Existence of Solution for a Nonlinear Fractional Order Differential Equation with a 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
$({}^c{D}_{0^{+}}^{\alpha }-\rho t{}^c{D}_{0^{+}}^{\beta })(\frac{x\left(t\right)}{f(t,x(t\left)\right)})=g(t,x(t\left)\right),t\in J=[0,1],1<\alpha <2,0<\beta <\alpha$
with conditions $x_0=\frac{x\left(0\right)}{f(0,x(0\left)\right)}$ and \\$x_1=\frac{x\left(1\right)}{f(1,x(1\left)\right)}$. 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.

Keywords

• Fractional differential equation
• Quadratic perturbations
• Dhage fixed point.

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