On the Existence and Uniqueness of Solutions for Neutral-Type Caputo Fractional Differential Equations with Iterated Delays: Hyers–Ulam–Mittag–Leffler Stability
Abstract
:1. Introduction
- i.
- ii.
2. Fractional Calculus: Definitions
- (i)
- The fractional order integral α of a function f with a lower limit of integration a can be determined by the formula
- (ii)
- The Riemann–Liouville fractional derivative is given by
- (iii)
- The Caputo fractional derivative for all of fractional order α is expressed asIf f is absolutely continuous on , then, the next formula gives a direct definition of the Caputo left-sided derivative:
- (a)
- (b)
- (c)
- (d)
- and (Euler’s reflection formula).
3. Existence and Uniqueness of Solutions: Ulam–Hyers–Mittag–Leffler Stability of the Solutions
3.1. Case 1 of the Main Results: Unified Lipschitz Constants
- The function f is defined and continuous in the domain and for (or ) satisfies
- (a)
- (b)
- where is positive constant;
- (c)
- The functions are defined and continuous in the domain and and for satisfies
- (a)
- (b)
- (c)
- where
- (d)
- .
- The inequality
- The inequalities expressed as
- By simple iteration,
- Obviously, , and for , we have
- (i)
- for all ;
- (ii)
- for all .
3.2. Case 2 of the Main Results: Distinct Lipschitz Constants
- The function f is defined and continuous in the domain and for (or ) satisfies
- (a)
- (b)
- where and are positive constants;
- (c)
- The functions are defined and continuous in the domain and and for satisfy
- (a)
- (b)
- (c)
- where and are positive constants;
- (d)
- The inequality
- The inequalities expressed as
4. Applications
5. Conclusions
- 1.
- Establishing sufficient conditions that guarantee the system is Ulam–Hyers–Mittag–Leffler stable over broader intervals for various initial functions;
- 2.
- Exploring classes of neutral-type fractional differential systems incorporating infinite iterative delays but subjected to less stringent constraints on their right-hand side components;
- 3.
- Investigating whether replacing traditional Lipschitz requirements with continuity matrices, such as [9], could lead to new insights in generalized norm applications within appropriate functional spaces.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
Iterated Delays | |
Caputo Fractional Differential Equation of Neutral Type | |
Initial Value Problem | |
Ulam–Hyers–Mittag–Leffler |
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Madamlieva, E.; Konstantinov, M. On the Existence and Uniqueness of Solutions for Neutral-Type Caputo Fractional Differential Equations with Iterated Delays: Hyers–Ulam–Mittag–Leffler Stability. Mathematics 2025, 13, 484. https://doi.org/10.3390/math13030484
Madamlieva E, Konstantinov M. On the Existence and Uniqueness of Solutions for Neutral-Type Caputo Fractional Differential Equations with Iterated Delays: Hyers–Ulam–Mittag–Leffler Stability. Mathematics. 2025; 13(3):484. https://doi.org/10.3390/math13030484
Chicago/Turabian StyleMadamlieva, Ekaterina, and Mihail Konstantinov. 2025. "On the Existence and Uniqueness of Solutions for Neutral-Type Caputo Fractional Differential Equations with Iterated Delays: Hyers–Ulam–Mittag–Leffler Stability" Mathematics 13, no. 3: 484. https://doi.org/10.3390/math13030484
APA StyleMadamlieva, E., & Konstantinov, M. (2025). On the Existence and Uniqueness of Solutions for Neutral-Type Caputo Fractional Differential Equations with Iterated Delays: Hyers–Ulam–Mittag–Leffler Stability. Mathematics, 13(3), 484. https://doi.org/10.3390/math13030484