  In this paper, we define weighted relative $p(.)$-capacity and discuss
properties of capacity in the space $W_{\vartheta }^{1,p(.)}(\mathbb{R}^{n}).$
Also, we investigate some properties of weighted variable Sobolev capacity. It
is shown that there is a relation between these two capacities. Moreover, we
introduce a thinness in sense to this new defined relative capacity and prove
an equivalence statement for this thinness.
