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Summary: ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR NEUTRAL
DYNAMIC EQUATIONS ON TIME SCALES
DOUGLAS R. ANDERSON
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
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