Periodic mild solutions for semilinear impulsive fractional evolution equations with memory

Ahmad Al-Omari; Hanan Al-Saadi; Ahmad Al-Omari; Hanan Al-Saadi

doi:10.3934/math.20261143

AIMS Mathematics

2026, Volume 11, Issue 9: 28710-28739. doi: 10.3934/math.20261143

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Periodic mild solutions for semilinear impulsive fractional evolution equations with memory

Ahmad Al-Omari ^{1
,},
Hanan Al-Saadi ^{2
,
,}

1.
Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq, Jordan; omarimutah1@yahoo.com
2.
Department of Mathematics, Faculty of Sciences, Umm Al-Qura University, Makkah 21955, Saudi Arabia

Received: 04 June 2026 Revised: 10 August 2026 Accepted: 19 August 2026 Published: 07 September 2026
MSC : 34A08, 34A37, 34C25

This work examines fractional evolution equations involving a piecewise Caputo derivative, periodic impulses, and a convolution memory kernel. For the associated linear impulsive periodic problem, we demonstrate under exponential stability of the semigroup the existence and uniqueness of a periodic mild solution and prove that the corresponding solution operator is linear and bounded. Turning to the semilinear setting, we establish the existence of at least one periodic mild solution under two distinct families of growth restrictions on the nonlinear terms: (i) linear growth, via Sadovskii's fixed point theorem; (ii) more general Osgood-type growth, via the Leray-Schauder alternative. The impulses are taken to be Lipschitz with a suitably small constant. Finally, we give an application to a fractional impulsive heat equation with Dirichlet boundary conditions.
- impulsive fractional evolution equations,
- periodic impulses,
- periodic mild solutions,
- existence,
- compact semigroup
Citation: Ahmad Al-Omari, Hanan Al-Saadi. Periodic mild solutions for semilinear impulsive fractional evolution equations with memory[J]. AIMS Mathematics, 2026, 11(9): 28710-28739. doi: 10.3934/math.20261143

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Abstract

This work examines fractional evolution equations involving a piecewise Caputo derivative, periodic impulses, and a convolution memory kernel. For the associated linear impulsive periodic problem, we demonstrate under exponential stability of the semigroup the existence and uniqueness of a periodic mild solution and prove that the corresponding solution operator is linear and bounded. Turning to the semilinear setting, we establish the existence of at least one periodic mild solution under two distinct families of growth restrictions on the nonlinear terms: (i) linear growth, via Sadovskii's fixed point theorem; (ii) more general Osgood-type growth, via the Leray-Schauder alternative. The impulses are taken to be Lipschitz with a suitably small constant. Finally, we give an application to a fractional impulsive heat equation with Dirichlet boundary conditions.

References

[1]	P. Chen, X. Zhang, Y. Li, Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators, Fract. Calc. Appl. Anal., 23 (2020), 268–291. https://doi.org/10.1515/fca-2020-0011
[2]	A. Al-Omari, M. H. M. Rashid, Semilinear Hadamard fractional integro-differential equations with nonlocal conditions and state-dependent delay under compact semigroup, Mathematics, 14 (2026), 3097. https://doi.org/10.3390/math14173097 doi: 10.3390/math14173097
[3]	Y. Li, Existence and asymptotic stability of periodic solution for evolution equations with delays, J. Funct. Anal., 261 (2011), 1309–1324. https://doi.org/10.1016/j.jfa.2011.05.001 doi: 10.1016/j.jfa.2011.05.001
[4]	Q. Li, M. Wei, Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay, Evol. Equ. Control Theory, 9 (2020), 753-772. https://doi.org/10.3934/eect.2020032 doi: 10.3934/eect.2020032
[5]	Z. Liu, N. S. Papageorgiou. Periodic systems with time dependent maximal monotone operators, Acta Math. Sci., 44 (2024), 1280–1300. https://doi.org/10.1007/s10473-024-0406-6 doi: 10.1007/s10473-024-0406-6
[6]	H. A. Hammad, D. A. Kattan, An optimal control for non-autonomous second-order stochastic differential equations with delayed arguments, Phys. Scr., 99 (2024), 125205. https://doi.org/10.1088/1402-4896/ad8604 doi: 10.1088/1402-4896/ad8604
[7]	L. Ren, J. Wang, M. Fečkan, Asymptotically periodic solutions for Caputo type fractional evolution equations, Fract. Calc. Appl. Anal., 21 (2018), 1294–1312. https://doi.org/10.1515/fca-2018-0068 doi: 10.1515/fca-2018-0068
[8]	H. Gou, Y. Li, A study on impulsive fractional hybrid evolution equations using sequence method, Comput. Appl. Math., 39 (2020), 225. https://doi.org/10.1007/s40314-020-01239-y doi: 10.1007/s40314-020-01239-y
[9]	X. B. Shu, Y. Shi, A study on the mild solution of impulsive fractional evolution equations, Appl. Math. Comput., 273 (2016), 465–476. https://doi.org/10.1016/j.amc.2015.10.020 doi: 10.1016/j.amc.2015.10.020
[10]	M. Fečkan, J. R. Wang, Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ., 8 (2011), 345–361. https://doi.org/10.4310/DPDE.2011.v8.n4.a3 doi: 10.4310/DPDE.2011.v8.n4.a3
[11]	H. A. Hammad, M. De la Sen, Existence of a mild solution and approximate controllability for fractional random integro-differential inclusions with non-instantaneous impulses, Alexandria Eng. J., 111 (2025), 306–328. https://doi.org/10.1016/j.aej.2024.10.017 doi: 10.1016/j.aej.2024.10.017
[12]	X. Pang, X. Li, Z. Liu, Decay mild solutions of Hilfer fractional differential variational–hemivariational inequalities, Nonlinear Anal. Real World Appl., 71 (2023), 103834. https://doi.org/10.1016/j.nonrwa.2023.103834 doi: 10.1016/j.nonrwa.2023.103834
[13]	M. Fečkan, J. R. Wang, Periodic impulsive fractional differential equations, Adv. Nonlinear Anal., 8 (2019), 482–496. https://doi.org/10.1515/anona-2017-0015 doi: 10.1515/anona-2017-0015
[14]	T. Zhang, L. Xiong, Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative, Appl. Math. Lett., 101 (2020), 106072. https://doi.org/10.1016/j.aml.2019.106072 doi: 10.1016/j.aml.2019.106072
[15]	L. Ren, J. R. Wang, M. Fečkan, Periodic mild solutions of impulsive fractional evolution equations, AIMS Math., 5 (2020), 497–506. https://doi.org/10.3934/math.2020033 doi: 10.3934/math.2020033
[16]	A. Al-Omari, M. H. M. Rashid, Local existence and regularity of mild solutions for Hadamard fractional semilinear integro-differential equations with compact semigroups, Axioms, 15 (2026), 350. https://doi.org/10.3390/axioms15050350 doi: 10.3390/axioms15050350
[17]	A. Al-Omari, H. Al-Saadi, $(\omega, \rho)$-BVP solution of impulsive Hadamard fractional differential equations, Mathematics, 11 (2023), 4370. https://doi.org/10.3390/math11204370 doi: 10.3390/math11204370
[18]	A. Al-Omari, H. Al-Saadi, Impulsive fractional order integrodifferential equation via fractional operators, PLoS ONE, 18 (2023), e0282665. https://doi.org/10.1371/journal.pone.0282665 doi: 10.1371/journal.pone.0282665
[19]	A. Al-Omari, H. Al-Saadi, F. Alharbi, $(\omega, c)$-periodic solution to semilinear integro differential equations with Hadamard derivatives, Fractal Fract., 8 (2024), 86. https://doi.org/10.3390/fractalfract8020086 doi: 10.3390/fractalfract8020086
[20]	A. Pazy, Semigroups of linear operators and applications to partial differential equations, Berlin: Springer-Verlag, 1983. https://doi.org/10.1007/978-1-4612-5561-1
[21]	J. Wang, Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl., 12 (2011), 262–272. https://doi.org/10.1016/j.nonrwa.2010.06.013 doi: 10.1016/j.nonrwa.2010.06.013
[22]	P. Chen, Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65 (2014), 711–728. https://doi.org/10.1007/s00033-013-0351-z doi: 10.1007/s00033-013-0351-z
[23]	C. T. Anh, K. T. Dinh, On nonlocal problems for retarded fractional differential equations in Banach spaces, Fixed Point Theory, 15 (2014), 373–392.
[24]	Y. Zhou, Fractional evolution equations and inclusions: analysis and control, London: Academic Press, 2016.
[25]	R. N. Wang, D. H. Chen, T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differ. Equations, 252 (2012), 202–235. https://doi.org/10.1016/j.jde.2011.08.048 doi: 10.1016/j.jde.2011.08.048
[26]	A. Kilbas, H. Srivastava, J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[27]	J. R. Wang, M. Fečkan, Y. Zhou, Nonexistence of periodic solutions and asymptotically periodic solutions for fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 246–256. https://doi.org/10.1016/j.cnsns.2012.07.004 doi: 10.1016/j.cnsns.2012.07.004
[28]	B. N. Sadovskii, A fixed-point principle, Funct. Anal. Appl., 1 (1967), 151–153. https://doi.org/10.1007/bf01076087
[29]	T. A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii Schaefer type, Math. Nachr., 189 (1998), 23–31. https://doi.org/10.1002/mana.19981890103 doi: 10.1002/mana.19981890103

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