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Fractal Fract
Volume 9
Issue 2
10.3390/fractalfract9020085
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Article

Asymptotic Periodicity of Bounded Mild Solutions for Evolution Equations with Non-Densely Defined and Fractional Derivative

by
Jiabin Zuo
1,*,
Abdellah Taqbibt
2,
Mohamed Chaib
2 and
M’hamed Elomari
2
1
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
2
Applied Mathematics and Scientific Computing Laboratory, Sultan Moulay Slimane University, P.O. Box 523, Beni Mellal 23000, Morocco
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(2), 85; https://doi.org/10.3390/fractalfract9020085
Submission received: 4 December 2024 / Revised: 20 January 2025 / Accepted: 24 January 2025 / Published: 26 January 2025
(This article belongs to the Special Issue Recent Advances in Nonlocal Problems Involving the Fractional Laplacian Operators)
Versions Notes

Abstract

In the present article, we establish conditions for the asymptotic periodicity of bounded mild solutions in two distinct cases of evolution equations. The first class involves non-densely defined operators, while the second class incorporates densely defined operators with fractional derivatives that generate a semigroup of contractions. Our method integrates the theory of spectral properties of uniformly bounded continuous functions defined on the positive real semi-axis. Additionally, we apply extrapolation theory to evolution equations with non-densely defined operators. To illustrate our main results, we provide a concrete example.
Keywords: eriodic solution; asymptotic periodic solution; spectrum of a function; semigroup of evolution; fractional derivative eriodic solution; asymptotic periodic solution; spectrum of a function; semigroup of evolution; fractional derivative

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MDPI and ACS Style

Zuo, J.; Taqbibt, A.; Chaib, M.; Elomari, M. Asymptotic Periodicity of Bounded Mild Solutions for Evolution Equations with Non-Densely Defined and Fractional Derivative. Fractal Fract. 2025, 9, 85. https://doi.org/10.3390/fractalfract9020085

AMA Style

Zuo J, Taqbibt A, Chaib M, Elomari M. Asymptotic Periodicity of Bounded Mild Solutions for Evolution Equations with Non-Densely Defined and Fractional Derivative. Fractal and Fractional. 2025; 9(2):85. https://doi.org/10.3390/fractalfract9020085

Chicago/Turabian Style

Zuo, Jiabin, Abdellah Taqbibt, Mohamed Chaib, and M’hamed Elomari. 2025. "Asymptotic Periodicity of Bounded Mild Solutions for Evolution Equations with Non-Densely Defined and Fractional Derivative" Fractal and Fractional 9, no. 2: 85. https://doi.org/10.3390/fractalfract9020085

APA Style

Zuo, J., Taqbibt, A., Chaib, M., & Elomari, M. (2025). Asymptotic Periodicity of Bounded Mild Solutions for Evolution Equations with Non-Densely Defined and Fractional Derivative. Fractal and Fractional, 9(2), 85. https://doi.org/10.3390/fractalfract9020085

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