Differential equations with tempered Ψ-Caputo fractional derivative

• Published in: Mathematical modelling and analysis Vol. 26; no. 4; pp. 631 - 650
• Main Authors: Medveď, Milan, Brestovanská, Eva
• Format: Journal Article
• Language: English
• Published: Vilnius Vilnius Gediminas Technical University 26.11.2021
• Subjects: Cauchy problems; Differential equations; Existence theorems; Fractional calculus; Inequality; Linear systems; Mathematical analysis; tempered riemann-liouville fractional derivative; tempered ψ−caputo fractional derivative; tempered Riemann-Liouville fractional derivative; tempered Ψ−Caputo fractional derivative
• ISSN: 1392-6292; 1648-3510
• DOI: 10.3846/mma.2021.13252

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.