×

zbMATH — the first resource for mathematics

Oscillation and asymptotic behavior of second order neutral differential equations. (English) Zbl 0647.34071
The authors consider the following second order differential equation \[ (1)\quad \frac{d^ 2}{dt^ 2}[y(t)+P(t)y(t-\tau)]+Q(t)y(t- \sigma)=0,\quad t\geq t_ 0,\quad \tau,\sigma \geq 0 \] with \(P,Q\in C([t_ 0,\infty],{\mathbb{R}})\), where the highest derivate of the unknown function appears both with delays \(\tau\) and \(\sigma\) in some terms and without delays in some others. Such equation is called neutral delay differential equation. In this paper the authors investigate the asymptotic behavior of the nonoscillatory solutions of (1) and find sufficient conditions for the oscillation of all solutions; of all bounded solutions; all unbounded solutions.
Reviewer: A.Laforgia

MSC:
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bellman, R.; Cooke, K. L., Differential-difference equations (1963), New York: Academic Press, New York · Zbl 0105.06402
[2] Brayton, R. K.; Willoughby, R. A., On the numerical integration of a symmetric system of difference-differential equations of neutral type, J. Math. Anal. Appl., 18, 182-189 (1967) · Zbl 0155.47302
[3] Driver, R. D., Existence and continuous dependence of solutions of a neutral functionaldifferential equation, Archs. Ration. Mech. Analysis, 19, 149-166 (1965) · Zbl 0148.05703
[4] Driver, R. D., A mixed neutral system, Nonlinear Analysis-TMA, 8, 155-158 (1984) · Zbl 0553.34042
[5] Grammatikopoulos, M. K.; Grove, E. A.; Ladas, G., Oscillations of first order neutral delay differential equations, J. Math. Anal. Appl., 120, 510-520 (1986) · Zbl 0566.34056
[6] Grammatikopoulos, M. K.; Grove, E. A.; Ladas, G., Oscillation and asymptotic behavior of neutral differential equations with deviating arguments, Applicable Analysis, 22, 1-19 (1986) · Zbl 0566.34057
[7] M. K.Grammatikopoulos - G.Ladas - Y. G.Sficas,Oscillation and asymptotic behavior of neutral equations with variable coefficients (to appear). · Zbl 0617.34067
[8] Hale, J., Theory of functional differential equations (1977), New York: Springer-Verlag, New York
[9] Ladas, G.; Sficas, Y. G., Oscillations of neutral delay differential equations, Canad. Math. Bull., 29, 438-445 (1986) · Zbl 0566.34054
[10] Ladas, G.; Stavroulakis, I. P., On delay differential inequalities of higher order, Canad. Math. Bull., 25, 348-354 (1982) · Zbl 0443.34064
[11] Slemrod, M.; Infante, E. F., Asymptotic stability criteria for linear systems of difference, differential equations of neutral type and their discrete analogues, J. Math. Anal. Appl., 38, 399-415 (1972) · Zbl 0202.10301
[12] W.Snow,Existence, uniqueness, and stability for nonlinear differential-difference equations in the neutral case, N.Y.U. Courant Inst. Math. Sci. Rep. IMM-NYU 328 (February 1965).
[13] Zahariev, A. I.; Bainov, D. D., Oscillating properties of the solutions of a class of neutral type functional differential equations, Bull. Austral. Math. Soc., 22, 365-372 (1980) · Zbl 0465.34042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.