The Role Fixed Point Theorems in Differential Equations

Abstract

The purpose of this paper is to explore fixed point theorem from application point of view of differential equation and fractional differential equation. From previous 13 application were explained with various fixed point theorem applicable to differential equation. Theorems are like Banach fixed point theorem, Brouwer fixed point theorem, Borel fixed point theorem, Schauder fixed point theorem, Krasnoselskii’s fixed point theorem etc. It is observed that differential equation such as first order ordinary differential equation, higher order, nonlinear differential differential equation order differential equation were solved with the help of above mentioned fixed point theorem. Common methodologies followed by researchers is selection of DE second by conversion of DE to integral equation then applying condition fixed point theorem and finally satisfying necessary condition. We can modify Lipschitz condition by adding constants and it became generalised. In fractional differential equation we can take review of L2 and Lp space and to solve the differential equation using modified fixed point theorem.