  In this paper, we introduce some new double sequence spaces
$\mathcal{M}_u(\Delta)$ and $\mathcal{C}_{\vartheta}(\Delta)$, where
$\vartheta\in\{bp,bp0,r,r0\}$ as the domains of the four-dimensional forward
difference matrix in the double sequence spaces $\mathcal{M}_u$ and
$\mathcal{C}_{\vartheta}$, respectively. Then we investigate some topological
and algebraic properties. Moreover, we determine the $\alpha-$,
$\beta(\vartheta)-$, and $\gamma-$duals of the new spaces
$\mathcal{M}_u(\Delta)$ and $\mathcal{C}_{\vartheta}(\Delta)$. Finally, we
characterize four-dimensional matrix classes $(\lambda(\Delta),\mu)$ and
$(\mu,\lambda(\Delta))$, where
$\lambda=\{\mathcal{M}_u,\mathcal{C}_{\vartheta}\}$ and
$\mu=\{\mathcal{M}_u,\mathcal{C}_{\vartheta}\}$.
