On a study of the representation of solutions of a $ \Psi $-Caputo fractional differential equations with a single delay

Mustafa Aydin; Nazim I. Mahmudov; Hüseyin Aktuğlu; Erdem Baytunç; Mehmet S. Atamert; Mustafa Aydin; Nazim I. Mahmudov; Hüseyin Aktuğlu; Erdem Baytunç; Mehmet S. Atamert

doi:10.3934/era.2022053

Electronic Research Archive

2022, Volume 30, Issue 3: 1016-1034. doi: 10.3934/era.2022053

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On a study of the representation of solutions of a $ \Psi $-Caputo fractional differential equations with a single delay

1.
Department of Mathematics, Eastern Mediterranean University, Famagusta 99628 T.R. North Cyprus, Mersin 10, Turkey
2.
Department of Mathematics, Istanbul Technical University, Sariyer, Istanbul 34469, Turkey

Academic Editor: Jean-Daniel Djida

Received: 15 December 2021 Revised: 31 January 2022 Accepted: 16 February 2022 Published: 07 March 2022

We give a representation of solutions to linear nonhomogeneous $ \Psi $-fractional delayed differential equations with noncommutative matrices. We newly define $ \Psi $-delay perturbation of Mittag-Leffler type matrix function with two parameters and apply the method of variation of constants to obtain the representation of the solutions. We investigate the existence and uniqueness of solutions for a class of $ \Psi $-fractional delayed semilinear differential equations by using Banach Fixed Point Theorem. Further, we establish the Ulam-Hyers stability result for the analyzed problem. Finally, we provide some examples to illustrate the applicability of our results.
Citation: Mustafa Aydin, Nazim I. Mahmudov, Hüseyin Aktuğlu, Erdem Baytunç, Mehmet S. Atamert. On a study of the representation of solutions of a $ \Psi $-Caputo fractional differential equations with a single delay[J]. Electronic Research Archive, 2022, 30(3): 1016-1034. doi: 10.3934/era.2022053

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Abstract

We give a representation of solutions to linear nonhomogeneous $ \Psi $-fractional delayed differential equations with noncommutative matrices. We newly define $ \Psi $-delay perturbation of Mittag-Leffler type matrix function with two parameters and apply the method of variation of constants to obtain the representation of the solutions. We investigate the existence and uniqueness of solutions for a class of $ \Psi $-fractional delayed semilinear differential equations by using Banach Fixed Point Theorem. Further, we establish the Ulam-Hyers stability result for the analyzed problem. Finally, we provide some examples to illustrate the applicability of our results.

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