  In this paper, we mainly discuss the $(p,q)$-frame in shift-invariant
subspace \begin{equation*}
V_{p,q}(\Phi)=\left\{\sum\limits_{i=1}^{r}\sum\limits_{j_{1}\in
\mathbf{Z}}\sum\limits_{j_{2}\in
\mathbf{Z}^{d}}d_{i}(j_{1},j_{2})\phi_{i}(\cdot-j_{1},\cdot-j_{2}):\Big(d_{i}(j_{1},j_{2})\Big)_{(j_{1},j_{2})\in
\mathbf{Z}\times\mathbf{Z}^{d}}\in
\ell^{p,q}(\mathbf{Z}\times\mathbf{Z}^d)\right\} \end{equation*} of mixed
Lebesgue space $L^{p,q}(\mathbf{R}\times \mathbf{R}^{d})$. Some equivalent
conditions for
$\{\phi_{i}(\cdot-j_{1},\cdot-j_{2}):(j_{1},j_{2})\in\mathbf{Z}\times\mathbf{Z}^d,1\leq
i\leq r\}$ to constitute a $(p,q)$-frame of $V_{p,q}(\Phi)$ are given.
Moreover, the result shows that $V_{p,q}(\Phi)$ is closed under these
equivalent conditions of $(p,q)$-frame for the family
$\{\phi_{i}(\cdot-j_{1},\cdot-j_{2}):(j_{1},j_{2})\in\mathbf{Z}\times\mathbf{Z}^d,1\leq
i\leq r\}$, although the general result is not correct.
