The main aim of this work is classified into four objects, these are summarized as follows: The first objective is to study the theory of existence and the uniqueness of the solutions for the periodic boundary value problems of the differential equations. The second objective is to devote the existence theorems of the extremal solutions of the above periodic boundary value problems. The third objective is to give the existence and the uniqueness theorems of the solutions for the periodic boundary value problems of the linear and nonlinear ordinary integro-differential equations. Also, the existence theorems of the extremal solutions for the above periodic boundary value problems is introduced. The fourth objective is to solve the periodic boundary value problems for ordinary integro-differential equations by using the expansion methods.