Approach of a class of discontinuous dynamical systems of fractional order: existence of the solutions

TL;DR

This paper investigates the existence of solutions for a class of discontinuous fractional-order dynamical systems by transforming the problem into a continuous fractional-order problem and applying fixed point theorems.

Contribution

It introduces a novel approach to establish the existence of solutions for discontinuous fractional systems using set-valued analysis and Cellina's Theorem.

Findings

Existence of solutions is proven for the considered class of systems.

Transformation into a single-valued continuous problem facilitates analysis.

A Pénon-like theorem is extended to fractional-order systems.

Abstract

In this letter we are concerned with the possibility to approach the existence of solutions to a class of discontinuous dynamical systems of fractional order. In this purpose, the underlying initial value problem is transformed into a fractional set-valued problem. Next, the Cellina's Theorem is applied leading to a single-valued continuous initial value problem of fractional order. The existence of solutions is assured by a P\'{e}ano like theorem for ordinary differential equations of fractional order.