Oscillations for first order neutral differential equations. (English) Zbl 0815.34062

Kozakiewicz, E.

Z. Angew. Math. Mech. 73, No. 7-8, T813-T815 (1993).

Oscillations of a first order neutral differential equation near the critical point \(1/e\) are studied. The main result of the paper is the following Theorem: Let \(x(t)\) be continuously differentiable on \(A - \tau \leq t < + \infty\), \(x'(t) - cx'(t - \tau) \leq - M(t) x(t-1)\), \(A \leq t < + \infty\), \(0 \leq c \leq 1\), \(0<\tau< + \infty\), \(M(t) \geq H(t) \geq 0\), \(H(t)\) locally summable, \(h(t) = \int_ t^{t+1} H(\sigma) d \sigma\) nonincreasing, \(h(t) > {1 \over e}\) and \(\prod^ \infty_{\nu = 0} eh(A + \nu) = + \infty\). Then \(x(t)\) is not eventually positive.

Reviewer: G.Derfel (Beer-Sheva)

MSC:

34K99	Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K40	Neutral functional-differential equations
34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

oscillations; first order neutral differential equation