Oscillation of second order nonlinear dynamic equations on time scales. (English) Zbl 1075.34028

Bohner, M.; Saker, S. H.

Rocky Mt. J. Math. 34, No. 4, 1239-1254 (2004).

The authors consider the nonlinear second order dynamic equation \[ (p(t)x^\Delta)^\Delta+q(t)(f\circ x^\sigma)=0,\tag{1} \] where \(p\) and \(q\) are positive, real-valued continuous functions, and the nonlinearity \(f:\mathbb{R}\to\mathbb{R}\) satisfies the sign condition \(xf(x)>0\) and the superlinearity condition \(f(x)>K x\) for some \(K>0\) and every \(x\neq 0\). Two cases, depending on the convergence of the integral \[ \int _1^\infty\frac 1{p(t)}\Delta t\tag{2} \] are discussed separately. New sufficient conditions involving the integral over the coefficients of equation (1) which guarantee that all solutions are oscillatory (in the case when (2) is divergent) or either oscillatory or convergent to zero (in the case of convergence of the integral (2)) are derived. The sharpness of these criteria is shown on the example of the Euler dynamic equation. The authors’ main tool is the Riccati transformation.

Reviewer: Robert Mařík (Brno)

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A12	Discrete version of topics in analysis
39A10	Additive difference equations

Keywords:

dynamic equation; Riccati transformation; Riccati equation; oscillation; Kamenev-type criteria; Euler dynamic equation; measure chain; time scales

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