Oscillation criteria for second-order nonlinear dynamic equations on time scales. (English) Zbl 1050.34042

Erbe, Lynn; Peterson, Allan; Saker, S. H.

J. Lond. Math. Soc., II. Ser. 67, No. 3, 701-714 (2003).

The authors consider the nonlinear dynamic equation \[ (p(t) x^\Delta(t))^\Delta+ q(t) (f\circ x^\sigma)= 0\tag{1} \] on time scales, where \(p\), \(q\) are positive, real-valued, right-dense, continuous functions, and \(f: \mathbb{R}\to\mathbb{R}\) is continuous and satisfies \(xf(x)> 0\) and \(| f(x)|\geq K| x|\) for \(x\neq 0\), for some \(K>0\). The authors consider the two cases \(\int^\infty_{t_0}{\Delta t\over p(t)}=\infty\), \(\int^\infty_{t_0}{\Delta t\over p(t)}<\infty\). They present oscillation criteria for (1) by using generalized Riccati transformation techniques and generalized exponential functions.

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A11	Stability of difference equations (MSC2000)

Keywords:

Riccati transformation; oscillation criteria; nonlinear dynamic equation