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Interval oscillation criteria for a forced second-order differential equation with nonlinear damping. (English) Zbl 1102.34022

The author studies the oscillatory behavior of a second-order nonlinear differential equation with a forcing term \[ \left[ r(t)k_{1}(x(t),x^{\prime}(t))\right] ^{\prime}+p(t)k_{2} (x(t),x^{\prime}(t))x^{\prime}(t) +q(t)f(x(t))=e(t),\qquad t\geq t_{0}, \tag{1} \] with \(p,q,e:[t_{0},\infty)\rightarrow \mathbb{R},\) \(r:[t_{0},\infty )\rightarrow(0,\infty),\) \(f:\mathbb{R}\rightarrow\mathbb{R},\) \(k_{1} ,k_{2}:\mathbb{R}^{2}\rightarrow\mathbb{R}.\) It is also assumed that (a) \(f(x)/x\geq K\left| x\right| ^{\gamma-1}\) for all \(x\neq0\) and for some \(K>0\) and \(\gamma\geq1;\) (b) \(k_{1}(u,v)\geq\alpha_{1}k_{1}^{2}(u,v),\) for some \(\alpha_{1}>0\) and for all \((u,v)\in\mathbb{R}^{2};\) (c) \(k_{2}(u,v)uv\geq\alpha_{2}k_{1}^{2}(u,v),\) for some \(\alpha_{2}\geq0\) and for all \((u,v)\in\mathbb{R}^{2}.\) New oscillation results are obtained for equation (1) and its particular case where \(k_{2}(u,v)v=k_{1}(u,v).\) Two illustrative examples are considered, the first one with the same selection of the functions \(k_{1}\) and \(k_{2}\) as in Example 5.3 from the recent paper by A. Tiryaki and A. Zafer [Nonlinear Anal., Theor. Methods Appl. 60, 49–63 (2005; Zbl 1064.34021)] and similar choice of coefficients.
Unfortunately, the author does not compare his results with those obtained for equation (1) in the cited paper as well as in the other work by the same authors [Math. Comput. Modeling 39, 197–208 (2004; Zbl 1049.34040)], where similar oscillation criteria have been obtained using both integral averaging technique and interval oscillation approach. Instead, the comparison is done with results obtained by J. S. W. Wong [J. Math. Anal. Appl. 231, 235–240 (1999; Zbl 0922.34029)] and Q. Yang [Appl. Math. Comput. 135, 49–64 (2003; Zbl 1030.34034)] for less general equations. Regretfully, one of the main assumptions that “for any \(T\geq t_{0}\) there exist \(a_{1},\) \(b_{1},\) \(a_{2},\) \(b_{2}\) such that \(T\leq a_{1}<b_{1}\leq a_{2}<b_{2}\) and \(p(t)\geq0\) and \(q(t)\geq0\) for \(t\in[ a_{1},b_{1} ]\cup[ a_{2},b_{2}]\)” excludes the possibility of application of new results to differential equations with a negative damping coefficient \(p(t)\) to which many known oscillation criteria do apply.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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References:

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