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An oscillation criterion for first order delay dynamic equations. (English) Zbl 1254.34126

Summary: The authors consider a first-order delay dynamic equation of the form

x Δ (t)+p(t)x(τ(t))=0

on a time scale 𝕋, where 𝕋 is unbounded above, p:𝕋 is rd-continuous and nonnegative, τ:𝕋𝕋 is nondecreasing and satisfies τ(t)<t for all t𝕋 and lim t τ(t)=. They show that the above equation is oscillatory provided there exists a constant M(0,1) such that

lim inf t τ(t) t p(s)Δs>Mandlim sup τ τ(t) t p(s)Δs>1-(1-1-M) 2 ·

This result improves some known results.

34N05Dynamic equations on time scales or measure chains
34K11Oscillation theory of functional-differential equations