Existence of Solutions for Fractional Langevin Equation

Abstract

In this research, we investigated nonlinear fractional Langevin equation involving two distinct generalized fractional derivatives. Existence of unique and at least one solution is demonstrated by making use of Krasnoselskii’s theorem and Banach contraction mapping principle. Pair of fractional Langevin equations with Caputo and Riemann- Liouville fractional derivatives were considered to check existence of positive solution. Green functions for related equations are found to verify existence of positive solution using fixed point theorems, upper and lower solution techniques. Also we have discussed a coupled system of fractional Langevin equations. Existence result is obtained by utilizing Schauder Fixed Point theorem and uniqueness result is proved by contraction mapping theorem. v