Е-Публикации
Технически университет - София

Abstract: In the present work we study the Hyers-Ulam and Hyers-Ulam-Rassias stability for inhomogeneous linear fractional systems with distributed delays as well as and for the corresponding homogeneous systems with nonlinear perturbation on arbitrary compact interval. We avoid any standard fixed-point argument by using a new approach, based on integral representation of the solutions of the initial problem for the linear fractional systems. This method allows to obtain sufficient conditions for stability in Hyers-Ulam and Hyers-Ulam-Rassias sense for the studied inhomogeneous delayed systems. Furthermore, we prove that the Hyers-Ulam type stability on some compact interval implies the finite-time stability on the same interval for the investigated homogeneous systems. As an application, sufficient conditions are established for the Hyers-Ulam and Hyers-Ulam-Rassias stability of nonlinear perturbed homogeneous fractional linear systems, under some natural conditions concerning the nonlinear perturbation term.

References

1. Kilbas A., H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science B.V, Amsterdam, 2006.
2. Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999.
3. Jiao Z., Y. Chen, I. Podlubny, Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives, Berlin: Springer-Verlag, 2012.
4. Stamova I., G. Stamov, Functional and Impulsive Differential Equations of Fractional Order, Qualitative analysis and applications, Boca Raton, CRC Press, 2017.
5. Li C.P., Zhang F.R., “A survey on the stability of fractional differential equations”, Eur. Phys. J.: Spec. Top. 193, pp. 27-47 (2011).
6. Gomoyunov M. I., “On representation formulas for solutions of linear differential equations with Caputo fractional derivatives”, Fract. Calc. Appl. Anal. 23, 4, pp. 1141-1160 (2020).
7. Matychyn I., “Analytical Solution of Linear Fractional Systems with Variable Coefficients Involving Riemann-Liouville and Caputo Derivatives”, Symmetry 11, 1366 (2019).
8. Krol K., “Asymptotic properties of fractional delay differential equations”, Appl. Math. Comput. 218, pp. 1515-1532 (2011).
9. Zhang H., D. Wu,“Variation of constant formulae for time invariant and time varying Caputo fractional delay differential systems”, J. Math. Res. Appl. 34, pp. 549-560 (2014).
10. Boyadzhiev D., H. Kiskinov, A. Zahariev,“Integral representation of solutions of fractional system with distributed delays”, Integral Transforms Spec. Funct. 29, 8 (2018).
11. Liang C., J. Wang, D. O’Regan,“Representation of a solution for a fractional linear system with pure delay”, Appl. Math. Lett. 77, pp. 72-78 (2018).
12. Zhang H., J. Cao, W. Jiang, “General solution of linear fractional neutral differential difference equations”, Discrete Dyn. Nat. Soc 2013 (2013).
13. Madamlieva E., M. Konstantinov, M. Milev, M. Petkova, “Integral Representation for the Solutions of Autonomous Linear Neutral Fractional Systems with Distributed Delay”, Mathematics 364, 8 (2020).
14. Kiskinov H., E. Madamlieva, M. Veselinova, A. Zahariev, “Integral Representation of the Solutions for Neutral Linear Fractional System with Distributed Delays”, Fractal Fract. 5, 4, p. 222 (2021).
15. Liu K., M. Fečkan, D. O’Regan, J. Wang, “Hyers-Ulam Stability and Existence of Solutions for Differential Equations with Caputo-Fabrizio Fractional Derivative”, Mathematics 7, 4, p. 333 (2019).
16. Liu K.; M. Fečkan; J. Wang, “A Fixed-Point Approach to the Hyers-Ulam Stability of Caputo-Fabrizio Fractional Differential Equations”, Mathematics 8, 4, p. 647 (2020).
17. Almarri B., X. Wang, A.M. Elshenhab, “Controllability and Hyers-Ulam Stability of Fractional Systems with Pure Delay”, Fractal Fract. 6, p. 611 (2022).
18. Chen C., Q. Dong, “Existence and Hyers-Ulam Stability for a Multi-Term Fractional Differential Equation with Infinite Delay”, Mathematics 10, p. 1013 (2022).
19. Kiskinov H.,E. Madamlieva, A. Zahariev, “Hyers-Ulam and Hyers-Ulam-Rassias types stability of Linear Fractional Systems with Riemann-Liouville Derivatives and Distributed Delays”, Axioms 12, p. 637 (2023).
20. Myshkis A., Linear Differential Equations with Retarded Argument, Nauka: Moscow, Russia, 1972. (In Russian)
21. Hale J., S. Lunel, Introduction to Functional Differential Equations, vol. 99, Springer Science & Business Media, New York, 1993.
22. Kolmanovskii V., A. Myshkis, Introduction to the Theory and Applications of Functional-Differential Equations, Mathematics and its Applications, 463, Kluwer Academic Publishers, Dordrecht, 1999.
23. Kiskinov H., M. Veselinova, E. Madamlieva, A. Zahariev, “A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis”, Axioms 10, p. 75 (2021).
24. Zahariev A., H. Kiskinov, E. Angelova, “Linear fractional system of incommensurate type with distributed delay and bounded Lebesgue measurable initial conditions”, Dyn. Syst. Appl. 28, pp. 491-506 (2019).
25. Zahariev A., H. Kiskinov, “Asymptotic stability of the solutions of neutral linear fractional system with nonlinear perturbation”, Mathematics 8, p. 390 (2020).
26. Kiskinov H., E. Madamlieva, M. Veselinova, A. Zahariev, “On Variation of Constant Formulae for Linear Fractional Delayed System with Lebesgue Integrable Initial Conditions”, AIP Conf. Proc. 2939, p. 040001 (2023), https://doi.org/10.1063/5.0178529.
27. Ye H., J. Gao, Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation”, J. Math. Anal. Appl. 328, pp. 1075-1081 (2007).

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