Abstract
For any linear delay-difference equation with constant coefficients, we show that the existence of an exponential dichotomy is equivalent to the Ulam–Hyers stability of the equation. This generalizes former work for scalar equations, with a substantial simplification of former arguments. We also show that the same equivalence holds for any linear delay-difference equation (with scalar coefficients replaced by arbitrary linear operators) such that the Jordan forms associated to central directions are diagonal. The proof depends on an explicit formula for the dynamics on each generalized eigenspace, which is of independent interest. Finally, we also consider Lipschitz perturbations of an exponential dichotomy, and we show that if the Lipschitz constant is sufficiently small, then the perturbed equation is Ulam–Hyers stable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and materials
Not applicable.
References
Alsina, C., Ger, R.: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 2, 373–380 (1998)
Barbu, D., Buşe, C., Tabassum, A.: Hyers–Ulam stability and discrete dichotomy. J. Math. Anal. Appl. 423, 1738–1752 (2015)
Barbu, D., Buşe, C., Tabassum, A.: Hyers–Ulam stability and exponential dichotomy of linear differential periodic systems are equivalent. Electron. J. Qual. Theory Differ. Equ. 58, 12 (2015)
Barreira, L., Valls, C.: Stability in delay difference equations with nonuniform exponential behavior. J. Differ. Equ. 238, 470–490 (2007)
Barreira, L., Valls, C.: Stability of delay equations. Electron. J. Qual. Theory Differ. Equ. 45, 24 (2022)
Buşe, C., Lupulescu, V., O’Regan, D.: Hyers–Ulam stability for equations with differences and differential equations with time-dependent and periodic coefficients. Proc. R. Soc. Edinb. Sect. A 150, 2175–2188 (2020)
Buşe, C., O’Regan, D., Saierli, O., Tabassum, A.: Hyers–Ulam stability and discrete dichotomy for difference periodic systems. Bull. Sci. Math. 140, 908–934 (2016)
Fukutaka, R., Onitsuka, M.: A necessary and sufficient condition for Hyers–Ulam stability of first-order periodic linear differential equations. Appl. Math. Lett. 100, 106040 (2020)
Huang, J., Li, Y.: Hyers–Ulam stability of linear functional differential equations. J. Math. Anal. Appl. 426, 1192–1200 (2015)
Hyers, D.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27, 222–224 (1941)
Hyers, D., Isac, G., Rassias, T.: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and their Applications, vol. 34. Birkhäuser Boston Inc., Boston (1998)
Jung, S.-M.: Hyers-Ulam stability of linear differential equations of first order. III. J. Math. Anal. Appl. 311, 139–146 (2005)
Jung, S.-M., Brzdȩk, J.: Hyers–Ulam stability of the delay equation \(y^{\prime }(t)=\lambda y(t-\tau )\). Abstr. Appl. Anal. 372176 (2010) https://doi.org/10.1155/2010/372176
Miura, T., Miyajima, S., Takahasi, S.-E.: Hyers–Ulam stability of linear differential operator with constant coefficients. Math. Nachr. 258, 90–96 (2003)
Öğrekçi, S.: Stability of delay differential equations in the sense of Ulam on unbounded intervals. Int. J. Optim. Control. Theor. Appl. 9, 125–131 (2019)
Otrocol, D., Ilea, V.: Ulam stability for a delay differential equation. Cent. Eur. J. Math. 11, 1296–1303 (2013)
Popa, D.: Hyers–Ulam–Rassias stability of a linear recurrence. J. Math. Anal. Appl. 309, 591–597 (2005)
Popa, D., Raşa, I.: On the Hyers–Ulam stability of the linear differential equation. J. Math. Anal. Appl. 381, 530–537 (2011)
Popa, D., Raşa, I.: Hyers–Ulam stability of the linear differential operator with nonconstant coefficients. Appl. Math. Comput. 219, 1562–1568 (2012)
Rassias, T.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)
Ulam, S.: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, vol. 8. Interscience Publishers, New York, London (1960)
Zada, A., Pervaiz, B., Alzabut, J., Shah, S.: Further results on Ulam stability for a system of first-order nonsingular delay differential equations. Demonstr. Math. 53, 225–235 (2020)
Funding
Supported by Fundação para a Ciência e a Tecnologia, Portugal through CAMGSD, IST-ID, Projects UIDB/04459/2020 and UIDP/04459/2020.
Author information
Authors and Affiliations
Contributions
All authors contributed equally to the manuscript and all authors reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
Not applicable.
Ethical approval
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Barreira, L., Valls, C. Delay-Difference Equations and Stability. J Dyn Diff Equat (2023). https://doi.org/10.1007/s10884-023-10304-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10884-023-10304-z