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Erbe, L.; Peterson, A.; Saker, S. H.

Oscillation criteria for second-order nonlinear delay dynamic equations. (English) Zbl 1125.34046

J. Math. Anal. Appl. 333, No. 1, 505-522 (2007).
The authors consider the second-order nonlinear delay dynamic equation \[ \left(r(t)x^\Delta(t)\right)^\Delta +p(t)f(x(\tau(t))=0 \] on a time scale. By employing a generalized Riccati transformation of the form \[ w(t):= \delta(t)\left[\frac{r(t)x^\Delta(t)}{x(t)} +r(t)a(t)\right], \] they establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. The obtained results improve the well-known oscillation results for dynamic equations and include as special cases the oscillation results for differential equations. Some applications to special time scales \(R, N, q^{N_{0}}\) with \(q>1\) and four examples are also included to illustrate the main results.
Reviewer: Qiru Wang (Guangzhou)

Cited in 80 Documents

MSC:

34K11 Oscillation theory of functional-differential equations
39A10 Additive difference equations

Keywords:

Second-order nonlinear delay dynamic equation; Time scales; Oscillation; Generalized Riccati technique
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References:

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