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Oscillation results for second order nonlinear differential equations. (English) Zbl 1062.34070
The authors are concerned with the oscillation of the second order neutral differential equation $\left( r(t)\psi(x(t))\left[ x(t)+p(t)x(\tau(t))\right] ^{\prime}\right) ^{\prime}+q(t)f\left( x\left[ \sigma(t)\right] \right) =0.\tag{*}$ Under the assumption that $$f^{\prime}(u)$$ is nondecreasing on $$(-\infty ,t^{\ast})$$ and nonincreasing on $$(t^{\ast},+\infty)$$ for some $$t^{\ast}\geq0$$ and several additional conditions, they obtain an oscillation criterion for equation (*) which generalizes Theorem 4.4.4 due to L. H. Erbe, Q. Kong and B. G. Zhang [Oscillation theory for functional differential equations. Pure and Applied Mathematics, Marcel Dekker. 190. New York: Marcel Dekker, Inc. (1994; Zbl 0821.34067)] and Theorem 1 due to P. Wang and Y. Yu [Math. J. Toyama Univ. 21, 55–66 (1998; Zbl 0983.34058)]. A number of interesting corollaries are derived for two particular cases of equation (*), namely, for the neutral differential equation $\left( r(t)\psi(x(t))\left[ x(t)+p(t)x(\tau(t))\right] ^{\prime}\right) ^{\prime}+q(t)\left| x\left[ \sigma(t)\right] \right| ^{\beta -1}x\left[ \sigma(t)\right] =0$ and the delay differential equation $\left( r(t)x^{\prime}(t)\right) ^{\prime}+q(t)f\left( x\left[ \sigma(t)\right] \right) =0.$ The paper concludes with an oscillation result, where the monotonicity restriction on $$f^{\prime}(u)$$ is relaxed.

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