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Oscillation theorems for certain nonlinear differential equations of second order. (English) Zbl 1057.34019
The authors are concerned with the general second-order nonlinear differential equation $$[ a(t)\Psi(x(t))k(x^{\prime}(t))] ^{\prime} +q(t)f(x(t))=r(t),\tag1$$ with $q,r\in C[ [t_{0},\infty),\Bbb{R}] ,$ $a\in C[ [t_{0},\infty),\Bbb{R}^{+}] ,$ $\Psi\in C[ \Bbb{R} ,\Bbb{R}^{+}] ,$ $k\in C[ \Bbb{R},\Bbb{R}] ,$ and $f\in C^{1}[ \Bbb{R},\Bbb{R}] .$ The research follows closely the recent paper by {\it J. V. Manojlovic} [Acta Sci. Math. 65, No. 3--4, 515--527 (1999; Zbl 0957.34037)], where under certain assumptions on the coefficients it has been shown that solutions of the nonlinear differential equation $$[ p(t)g(x(t))x^{\prime}(t)] ^{\prime}+q(t)f(x(t))=r(t)\tag2$$ are either oscillatory or satisfy $$\liminf_{t\to\infty}\vert x(t)\vert =0.$$ The main goal of the paper under review, according to the authors, is to present new oscillation criteria and to show that some of Manojlovic’s results contain the superfluous condition. Results of the paper are proved using the so-called integral averaging technique similar to that developed by {\it H. J. Li} [J. Math. Anal. Appl. 194, No. 1, 217--234 (1995; Zbl 0836.34033)]. However, it has been pointed out by O. Došlý in his review on the paper by {\it J. V. Manojlovic} [Acta Sci. Math. 65, No. 3--4, 515--527 (1999; Zbl 0957.34037)] that a certain assumption in the oscillation criteria of Li is not necessary as observed by {\it Yu. V. Rogovchenko} [J. Math. Anal. Appl. 203, No. 2, 560--563 (1996; Zbl 0862.34024)]. The criteria of the reviewed paper contain a similar assumption and it seems that this assumption can be removed without affecting the results of the paper. The authors just repeat these comments on several occasions referring to the assumption $$\int_{t_{0}}^{t}a(s)h^{2}(t,s)\,ds<\infty\qquad\text{for }t\geq t_{0}.$$ It is not clear whether the study of equation (1) has been motivated by any real-world applications since all examples in the paper are artificial, while the choice of the functions which appear in all examples illustrating the main results reduces to trivial: $\Phi(s)=1,$ $R(t)=0$ (examples 1, 2 and 4) or $\Phi(s)=s,$ $R(t)=0$ (example 3). The authors could not exhibit any exact oscillatory solution to illustrative examples. All these facts make the usefulness of extension from equation (2) to equation (1) questionable. We conclude by noticing that new oscillation criteria extending Theorems 6-9 in the paper under review have been reported for a nonlinear differential equation with damping $$[ a(t)\Psi(x(t))k(x^{\prime}(t))] ^{\prime}+p(t)k(x^{\prime }(t))+q(t)f(x(t))=0$$ in the recent paper by {\it Q.-R. Wang} [Acta Math. Hung. 102, No. 1--2, 117--139 (2004; Zbl 1052.34040)].

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
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