# Nonlinear Volterra Integrodifferential Equations from above on Unbounded Time Scales

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Notations and Preliminaries

## 3. Existence and Uniqueness

**Theorem**

**1.**

**Proof.**

**Example**

**1.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Hilger, S. Analysis on measure chains. A unified approach to continuous and discrete calculus. Results Math.
**1990**, 18, 18–56. [Google Scholar] [CrossRef] - Bohner, M.; Peterson, A. Dynamic Equations on Time Scales. An Introduction with Applications; Birkhäuser: Boston, MA, USA; Basel, Switzerland; Berlin, Germany, 2001. [Google Scholar]
- Bohner, M.; Peterson, A. Advances in Dynamic Equations on Time Scales; Birkhäuser: Boston, MA, USA; Basel, Switzerland; Berlin, Germany, 2003. [Google Scholar]
- Georgiev, S. Integral Equation on Time Scales; Atlamtis Press: Paris, France, 2016. [Google Scholar]
- Adivar, M.; Raffoul, Y.N. Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales; Springer: Berlin/Heidelberg, Germany, 2020. [Google Scholar]
- Karpuz, B. Volterra theory on time scales. Results Math.
**2014**, 65, 263–292. [Google Scholar] [CrossRef] - Atici, F.M.; Biles, D.C.; Lebedinsky, A. An application of time scales to economics. Math. Comput. Model.
**2006**, 43, 718–726. [Google Scholar] [CrossRef] - Ferguson, B.S.; Lim, G.C. Dynamic Economic Models in Discrete Time. Theory and Empirical Applications; Routledge: London, UK, 2003. [Google Scholar]
- Tisdell, C.C.; Zaidi, A. Basic qualitative and quantitative results for solutions to nonlinear dynamic equations on time scales with an application to economic modeling. Nonlinear Anal.
**2008**, 68, 3504–3524. [Google Scholar] [CrossRef] - Messina, E.; Pezzalla, M.; Vecchio, A. A non-standard numerical scheme for age-of-infection epidemic model. J. Comput. Dyn.
**2022**, 9, 239–252. [Google Scholar] [CrossRef] - Sikorska-Nowak, A. Integrodifferential equation on time scales with Henstock-Kurzweil delta integrals. Discuss. Math. Differ. Incl. Control Opt.
**2011**, 31, 71–90. [Google Scholar] [CrossRef] - Georgiev, S. Volterra integrodifferential equation on time scales. Int. J. Appl. Math. Comput. Sci.
**2017**, 3, 1577–1587. [Google Scholar] [CrossRef] - Kulik, T.; Tisdell, C.C. Volterra integral equations on time scales. Basic qualitative and quantitative results with applications to initial value problems on unbounded domains. Int. J. Differ. Equ.
**2008**, 3, 103–133. [Google Scholar] - Reinfelds, A.; Christian, S. Volterra integral equations on unbounded time scales. Int. J. Differ. Equ.
**2019**, 14, 169–177. [Google Scholar] [CrossRef] - Reinfelds, A.; Christian, S. A nonstandard Volterra integral equation on time scales. Demonstr. Math.
**2019**, 52, 503–510. [Google Scholar] [CrossRef] - Reinfelds, A.; Christian, S. Hyers-Ulam stability of Volterra type integral equations on time scales. Adv. Dyn. Syst. Appl.
**2020**, 15, 39–48. [Google Scholar] - Pachpatte, B.G. On certain Volterra integral and integrodifferential equations. Facta Univ. Ser. Math. Inform.
**2008**, 23, 1–12. [Google Scholar] - Pachpatte, B.G. Implict type Volterra integrodifferential equation. Tamkang J. Math.
**2010**, 41, 97–107. [Google Scholar] [CrossRef] - Pachpatte, D.B. Fredholm type integrodifferential equation on time scales. Electron. J. Differ. Equ.
**2010**, 140, 1–10. [Google Scholar] - Pachpatte, D.B. Properties of solutions to nonlinear dynamic integral equations on time scales. Electron. J. Differ. Equ.
**2008**, 136, 1–8. [Google Scholar] - Pachpatte, D.B. On a nonlinear dynamic integrodifferential equation on time scales. J. Appl. Anal.
**2010**, 16, 279–294. [Google Scholar] [CrossRef] - Christian, S. Volterra Integral Equation on Time Scales. Ph.D. Thesis, University of Latvia, Riga, Latvia, 27 November 2020. [Google Scholar]
- Bohner, M. Some oscillation criteria for first order delay dynamic equations. Far East J. Appl. Math.
**2005**, 18, 289–304. [Google Scholar] - Corduneanu, C. Integral Equations and Applications; Cambridge University Press: Cambridge, UK, 1991. [Google Scholar]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Reinfelds, A.; Christian, S.
Nonlinear Volterra Integrodifferential Equations from above on Unbounded Time Scales. *Mathematics* **2023**, *11*, 1760.
https://doi.org/10.3390/math11071760

**AMA Style**

Reinfelds A, Christian S.
Nonlinear Volterra Integrodifferential Equations from above on Unbounded Time Scales. *Mathematics*. 2023; 11(7):1760.
https://doi.org/10.3390/math11071760

**Chicago/Turabian Style**

Reinfelds, Andrejs, and Shraddha Christian.
2023. "Nonlinear Volterra Integrodifferential Equations from above on Unbounded Time Scales" *Mathematics* 11, no. 7: 1760.
https://doi.org/10.3390/math11071760