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Oscillation of second-order nonlinear differential equations with nonlinear damping. (English) Zbl 1049.34040
The paper contains several oscillation criteria for the nonlinear differential equation $$(r(t)k_1(x,x^{\prime }))^{\prime }+p(t)k_2(x,x^{\prime })x^{\prime }+q(t)f(x)=0,$$ where $t\geq t_0\geq 0$, under one of the leading assumptions: (1) $k_1^2(u,v)\leq \alpha _1vk_1(u,v)$, $uvk_2(u,v)\geq \alpha _2k_1^2(u,v)$, $p(t)\geq 0$, $\alpha _1>0$, $\alpha _2\geq 0$; (2) $k_1^2(u,v)\leq \alpha _1vk_1(u,v)$, $uvk_2(u,v)=\alpha _2k_1^2(u,v)$, $\alpha _1\alpha _2p(t)+r(t)>0$, $\alpha _1$, $\alpha _2>0$; (3) $k_1^2(u,v)\leq \alpha _1vk_1(u,v)$, $k_1(u,v)=vk_2(u,v)$, $\alpha _1>0$. The Philos class of test functions $H(t,s)$ introduced here (p. 198) is governed by the formula $-\partial H(t,s)/\partial s=h(t,s)\sqrt{H(t,s)}$. The proofs of the results rely on an averaging technique of Kamenev-type for certain Riccati substitutions. In case (1), one of the results reads as: Suppose that $f(x)/x\geq K>0$ for all $x\neq 0$ and $q(t)\geq 0$. Assume also that $$\limsup_{t\rightarrow +\infty} H(t,t_0)^{-1}\int_{t_0}^t[H(t,s)\rho (s)q(s)-\frac{\alpha _1r^2(s)\rho (s)}{4K(\alpha _1\alpha _2p(s)+r(s))}Q^2(t,s)]ds=+\infty,$$ where $Q(t,s)=h(t,s)-[\rho ^{\prime }(s)/\rho (s)]\sqrt{H(t,s)}$, for a certain positive, continuously differentiable function $\rho $. Then, the equation is oscillatory. In case (2), the same conclusion is valid provided that the above condition is fulfilled. In case (3), we have: Suppose that $xf(x)\neq 0$ and $f^{\prime }(x)\geq K>0$ for all $x\neq 0$. Assume also that $$\limsup_{t\rightarrow +\infty } H(t,t_0)^{-1}\int_{t_0}^t[H(t,s)\rho (s)q(s)-(\alpha _1/4K)r(s)\rho (s)Q^2(t,s)]ds=+\infty ,$$ where $Q(t,s)=h(t,s)+[p(s)/r(s)-\rho ^{\prime }(s)/\rho (s)]\sqrt{H(t,s)}$, for a certain positive, continuously differentiable function $\rho $. Then, the equation is oscillatory. Another result deals with a nondifferentiable function $f$ in the case of $p$ with varying sign by imposing a differentiability condition on $p$ (Theorem 3.5). The paper is elegantly written and an elaborated discussion of the relevant literature accompanies the computations.

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
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References:
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