Four-dimensional generalized difference matrix and some double sequence spaces

In this study, I introduce some new double sequence spaces [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] as the domain of four-dimensional generalized difference matrix [Formula: see text] in the spaces [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], respectively. I show that the double sequence spaces [Formula: see text], [Formula: see text] and [Formula: see text] are the Banach spaces under some certain conditions. I give some inclusion relations with some topological properties. Moreover, I determine the α-dual of the spaces [Formula: see text] and [Formula: see text], the [Formula: see text]-duals of the spaces [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text], and the γ-dual of the spaces [Formula: see text], [Formula: see text] and [Formula: see text]. Finally, I characterize the classes of four-dimensional matrix mappings defined on the spaces [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of double sequences.