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.


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