## Non-oscillatory behaviour of higher order functional differential equations of neutral type.(English)Zbl 1138.34031

Summary: We obtain sufficient conditions for the neutral functional differential equation
$\displaylines{ \big[r(t) [y(t)-p(t)y(\tau (t))]'\big]^{(n-1)} + q(t) G(y(h(t))) = f(t) }$
to have a bounded and positive solution. Here, $$n\geq 2$$; $$q,\tau, h$$ are continuous functions with $$q(t) \geq 0$$; $$h(t)$$ and $$\tau(t)$$ are increasing functions which are less than $$t$$, and approach infinity as $$t \to \infty$$. In our work, $$r(t) \equiv 1$$ is admissible, and neither we assume that $$G$$ is non-decreasing, that $$xG(x) > 0$$ for $$x \neq 0$$, nor that $$G$$ is Lipschitzian. Hence, the results of this paper generalize many known results.

### MSC:

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