On Impulsive Implicit ψ-Caputo Hybrid Fractional Differential Equations with Retardation and Anticipation
Abstract
:1. Introduction
- Given the varied conditions we imposed on problem (1), our study may be viewed as a partial continuation of the ones in the aforementioned studies.
- The -fractional derivative unifies a larger number of fractional derivatives in a single fractional operator and opens the door to new applications.
- If we take , , and remove the impulses, and the retarded and advanced arguments, we then obtain the problem studied in [20].
- We weaken the several conditions imposed in the study of [20], such as the requirement that functions and are identically zero at the origin and .
- We study the Ulam–Hyers–Rassias stability of an implicit problem with non-instantaneous impulses, delay, and anticipation.
2. Preliminaries
3. Main Results
- (A1)
- Function is continuous.
- (A2)
- There exist constants and such that
- (A3)
- Function is continuous on , and there exists a positive real constant such that
4. Ulam–Hyers–Rassias Stability
- Definition 3 ⟹ Definition 4.
- Definition 5 ⟹ Definition 6.
- Definition 5 for ⟹ Definition 3.
- , , and .
5. Examples
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Salim, A.; Alzabut, J.; Sudsutad, W.; Thaiprayoon, C. On Impulsive Implicit ψ-Caputo Hybrid Fractional Differential Equations with Retardation and Anticipation. Mathematics 2022, 10, 4821. https://doi.org/10.3390/math10244821
Salim A, Alzabut J, Sudsutad W, Thaiprayoon C. On Impulsive Implicit ψ-Caputo Hybrid Fractional Differential Equations with Retardation and Anticipation. Mathematics. 2022; 10(24):4821. https://doi.org/10.3390/math10244821
Chicago/Turabian StyleSalim, Abdelkrim, Jehad Alzabut, Weerawat Sudsutad, and Chatthai Thaiprayoon. 2022. "On Impulsive Implicit ψ-Caputo Hybrid Fractional Differential Equations with Retardation and Anticipation" Mathematics 10, no. 24: 4821. https://doi.org/10.3390/math10244821
APA StyleSalim, A., Alzabut, J., Sudsutad, W., & Thaiprayoon, C. (2022). On Impulsive Implicit ψ-Caputo Hybrid Fractional Differential Equations with Retardation and Anticipation. Mathematics, 10(24), 4821. https://doi.org/10.3390/math10244821