Abstract
For a delay differential equation
a generalized Riccati inequality is constructed for nonoscillation and exponential stability of the differential equation.
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
© 2017 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Outer measure on effect algebras
- Hamiltonian ordered algebras and congruence extension
- Automorphism groups with some finiteness conditions
- Only finitely many Tribonacci Diophantine triples exist
- Abundant semigroups with a *-normal idempotent
- Stability results for fractional differential equations with state-dependent delay and not instantaneous impulses
- Monotonicity results for delta fractional differences revisited
- M-Cantorvals of Ferens type
- Comparison of ψ-porous topologies
- On certain generalized matrix methods of convergence in (ℓ)-groups
- Some applications of first-order differential subordinations
- Hankel determinant for a class of analytic functions involving conical domains defined by subordination
- Nonoscillation and exponential stability of the second order delay differential equation with damping
- Positive solutions of perturbed nonlinear hammerstein integral equation
- Ricci solitons on 3-dimensional cosymplectic manifolds
- Probabilistic uniformization and probabilistic metrization of probabilistic convergence groups
- The super socle of the ring of continuous functions
- On the internal approach to differential equations 2. The controllability structure
- Finiteness of the discrete spectrum in a three-body system with point interaction
- On a subclass of Bazilevic functions