Abstract
In this paper, a nonlinear Volterra integro-differential equation with Caputo fractional derivative, multiple kernels, and multiple constant delays is considered. The aim of this paper is to investigate qualitative properties of solutions of this equation such as uniform stability, asymptotic stability, and Mittag-Leffler stability of the zero solution as well as boundedness of nonzero solutions. Here, we prove four new theorems on the mentioned properties of the solutions of the considered fractional integro-differential equation. The technique used in the proofs of these theorems includes defining an appropriate Lyapunov function and applying the Lyapunov–Razumikhin method. To illustrate the obtained results, two examples are provided, one of them related to an RLC circuit, to illustrate and show applications of the given results. The obtained results are new, original, and they can be useful for applied researchers in sciences and engineering.
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Acknowledgements
The authors would like to thank the two anonymous referees and the handling Editor Professor José Tenreiro Machado for many useful comments and suggestions, leading to a substantial improvement of the presentation of this article.
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Bohner, M., Tunç, O. & Tunç, C. Qualitative analysis of caputo fractional integro-differential equations with constant delays. Comp. Appl. Math. 40, 214 (2021). https://doi.org/10.1007/s40314-021-01595-3
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DOI: https://doi.org/10.1007/s40314-021-01595-3
Keywords
- Lyapunov–Razumikhin method
- Lyapunov function
- Caputo fractional Volterra integro-differential equation
- Stability
- Boundedness
- Mittag-Leffler stability