Abstract
Considering some linear autonomous differential equation with distributed delay that has a positive fundamental solution, we develop a method for obtaining effective exponential stability tests and two-sided estimates of the fundamental solution in the form of two exponential functions with exponents and coefficients determined exactly. A few examples illustrate the use of the method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Demidenko G. V. and Matveeva I. I., “Asymptotic properties of solutions to delay differential equations,” Vestnik Novosibirsk Univ. Ser. Mat. Mekh. Inform., vol. 5, no. 3, 20–28 (2005).
Liz E. and Pituk M., “Exponential stability in a scalar functional differential equation,” J. Inequal. Appl., vol. 2006 (2006) (Article no. 37195).
Pertsev N. V., “Application of \( M \)-matrices in construction of exponential estimates for solutions to the Cauchy problem for systems of linear difference and differential equations,” Siberian Adv. Math., vol. 24, no. 4, 240–260 (2014).
Myshkis A. D., Linear Differential Equations with Retarded Argument, Nauka, Moscow (1972) [Russian].
Ryabov Yu. A., “Some asymptotic properties of linear systems with a small delay,” Dokl. Akad. Nauk SSSR, vol. 151, no. 1, 52–54 (1963).
El’sgol’ts L. E. and Norkin S. B., Introduction to the Theory and Application of Differential Equations with Deviating Argument, Academic, New York (1973).
Hale J. K., Theory of Functional Differential Equations, Springer, New York, Heidelberg, and Berlin (1977).
Azbelev N., Maksimov V., and Rakhmatullina L., Introduction to the Theory of Linear Functional Differential Equations. Advanced Series in Mathematical Science and Engineering, World Federation, Atlanta (1995).
Azbelev N. V. and Simonov P. M., Stability of Differential Equations with Aftereffect, Taylor and Francis, London (2002).
Agarwal R. P., Berezansky L., Braverman E., and Domoshnitsky A., Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York (2012).
Győri I. and Ladas G., Oscillation Theory of Delay Differential Equations with Applications, Clarendon, Oxford (1991).
Sabatulina T. and Malygina V., “On positiveness of the fundamental solution for a linear autonomous differential equation with distributed delay,” Electron. J. Qual. Theory Differ. Equ., vol. 61, 1–16 (2014).
Lavrentev M. A. and Shabat B. V., Methods of the Theory of Functions of Complex Variables, Nauka, Moscow (1973) [Russian].
Daleckii Ju. L. and Krein M. G., Stability of Solutions to Differential Equations in Banach Space, Amer. Math. Soc., Providence (1974).
Sabatulina T. L., “On sufficient conditions of exponential stability for systems of linear autonomous differential equations with aftereffect,” Funct. Differ. Equ., vol. 28, no. 1, 73–83 (2021).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 2, pp. 360–378. https://doi.org/10.33048/smzh.2022.63.208
Rights and permissions
About this article
Cite this article
Malygina, V.V., Chudinov, K.M. About Exact Two-Sided Estimates for Stable Solutions to Autonomous Functional Differential Equations. Sib Math J 63, 299–315 (2022). https://doi.org/10.1134/S0037446622020082
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446622020082