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On oscillation and asymptotic behaviour of a neutral differential equation of first order with positive and negative coefficients. (English) Zbl 1118.34054

Summary: Sufficient conditions are obtained such that every solution of \[ (y(t)- p(t)y(t-\tau))'+ Q(t)G(y(t-\sigma))-U(t)G(y(t-\alpha)) = f(t) \] tends to zero or to \(\pm \infty\) as \(t\) tends to \(\infty\), where \(\tau ,\sigma ,\alpha\) are positive real numbers, \(p,f\in C([0,\infty),\mathbb{R}),Q,U\in C([0,\infty),[0,\infty))\), and \(G\in C(\mathbb{R},\mathbb{R})\), \(G\) is nondecreasing with \(xG(x)>0\) for \( x\neq 0\). The two primary assumptions in this paper are \(\int_{t_0}^{\infty}Q(t)=\infty\) and \(\int_{t_0}^{\infty}U(t)<\infty\). The results hold when \(G\) is linear, super linear,or sublinear and also hold when \(f(t) \equiv 0\). This paper generalizes and improves some of the recent results.

MSC:

34K11 Oscillation theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
34K40 Neutral functional-differential equations
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