  In this paper we study the existence and uniqueness of a solution and propose
an iterative method for solving a beam problem which is described by the fully
fourth 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 was
studied by several authors. Here we propose a novel approach by the reduction
of 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 a
specified bounded domain, we prove the contraction of the operator. This
guarantees the existence and uniqueness of a solution and the convergence of an
iterative method for finding it. Many examples demonstrate the applicability of
the theoretical results and the efficiency of the iterative method. The
advantages of the obtained results over those of Agarwal are shown on some
examples.
