 
Summary: NABLA DYNAMIC EQUATIONS ON TIME SCALES
DOUGLAS ANDERSON, JOHN BULLOCK1
AND LYNN ERBE1
, ALLAN PETERSON1
, HOAINAM
TRAN1
1. preliminaries about time scales
The following definitions can be found in Bohner and Peterson [4] and Agarwal and
Bohner [2]. A time scale T is defined to be any closed subset of R. Then the forward and
backwards jump operators , : T T
(t) = inf{s T : s > t} and (t) = sup{s T : s < t}
(supplemented by inf := sup T and sup := inf T) are well defined. A point t T is
called leftdense if t > inf T and (t) = t, leftscattered if (t) < t, rightdense if t < sup T
and (t) = t, rightscattered if (t) > t. If T has a rightscattered minimum m, define
T := T  {m}; otherwise, set T = T. The backwards graininess : T R+
0 is defined
by
(t) = t  (t).
For f : T R and t T, define the nabla derivative [3] of f at t, denoted f (t),
to be the number (provided it exists) with the property that given any > 0, there is a
