This paper reviews some earlier results of the authors given in the papers by the second and third author [J. Difference Equ. Appl. 7, No. 1, 21–50 (2001; Zbl 0972.39004)], and the first and the third author [Dyn. Syst. Appl. 12, No. 1–2, 23–43 (2003; Zbl 1053.39029)]. They address linear dynamic equations on time scales , which include ordinary differential equations (), ordinary difference equations () or -difference equations (, ) as special cases. More detailed, the following results are discussed:
(1) If the linear dynamic equation
in possesses an ordinary dichotomy, then there exists a homeomorphism between the bounded solutions of and of the perturbed equation
provided the perturbation matrix is of class w.r.t. the -integral.
(2) In order to apply this result to the special case of a diagonal matrix , sufficient criteria for the existence of an ordinary dichotomy are given in terms of the functions . They are illustrated for the classical examples and .
(3) The authors discuss possible strategies to transform equation into
with a diagonal matrix using an invertible linear transformation . This yields a representation for of a fundamental matrix for , with the Hardy-Littlewood symbol and a diagonal matrix . Such an approach is illustrated for the special case with a diagonalizable constant matrix . Then it is possible to establish a representation of as above.
Finally, results on the asymptotic behavior of the scalar exponential function on time scales from Bodine and Lutz [loc cit] conclude the paper.