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Zbl 1116.34333
Wang, Peiguang; Wang, Min
Oscillation of a class of higher order neutral differential equations.
(English)
[J] Arch. Math., Brno 40, No. 2, 201-208 (2004). ISSN 0044-8753; ISSN 1212-5059/e

The authors consider a higher order nonlinear neutral differential equation of the form $$\big (x(t)-p(t)x(t-\tau )\big )^{(n)}+q(t)f\big (x(g_1(t)),\dots ,x(g_m(t))\big )=0, \quad t\geq t_0, \tag {1}$$ where $n\geq 2$ is even, $\tau >0$ is a constant; $p\in C([t_0,+\infty [\,;\bbfR)$, $0\leq p(t)\leq 1$; $q\in C([t_0,+\infty [\,;\bbfR_+)$ is not identically zero on any ray $[t_1,+\infty [\,$, $t_1>t_0$; $g_i\in C([t_0,+\infty [\,;\bbfR)$, and $\lim \limits _{t\to +\infty } g_i(t)=+\infty$; $f\in C(\bbfR^m;\bbfR)$ is nondecreasing in each of the variables. \par In Theorem 1, there are established conditions under which every proper solution of (1) is oscillatory. This result is reformulated as Theorem 2 for the special case, where $$f\big (x(g_1(t)),\dots ,x(g_m(t))\big )=\sum _{i=1}^m f_i\big (x(g_i(t))\big ),$$ $f_i\in C(\bbfR,\bbfR)$, $xf_i(x)>0$ for $x\not =0$ $(i=1,\dots ,m)$. The result of Theorem 1 is illustrated by an example.
[Robert Hakl (Brno)]
MSC 2000:
*34K11 Oscillation theory of functional-differential equations
34K40 Neutral equations

Keywords: neutral equation; oscillation

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