Existence and Uniqueness of the Solution to a Nonlinear Differential Equation with Caputo Fractional Derivative in the Space of Continuously Differentiable Functions

TL;DR

This paper establishes the existence and uniqueness of solutions for a nonlinear differential equation involving Caputo fractional derivatives, using integral equations and Banach fixed point theorem.

Contribution

It introduces a method to prove existence and uniqueness for nonlinear fractional differential equations in a space of continuously differentiable functions.

Findings

Proves equivalence between the differential and integral form of the equation.

Applies Banach fixed point theorem to establish solution uniqueness.

Provides conditions for the existence of solutions in the specified function space.

Abstract

In the paper, we considered the existence and uniqueness of the global solution in the space of continuously differentiable functions for a nonlinear differential equation with the Caputo fractional derivative of general form. Our main method is to derive an integral equation corresponding to the original nonlinear fractional differential equation and to prove their eqivalence. Once we prove the equivalence, then the proof of existence and uniquness of solution can be done by standard way of using Banach fixed point theorem.