Ulam-Type Stability for a Boundary-Value Problem for Multi-Term Delay Fractional Differential Equations of Caputo Type
Abstract
:1. Introduction
2. Preliminary Notes on Generalized Proportional Fractional Derivatives
3. Statement of the Boundary-Value Problem for Multi-Term Fractional Problem
3.1. Generalized Proportional Caputo Fractional Derivatives
- 1.
- The constants , , , and condition A1 is satisfied.
- 2.
- There exist constants such that for the inequalities
- 3.
- The inequalities
3.2. Caputo Fractional Derivatives
- 1.
- The constants , , and .
- 2.
- There exist constants such that for the inequalities
- 3.
- The inequalities
4. Ulam-Type Stability for Boundary-Value Problems
4.1. Some Comments and Remarks about Ulam-Type Stability
- -
- initial value condition
- -
- boundary-value condition
4.2. Ulam-Type Stability for Multi-Term Couple of Generalized Proportional Caputo Fractional Derivatives
- (i)
- (ii)
- The function for , is nondecreasing, and there exists a constant
- (iii)
- (i)
- (iii)
4.3. Ulam-Type Stability for a Multi-Term Couple of Caputo Fractional Derivatives
- (i)
- (ii)
- The function for , is nondecreasing, and there exists a constant
- (iii)
- (i)
5. Example
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Agarwal, R.P.; Hristova, S. Ulam-Type Stability for a Boundary-Value Problem for Multi-Term Delay Fractional Differential Equations of Caputo Type. Axioms 2022, 11, 742. https://doi.org/10.3390/axioms11120742
Agarwal RP, Hristova S. Ulam-Type Stability for a Boundary-Value Problem for Multi-Term Delay Fractional Differential Equations of Caputo Type. Axioms. 2022; 11(12):742. https://doi.org/10.3390/axioms11120742
Chicago/Turabian StyleAgarwal, Ravi P., and Snezhana Hristova. 2022. "Ulam-Type Stability for a Boundary-Value Problem for Multi-Term Delay Fractional Differential Equations of Caputo Type" Axioms 11, no. 12: 742. https://doi.org/10.3390/axioms11120742