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A necessary and sufficient condition for the oscillation of higher-order neutral equations. (English) Zbl 0684.34068
Summary: Consider the higher-order neutral delay differential equation $$(*)\quad d\sp n/dt\sp n(x(t)+\sum\sp{L}\sb{i=1}p\sb ix(t-\tau\sb i)- \sum\sp{M}\sb{j=1}r\sb jx(t-\rho\sb j))+\sum\sp{N}\sb{k=1}q\sb kx(t-u\sb k)=0,$$ where the coefficients and the delays are nonnegative constants with $n\ge 1$ odd. Then a necessary and sufficient condition for the oscillation of (*) is that the characteristic equation $$F(\lambda):=\lambda\sp n+\lambda\sp n\sum\sp{L}\sb{i=1}p\sb ie\sp{- \lambda \tau\sb i}-\lambda\sp n\sum\sp{M}\sb{j=1}r\sb je\sp{-\lambda \rho\sb j}+\sum\sp{N}\sb{k=1}q\sb ke\sp{-\lambda u\sb k}=0$$ has no real roots.

MSC:
 34K99 Functional-differential equations 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
Full Text:
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 · doi:10.1007/BF00282279 [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 · doi:10.1016/0022-0396(88)90077-0 [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 · doi:10.1017/S0334270000005105 [7] G. LADAS AND I. P. STAVROULAKIS, On delay differential inequalities of higher order, Canad. Math Bull. 25 (1982), 348-354. · Zbl 0443.34064 · doi:10.4153/CMB-1982-049-8 [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 · doi:10.2307/2323283 · http://links.jstor.org/sici?sici=0002-9890%28198311%2990%3A9%3C637%3ANASCFO%3E2.0.CO%3B2-K&origin=euclid [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 · doi:10.1016/0022-247X(87)90326-X [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 · doi:10.1016/0022-247X(72)90098-4