Abstract
Equations of the second order are popular due to their numerous applications. For equations of the second order with several delays and not including explicitly the first derivative, Chap. 7 presents nonoscillation criteria in the terms of the fundamental function of the equation and the generalized Riccati inequality. In addition, the chapter includes comparison results, explicit nonoscillation and oscillation conditions, the result that the equation is oscillatory if and only if it has a slowly oscillating solution, and explicit nonoscillation conditions which are obtained by substituting specific solutions of the generalized Riccati inequality. Sufficient conditions for positivity of a solution of the initial value problem are also presented in Chap. 7.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Azbelev, N.V.: The zeros of the solutions of a second order linear differential equation with retarded argument. Differ. Uravn. 7, 1147–1157 (1971), 1339 (in Russian)
Berezansky, L., Braverman, E.: Some oscillation problems for a second order linear delay differential equation. J. Math. Anal. Appl. 220, 719–740 (1998)
Berezansky, L., Larionov, A.: Positivity of the Cauchy matrix of a linear functional-differential equation. Differ. Equ. 24, 1221–1230 (1988)
Brands, J.J.A.M., Oscillation theorems for second-order functional differential equations. J. Math. Anal. Appl. 63, 54–64 (1978)
Domshlak, Y.: Sturmian Comparison Method in Investigation of Behavior of Solutions for Differential-Operator Equations. Elm, Baku (1986) (in Russian)
Domshlak, Y.: Comparison theorems of Sturm type for first and second order differential equations with sign variable deviations of the argument. Ukr. Mat. Zh. 34, 158–163 (1982)
Erbe, L.H., Kong, Q., Zhang, B.G.: Oscillation Theory for Functional Differential Equations. Dekker, New York (1995)
Gil’, M.I.: On Aizerman-Myshkis problem for systems with delay. Automatica 36, 1669–1673 (2000)
Gil’, M.I.: Boundedness of solutions of nonlinear differential delay equations with positive Green functions and the Aizerman-Myshkis problem. Nonlinear Anal. 49, 1065–1078 (2002)
Gil’, M.I.: The Aizerman-Myshkis problem for functional-differential equations with causal nonlinearities. Funct. Differ. Equ. 11, 445–457 (2005)
Gil’, M.I.: Lower bounds and positivity conditions for Green’s functions to second order differential-delay equations. Electron. J. Qual. Theory Differ. Equ. 2009(65) (2009), 11 pp.
Grace, S.R., Lalli, B.S.: Oscillation theory for damped differential equations of even order with deviating argument. SIAM J. Math. Anal. 15, 308–316 (1984)
Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, New York (1991)
Győri, I., Pituk, M.: Comparison theorems and asymptotic equilibrium for delay differential and difference equations. Dyn. Syst. Appl. 5, 277–303 (1996)
Hille, E.: Nonoscillation theorems. Trans. Am. Math. Soc. 64, 234–252 (1948)
Kartsatos, A.G.: Recent results in oscillation of solutions of forced and perturbed nonlinear differential equations of even order. In: Stability of Dynamic Systems: Theory and Applications. Lecture Notes in Pure and Applied Mathematics. Springer, New York (1977)
Kiguradze, I.T., Partsvaniya, N.L., Stavroulakis, I.P.: On the oscillatory properties of higher-order advance functional-differential equations. Differ. Equ. 38, 1095–1107 (2002)
Koplatadze, R., Kvinikadze, G., Stavroulakis, I.P.: Oscillation of second order linear delay differential equations. Funct. Differ. Equ. 7, 121–147 (2000)
Ladde, G.S., Lakshmikantham, V., Zhang, B.G.: Oscillation Theory of Differential Equations with Deviating Arguments. Dekker, New York (1987)
Li, M., Wang, M., Yan, J.: On oscillation of nonlinear second order differential equation with damping term. J. Appl. Math. Comput. 13, 223–232 (2003)
Manfoud, W.E.: Comparison theorems for delay differential equations. Pac. J. Math. 83, 187–197 (1979)
Myshkis, A.D.: Linear Differential Equations with Retarded Argument. Nauka, Moscow (1972) (in Russian)
Norkin, S.B.: Differential Equations of the Second Order with Retarded Argument. Translations of Mathematical Monographs, vol. 31. Am. Math. Soc., Providence (1972)
Parhi, N.: Sufficient conditions for oscillation and nonoscillation of solutions of a class of second order functional-differential equations. Analysis 13, 19–28 (1993)
Philos, Ch.G.: Oscillatory and asymptotic behavior of the bounded solutions of differential equations with deviating arguments. Hiroshima Math. J. 8, 31–48 (1978)
Philos, Ch.G.: A comparison result in oscillation theory. J. Pure Appl. Math. 11, 1–7 (1980)
Philos, C.G., Purnaras, I.K., Sficas, Y.G.: Oscillations in higher-order neutral differential equations. Can. J. Math. 45, 132–158 (1993)
Philos, C.G., Sficas, Y.G.: Oscillatory and asymptotic behavior of second and third order retarded differential equations. Czechoslov. Math. J. 32, 169–182 (1982)
Tiryaki, A., Zafer, A.: Oscillation criteria for second order nonlinear differential equations with damping. Turk. J. Math. 24, 185–196 (2000)
Yan, J.R.: Oscillation theory for second order linear differential equations with damping. Proc. Am. Math. Soc. 98, 276–284 (1986)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Agarwal, R.P., Berezansky, L., Braverman, E., Domoshnitsky, A. (2012). Second-Order Delay Differential Equations. In: Nonoscillation Theory of Functional Differential Equations with Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3455-9_7
Download citation
DOI: https://doi.org/10.1007/978-1-4614-3455-9_7
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-3454-2
Online ISBN: 978-1-4614-3455-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)