Existence of Solutions for Nonlinear Fractional Order Differential Equations with Quadratic Perturbations

Abstract

In this work, we prove the existence of a solution for the initial value problem of nonlinear fractional differential equation with quadratic perturbations involving the Caputo fractional derivative cDα0+ − ρtcDβ0+x(t)f(t,x(t))!= g(t, x(t)), t ∈ J = [0, 1],1 < α < 2, 0 < β < α with conditions x0 =x(0)f(0, x(0)) and x1 =x(1) f(1, x(1)). Dhage’s fixed-point the theorem was used to establish this existence. As an application, we havegiven example to demonstrate the effectiveness of our main result.