Four-dimensional generalized difference matrix and some double sequence spaces

In this study, I introduce some new double sequence spaces B(M-u), B(C-p), B(C-bp), B(C-r) and B(L-q) as the domain of four-dimensional generalized difference matrix B(r, s, t, u) in the spaces M-u, C-p, C-bp, C-r and L-q, respectively. I show that the double sequence spaces B(M-u), B(C-bp) and B(C-r) are the Banach spaces under some certain conditions. I give some inclusion relations with some topological properties. Moreover, I determine the alpha-dual of the spaces B(M-u) and B(C-bp), the beta(V)-duals of the spaces B(M-u), B(C-p), B(C-bp), B(C-r) and B(L-q), where V is an element of{p, bp, r}, and the gamma-dual of the spaces B(M-u), B(C-bp) and B(L-q). Finally, I characterize the classes of four-dimensional matrix mappings defined on the spaces B(M-u), B(C-p), B(C-bp), B(C-r) and B(L-q) of double sequences.