x: T → R: [x(t) − p(t)x(k(t))] ∆ ∈ Crd([t0,∞)T;R)

Abstract. We investigate the boundedness and asymptotic behavior of a first-order neutral delay dynamic equation on arbitrary time scales, extending some results from difference equations. 1. Neutral Delay Dynamic Equation We consider, on arbitrary time scales, the neutral delay dynamic equation (1.1) [x(t) − p(t)x(k(t))] ∆ + q(t)x(`(t)) = 0, t ∈ [t0,∞)T, where T is a time scale unbounded above, the variable delays k, ` : [t0,∞)T → T are nondecreasing with k(t), `(t) < t for all t ∈ [t0,∞)T such that limt→ ∞ k(t), `(t) =∞. The coefficient functions p, q: T → R are right-dense continuous with p bounded and q ≥ 0. To clarify some notation, take `−1(t): = sup{s: `(s) ≤ t}, `−(n+1)(t) = `−1(`−n(t)) for t ∈ [`(t0),∞)T, and `n+1(t) = `(`n(t)) for t ∈ [`−3(t0),∞)T. For p and k above, let Ω be the linear set of all functions given b