Abstract
Equations with advanced and mixed (both advanced and delayed) terms occur when the rate of change may depend not only on the state in the past, but also on the future state of the system. Such equations arise, for example, in mathematical economics. For equations with several advanced terms and positive coefficients, sufficient nonoscillation conditions are obtained using the methods similar to the previous chapters (excluding, however, solution representations which are unknown in this case). In addition, results on the asymptotics of nonoscillatory solutions are presented.
For the study of nonoscillation of equations with delayed and advanced terms, the main investigation method is the fixed point theory. Equations with two positive terms, two negative terms, and the terms of different signs are considered. For mixed equations with two coefficients of different signs, the asymptotics of nonoscillatory solutions can be defined, once the relation between the two coefficients is known; it will also depend on the deviation (delay or advanced) of the positive term.
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
Agarwal, R.P., Bohner, M., Li, W.-T.: Nonoscillation and Oscillation: Theory for Functional Differential Equations. Monographs and Textbooks in Pure and Applied Mathematics, vol. 267. Dekker, New York (2004)
Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht (2000)
Agarwal, R.P., Grace, S.R., O’Regan, D.: On the oscillation of certain advanced functional differential equations using comparison methods. Fasc. Math. 35, 5–22 (2005)
Berezansky, L., Braverman, E.: On nonoscillation of advanced differential equations with several terms. Abstr. Appl. Anal. 2011 (2011). Art.ID 637142, 14 pp.
Berezansky, L., Braverman, E., Pinelas, S.: On nonoscillation of mixed advanced-delay differential equations with positive and negative coefficients. Comput. Math. Appl. 58, 766–775 (2009)
Berezansky, L., Domshlak, Y.: Differential equations to several delays: Sturmian comparison method in oscillation theory, II. Electron. J. Differ. Equ. 2002(31), 1–18 (2002)
Džurina, J.: Oscillation of second-order differential equations with mixed argument. J. Math. Anal. Appl. 190, 821–828 (1995)
Erbe, L.H., Kong, Q., Zhang, B.G.: Oscillation Theory for Functional Differential Equations. Dekker, New York (1995)
Gopalsamy, K.: Nonoscillatory differential equations with retarded and advanced arguments. Q. Appl. Math. 43, 211–214 (1985)
Gopalsamy, K.: Stability and Oscillation in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht, Boston, London (1992)
Grammatikopoulos, M.K., Stavroulakis, I.P.: Oscillations of neutral differential equations. Rad. Mat. 7, 47–71 (1991)
Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, New York (1991)
Hakl, R., Lomtatidze, A., Puža, B.: On nonnegative solutions of first order scalar functional differential equations. Mem. Differ. Equ. Math. Phys. 23, 51–84 (2001)
Kordonis, I.-G.E., Philos, Ch.G.: Oscillation and nonoscillation in delay or advanced differential equations and in integrodifferential equations. Georgian Math. J. 6, 263–284 (1999)
Kusano, T.: On even-order functional-differential equations with advanced and retarded arguments. J. Differ. Equ. 45, 75–84 (1982)
Ladas, G., Stavroulakis, I.P.: Oscillations of differential equations of mixed type. J. Math. Phys. Sci. 18, 245–262 (1984)
Ladde, G.S., Lakshmikantham, V., Zhang, B.G.: Oscillation Theory of Differential Equations with Deviating Arguments. Dekker, New York (1987)
Li, X., Zhu, D., Wang, H.: Oscillation for advanced differential equations with oscillating coefficients. Int. J. Math. Math. Sci. 28, 2109–2118 (2003)
Litsyn, E., Stavroulakis, I.P.: On the oscillation of solutions of higher order Emden-Fowler state dependent advanced differential equations. Proceedings of the Third World Congress of Nonlinear Analysts, Part 6, Catania, 2000. Nonlinear Anal. 47, 3877–3883 (2001)
Meng, Q., Yan, J.: Nonautonomous differential equations of alternately retarded and advanced type. Int. J. Math. Math. Sci. 26, 597–603 (2001)
Nadareishvili, V.A.: Oscillation and nonoscillation of first order linear differential equations with deviating arguments. Differ. Equ. 25, 412–417 (1989)
Yan, J.R.: Comparison theorems for differential equations of mixed type. Ann. Differ. Equ. 7, 316–322 (1991)
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). Scalar Advanced and Mixed Differential Equations on Semiaxes. In: Nonoscillation Theory of Functional Differential Equations with Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3455-9_5
Download citation
DOI: https://doi.org/10.1007/978-1-4614-3455-9_5
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)