In this study, I introduce some new double sequence spaces , , , and as the domain of four-dimensional generalized difference matrix in the spaces , , , and , respectively. I show that the double sequence spaces , and 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 and , the -duals of the spaces , , , and , where , and the γ-dual of the spaces , and . Finally, I characterize the classes of four-dimensional matrix mappings defined on the spaces , , , and of double sequences.

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The concept of a four-dimensional generalized difference matrix and its domain on some double sequence spaces was recently introduced and studied by Tuğ and Başar (AIP Conference Proceedings, vol. 1759, 2016) and Tuğ (J. Inequal. Appl. 2017(1):149, 2017). In this present paper, as a natural continuation of (J. Inequal. Appl. 2017(1):149, 2017), we introduce new almost null and almost convergent double sequence spaces and as the four-dimensional generalized difference matrix domain in the spaces and , respectively. Firstly, we prove that the spaces and of double sequences are Banach spaces under some certain conditions. Then we give an inclusion relation of these new almost convergent double sequence spaces. Moreover, we identify the α-dual, -dual and γ-dual of the space . Finally, we characterize some new matrix classes , , and we complete this work with some significant results.