The main objective of this project work is to present, the Banach contraction fixed point theorem ' , . and its application to the existence of the solution of differential equations. We shall use Banach's fixed point theorem to prove the famous Picard's theorem which plays a vital role in the theory of ordinary differential equation. The idea of the application is the differential equation will be converted to an integral equation, we defines a mapping T ,and the condition of the theorem will imply that T is a contraction such that its fixed point becomes the solution of our problem

Project Work Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics