Existence and uniqueness of solutions to two point boundary value problems for ordinary differential equations

In this paper, using the optimal control method, we deal with the following boundary value problem y '' + f(x, y) = 0, y(0) = c, y(1) = d, under new nonresonance conditions of the form -A less than or equal to f(y)'(x, y) less than or equal to beta(x) less than or equal to B, where A > 0. We obtain the existence and uniqueness of solutions of the BVP (1).