Existence of $ S $-asymptotically $ \omega $-periodic solutions for non-instantaneous impulsive semilinear differential equations and inclusions of fractional order $ 1 &lt; \alpha &lt; 2 $

Zainab Alsheekhhussain; Ahmed Gamal Ibrahim; Rabie A. Ramadan; Zainab Alsheekhhussain; Ahmed Gamal Ibrahim; Rabie A. Ramadan

doi:10.3934/math.2023004

AIMS Mathematics

2023, Volume 8, Issue 1: 76-101. doi: 10.3934/math.2023004

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Existence of $ S $-asymptotically $ \omega $-periodic solutions for non-instantaneous impulsive semilinear differential equations and inclusions of fractional order $ 1 < \alpha < 2 $

1.
Department of Mathematics, Faculty of Science, University of Ha'il, Hail Saudi Arabia
2.
Department of Mathematics, College of Sciences, King Faisal University, Saudi Arabia
3.
Cairo University, Department of Computer Engineering, Faculty of Engineering, Cairo, 12613, Egypt

Received: 11 August 2022 Revised: 11 September 2022 Accepted: 15 September 2022 Published: 27 September 2022
MSC : 26A33, 34A08, 34A60

It is known that there is no non-constant periodic solutions on a closed bounded interval for differential equations with fractional order. Therefore, many researchers investigate the existence of asymptotically periodic solution for differential equations with fractional order. In this paper, we demonstrate the existence and uniqueness of the $ S $-asymptotically $ \omega $-periodic mild solution to non-instantaneous impulsive semilinear differential equations of order $ 1 < \alpha < 2 $, and its linear part is an infinitesimal generator of a strongly continuous cosine family of bounded linear operators. In addition, we consider the case of differential inclusion. Examples are given to illustrate the applicability of our results.
- non-instantaneous impulses,
- cosine family,
- asymptotically periodic solutions,
- differential inclusions
Citation: Zainab Alsheekhhussain, Ahmed Gamal Ibrahim, Rabie A. Ramadan. Existence of $ S $-asymptotically $ \omega $-periodic solutions for non-instantaneous impulsive semilinear differential equations and inclusions of fractional order $ 1 < \alpha < 2 $[J]. AIMS Mathematics, 2023, 8(1): 76-101. doi: 10.3934/math.2023004

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Abstract

It is known that there is no non-constant periodic solutions on a closed bounded interval for differential equations with fractional order. Therefore, many researchers investigate the existence of asymptotically periodic solution for differential equations with fractional order. In this paper, we demonstrate the existence and uniqueness of the $ S $-asymptotically $ \omega $-periodic mild solution to non-instantaneous impulsive semilinear differential equations of order $ 1 < \alpha < 2 $, and its linear part is an infinitesimal generator of a strongly continuous cosine family of bounded linear operators. In addition, we consider the case of differential inclusion. Examples are given to illustrate the applicability of our results.

References

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