Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1049.34040
Tiryaki, A.; Zafer, A.
Oscillation of second-order nonlinear differential equations with nonlinear damping. (English)
[J] Math. Comput. Modelling 39, No. 2-3, 197-208 (2004). ISSN 0895-7177

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: \par (1) $k_1^2(u,v)\leq \alpha _1vk_1(u,v)$, $uvk_2(u,v)\geq \alpha _2k_1^2(u,v)$, $p(t)\geq 0$, $\alpha _1>0$, $\alpha _2\geq 0$; \par (2) $k_1^2(u,v)\leq \alpha _1vk_1(u,v)$, $uvk_2(u,v)=\alpha _2k_1^2(u,v)$, $\alpha _1\alpha _2p(t)+r(t)>0$, $\alpha _1$, $\alpha _2>0$; \par (3) $k_1^2(u,v)\leq \alpha _1vk_1(u,v)$, $k_1(u,v)=vk_2(u,v)$, $\alpha _1>0$. \par 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. \par 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. \par In case (2), the same conclusion is valid provided that the above condition is fulfilled. \par 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. \par 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.
[Octavian Mustafa (Craiova)]

MSC 2000:: *34C10 Qualitative theory of oscillations of ODE: Zeros, etc.

Keywords: nonlinear damping; oscillation

Cited in: Zbl 1102.34022

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]