Existence of Solution for a Nonlinear Fractional Order Differential Equation with a Quadratic Perturbations

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+−ρt cDβ0+)(x(t)f(t,x(t)))=g(t,x(t)),t∈J=[0,1],1<α<2,0<β<α( cD0+α−ρt cD0+β)(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))x0=x(0)f(0,x(0)) and \\x1=x(1)f(1,x(1))x1=x(1)f(1,x(1)). Dhage's fixed-point the theorem was used to establish this existence. As an application, we have given example to demonstrate the effectiveness of our main result.