In this study, I introduce some new double sequence spaces B ( M u ) $B(\mathcal {M}_{u})$ , B ( C p ) $B(\mathcal{C}_{p})$ , B ( C b p ) $B(\mathcal{C}_{bp})$ , B ( C r ) $B(\mathcal {C}_{r})$ and B ( L q ) $B(\mathcal{L}_{q})$ as the domain of four-dimensional generalized difference matrix B ( r , s , t , u ) $B(r,s,t,u)$ in the spaces M u $\mathcal {M}_{u}$ , C p $\mathcal{C}_{p}$ , C b p $\mathcal{C}_{bp}$ , C r $\mathcal{C}_{r}$ and L q $\mathcal{L}_{q}$ , respectively. I show that the double sequence spaces B ( M u ) $B(\mathcal{M}_{u})$ , B ( C b p ) $B(\mathcal{C}_{bp})$ and B ( C r ) $B(\mathcal{C}_{r})$ 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 B ( M u ) $B(M_{u})$ and B ( C b p ) $B(\mathcal{C}_{bp})$ , the β ( ϑ ) $\beta(\vartheta)$ -duals of the spaces B ( M u ) $B(M_{u})$ , B ( C p ) $B(\mathcal{C}_{p})$ , B ( C b p ) $B(\mathcal{C}_{bp})$ , B ( C r ) $B(\mathcal{C}_{r})$ and B ( L q ) $B(\mathcal{L}_{q})$ , where ϑ ∈ { p , b p , r } $\vartheta\in\{p,bp,r\}$ , and the γ-dual of the spaces B ( M u ) $B(\mathcal{M}_{u})$ , B ( C b p ) $B(\mathcal{C}_{bp})$ and B ( L q ) $B(\mathcal{L}_{q})$ . Finally, I characterize the classes of four-dimensional matrix mappings defined on the spaces B ( M u ) $B(\mathcal{M}_{u})$ , B ( C p ) $B(\mathcal{C}_{p})$ , B ( C b p ) $B(\mathcal{C}_{bp})$ , B ( C r ) $B(\mathcal{C}_{r})$ and B ( L q ) $B(\mathcal{L}_{q})$ of double sequences.

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 $B(\mathcal{C}_{f})$ and $B(\mathcal{C}_{f_{0}})$ as the four-dimensional generalized difference matrix $B(r,s,t,u)$ domain in the spaces $\mathcal{C}_{f}$ and $\mathcal{C}_{f_{0}}$ , respectively. Firstly, we prove that the spaces $B(\mathcal{C}_{f})$ and $B(\mathcal{C}_{f_{0}})$ 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, $\beta(bp)$ -dual and γ-dual of the space $B(\mathcal{C}_{f})$ . Finally, we characterize some new matrix classes $(B(\mathcal{M}_{u}):\mathcal{C}_{f})$ , $(\mathcal{M}_{u}:B(\mathcal {C}_{f}))$ , and we complete this work with some significant results.

In this article, domain of the third order generalized difference operator ${\mathrm{\Delta }}_i^3$ in Hahn sequence space h is introduced. Some topological properties and inclusion relations are shown. Additionally, Schauder basis of the new Hahn sequence space $h({\mathrm{\Delta }}_i^3)$ is calculated and $\alpha -$, $\beta -$ and $\gamma -$dual of the space $h({\mathrm{\Delta }}_i^3)$ are computed. In the last section, some new matrix classes $(h;bv)$, $(h;b{v}_p)$, $(h;b{v}_{\mathrm{\infty }})$, $(h;bs)$, $(h;cs)$, $(bv;h)$, $(b{v}_0;h)$, $(bs;h)$, $(c{s}_0;h)$, $(cs;h)$ are characterized. We conclude the paper with characterization of the classes $(h({\mathrm{\Delta }}_i^3);\mu )$ and $(\mu ;h({\mathrm{\Delta }}_i^3))$ where $\mu =\{c,c_0,{\ell }_1,{\ell }_p,{\ell }_{\mathrm{\infty }},h,bs,cs,c{s}_0,bv,b{v}_0\}$.

Let E represent any of the spaces ${\mathcal{M}}_u$ , ${\mathcal{C}}_{\vartheta }$ $(\vartheta =\{b,bp,r\left\}\right)$ , and ${\mathcal{L}}_q$ $(0<q<\infty )$ of bounded, $\vartheta$ -convergent, and q-absolutely summable double sequences, respectively, and $\tilde{E}$ be the domain of the four-dimensional (4D) infinite sequential band matrix $B(\tilde{r},\tilde{s},\tilde{t},\tilde{u})$ in the double sequence space E, where $\tilde{r}={\left(r_m\right)}_{m=0}^{\infty },\tilde{s}={\left(s_m\right)}_{m=0}^{\infty },\tilde{t}={\left(t_n\right)}_{n=0}^{\infty }$ , and $\tilde{u}={\left(u_n\right)}_{n=0}^{\infty }$ are given sequences of real numbers in the set $c\setminus c_0$ . In this paper, we investigate the double sequence spaces $\tilde{E}$ . First, we determine some topological properties and prove several inclusion relations under some strict conditions. Then, we examine $\alpha$ -, $\beta \left(\vartheta \right)$ -, and $\gamma$ -duals of $\tilde{E}$ . Finally, we characterize some new classes of 4D matrix mappings related to our new double sequence spaces and conclude the paper with some significant consequences.