Nonlinear generalized fractional differential equations with generalized fractional integral conditions
Samiha Belmor, C. Ravichandran, Fahd Jarad
Abstract
This research work is dedicated to an investigation of the existence and uniqueness of a class of nonlinear ψ-Caputo fractional differential equation on a finite interval $[0, T] $, equipped with nonlinear ψ-Riemann–Liouville fractional integral boundary conditions of different orders $0 \lt \alpha , \beta \lt 1 $, we deal with a recently introduced ψ-Caputo fractional derivative of order $1 \lt q\leq ~2 $. The formulated problem will be transformed into an integral equation with the help of Green function. A full analysis of existence and uniqueness of solutions is proved using fixed point theorems: Leray–Schauder nonlinear alternative, Krasnoselskii and Schauder's fixed point theorems, Banach's and Boyd–Wong's contraction principles. We show that this class generalizes several other existing classes of fractional-order differential equations, and therefore the freedom of choice of the standard fractional operator. As an application, we provide an example to demonstrate the validity of our results.