Discrete Gronwall’s inequality for Ulam stability of delay fractional difference equations
Abstract
This paper investigates Ulam stability of delay fractional difference equations. First, a useful equality of double fractional sums is employed and discrete Gronwall’s inequality of delay type is provided. A delay discrete-time Mittag-Leffler function is used and its non-negativity condition is given. With the solutions’ existences, Ulam stability condition is presented to discuss the error estimation of exact and approximate solutions.
Keyword : delay fractional difference equations, discrete Gronwall’s inequality, uniqueness of solution, Ulam stability
How to Cite
Yang, S.-Y., & Wu, G.-C. (2025). Discrete Gronwall’s inequality for Ulam stability of delay fractional difference equations. Mathematical Modelling and Analysis, 30(1), 169–185. https://doi.org/10.3846/mma.2025.20017
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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H. Rezaei, S.-M. Jung and Th.M. Rassias. Laplace transform and Hyers–Ulam stability of linear differential equations. Journal of Mathematical Analysis and Applications, 403(1):244–251, 2013. https://doi.org/10.1016/j.jmaa.2013.02.034
S.M. Ulam. A Collection of Mathematical Problems. Interscience Publisher, New York, 1960.
J. Wang, L. Lv and Y. Zhou. Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electronic Journal of Qualitative Theory of Differential Equations, 2011(63):1–10, 2011. https://doi.org/10.14232/ejqtde.2011.1.63
G.-C. Wu, D. Baleanu and S.D. Zeng. Finite-time stability of discrete fractional delay systems: Gronwall inequality and stability criterion. Communications in Nonlinear Science and Numerical Simulation, 57:299–308, 2018. https://doi.org/10.1016/j.cnsns.2017.09.001
G.-C. Wu, J.-L. Wei and M. Luo. Right fractional calculus to inverse-time chaotic maps and asymptotic stability analysis. Journal of Difference Equations and Applications, 29(9-12):1140–1155, 2023. https://doi.org/10.1080/10236198.2023.2198043
H.P. Ye, J.M. Gao and Y.S. Ding. A generalized Gronwall inequality and its application to a fractional differential equation. Journal of Mathematical Analysis and Applications, 328(2):1075–1081, 2010. https://doi.org/10.1016/j.jmaa.2006.05.061
R. Abu-Saris and Q. Al-Mdallal. On the asymptotic stability of linear system of fractional-order difference equations. Fractional Calculus and Applied Analysis, 16(3):613–629, 2013. https://doi.org/10.2478/s13540-013-0039-2
J. Alzabut and T. Abdeljawad. A generalized discrete fractional Gronwall inequality and its application on the uniqueness of solutions for nonlinear delay fractional difference system. Applicable Analysis and Discrete Mathematics, 12(1):36–48, 2018. https://doi.org/10.2298/AADM1801036A
F. Atici and P.W. Eloe. Initial value problems in discrete fractional calculus. Proceedings of the American mathematical society, 137(3):981–989, 2009. https://doi.org/10.1090/S0002-9939-08-09626-3
F.M. Atıcı and P.W. Eloe. Gronwall’s inequality on discrete fractional calculus. Computers & Mathematics with Applications, 64(10):3193–3200, 2012. https://doi.org/10.1016/j.camwa.2011.11.029
D. Baleanu and G.-C. Wu. Some further results of the Laplace transform for variable–order fractional difference equations. Fractional Calculus and Applied Analysis, 22(6):1641–1654, 2019. https://doi.org/10.1515/fca-2019-0084
J. Brzde¸k, J. Chudziak and Z. Pa´les. A fixed point approach to stability of functional equations. Nonlinear Analysis: Theory, Methods & Applications, 74(17):6728–6732, 2011. https://doi.org/10.1016/j.na.2011.06.052
C.R. Chen, R. Mert, B.G. Jia, L. Erbe and A. Peterson. Gronwall’s inequality for a nabla fractional difference system with a retarded argument and an application. Journal of Difference Equations and Applications, 25(6):855–868, 2019. https://doi.org/10.1080/10236198.2019.1581180
F.L. Chen. Fixed points and asymptotic stability of nonlinear fractional difference equations. Electronic Journal of Qualitative Theory of Differential Equations, 2011(39):1–18, 2011. https://doi.org/10.14232/ejqtde.2011.1.39
F.L. Chen and Y. Zhou. Existence and Ulam stability of solutions for discrete fractional boundary value problem. Discrete Dynamics in Nature and Society, 2013(1):459161, 2013. https://doi.org/10.1155/2013/459161
F.F. Du and B.G. Jia. A generalized fractional (q, h)–Gronwall inequality and its applications to nonlinear fractional delay (q, h)–difference systems. Mathematical Methods in the Applied Sciences, 44(13):10513–10529, 2021. https://doi.org/10.1002/mma.7426
A.C. Ferreira. A discrete fractional Gronwall inequality. Proceedings of the American Mathematical Society, 140(5):1605–1612, 2012. https://doi.org/10.1090/S0002-9939-2012-11533-3
C. Goodrich and A.C. Peterson. Discrete Fractional Calculus, volume 10. Springer, Cham, 2015. https://doi.org/10.1007/978-3-319-25562-0
S.S. Haider, M.U. Rehman and T. Abdeljawad. On Hilfer fractional difference operator. Advances in Difference Equations, 2020(122):1–20, 2020. https://doi.org/10.1186/s13662-020-02576-2
M. Holm. The Theory of Discrete Fractional Calculus: Development and Application. The University of Nebraska, Lincoln, 2011.
L.-L. Huang, J.H. Park, G.-C. Wu and Z.-W. Mo. Variable-order fractional discrete-time recurrent neural networks. Journal of Computational and Applied Mathematics, 370:112633, 2020. https://doi.org/10.1016/j.cam.2019.112633
L.-L. Huang, G.-C. Wu, D. Baleanu and H.-Y. Wang. Discrete fractional calculus for interval–valued systems. Fuzzy Sets and Systems, 404:141–158, 2021. https://doi.org/10.1016/j.fss.2020.04.008
D.H. Hyers. On the stability of the linear functional equation. Proceedings of the National Academy of Sciences, 27(4):222–224, 1941. https://doi.org/10.1073/pnas.27.4.222
H. Kong, G. Yang and C. Luo. Modelling aftershcoks by fractional calculus: Exact discretization versus approximation discretization. Fractals, 29(08):2140038, 2021. https://doi.org/10.1142/S0218348X21400387
X. Liu, A. Peterson, B.G. Jia and L. Erbe. A generalized h-fractional Gronwall’s inequality and its applications for nonlinear h-fractional difference systems with ‘maxima’. Journal of Difference Equations and Applications, 25(6):815–836, 2019. https://doi.org/10.1080/10236198.2018.1551382
D. Mozyrska and E. Girejko. Overview of fractional h-difference operators. In A. Almeida, L. Castro and F.-O. Speck(Eds.), Advances in Harmonic Analysis and Operator Theory, volume 229, pp. 253–268, Basel, 2013. Sirkhäuser, Basel, Springer, Basel. https://doi.org/10.1007/978-3-0348-0516-2_14
Th.M. Rassias. On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society, 72(2):297–300, 1978. https://doi.org/10.1090/S0002-9939-1978-0507327-1
H. Rezaei, S.-M. Jung and Th.M. Rassias. Laplace transform and Hyers–Ulam stability of linear differential equations. Journal of Mathematical Analysis and Applications, 403(1):244–251, 2013. https://doi.org/10.1016/j.jmaa.2013.02.034
S.M. Ulam. A Collection of Mathematical Problems. Interscience Publisher, New York, 1960.
J. Wang, L. Lv and Y. Zhou. Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electronic Journal of Qualitative Theory of Differential Equations, 2011(63):1–10, 2011. https://doi.org/10.14232/ejqtde.2011.1.63
G.-C. Wu, D. Baleanu and S.D. Zeng. Finite-time stability of discrete fractional delay systems: Gronwall inequality and stability criterion. Communications in Nonlinear Science and Numerical Simulation, 57:299–308, 2018. https://doi.org/10.1016/j.cnsns.2017.09.001
G.-C. Wu, J.-L. Wei and M. Luo. Right fractional calculus to inverse-time chaotic maps and asymptotic stability analysis. Journal of Difference Equations and Applications, 29(9-12):1140–1155, 2023. https://doi.org/10.1080/10236198.2023.2198043
H.P. Ye, J.M. Gao and Y.S. Ding. A generalized Gronwall inequality and its application to a fractional differential equation. Journal of Mathematical Analysis and Applications, 328(2):1075–1081, 2010. https://doi.org/10.1016/j.jmaa.2006.05.061