Ulam Type Stability Results of Nonlinear Impulsive Volterra–Fredholm Integro-Dynamic Adjoint Equations on Time Scale
Abstract
:1. Introduction
Symbol | Interpretation |
The set of time scale | |
The set of real numbers | |
Forward jump operator | |
Backward jump operator | |
Graininess function | |
The set of right-dense-continuous functions | |
The set of regressive functions | |
The set of positively regressive functions | |
Derived form of time scale | |
The Banach space of continuous functions | |
The Banach space of piecewise continuous functions | |
Unknown function | |
Delta derivative of | |
Perturbed function | |
Nondecreasing function | |
Variables | |
Points of impulses |
2. Basic Concepts and Remarks
3. Existence and Uniqueness of Solutions
- (C)
- For , there exists such that
- (C)
- For , there exists such that
- (C)
- .
- (C)
- For a nondecreasing , there exists such that
4. Ulam-Type Stability Results
5. Example
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Shah, S.O.; Tikare, S.; Osman, M. Ulam Type Stability Results of Nonlinear Impulsive Volterra–Fredholm Integro-Dynamic Adjoint Equations on Time Scale. Mathematics 2023, 11, 4498. https://doi.org/10.3390/math11214498
Shah SO, Tikare S, Osman M. Ulam Type Stability Results of Nonlinear Impulsive Volterra–Fredholm Integro-Dynamic Adjoint Equations on Time Scale. Mathematics. 2023; 11(21):4498. https://doi.org/10.3390/math11214498
Chicago/Turabian StyleShah, Syed Omar, Sanket Tikare, and Mawia Osman. 2023. "Ulam Type Stability Results of Nonlinear Impulsive Volterra–Fredholm Integro-Dynamic Adjoint Equations on Time Scale" Mathematics 11, no. 21: 4498. https://doi.org/10.3390/math11214498
APA StyleShah, S. O., Tikare, S., & Osman, M. (2023). Ulam Type Stability Results of Nonlinear Impulsive Volterra–Fredholm Integro-Dynamic Adjoint Equations on Time Scale. Mathematics, 11(21), 4498. https://doi.org/10.3390/math11214498