  Based on the rapid development of dyadic analysis and the theory of variable
weighted function spaces over the spaces of homogeneous type $(X,d,\mu)$ in
recent years, we systematically consider the quantitative variable weighted
characterizations for fractional maximal operators. On the one hand, a new
class of variable multiple weight $A_{\vec{p}(\cdot),q(\cdot)}(X)$ is
established, which enables us to prove the strong and weak type variable
multiple weighted estimates for multilinear fractional maximal operators
${{{\mathscr M}_{\eta }}}$. More precisely, \[ {\left[ {\vec \omega }
\right]_{{A_{\vec p( \cdot ),q( \cdot )}}(X)}} \lesssim {\left\|
\mathscr{M}_\eta \right\|_{\prod\limits_{i = 1}^m {{L^{p_i( \cdot )}}({X,\omega
_i})} \to {L^{q( \cdot )}}(X,\omega )({WL^{q( \cdot )}}(X,\omega ))}} \le
{C_{\vec \omega ,\eta ,m,\mu ,X,\vec p( \cdot )}}. \]
  On the other hand, on account of the classical Sawyer's condition
$S_{p,q}(\mathbb{R}^n)$, a new variable testing condition
$C_{{p}(\cdot),q(\cdot)}(X)$ also appears in here, which allows us to obtain
quantitative two-weighted estimates for fractional maximal operators
${{{M}_{\eta }}}$. To be exact, \begin{align*}
  \|M_{\eta}\|_{L^{p(\cdot)}(X,\omega)\rightarrow L^{q(\cdot)}(X,v)} \lesssim
\sum_{\theta=\frac{1}{{{p_{\rm{ - }}}}},\frac{1}{{{p_{\rm{ + }}}}}}
\left([\omega ]_{C_{p( \cdot ),q( \cdot )}^1(X)} [\omega, v]_{C_{p(\cdot),
q(\cdot)}^2(X)}\right)^{\theta}, \end{align*} The implicit constants mentioned
above are independent on the weights.
