Existence Results and Numerical Method for a Fourth Order Nonlinear Problem

In this paper we consider a fully fourth order nonlinear boundary value problem which models the bending equilibrium of an extensible beam. Firstly we establish the existence and uniqueness of solution under some easily verified conditions. Next, for finding the approximate solution we propose an iterative method at continuous level and prove its convergence. To numerically realize the iterative method we carry out its discretization with the use of difference schemes of the fourth order of accuracy. Finally, some examples demonstrate the applicability of the theoretical results and the efficiency of the iterative method.