The difference sequence spaces ~ (Δ),c(Δ) and c 0 (Δ) were studied by Kizmaz [Canad. Math. Bull. 24 (2) (1981) 169]. The sequence spaces a 0 r and a c r have been recently defined and examined by Aydin and Basar in [Hokkaido Math. J., in press]. The main purpose of the present paper is to introduce the spaces a r 0 (Δ) and a r c (Δ) of difference sequences. Moreover, it is proved that the spaces a r 0 (Δ) and a r c (Δ) are the BK-spaces including the spaces c 0 and c, and some inclusion relations have been given. It is also proved that the sequence space a 0 r has AD property while the space a 0 r (Δ) has not. Furthermore, the basis and the α-, β- and γ-duals of the spaces a r 0 (Δ) and a r c (Δ) have been determined. The last section of the paper has been devoted to theorems on the characterizations of the matrix classes (a r c (Δ): p ) and (a r c (Δ):c), and the characterizations of some other matrix classes have been obtained by means of a given basic lemma, where 1=<p=<~.