Existence results and iterative method for solving the cantilever beam equation with fully nonlinear term. (August 2017) - Dang, Quang A.; Ngo, Thi Kim Quy
Abstract: In this paper we study the solvability and iterative solution of the fully fourth order nonlinear boundary value problem u ( 4 ) ( x ) = f ( x, u ( x ), u ′ ( x ), u ″ ( x ), u ‴ ( x ) ), 0 < x < 1, u ( 0 ) = u ′ ( 0 ) = u ″ ( 1 ) = u ‴ ( 1 ) = 0, where f : [ 0, 1 ] × R 4 → R is continuous. Differently from the approaches of the other authors to fully fourth order nonlinear boundary value problems, where they established the existence of solution for the problems under the restriction of the growth including a Nagumo-type condition of the nonlinear function f, in this paper, we propose a novel approach by the reduction of it to an operator equation for the function φ ( x ) = f ( x, u ( x ), u ′ ( x ), u ″ ( x ), u ‴ ( x ) ) . Due to this, we free the above restriction. In result we established the existence and uniqueness of a solution and the convergence of an iterative process. In addition, we show the monotonicity of the iterative sequence. Some examples demonstrate the applicability of the proposed approach and iterative method.