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Oscillatory and nonoscillatory solutions of neutral differential equations. (English) Zbl 0971.34056

Consider the neutral differential equation \[ {d^n\over dt^n} \bigl[x(t)+ \lambda x(t-\tau) \bigr]+f\biggl( t,x\bigl(g(t) \bigr)\biggr) =0 \] with \(\lambda>0\), \(\tau>0\), \(g\in C([t_0,\infty))\), \(\lim_{t\to \infty} g(t)= \infty\), \(f\in C([t_0,\infty)\times\mathbb{R})\) and \(|f(t,u) |\leq F(t, |u|)\) where \(F\) is a continuous and nondecreasing function in \(u\in [0,\infty)\) for each fixed \(t\geq t_0\).
Here, some sufficient conditions for the existence of oscillatory (or nonoscillatory) solutions are proved. The proofs of the main results are based on the Schauder-Tikhonov fixed-point theorem.

MSC:

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