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Oscillation results for second-order linear equations on a time scale. (English) Zbl 1021.34012

The author considers the second-order scalar dynamic equation

(r(t)x Δ (t)) Δ +q(t)x σ (t)=0(*)

on a time scale 𝕋, where sup𝕋=, r:𝕋 + is assumed to be rd-continuously differentiable, q:𝕋 is rd-continuous and the graininess μ is not identically zero on [τ,)𝕋, for large τ𝕋. Using averaging techniques, criteria for the oscillatory behavior of (*) are deduced, which extend earlier results by the author for difference equations in [J. Math. Anal. Appl., 142, No. 2, 468–487 (1989, Zbl 0686.39001)]. In particular, these criteria indicate how oscillation depends on the graininess of the time scale. Finally, two examples illustrate the obtained results.

MSC:
34B10Nonlocal and multipoint boundary value problems for ODE
39A13Difference equations, scaling (q-differences)