β–Ulam–Hyers Stability and Existence of Solutions for Non-Instantaneous Impulsive Fractional Integral Equations
Abstract
:1. Introduction and Preliminaries
- (i)
- if and only if ;
- (ii)
- for all and ;
- (iii)
- .
- (Ulam)
- Assume that is a group and is a metric group with the metric . Given , does there exist a such that if a function satisfies the inequalityfor all , then there exists a homomorphism with
2. The Existence and Uniqueness of Solutions
- (C1)
- The function is measurable in for all and there is a constant , such that for all and all ;
- (C2)
- The function is measurable with respect to , for all and there are constants , such that for all , and all ;
- (C3)
- Suppose is continuous with the respect to , for all and there is a constant , such that for all and all ;
- (C4)
- Suppose is measurable with respect to for all and continuous with respect to for all . Additionally, there exists a bounded measurable function and a constant , such that , where ;
- (C5)
- Suppose is measurable with respect to for all , continuous with respect to for all and is measurable in , for all and continuous with respect to for all , . Additionally, there exists a bounded measurable function and a real number and there are bounded measurable functions and a constant , such that , where and for all ;
- (C6)
- Suppose is measurable with respect to for all and continuous with respect to for all . Additionally, there exists a bounded measurable function and a constant , such that , for all , , where ;
- (C7)
- The number is a positive solution of the equation
- (C8)
- It holds .
- Case 1.
- Let and . Then,
- Case 2.
- Let and , . Since
- Case 3.
- Let and , . Since
- Case 1.
- Let . Then, we have
- Case 2.
- Let , . Then,
- Case 3.
- Let , for . Then, we have
- Case (a).
- Let . Then,
- Case (b).
- Let , . So,
- Case (c).
- Let , . Since
- Case (i).
- Let . Then, we have
- Case (ii).
- Let , . Then,
- Case (iii).
- Let , . Hence, we have
3. –Ulam–Hyers Stability of Solutions
- (i)
- ;
- (ii)
- ;
- (iii)
- (iv)
- , , , where
- (1).
- For ,
- (2).
- For , ,
- (3).
- For , ,
- Case 1.
- Let . Then,
- Case 2.
- Let , . Since
- Case 3.
- Let , . Then,
4. Nontrivial Example of Application of Theorem 1
- Here, we set
- Now,
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Du, W.-S.; Fečkan, M.; Kostić, M.; Velinov, D. β–Ulam–Hyers Stability and Existence of Solutions for Non-Instantaneous Impulsive Fractional Integral Equations. Fractal Fract. 2024, 8, 469. https://doi.org/10.3390/fractalfract8080469
Du W-S, Fečkan M, Kostić M, Velinov D. β–Ulam–Hyers Stability and Existence of Solutions for Non-Instantaneous Impulsive Fractional Integral Equations. Fractal and Fractional. 2024; 8(8):469. https://doi.org/10.3390/fractalfract8080469
Chicago/Turabian StyleDu, Wei-Shih, Michal Fečkan, Marko Kostić, and Daniel Velinov. 2024. "β–Ulam–Hyers Stability and Existence of Solutions for Non-Instantaneous Impulsive Fractional Integral Equations" Fractal and Fractional 8, no. 8: 469. https://doi.org/10.3390/fractalfract8080469