On The Bounded Oscillation of Certain Second-order Nonlinear Neutral Delay Dynamic Equations with Oscillating Coefficients

Received: November 24, 2010 Accepted: December 10, 2010 doi:10.5539/jmr.v3n2p193 This work was supported by the Science and Technology Program of Hunan Province of P. R. China (Grant No. 2010FJ6021). Abstract We investigate the bounded oscillation of the second-order nonlinear neutral delay dynamic equation with oscillating coefficients ( r(t) ∣∣∣∣[x(t) + p(t)x(τ(t))]Δ∣∣∣∣α−1[x(t) + p(t)x(τ(t))]Δ)Δ + q(t)|x(t)|β−1x(t) = 0 on an arbitrary time scale T, where p is an oscillating function defined on T and α, β > 0 are constants, and obtain several sufficient conditions for the oscillation of all bounded solutions of the equation when β > α, β = α and β < α, respectively. Our results extend and complement some known results where p(t) ≡ 0 and α, β are quotients of odd positive integers.