Solutions and anti-periodic solutions for impulsive differential equations and inclusions containing Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2)  $ in infinite dimensional Banach spaces

Muneerah Al Nuwairan; Ahmed Gamal Ibrahim; Muneerah Al Nuwairan; Ahmed Gamal Ibrahim

doi:10.3934/math.2024508

AIMS Mathematics

2024, Volume 9, Issue 4: 10386-10415. doi: 10.3934/math.2024508

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Solutions and anti-periodic solutions for impulsive differential equations and inclusions containing Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $ in infinite dimensional Banach spaces

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: 23 December 2023 Revised: 28 February 2024 Accepted: 05 March 2024 Published: 15 March 2024
MSC : 26A33, 34A08

In this paper, we improved recent results on the existence of solutions for nonlinear fractional boundary value problems containing the Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $. We also derived the exact relations between these fractional boundary value problems and the corresponding fractional integral equations in infinite dimensional Banach spaces. We showed that the continuity assumption on the nonlinear term of these equations is insufficient, give the derived expression for the solution, and present two results about the existence and uniqueness of the solution. We examined the case of impulsive impact and provide some sufficiency conditions for the existence and uniqueness of the solution in these cases. We also demonstrated the existence and uniqueness of anti-periodic solution for the studied problems and considered the problem when the right-hand side was a multivalued function. Examples were given to illustrate the obtained results.
- AB fractional derivative,
- fractional differential inclusions,
- instantaneous impulses,
- solutions and anti-periodic solutions
Citation: Muneerah Al Nuwairan, Ahmed Gamal Ibrahim. Solutions and anti-periodic solutions for impulsive differential equations and inclusions containing Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $ in infinite dimensional Banach spaces[J]. AIMS Mathematics, 2024, 9(4): 10386-10415. doi: 10.3934/math.2024508

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Abstract

In this paper, we improved recent results on the existence of solutions for nonlinear fractional boundary value problems containing the Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $. We also derived the exact relations between these fractional boundary value problems and the corresponding fractional integral equations in infinite dimensional Banach spaces. We showed that the continuity assumption on the nonlinear term of these equations is insufficient, give the derived expression for the solution, and present two results about the existence and uniqueness of the solution. We examined the case of impulsive impact and provide some sufficiency conditions for the existence and uniqueness of the solution in these cases. We also demonstrated the existence and uniqueness of anti-periodic solution for the studied problems and considered the problem when the right-hand side was a multivalued function. Examples were given to illustrate the obtained results.

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