A note on the existence of nonoscillatory solutions of neutral differential equations
The author considers the neutral differential equation \[ {d^n \over dt^n} \bigl[ y(t) - p(t) y(t - \tau) \bigr] + f \biggl( t,y \bigl( \sigma (t) \bigr) \biggr) = 0 \tag{1} \] where \(n \geq 1\) and the following conditions are assumed: a) \(p \in C [a, \infty)\), \(p(t) \geq 0\) for \(t \geq a > 0\) and \(\tau\) is a positive constant, b) \(f \in C ([a, \infty) \times \mathbb{R})\), and \(|f(t,u) |\leq F(t, |u |)\), \((t,u) \in [a, \infty) \times \mathbb{R}\), for some continuous function \(F(t,u)\) on \([a, \infty) \times [0, \infty)\) which is nondecreasing in \(u\) for each fixed \(t \geq a\), c) \(\sigma \in C[a, \infty)\), \(\lim_{t \to \infty} \sigma (t) = \infty\). Under the assumption \(0 \leq p_0 \leq p(t) \leq p_1\) for \(t \geq a\), where \(p_0\) and \(p_1\) are constants, the author presents sufficient conditions for (1) to have an eventually positive solution \(y\) satisfying \[ \lim_{t \to \infty} \inf {y(t) \over u(t;k,p_0)} > 0 \quad \text{and} \quad\lim_{t \to \infty} \sup {y(t) \over u(t; k,p_1)} < \infty. \]
scientific article; zbMATH DE number 863647