Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations

In this paper, we describe a systematic development of the different types of perturbations methods in the theory of differential and integral equations. A special quadratic perturbation of the periodic boundary value problems of second order ordinary differential equations is studied in detail for different aspects of the solutions. An existence theorem is proved under mixed generalized Lipschitz and Carathodory conditions and the existence results for extremal positive solutions are established for Carathodory as well as discontinuity conditions. Our results include some known existence results for periodic boundary value problems of second order ordinary nonlinear differential equations as special cases.