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

Saker, S. H.

Appl. Math. Comput. 148, No. 1, 81-91 (2004).

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 \(\limsup_{t\to \infty} \int^t q(s)\,ds=\infty\).

Reviewer: Ondřej Došlý (Brno)

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)

Keywords:

time scale; Riccati transformation; nonlinear second order dynamic equation; Sturm-Liouville differential equation; Euler differential equation; Leightnon-Wintner criterion

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