Existence and Ulam-Type Stability for Fractional Multi-Delay Differential Systems
Abstract
1. Introduction
2. Preliminaries
- (i)
- For all , we have
- (ii)
- For all , we have
- (iii)
- For all , we have
3. Existence and Uniqueness Results
- Step 1.
- We show that . For any and , by Lemmas 1 and 2, we have
- Step 2.
- We prove is continuous.
- Step 3.
- We show is equicontinuous.
- (i)
- When , We have . Thus .
- (ii)
- When , We have . So , which deduce . Thus,
- (iii)
- When , we have . So , which deduce . Thus,As a result, we immediately obtain that as . Thus, is equicontinuous. From the Arzela-Ascoli theorem, is relatively compact. Schauder fixed point theorem guarantees that has a fixed point in .
4. Ulam-Type Stability Analysis
4.1. Uniqueness of the Solution for System (3)
4.2. Ulam-Hyers Stablity
- (i)
- ;
- (ii)
- ;
- (iii)
- ;
- (iv)
- .
4.3. Ulam-Hyers-Rassias Stability
- (i)
- ;
- (ii)
- ;
- (iii)
- ;
- (iv)
- .
5. An Example
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Zhang, X.; Li, M. Existence and Ulam-Type Stability for Fractional Multi-Delay Differential Systems. Fractal Fract. 2025, 9, 288. https://doi.org/10.3390/fractalfract9050288
Zhang X, Li M. Existence and Ulam-Type Stability for Fractional Multi-Delay Differential Systems. Fractal and Fractional. 2025; 9(5):288. https://doi.org/10.3390/fractalfract9050288
Chicago/Turabian StyleZhang, Xing, and Mengmeng Li. 2025. "Existence and Ulam-Type Stability for Fractional Multi-Delay Differential Systems" Fractal and Fractional 9, no. 5: 288. https://doi.org/10.3390/fractalfract9050288
APA StyleZhang, X., & Li, M. (2025). Existence and Ulam-Type Stability for Fractional Multi-Delay Differential Systems. Fractal and Fractional, 9(5), 288. https://doi.org/10.3390/fractalfract9050288