On the lower and upper solutions method for the problem of elastic beam with hinged ends

We establish the existence theorem for solutions of fourth-order nonlinear equations with the Lidstone boundary conditions which model deflections of an elastic beam (curve) with the hinged ends. The new lower and upper solutions method is formulated and then applied to a class of singularly perturbed problems and here is shown that the solutions uniformly converge to the solution of a reduced problem in the interior points of elastic curve and formula for an approximate solution without the use of numerical or iterative methods is derived.