Stability of mild solutions of the fractional nonlinear abstract Cauchy problem

J. Vanterler da C. Sousa; Kishor D. Kucche; E. Capelas de Oliveira; J. Vanterler da C. Sousa; Kishor D. Kucche; E. Capelas de Oliveira

doi:10.3934/era.2022015

Electronic Research Archive

2022, Volume 30, Issue 1: 272-288. doi: 10.3934/era.2022015

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Research article

Stability of mild solutions of the fractional nonlinear abstract Cauchy problem

1.
Department of Applied Mathematics, State University of Campinas, Imecc, 13083-859, Campinas, SP, Brazil
2.
Department of Mathematics, Shivaji University, Kolhapur, Maharashtra 416 004, India

Received: 08 October 2021 Revised: 15 December 2021 Accepted: 20 December 2021 Published: 30 December 2021

Since the first work on Ulam-Hyers stabilities of differential equation solutions to date, many important and relevant papers have been published, both in the sense of integer order and fractional order differential equations. However, when we enter the field of fractional calculus, in particular, involving fractional differential equations, the path that is still long to be traveled, although there is a range of published works. In this sense, in this paper, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of mild solutions for fractional nonlinear abstract Cauchy problem in the intervals $ [0, T] $ and $ [0, \infty) $ using Banach fixed point theorem.
- fractional nonlinear abstract Cauchy,
- Ulam-Hyers stabilities,
- semigroup theory,
- mild solution,
- Banach fixed point theorem
Citation: J. Vanterler da C. Sousa, Kishor D. Kucche, E. Capelas de Oliveira. Stability of mild solutions of the fractional nonlinear abstract Cauchy problem[J]. Electronic Research Archive, 2022, 30(1): 272-288. doi: 10.3934/era.2022015

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Abstract

Since the first work on Ulam-Hyers stabilities of differential equation solutions to date, many important and relevant papers have been published, both in the sense of integer order and fractional order differential equations. However, when we enter the field of fractional calculus, in particular, involving fractional differential equations, the path that is still long to be traveled, although there is a range of published works. In this sense, in this paper, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of mild solutions for fractional nonlinear abstract Cauchy problem in the intervals $ [0, T] $ and $ [0, \infty) $ using Banach fixed point theorem.

References

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