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Oscillation criteria for half-linear delay dynamic equations on time scales. (English) Zbl 1173.34037

Summary: This paper is concerned with oscillation of the second-order halflinear delay dynamic equation

(r(t)(x Δ ) γ ) Δ +p(t)x γ (τ(t))=0,

on a time scale 𝕋, where γ1 is the quotient of odd positive integers, p(t), and τ:𝕋𝕋 are positive rd-continuous functions on 𝕋, r(t) is positive and (delta) differentiable, τ(t)t, and lim t r(t)=. We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results in the special cases when 𝕋= and 𝕋= involve and improve some oscillation results for second-order differential and difference equations; and when 𝕋=h, 𝕋=q 0 and 𝕋= 2 our oscillation results are essentially new. Some examples illustrating the importance of our results are also included.

MSC:
34K11Oscillation theory of functional-differential equations
39A10Additive difference equations