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Oscillation of delay differential equations on time scales. (English) Zbl 1034.34080

The authors consider linear delay dynamic equations

x Δ (t)+p(t)x(τ(t))=0(*)

on a time scale 𝕋, i.e., a closed subset of the reals, which is assumed to be unbounded above. Moreover, -p:𝕋 is supposed to be regressive, rd-continuous, and τ:𝕋𝕋 satisfies τ(t)<t for all t𝕋. Using a necessary condition for the existence of an eventually periodic solution, an oscillation criterion for (*) is obtained. Furthermore, the authors deduce sufficient conditions for the existence of positive solutions, and for the oscillatory behavior of a nonhomogeneous version of (*).

These criteria generalize well known results for ordinary differential and difference equations to dynamic equations on time scales.

34K11Oscillation theory of functional-differential equations
39A13Difference equations, scaling (q-differences)
39A14Partial difference equations
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory