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In this paper we study the existence and uniqueness of a solution and proposean iterative method for solving a beam problem which is described by the fullyfourth order equation $$u^{(4)}(x)=f(x,u(x),u'(x),u'''(x),u'''(x)), \quad 0 < x< 1$$ associated with the Dirichlet boundary conditions. This problem wasstudied by several authors. Here we propose a novel approach by the reductionof the problem to an operator equation for the triplet of the nonlinear term$\varphi (x)=f(x,u(x),u'(x),u''(x),u'''(x))$ and the unknown values $u''(0),u''(1).$ Under some easily verified conditions on the function $f$ in aspecified bounded domain, we prove the contraction of the operator. Thisguarantees the existence and uniqueness of a solution and the convergence of aniterative method for finding it. Many examples demonstrate the applicability ofthe theoretical results and the efficiency of the iterative method. Theadvantages of the obtained results over those of Agarwal are shown on someexamples.

Existence results and numerical solution for the Dirichlet problem for fully fourth order nonlinear equation