Abstract
We consider a class of scalar linear differential equations with several variable delays and constant coefficients. A family of equations of the class is defined by coefficients and maximum admissible values of delays. We obtain conditions that are necessary and sufficient for the stability of solutions to all equations of the family. It is ascertained that the conditions are determined entirely by properties of the solution to the initial problem for an autonomous equation that belongs to the family. Some alternatives of required conditions are obtained in the form of estimates for solutions to autonomous equations in a finite interval.
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References
R. Bellman and K. L. Cooke, Differential-Difference Equations (Academic Press, New York/London, 1963; Mir, Moscow, 1967).
A. M. Zverkin, “On the Theory of Linear Differential Equations with a Delayed Argument and Periodic Coefficients,” Sov. Phys. Dokl. 128(5), 882–885 (1959).
S.N. Shimanov and Yu. F. Dolgii, “About the Existence of a Stability Domain forOne Equation with a Delay,” in Stability and Nonlinear Oscillations (Sverdlovsk, 1989), pp. 11–18.
A. I. Bashkirov, “Stability of Equations of Delayed Type with Periodic Parameters,” Differents. Uravn. 22(11), 1994–1997 (1986).
A. D. Myshkis, “On Solutions of Linear Homogeneous Differential Equations of the First Order of Stable Type with a Retarded Argument,” Matem. Sb. 28(70), No. 3, 641–658 (1951).
J. A. Yorke, “Asymptotic Stability for One Dimensional Differential-Delay Equations,” J. Different. Equat., No. 7, 189–202 (1970).
T. Yoneyama, “On the 3/2 Stability Theorem for one Dimensional Delay-Differential Equations,” J. Math. Anal. Appl. 125(1), 161–173 (1987).
V. V. Malygina, “Some Criterions of Stability of Equations with Delayed Argument,” Differents. Uravn. 28(10), 1716–1723 (1992).
T. Amemiya, “On the Delay-Independent Stability of a Delayed Differential Equation of 1st Order,” J. Math. Anal. Appl. 142(1), 13–25 (1989).
V. V. Malygina, “On Stability of Solutions to Certain Linear Differential Equations With Aftereffect,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 72–85 (1993) [RussianMathematics (Iz. VUZ) 37 (5), 63–75 (1993)].
A. D. Myshkis, Linear Differential Equations with Delayed Argument (Nauka, Moscow, 1972) [in Russian].
J. Hale, Theory of Functional Differential Equations (Springer-Verlag, New York, 1977; Mir, Moscow, 1984).
V. P. Maksimov and Rakhmatullina, “About the Presentation of Solution to Linear Functional-Differential Equation,” Differents. Uravn. 9(6), 1026–1036 (1973).
N. V. Azbelev and P.M. Simonov, “Stability of the Equations with Delay,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 6, 3–16 (1997) [Russian Mathematics (Iz. VUZ) 41 (6), 1–14 (1997)].
N. V. Azbelev and P. M. Simonov, Stability of Equations with Ordinary Derivatives (Perm Univ. Press, Perm, 2001) [in Russian].
N. V. Azbelev and L. F. Rakhmatullina, “The Cauchy Problem for Differential Equation with a Delayed Argument,” Differents. Uravn. 8(9), 1542–1552 (1972).
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Original Russian Text © V.V. Malygina and K.M. Chudinov, 2013, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2013, No. 6, pp. 25–36.
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Malygina, V.V., Chudinov, K.M. Stability of solutions to differential equations with several variable delays. I. Russ Math. 57, 21–31 (2013). https://doi.org/10.3103/S1066369X13060030
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DOI: https://doi.org/10.3103/S1066369X13060030