Oscillation criteria for second-order nonlinear delay dynamic equations
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.
Second-order nonlinear delay dynamic equation
scientific article; zbMATH DE number 5167530