Abstract
We discuss the well-known Myshkis result on the stability of nonautonomous first-order delay differential equations, providing an extension to the general differential equations with aftereffect, and make comparison with available results.
References
Myshkis A.D., “On solutions of linear homogeneous differential equations of the second order of periodic type with a retarded argument,” Mat. Sb., vol. 28, no. 3, 641–658 (1951).
Azbelev N.V. and Simonov P.M., Stability of Differential Equations with Aftereffect, Taylor and Francis, London (2002).
Yorke J.A., “Asymptotic stability for one dimensional differential-delay equations,” J. Differ. Equ., vol. 7, 189–202 (1970).
Amemiya T., “On the delay-independent stability of a delayed differential equation of \( 1 \)st order,” J. Math. Anal. Appl., vol. 142, no. 1, 13–25 (1989).
Malygina V.V., “Stability of solutions of some linear differential equations with aftereffect,” Russian Math. (Iz. VUZ. Matematika), vol. 37, no. 5, 63–75 (1993).
Malygina V.V. and Chudinov K.M., “Stability of solutions to differential equations with several variable delays. III,” Russian Math. (Iz. VUZ. Matematika), vol. 57, no. 8, 37–48 (2013).
Demidovich B.P., Lectures on the Mathematical Theory of Stability, Nauka, Moscow (1967) [Russian].
Andronov A.A. and Maier A.G., “Simple linear systems with delay,” Avtomat. i Telemekh., vol. 7, no. 2, 95–106 (1946).
Bellman R.E. and Cooke K.L., Differential-Difference Equations, Academic, London (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).
Daleckii Ju.L. and Krein M.G., Stability of Solutions to Differential Equations in Banach Space, Amer. Math. Soc., Providence (1974).
Krisztin T., “On stability properties for one-dimensional functional differential equations,” Funkcial. Ekvac., vol. 34, no. 2, 241–256 (1991).
Ladas G., Sficas Y.G., and Stavroulakis I.P., “Asymptotic behavior of solutions of retarded differential equations,” Proc. Amer. Math. Soc., vol. 88, no. 2, 247–253 (1983).
Yoneyama T., “On the \( {3/2} \) stability theorem for one-dimensional delay-differential equations,” J. Math. Anal. Appl., vol. 125, no. 1, 161–173 (1987).
Yoneyama T., “The \( 3/2 \) stability theorem for one-dimensional delay-differential equations with unbounded delay,” J. Math. Anal. Appl., vol. 165, no. 1, 133–143 (1992).
Gusarenko S.A., “Solvability criteria for the problems of the accumulation of perturbations of functional differential equations,” in: Functional Differential Equations, Perm Polytechnic University, Perm (1987), 30–40 [Russian].
Gusarenko S.A. and Domoshnitskii A.I., “Asymptotic and oscillation properties of first-order linear scalar functional-differential equations,” Differ. Equ., vol. 25, no. 12, 1480–1491 (1989).
Győri I. and Hartung F., “Stability in delayed perturbed differential and difference equations,” Fields Institute Communications, vol. 29, 181–194 (2001).
Yoneyama T. and Sugie J., “Perturbing uniformly stable nonlinear scalar delay-differential equations,” Nonlinear Anal., vol. 12, no. 3, 303–311 (1988).
So J.W.-H., Yu J.S., and Chen M.P., “Asymptotic stability for scalar delay differential equations,” Funkcial. Ekvac., vol. 39, no. 1, 1–17 (1996).
Cooke K.L. and Yorke J.A., “Some equations modelling growth processes and gonorrhea epidemics,” Math. Biosci., vol. 16, no. 1, 75–101 (1973).
Yoneyama T., “On the stability for the delay-differential equation \( \dot{x}(t)=-a(t)f(x(t-r(t))) \),” J. Math. Anal. Appl., vol. 120, no. 1, 271–275 (1986).
Funding
The author was supported by the Ministry for Science and Higher Education of the Russian Federation (Grant no. FSNM–2023–0005).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
As author of this work, I declare that I have no conflicts of interest.
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 6, pp. 1248–1262. https://doi.org/10.33048/smzh.2023.64.611
Publisher's Note
Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Malygina, V.V. The Myshkis 3/2 Theorem and Its Generalizations. Sib Math J 64, 1372–1384 (2023). https://doi.org/10.1134/S0037446623060113
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446623060113