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Zbl 1059.34039
Wang, Qi-Ru; Wu, Xiao-Ming; Zhu, Si-Ming
Oscillation criteria for second-order nonlinear damped differential equations. (English)
[J] Comput. Math. Appl. 46, No. 8-9, 1253-1262 (2003). ISSN 0898-1221

The paper deals with the oscillatory nature of solutions of the second-order nonlinear functional-differential equation $$\multline \left( r(t)\psi(x(t))x^{\prime}(t)\right) ^{\prime}+p(t)k(t,x(t),x^{\prime }(t))x^{\prime}(t)\\ +f\left( t,x\left[ \tau_{01}(t)\right] ,\ldots,x\left[ \tau_{0m} (t)\right] ,x^{\prime}\left[ \tau_{11}(t)\right] ,\ldots,x^{\prime}\left[ \tau_{1m}(t)\right] \right) =0.\endmultline\tag{1}$$ Several oscillation theorems for (1) and for its particular case when $k(t,x,x^{\prime})=1$ are established. Since (1) has a very general form, it might be very hard to satisfy conditions of the theorems. Some assumptions on the functions in (1) like $k(t,x,y)\leq\left\vert y\right\vert ^{\alpha}$ for some $\alpha\geq0$ and for all $x,y$ real, $p(t)\geq0,$ or $f(t,x_{01},\ldots,x_{0m},x_{11},\ldots ,x_{1m}) \operatorname{sgn} x_{01}\geq q(t)\sum_{i=1}^{m}\alpha_{i}\left\vert x_{0i}\right\vert $ for $x_{0i}x_{01}>0,$ $i=1,2,\ldots,m,$ restrict significantly possible applications of oscillation criteria. The proofs of all results are based on the fundamental Lemma 1 whose proof is not provided, although all the routine computations which can be found in numerous related papers on oscillation are meticulously carried out. The paper concludes with a quite general example of oscillatory equation (1) with $k(t,x,x^{\prime})\equiv 1.$
[Yuri V. Rogovchenko (Famagusta)]

MSC 2000:: *34K11 Oscillation theory of functional-differential equations

Keywords: nonlinear functional-differential equation; oscillation; integral averaging

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Zentralblatt MATH Berlin [Germany]