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Zbl 1125.34046
Erbe, L.; Peterson, A.; Saker, S.H.
Oscillation criteria for second-order nonlinear delay dynamic equations.
(English)
[J] J. Math. Anal. Appl. 333, No. 1, 505-522 (2007). ISSN 0022-247X

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.
[Qiru Wang (Guangzhou)]
MSC 2000:
*34K11 Oscillation theory of functional-differential equations
39A10 Difference equations

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

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