A necessary and sufficient condition for the oscillation of higher-order neutral equations. (English) Zbl 0684.34068

Wang, Zhicheng

Tôhoku Math. J., II. Ser. 41, No. 4, 575-588 (1989).

Summary: Consider the higher-order neutral delay differential equation \[ (*)\quad d^ n/dt^ n(x(t)+\sum^{L}_{i=1}p_ ix(t-\tau_ i)- \sum^{M}_{j=1}r_ jx(t-\rho_ j))+\sum^{N}_{k=1}q_ kx(t-u_ k)=0, \] where the coefficients and the delays are nonnegative constants with \(n\geq 1\) odd. Then a necessary and sufficient condition for the oscillation of (*) is that the characteristic equation \[ F(\lambda):=\lambda^ n+\lambda^ n\sum^{L}_{i=1}p_ ie^{- \lambda \tau_ i}-\lambda^ n\sum^{M}_{j=1}r_ je^{-\lambda \rho_ j}+\sum^{N}_{k=1}q_ ke^{-\lambda u_ k}=0 \] has no real roots.

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

Keywords:

higher-order neutral delay differential equation; characteristic equation

References:

[1]	R. BELLMAN AND K. L. COOKE, Differential-Difference Equations, Academic Press, New York, 1963. · Zbl 0105.06402
[2]	R. D. DRIVER, Existence and continuous dependence of solutions of a neutral functional-differentia equations, Arch. Ration. Mech. Anal. 19 (1965), 149-166. · Zbl 0148.05703
[3]	J. K. HALE, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977 · Zbl 0352.34001
[4]	M. K. GRAMMATIKOPOULOS, G. LADAS AND A. MEIMARIDOU, Oscillation and asymptotic behavior o higher order neutral equations with variable coefficients, · Zbl 0672.34066
[5]	M. K. GRAMMATIKOPOULOS, Y. G. SFICAS AND I. P. STAVROULAKIS, Necessary and suffiicent condition for oscillations of neutral equations with several coefficients, · Zbl 0669.34069
[6]	G. LADAS AND Y. G. SFICAS, Oscillation of higher-order neutral equations, J. Austral. Math. Soc.Se B, 27(1986), 502-511. · Zbl 0566.34055
[7]	G. LADAS AND I. P. STAVROULAKIS, On delay differential inequalities of higher order, Canad. Math Bull. 25 (1982), 348-354. · Zbl 0443.34064
[8]	G. LADAS, Y. G. SFICAS AND I. P. STAVROULAKIS, Necessary and sufficient conditions for oscillations, Amer. Math. Monthly 90 (1983), 637-640. 588Z-C. WANG JSTOR: · Zbl 0526.34054
[9]	Y. G. SFICAS, AND I. P STAVROULAKIS, Necessary and sufficient conditions for oscillations of neutra differential equations, J Math. Anal. Appl. 123 (1987), 494^507 · Zbl 0631.34074
[10]	M. SLEMROD AND E. F. INFANTE, Asymptotic stability ceiteria for linear systems of difference-differentia equations of neutral type and their discrete analogues, J. Math. Anal Appl 38 (1972), 399-415 · Zbl 0202.10301