## Oscillation of nonlinear dynamic equations on time scales.(English)Zbl 1045.39012

The author investigates oscillatory properties of the nonlinear second order dynamic equation on a time scale $(p(t)x^\Delta(t))^\Delta+q(t)f(x(\sigma(t)))=0, \tag{*}$ where $$p,q,f$$ are rd-continuous positive functions, $$\sigma$$ is the time scale right jump operator, and the nonlinearity $$f$$ satisfies $$xf(x)>0$$ and $$f(x)>Kx$$ for $$x\neq 0$$. Under the last assumption, the nonlinear equation (*) is a Sturmian majorant, in a certain sense, of the linear equation $(p(t)x^\Delta(t))^\Delta+Kq(t)x(\sigma(t))=0. \tag{**}$ The standard Riccati technique is applied to (**) and this approach then implies oscillation of the nonlinear equation (*). A typical result is the following statement.
Theorem. Suppose that $$\int^\infty p^{-1}(t)\Delta t=\infty$$. Further, suppose that there exist a positive function $$\delta$$ and a positive constant $$M$$ such that $\limsup_{t\to \infty}\int^t \left[ K\delta(\sigma (s)) q(s)- \frac{(\delta^\Delta (s))^2p(s)(s+M\mu(s))}{4s\delta(\sigma(s))}\right] \Delta s=\infty,$ where $$\mu(t)=\sigma(t)-t$$ is the gaininess of the time scale under consideration, then every solution of (*) oscillates.
Applying this criterion with $$\delta(t)\equiv t$$ to the Sturm-Liouville differential equation $$x''+q(t)x=0$$ considered as a perturpation of the Euler differential equation $$y''+\frac{1}{4t^2}y=0$$, i.e. the former equation is written in the form $x''+\frac{1}{4t^2}x +\left[q(t)-\frac{1}{4t^2}\right]x=0, \tag{***}$ claims that this equation is oscillatory provided $\limsup_{t\to \infty} \int^t \left[q(s)-\frac{1}{4s^2}\right]s\,ds=\infty.$ The last condition is interesting from the following point of view. The transformation $$x=\sqrt t y$$ transforms (***) into the equation $(ty')'+ \left[q(t)-\frac{1}{4t^2}\right]ty=0$ and there exist counterexamples to the classical Leightnon-Wintner criterion ($$\int^\infty p^{-1}(t)\,dt=\infty=\int^\infty q(t)\,dt$$ $$\implies$$ $$(p(t)x')'+q(t)x=0$$ is ocillatory) that the condition $$\int^\infty q(t)\,dt=\infty$$ cannot be replaced by a weaker condition $$\limsup_{t\to \infty} \int^t q(s)\,ds=\infty$$.

### MSC:

 39A12 Discrete version of topics in analysis 93C70 Time-scale analysis and singular perturbations in control/observation systems 34B24 Sturm-Liouville theory 39A11 Stability of difference equations (MSC2000)
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### References:

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