Oscillation in neutral equations with an ``integrally small'' coefficient
The first order neutral delay differential equation \[ {d \over dt} \bigl[ x(t)-P(t) x(t-\tau) \bigr] + Q(t)x(t-\sigma) = 0 \quad t \geq t_ 0, \tag{1} \] is considered where \[ \tau \in (0,\infty),\;\sigma \in R^ + \quad \text{ and } \quad P,Q \in C \bigl( [t_ 0,\infty),R^ +\bigr). \tag{2} \] The aim is to establish sufficient conditions for the oscillation of all solutions of equation (1), which do not require that \[ \int_{t_ 0}^ \infty Q(s)ds = \infty, \tag{3} \] (a hypothesis assumed by many authors). Some assumptions that are used are: \((Y_ 1)\) There exists a \(t^*\geq t_ 0\) such that \[ P(t^*+ i \tau) \leq 1 \quad \text{ for }\quad i=1,2,\dots,k, \] \((Y_ 2)\) There exists a nonnegative integer \(k\) such that the functions \[ H_ m(t) = \int_ t^ \infty sQ(s) H_{m-1} ds, \quad m=1,2,\dots,k \] exist and \[ \int_{t_ 0}^ \infty sQ(s) H_ k ds = \infty. \] \[ P(t) \geq 1 \quad \text{ for } \quad t \geq t_ 0. \tag{4} \] The main results are formulated as follows: Theorem 1. Assume that (2) and \((Y_ 2)\) hold with \(P(t) \equiv 1\). Then every solution of equation (1) oscillates. Theorem 2. Assume that (2), \((Y_ 1)\) and \((Y_ 2)\) hold. Suppose also that \[ P(t-\delta) Q(t) \geq Q (t - \tau) \quad \text{ for } \quad t \geq t_ 0 + \rho. \] Then every solution of equation (1) oscillates. Theorem 3. Assume that (2), \((Y_ 2)\) and (4) hold. Suppose also that \[ P(t-\delta) Q(t) \leq Q(t-\tau) \quad \text{ for } \quad t \geq t_ 0 + \tau + \delta. \] Then every solution of equation (1) oscillates. Theorem 4. Assume that (2), \((Y_ 1)-(Y_ 2)\) and (4) hold. Then every solution of equation (1) oscillates.
first order neutral delay differential equation
scientific article; zbMATH DE number 569405