Unique Solutions for Caputo Fractional Differential Equations with Several Delays Using Progressive Contractions
Abstract
:1. Introduction
2. Preliminaries
3. Progressive Contractions and Existence of Solutions
4. Numerical Applications of Theorem 1
5. The Outcomes and Contributions
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- In view of the literature review that is provided in this study’s introduction and available in the pertinent literature in order to investigate the existence and uniqueness of solutions for a variety of mathematical models, including IDEs, delay IEs, a scalar FDE of the Riemann–Liouville type, scalar nonlinear IEs, Hammerstein-type FIEs, nonlinear IEs including several variable time delays and nonlinear IDEs without delay on the intervals and , the method of progressive contractions, which belongs to T.A. Burton, has effectively been applied and very interesting results have been obtained in the pertinent literature by this time. Nevertheless, to the best of our information from the relevant literature, there is no result on the same topic for CFDEs, including time delay(s) and without time delay(s), where the method of progressive contractions is used as a basic technique on the subject. Therefore, the aim of this study is to achieve the application of the progressive contractions technique to the existence and uniqueness of solutions for CFDEs, including several variable time delays. This fact represents the paper’s novelty and originality.
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- Burton [8], Theorem 1.2 dealt with a scalar FDE of the Riemann–Liouville type without delay. To the best of our knowledge, the fractional differential equation considered in Burton [8] has a simple form and does not include any time delay. Moreover, Burton [8] provided no numerical application to validate the primary outcome of [8], Theorem 1.2. Despite this case, in this paper, we examine a distinct kind of fractional differential equation, namely CFDE (1) with time delays in multiple variables, and also offer two new two examples as numerical applications of our primary finding, i.e., Theorem 1.
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- Regarding the benefits of the technique called progressive contractions, which was used in the proof of this paper, the existence and the uniqueness of solutions of different kinds of nonlinear mathematical models, as well as FDEs of various kinds, for example, FDEs of the Riemann–Liouville type with time delay and without time delay, CFDEs with time delay and without time delay, Hilfer type FDEs with time delay and without time delay, etc. can be discussed via the progressive contractions throughout very simple and short steps for the finite and infinite interval cases.
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
DE | Differential equation |
DEs | Differential equations |
DDEs | Delay differential equations |
CFDE | Caputo fractional-order differential equation |
CFDEs | Caputo fractional-order differential equations |
CFIDEs | Caputo fractional-order integro-differential equations |
IE | Integral equation |
IEs | Integral equations |
IDE | Integro-differential equation |
IDEs | Integro-differential equations |
FDE | Fractional differential equation |
FDEs | Fractional differential equations |
FIE | Functional integral equation |
FIEs | Functional integral equations |
UHML stability | Ulam–Hyers–Mittag–Leffler stability |
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Tunç, C.; Akyildiz, F.T. Unique Solutions for Caputo Fractional Differential Equations with Several Delays Using Progressive Contractions. Mathematics 2024, 12, 2799. https://doi.org/10.3390/math12182799
Tunç C, Akyildiz FT. Unique Solutions for Caputo Fractional Differential Equations with Several Delays Using Progressive Contractions. Mathematics. 2024; 12(18):2799. https://doi.org/10.3390/math12182799
Chicago/Turabian StyleTunç, Cemil, and Fahir Talay Akyildiz. 2024. "Unique Solutions for Caputo Fractional Differential Equations with Several Delays Using Progressive Contractions" Mathematics 12, no. 18: 2799. https://doi.org/10.3390/math12182799