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.


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