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Nonoscillation and exponential stability of the second order delay differential equation with damping

  • Leonid Berezansky EMAIL logo , Alexander Domoshnitsky , Mikhail Gitman and Valery Stolbov
Published/Copyright: July 14, 2017
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Abstract

For a delay differential equation

x¨(t)+a(t)x˙(t)+k=1mbk(t)x(gk(t))=0,gk(t)t,

a generalized Riccati inequality is constructed for nonoscillation and exponential stability of the differential equation.

MSC 2010: Primary 34K20; 34K11

(Communicated by Michal Fečkan)

This paper appeared as a result of theoretical discussions around the project “Exploitation of a Synergetic Model for Development of Business of Innovation Type” on the Topic no. 2013/276-C “Development of a Model of Operation of Innovation Business as Dynamic Model with Memory Effect,” supported by Perm National Research Polytechnic University with financial support of Ministry of Science and Education of Russian Federation (Agreement no. 02.G25.31.0068 of 23.05.2013).

References

[1] Agarwal, R. P.— Berezansky, L.—Braverman, E.— Domoshnitsky, A.: Theory of FunctionalDifferential Equations with Applications, Springer, New York, 2012.10.1007/978-1-4614-3455-9Search in Google Scholar

[2] Agarwal, R. P.—Domoshnitsky, A.—Maghakyan, A.: On exponential stability of second order delay differential equations, Czechoslovak Math. J. 65 (2015), 1047–1068.10.1007/s10587-015-0227-9Search in Google Scholar

[3] Berezansky, L.—Domoshnitsky, A.—Gitman, M.—Stolbov, V.: Exponential stability of a second order delay differential equation without damping term, Appl. Math. Comput. 258 (2015), 483–488.10.1016/j.amc.2015.01.114Search in Google Scholar

[4] Azbelev, N. V.—Simonov, P. M.: Stability of Differential Equations with Aftereffect. Stability and Control: Theory, Methods and Applications 20, Taylor & Francis, London, 2003.10.1201/9781482264807Search in Google Scholar

[5] Berezansky, L.—Braverman, E.: Oscillation of a second-order delay differential equation with a middle term, Appl. Math. Lett. 13 (2000), 21–25.10.1016/S0893-9659(99)00160-3Search in Google Scholar

[6] Berezansky, L.—Braverman, E.—Domoshnitsky, A.: Stability of the second order delay differential equations with a damping term, Differ. Equ. Dyn. Syst. 16 (2008), 185–205.10.1007/s12591-008-0012-4Search in Google Scholar

[7] Berezansky, L.—Braverman, E.—Idels, L.: Stability tests for second order linear and nonlinear delayed models, NoDEA Nonlinear Differential Equations Appl. 22 (2015), 1523–1543.10.1007/s00030-015-0334-1Search in Google Scholar

[8] Berezansky, L.—Diblik, J.—Smarda, Z.: Positive solutions of second-order delay differential equations with a damping term, Comput. Math. Appl. 60 (2010), 1332–1342.10.1016/j.camwa.2010.06.014Search in Google Scholar

[9] Burton, T.— Zhang, B.: Boundedness, periodicity, and convergence of solutions in a retarded Lienard equation, Ann. Mat. Pura Appl. 165 (1993), 351–368.10.1007/BF01765856Search in Google Scholar

[10] Burton, T.: Fixed points, stability, and exact linearization, Nonlinear Anal. 61 (2005), 857–870.10.1016/j.na.2005.01.079Search in Google Scholar

[11] Dunford, N.—Schwartz, J. T.: Linear Operators, Part 1: General Theory, Interscience Publishers, Inc., New York, 1958.Search in Google Scholar

[12] Domoshnitsky, A.: About applicability of Chaplygin’s theorem to one component of the solution vector, Differ. Equ. (Transl. from Differ. Uravn.) 26 (1990), 1699–1705.Search in Google Scholar

[13] Domoshnitsky A.: Unboundedness of solutions and instability of second order equations with delayed argument, Differential Integral Equations 14 (2001), 559–576.10.57262/die/1356123256Search in Google Scholar

[14] Domoshnitsky, A.: Nonoscillation, maximum principles, and exponential stability of second order delay differential equations without damping term, J. Inequal. Appl. 2014:361 (2014), 26 pp.10.1186/1029-242X-2014-361Search in Google Scholar

[15] Domoshnitsky, A.–Maghakyan, A.—Berezansky, L.: W-transform for exponential stability ofsecond order delay dierential equations without damping terms, J. Inequal. Appl. 20 (2017), 10.1186/s13660-017-1296-0.Search in Google Scholar PubMed PubMed Central

[16] Erbe, L. N.—Kong, Q.—Zhang, B. G.: Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, Basel, 1995.Search in Google Scholar

[17] Györi, I.—Ladas, J.: Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991.10.1093/oso/9780198535829.001.0001Search in Google Scholar

[18] Kolmanovskii, V.—Myshkis, A. Applied Theory of Functional-Differential Equations. Mathematics and its Applications (Soviet Series) 85, Kluwer Academic Publishers Group, Dordrecht, 1992.10.1007/978-94-015-8084-7Search in Google Scholar

[19] Krasovski, N. N.: Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay Translated by J. L. Brenner, Stanford University Press, Stanford, Calif. 1963.Search in Google Scholar

[20] Ladde, G. S.—Lakshmikantham, V.—Zhang, B. G.: Oscillation Theory of Differential Equations with Deviating Argument, Marcel Dekker, New York, Basel, 1987.Search in Google Scholar

[21] Myshkis, A. D.: Linear Differential Equations with Retarded Argument, Nauka, Moscow, 1972 (in Russian).Search in Google Scholar

[22] Norkin, S. B.: Differential Equations of the Second Order with Retarded Argument, Translation of Mathematical Monographs 31, AMS, Providence, R.I., 1972.10.1090/mmono/031Search in Google Scholar

[23] Zabreiko, P. P.— Koshelev, A. I. et al: Integral Equations, Noordhoff International Publishing, Leyden, Netherlands, 1975.Search in Google Scholar

Received: 2015-4-22
Accepted: 2016-4-10
Published Online: 2017-7-14
Published in Print: 2017-8-28

© 2017 Mathematical Institute Slovak Academy of Sciences

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