum theorem due Bonanno and mountain pass theorem, we look into the existence results for our problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by combining two algebraic conditions on the nonlinear term, employing two consequences of the local minimum theorem due Bonanno we guarantee the existence of two solutions, and applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for the problem