Hyers-Ulam and exponential stabilities of autonomous and non-autonomous difference equations

Document Type : Research Paper

Authors

1 Department of Mathematics, Islamia College Peshawar, Pakistan.

2 1. Department of Mathematics, University of Malakand, Dir Lower, Pakistan.\\ 2. Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia.

3 Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia.

4 School of Engineering and Natural Sciences, Istanbul Medipol University, 34810, Istanbul, Turkey.

Abstract

 In this manuscript, we studied the Hyers-Ulam and exponential stabilities of autonomous and non-autonomous difference equations of first and second order.  Ultimately, we provide some examples to support our results.

Keywords

Main Subjects


  • [1] A. R. Aruldass, D. Pachaiyappan, and C. Park, Hyers–Ulam stability of second-order differential equations using Mahgoub transform, Advances in Difference Equations, 2021(1) (2021), 1-10.
  • [2] Y. Almalki, G. Rahmat, A. Ullah, F. Shehryar, M. Numan, and M. U. Ali, Generalized β-Hyers-Ulam-Rassias stability of impulsive difference equations, Computational Intelligence and Neurosciences, 2022 (2022), 9462424.
  • [3] M. I. Ayari and S. T. M. Thabet, Qualitative properties and approximate solutions of thermostat fractional dynamics system via a nonsingular kernel operator, Arab Journal of Mathematical Sciences, (2023).
  • [4] A. Boutiara, S. Etemad, A. Hussain, and S. Rezapour, The generalized Ulam-Hyer and Ulam–Hyer stability and existence analysis of a coupled hybrid system of integro-differential IVPs involving ϕ-Caputo fractional operators, Advances in Difference Equations, 2021(1) (2021), 1-21.
  • [5] A. Boutiara, M. Benbachir, M. K. A. Kaabar, F. Mart´ınez, M. Samei, and M. Kaplan, Explicit iteration and unbounded solutions for fractional q–q-difference equations with boundary conditions on an infinite interval, J. Inequal Appl, 2022 (2022). Abdellatif Boutiara, Maamar Benbachir, Mohammed K. A. Kaabar,
  • [6] S. Brianzoni, C. Mammana, E. Michetti, and F. Zirilli, A stochastic cobweb dynamical model, Discrete Dynamics in Nature and Society, 2008 (2008).
  • [7] C. Buse, A. Khan, G. Rahmat, and A. Tabassum, A new estimation of the growth bound of a periodic evolution family on Banach spaces, Journal of Function Spaces and Applications (Journal of function spaces), 2013 (2013), 6.
  • [8] J. Diblık, I. Dzhalladova, and M. Ruzickova, Stabilization of company’s income modeled by a system of discrete stochastic equations, Advances in Difference Equations, 2014(1) (2014), 1-8.
  • [9] S. Frassu and G. Viglialoro, Boundedness for a fully parabolic Keller-Segel model with sublinear segregation and superlinear aggregation, Acta Applicandae Mathematicae, 171(1) (2021), 1-20.
  • [10] Z. Gao, X. Yu, and J. Wang, Exp-type Ulam-Hyers stability of fractional differential equations with positive constant coefficient, Advances in Difference Equations, 2015(1) (2015), 1-10.
  • [11] D. H. Hyers, On the stability of the linear functional equation, Proceedings of the national academy of sciences of the United States of America, 27(4) (1941), 222.
  • [12] S. Hussain, M. Sarwar, G. Rahmat, H. Aydi, and M. D. L Sen, Mild solutions and controllability of fractional evolution inclusions of Clarke’s subdifferential type with nonlocal conditions in Hilbert spaces, Alexandria Engineering Journal, 80 (2023), 58-73.
  • [13] S. M. Jung, Hyers-Ulam stability of the first order matrix difference equations, Advances in Difference Equations, 2015(1) (2015), 1-13.
  • [14] A. Khan, G. Rahmat, and A. Zada, On uniform exponential stability and exact admissibility of discrete semigroups, International Journal of Differential Equations, 2013 (2013).
  • [15] A. Lachouri, M. E. Samei, and A. Ardjouni, Existence and stability analysis for a class of fractional pantograph q-difference equations with nonlocal boundary conditions, Boundary Value Problems, 2023 (2023).
  • [16] T. Li and G. Viglialoro, Analysis and explicit solvability of degenerate tonsorial problems, Boundary Value Problems, 2018(1) (2018), 1-13.
  • [17] T. Li, N. Pintus, and G. Viglialoro, Properties of solutions to porous medium problems with different sources and boundary conditions, Zeitschrift fur angewandte Mathematik und Physik, 70(3) (2019), 1-18.
  • [18] T. Li and G. Viglialoro, Boundedness for a non local reaction chemotaxis model even in the attraction-dominated regime, Differential and Integral Equations, 34 (2021), 315-336.
  • [19] S. Moonsuwan, G. Rahmat, A. Ullah, M. Y. Khan, Kamran, and K. Shah, Hyers-Ulam stability, exponential stability and relative controllability of non-singular delay difference equations, Complexity, 2022 (2022), 8911621.
  • [20] M. Obloza, Connections between Hyers and Lyapunov stability of the ordinary differential equations, (1997).
  • [21] T. Rassias, On the stability of the linear mapping in Banach spaces, Proceedings of the American mathematical society, 72(2) (1978), 297-300.
  • [22] S. O. Shah, A. Zada, M. Muzammil, M. Tayyab, and R. Rizwan, On the Bielecki-Ulam’s type stability results of first order non-linear impulsive delay dynamic systems on time scales, Qualitative Theory of Dynamical Systems, 19(3) (2020), 1-18.
  • [23] S. T. M. Thabet, M. V. Cortez, I. Kedim, M. E. Samei, and M. I. Ayari, Solvability of ρ-Hilfer fractional snap dynamic system on unbounded domains, Fractals and Fractional, (2023).
  • [24] O. Tunc and C. Tunc, Ulam stabilities of nonlinear iterative integro-differential equations, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 117(3) (2023).
  • [25] O. Tunc and C.Tunc, On Ulam stabilities of delay Hammerstein integral equation, Symmetry, 2023(15), 1736.
  • [26] O. Tunc, D. R. Sahu, and C. Tun¸c, On the Ulam type stabilities of a general iterative integro-differential equation including a variable delay, J. Nonlinear Convex Anal., 25(2) (2024), 399–417.
  • [27] O. Tunc, C. Tunc, and J. C. Yao, New results on Ulam stabilities of nonlinear integral equations, Mathematics, 2024(12) (2024), 682.
  • [28] O. Tunc, C. Tunc, G. Petrusel, and J. C. Yao, On the Ulam stabilities of nonlinear integral equations and integrodifferential equations, Math. Meth. Appl. Sci., 47(6) (2024), 1–15.
  • [29] O. Tunc and C. Tunc, On Ulam stabilities of iterative Fredholm and Volterra integral equations with multiple time-varying delays, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 118(3) (2024), 20.
  • [30] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, 8 (1960).