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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).]

34K11 Oscillation theory of functional-differential equations
34K40 Neutral functional-differential equations
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