Periodic solutions of a certain nonlinear boundary value problem (BVP) of a fourth order differential equation

The consideration of the eigenvalue approach and a comparison between thelinear and nonlinear fourth order differential equation formed the basis for a theorem for existence of periodic solutions for the nonlinear boundary value problem of a fourth order differential equation. The proof of the theorem is by the Leray-Schauder fixed point technique with the use of integrated equation as the mode for estimating the a priori bounds