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Oscillation of third order nonlinear delay dynamic equations on time scales. (English) Zbl 1175.34086

Summary: This paper gives oscillation criteria for the third order nonlinear delay dynamic equation

a(t)r(t)x Δ (t) Δ γ Δ +f(t,x(τ(t)))=0

on a time scale 𝕋 where γ1 is the quotient of odd positive integers, a and r are positive rd-continuous functions on 𝕋, and the so-called delay function τ:𝕋𝕋 satisfies τ(t)t for t𝕋 and lim t τ(t)= and fC(𝕋×,). Our results are new for third order delay dynamic equations and extend many known results for oscillation of third order dynamic equation. These results in the special cases when 𝕋= and 𝕋= involve and improve some oscillation results for third order delay differential and difference equations; when 𝕋=h, 𝕋=q 0 and 𝕋= 2 our oscillation results are essentially new. Some examples are given to illustrate the main results.

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