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

Year 2022, Volume: 5 Issue: 3, 360 - 371, 30.09.2022

Ahmed Kajounı Najat Chefnaj Khalid Hilal

https://doi.org/10.53006/rna.1124961

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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