NABLA DYNAMIC EQUATIONS ON TIME SCALES DOUGLAS ANDERSON, JOHN BULLOCK1 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 left-dense if t > inf T and (t) = t, left-scattered if (t) < t, right-dense if t < sup T and (t) = t, right-scattered if (t) > t. If T has a right-scattered 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 Collections: Mathematics