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Necessary and sufficient conditions for oscillatory behaviour of solutions of a forced nonlinear neutral equation of first order with positive and negative coefficients. (English) Zbl 1078.34528
Summary: Necessary and sufficient conditions are obtained such that every nonoscillatory solution of $\bigl (y(t)-p(t)y(t-\tau )\bigr )' +Q(t)G \bigl (y(t-\sigma )\bigr ) -R(t)G \bigl (y(t-\alpha )\bigr )=f(t)$ tends to zero or to $$\infty$$ as $$t \to \infty$$, where $$p, f \in C \bigl ([0,\infty ),\mathbb R\bigr )$$, $$Q, R \in C\bigl ([0,\infty ),[0,\infty )\bigr )$$, $$G \in C(\mathbb R,\mathbb R)$$, $$\tau , \sigma , \alpha \geq 0$$. $$p(t)$$ is considered in various ranges and the nonlinear function $$G$$ could be linear, sublinear, or superlinear. The results also hold when $$f(t) \equiv 0$$. This paper improves and generalizes some recent results. [See P. Das and N. Misra, J. Math. Anal. Appl. 204, 78–87 (1997; Zbl 0874.34058), N. Parhi and S. Chand, Math. Slovaca 50, 81–94 (2000; Zbl 0959.34051), N. Parhi and R. N. Rath, Bull. Inst. Math., Acad. Sin. 28, 59–70 (2000; Zbl 0961.34059), and N. Parhi and R. N. Rath, J. Math. Anal. Appl. 256, 525–541 (2001; Zbl 0982.34057).]

##### MSC:
 34K11 Oscillation theory of functional-differential equations 34K40 Neutral functional-differential equations
##### Keywords:
oscillation; non-oscillation; neutral equation
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##### References:
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