DIFFERENTIAL EQUATIONS WITH TEMPERED Ψ-CAPUTO FRACTIONAL DERIVATIVE

Milan Medveď,Eva Brestovanská

doi:10.3846/mma.2021.13252

Abstract

In this paper we define a new type of the fractional derivative, which we call tempered Ψ−Caputo fractional derivative. It is a generalization of the tempered Caputo fractional derivative and of the Ψ−Caputo fractional derivative. The Cauchy problem for fractional differential equations with this type of derivative is discussed and some existence and uniqueness results are proved. We present a Henry-Gronwall type inequality for an integral inequality with the tempered Ψ−fractional integral. This inequality is applied in the proof of an existence theorem. A result on a representation of solutions of linear systems of Ψ−Caputo fractional differential equations is proved and in the last section an example is presented.

Highlights

Inspired by the papers [10,39,41], we define a modification of the Ψ −Caputo derivative, which we call the tempered Ψ −Caputo derivative and we study the Cauchy problem for fractional differential equations with this type of fractional derivative
The main aim of this paper is to present some results on the existence and uniqueness of solutions of fractional differential equations with the tempered Ψ −Caputo derivative
We proposed a new type of fractional derivative, which we call the tempered Ψ −Caputo fractional derivative

Summary

Introduction

Brestovanska have been defined in recent decades They are more general and more suitable for applications than the Riemann-Liouville fractional derivative. In the recently published papers [12, 23, 37] so-called tempered Riemann-Liouville and tempered Caputo fractional derivatives are defined. Inspired by the papers [10,39,41], we define a modification of the Ψ −Caputo derivative, which we call the tempered Ψ −Caputo derivative and we study the Cauchy problem for fractional differential equations with this type of fractional derivative This derivative includes as special cases the tempered Caputo [23]. The main aim of this paper is to present some results on the existence and uniqueness of solutions of fractional differential equations with the tempered Ψ −Caputo derivative

Preliminaries

Example

Conclusions and a future work