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Necessary and sufficient conditions for the nonoscillation of a first order neutral equation with several delays. (English) Zbl 1057.34081
Summary: Necessary and sufficient conditions are obtained so that every solution of the neutral delay differential equation (NDDE) $\biggl (y(t)-\sum \limits _{j = 1}^k p_j y({t - \tau _j})\biggr )' + Q(t)G\bigl (y({t-\sigma })\bigr ) = f(t)$ is oscillatory or tends to zero as $$t\to \infty$$ for different ranges of $$\sum_{j = 1}^k p_j$$. This paper improves and generalizes two recent works [P. Das and N. Misra, J. Math. Anal. Appl. 205, 78–87 (1997; Zbl 0874.34058)] and [N. Parhi and R. N. Rath, Bull. Inst. Math. Acad. Sin. 28, 59–70 (2000; Zbl 0961.34059)]. The results of this paper hold for linear, sublinear and superlinear equations. Also, they are valid for homogeneous equations. The results can be extended to NDDE with variable coefficients without assumption of any further condition on the coefficient functions.

##### MSC:
 34K11 Oscillation theory of functional-differential equations 34K40 Neutral functional-differential equations 34K25 Asymptotic theory of functional-differential equations
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##### References:
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