Existence theory and generalized Mittag-Leffler stability for a nonlinear Caputo-Hadamard FIVP via the Lyapunov method

Hadjer Belbali; Maamar Benbachir; Sina Etemad; Choonkil Park; Shahram Rezapour; Hadjer Belbali; Maamar Benbachir; Sina Etemad; Choonkil Park; Shahram Rezapour

doi:10.3934/math.2022794

AIMS Mathematics

2022, Volume 7, Issue 8: 14419-14433. doi: 10.3934/math.2022794

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Existence theory and generalized Mittag-Leffler stability for a nonlinear Caputo-Hadamard FIVP via the Lyapunov method

1.
Laboratory of Mathematics and Applied Sciences, University of Ghardaia, Algeria
2.
Faculty of Sciences, Saad Dahlab University, Blida 1, Algeria
3.
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
4.
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea
5.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 02 April 2022 Revised: 19 May 2022 Accepted: 23 May 2022 Published: 06 June 2022
MSC : 26A33, 34A08

This paper discusses the existence, uniqueness and stability of solutions for a nonlinear fractional differential system consisting of a nonlinear Caputo-Hadamard fractional initial value problem (FIVP). By using some properties of the modified Laplace transform, we derive an equivalent Hadamard integral equation with respect to one-parametric and two-parametric Mittag-Leffer functions. The Banach contraction principle is used to give the existence of the corresponding solution and its uniqueness. Then, based on a Lyapunov-like function and a $ \mathcal{K} $-class function, the generalized Mittag-Leffler stability is discussed to solve a nonlinear Caputo-Hadamard FIVP. The findings are validated by giving an example.
- Caputo-Hadamard derivative,
- Lyapunov direct method,
- $ \mathcal{K} $-class function,
- fixed point,
- Mittag-Leffler stability
Citation: Hadjer Belbali, Maamar Benbachir, Sina Etemad, Choonkil Park, Shahram Rezapour. Existence theory and generalized Mittag-Leffler stability for a nonlinear Caputo-Hadamard FIVP via the Lyapunov method[J]. AIMS Mathematics, 2022, 7(8): 14419-14433. doi: 10.3934/math.2022794

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Abstract

This paper discusses the existence, uniqueness and stability of solutions for a nonlinear fractional differential system consisting of a nonlinear Caputo-Hadamard fractional initial value problem (FIVP). By using some properties of the modified Laplace transform, we derive an equivalent Hadamard integral equation with respect to one-parametric and two-parametric Mittag-Leffer functions. The Banach contraction principle is used to give the existence of the corresponding solution and its uniqueness. Then, based on a Lyapunov-like function and a $ \mathcal{K} $-class function, the generalized Mittag-Leffler stability is discussed to solve a nonlinear Caputo-Hadamard FIVP. The findings are validated by giving an example.

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