The weighted generalized Atangana-Baleanu fractional derivative in banach spaces- definition and applications

Muneerah AL Nuwairan; Ahmed Gamal Ibrahim; Muneerah AL Nuwairan; Ahmed Gamal Ibrahim

doi:10.3934/math.20241722

AIMS Mathematics

2024, Volume 9, Issue 12: 36293-36335. doi: 10.3934/math.20241722

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Research article

The weighted generalized Atangana-Baleanu fractional derivative in banach spaces- definition and applications

Muneerah AL Nuwairan ^{1
,
,},
Ahmed Gamal Ibrahim ²

1.
Department of Mathematics, College of Sciences, King Faisal university, P.O.Box. 400, Al-Ahsa 31982, Saudi Arabia
2.
Department of Mathematics, College of Sciences, Cairo University, Egypt

Received: 11 October 2024 Revised: 05 December 2024 Accepted: 13 December 2024 Published: 30 December 2024
MSC : 34A08, 26A33

In this paper, we introduce the concept of the weighted generalized Atangana-Baleanu fractional derivative. We prove the existence of the stability of solutions of non-local differential equations and non-local differential inclusions, in Banach spaces, with this new fractional derivative in the presence of instantaneous and non-instantaneous impulses. We considered the case in which the lower limit of the fractional derivative was kept at the initial point and where it was changed to the impulsive points. To prove our results, we established the relationship between solutions to each of the four studied problems and those of the corresponding fractional integral equation. There has been no previous study of the weighted generalized Atangana-Baleanu fractional derivative, and so, our findings are new and interesting. The technique we used based on the properties of this new fractional differential operator and suitable fixed point theorems for single valued and set valued functions. Examples are given to illustrate the theoretical results.
- fractional differential equations,
- inclusions, instantaneous impulses,
- measure of noncompactness
Citation: Muneerah AL Nuwairan, Ahmed Gamal Ibrahim. The weighted generalized Atangana-Baleanu fractional derivative in banach spaces- definition and applications[J]. AIMS Mathematics, 2024, 9(12): 36293-36335. doi: 10.3934/math.20241722

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Abstract

In this paper, we introduce the concept of the weighted generalized Atangana-Baleanu fractional derivative. We prove the existence of the stability of solutions of non-local differential equations and non-local differential inclusions, in Banach spaces, with this new fractional derivative in the presence of instantaneous and non-instantaneous impulses. We considered the case in which the lower limit of the fractional derivative was kept at the initial point and where it was changed to the impulsive points. To prove our results, we established the relationship between solutions to each of the four studied problems and those of the corresponding fractional integral equation. There has been no previous study of the weighted generalized Atangana-Baleanu fractional derivative, and so, our findings are new and interesting. The technique we used based on the properties of this new fractional differential operator and suitable fixed point theorems for single valued and set valued functions. Examples are given to illustrate the theoretical results.

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