A Positive Solution to the Boundary Value Problem for One Fourth-Order Nonlinear Ordinary Differential Equation

A boundary value problem for a nonlinear ordinary differential equation of the fourth order is considered. Using the Green function, the boundary value problem si reduced to an equivalent integral equation. In the sublinear case, after identifying the properties of the Green function necessary for further study, the existence of at least one positive solution to the problem under consideration is proved using Krasnosel’skii theorem on the contraction of a cone in semiordered spaces. The uniqueness of such a solution is established by topological methods.