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Oscillation criteria for second order nonlinear retarded differential equations. (English) Zbl 1080.34556
The author studies the oscillatory behaviour of the nonlinear delay differential equation \[ \bigl (| u'(t)| ^{\alpha -1}u'(t)\bigr )'+p(t)f\Bigl [u\bigl (\tau (t)\bigr )\Bigr ]=0 \] under the assumptions \[ \begin{aligned} \alpha >& 0, \tag{1} \\ p\in C[t_0,\infty )&,\;p(t)>0, \tag{2} \\ \tau \in C^1[t_0,\infty ), \;\tau '(t)>0,&\, \tau (t)\leq t,\;\lim _{t\to \infty }\tau (t)=\infty , \tag{3} \\ f\in C(-\infty ,\infty ), \;f\;\text{nondecreasing on}\quad & (-\infty ,\infty ), \;f\in C^1(M), \tag{4}\\\;M=(-\infty ,0)\cup (0,\infty ), \quad & uf(u)>0, \;\text{for}\;u\neq 0. \end{aligned} \] The author generalizes the well-known condition \(\int ^\infty p(s)ds =\infty \) sufficient for the oscillation of a solution. He assumes, for \(\alpha \geq 1, \,f'(u)\) nondecreasing on \((-\infty ,0)\) and nonincreasing on \((0,\infty )\), and \(\int ^\infty p(s)| f[c\tau (s)]| ds=\infty \) for all \(c\neq 0.\) Other oscillation criteria are derived, too.

MSC:
34K11 Oscillation theory of functional-differential equations
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References:
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