On non-empty cross-t-intersecting families

Let $A_1,A_2,\dots ,A_m$ be families of k-element subsets of a n-element set. We call them cross-t-intersecting if $|A_i\cap A_j|\ge t$ for any $A_i\in A_i$ and $A_j\in A_j$ with $i\ne j$. In this paper we will prove that, for $n\ge 2k-t+1$, if $A_1,A_2,\dots ,A_m$ are non-empty cross-t-intersecting families, then$\sum _{1\le i\le m}\left|A_i\right|\le \max \{\left(\begin{array}{c}n\\ k\end{array}\right)-\sum _{1\le i\le t-1}\left(\begin{array}{c}k\\ i\end{array}\right)\left(\begin{array}{c}n-k\\ k-i\end{array}\right)+m-1,mM(n,k,t\left)\right\},$ where $M(n,k,t)$ is the size of the maximum t-intersecting family of $\left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$. Moreover, the extremal families attaining the upper bound are characterized.