# Article

Full entry | PDF   (0.2 MB)
Keywords:
exponential stability; nonoscillation; explicit stability condition; perturbation
Summary:
We consider preservation of exponential stability for the scalar nonoscillatory linear equation with several delays $$\dot {x}(t) + \sum _{k=1}^m a_k(t) x(h_k(t)) = 0, \quad a_k(t) \geq 0$$ under the addition of new terms and a delay perturbation. We assume that the original equation has a positive fundamental function; our method is based on Bohl-Perron type theorems. Explicit stability conditions are obtained.
References:
[1] Györi, I., Hartung, F., Turi, J.: Preservation of stability in delay equations under delay perturbations. J. Math. Anal. Appl. 220 (1998), 290-312. DOI 10.1006/jmaa.1997.5883 | MR 1613964
[2] Berezansky, L., Braverman, E.: Preservation of the exponential stability under perturbations of linear delay impulsive differential equations. Z. Anal. Anwendungen 14 (1995), 157-174. DOI 10.4171/ZAA/668 | MR 1327497 | Zbl 0821.34072
[3] Hale, J. K., Lunel, S. M. Verduyn: Introduction to Functional Differential Equations. Applied Mathematical Sciences, Vol. 99, Springer, New York (1993). DOI 10.1007/978-1-4612-4342-7_3 | MR 1243878
[4] Azbelev, N. V., Berezansky, L., Rakhmatullina, L. F.: A linear functional-differential equation of evolution type. Differ. Equations 13 (1977), 1331-1339.
[5] Azbelev, N. V., Berezansky, L., Simonov, P. M., Chistyakov, A. V.: The stability of linear systems with aftereffect I. Differ. Equations 23 (1987), 493-500; Differ. Equations 27 (1991), 383-388; Differ. Equations 27 (1991), 1165-1172; Differ. Equations 29 (1993), 153-160. MR 1236101
[6] Azbelev, N. V., Simonov, P. M.: Stability of Differential Equations with Aftereffect. Stability and Control: Theory, Methods and Applications, Vol. 20. Taylor & Francis, London (2003). MR 1965019 | Zbl 1049.34090
[7] Berezansky, L., Braverman, E.: Nonoscillation and exponential stability of delay differential equations with oscillating coefficients. J. Dyn. Control Syst. 15 (2009), 63-82. DOI 10.1007/s10883-008-9058-4 | MR 2475661 | Zbl 1203.34103
[8] Györi, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, Oxford University Press, New York (1991). MR 1168471

Partner of