an:02114120
Zbl 1062.34070
D??urina, J.; Lackov??, D.
Oscillation results for second order nonlinear differential equations
EN
Stud. Univ. ??ilina, Math. Ser. 17, No. 1, 79-86 (2003).
00109939
2003
j
34K11
neutral differential equation; delay differential equation; oscillation
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 \textit{L. H. Erbe}, \textit{Q. Kong} and \textit{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 \textit{P. Wang} and \textit{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.
Svitlana P. Rogovchenko (Famagusta)
Zbl 0821.34067; Zbl 0983.34058