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

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.


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