Solvability of Third-Order Three-Point Boundary Value Problems with Carathéodory Nonlinearity

Abstract

A class of third-order three-point boundary value problems is considered, where the nonlinear term is a Carathéodory function. By introducing a height function and considering the integration of this height function, an existence theorem of solution is proved when the limit growth function exists. The main tools are the Lebesgue dominated convergence theorem and the Schauder fixed point theorem.