Electron. J. Differential Equations, Vol. 2017 (2017), No. 28, pp. 1-10.

Criteria and estimates for decaying oscillatory solutions for some second-order quasilinear ODEs


Oscillation criteria for the solutions of quasilinear second order ODE are revisited. In our early works [6,7], we obtained basic oscillation criteria for
 \big\{ \phi_\alpha(u'(t))\big\}' + \alpha c(t) \phi_\beta(u(t)) =0
by estimating of the diameters of the nodal sets of the solutions. The focus of this work is to estimate the decay of the oscillatory solutions. Let u be a strongly oscillatory solution, $(t_m)$ the increasing sequence of zeros of u', and $D_m$ the nodal set of u that contains $t_m$. We estimate $|u(t_m)|_\infty:=\max_{t\in D_m} |u(t)|$ and the diameter of $D_m$ as $m\to \infty$.

Submitted September 14, 2016. Published January 24, 2017.
Math Subject Classifications: 34C10, 34K15
Key Words: Oscillation criteria for ODE and Estimates of decaying solutions

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