Oscillation of third order nonlinear delay dynamic equations on time scales

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Abstract

It is the purpose of this paper to give oscillation criteria for the third order nonlinear delay dynamic equation (a(t){[r(t)xΔ(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 andr are positive rd-continuous functions on T, and the so-called delay function τ:TT satisfies τ(t)t for tT and limtτ(t)= and fC(T×R,R). 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=R and T=N involve and improve some oscillation results for third order delay differential and difference equations; when T=hN, T=qN0 and T=N2 our oscillation results are essentially new. Some examples are given to illustrate the main results.

Keywords

Oscillation
Delay nonlinear dynamic equations
Time scales

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