Abstract
We obtain new sufficient conditions for the oscillation of all solutions of first-order delay dynamic equations on arbitrary time scales, hence combining and extending results for corresponding differential and difference equations. Examples, some of which coincide with well-known results on particular time scales, are provided to illustrate the applicability of our results.
Highlights
Oscillation theory on Z and R has drawn extensive attention in recent years
Most of the results on Z have corresponding results on R and vice versa because there is a very close relation between Z and R. This relation has been revealed by Hilger in 1, which unifies discrete and continuous analysis by a new theory called time scale theory
A corresponding result of 1.4 for 1.3 has been given in 6, Corollary 1, which coincides in the discrete case with our main result as lim inf t→∞
Summary
Most of the results on Z have corresponding results on R and vice versa because there is a very close relation between Z and R This relation has been revealed by Hilger in 1 , which unifies discrete and continuous analysis by a new theory called time scale theory. A corresponding result of 1.4 for 1.3 has been given in 6, Corollary 1 , which coincides in the discrete case with our main result as lim inf t→∞. We consider the first-order delay dynamic equation xΔ tptxτt 0, 1.9 where t ∈ T, T is a time scale i.e., any nonempty closed subset of R with sup T ∞, p ∈ Crd T, R , the delay function τ : T → T satisfies limt → ∞τ t ∞ and τ t ≤ t for all t ∈ T.
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