OSCILLATION OF SECOND-ORDER DELAY AND NEUTRAL DELAY DYNAMIC EQUATIONS ON TIME SCALES

In this paper, we consider the second-order linear delay dynamic equation x ∆∆(t) + q(t)x(τ (t)) = 0, on a time scale T. We will study the properties of the solutions and establish some sufficient conditions for oscillations. In the special case when T = R and τ (t) = t, our results include some well-known results in the literature for differential equations. When, T = Z, T = hZ, for h > 0 and T = Tn = {tn : n ∈ N0} where tn} is the set of the harmonic numbers defined by t0 = 0, tn = Pn k=1 1 k for n ∈ N0 our results are essentially new. The results will be applied on second-order neutral delay dynamic equations in time scales to obtain some sufficient conditions for oscillations. An example is considered to illustrate the main results.