The author analyzes some algorithms for solving a monotone mixed variational inequality

$\phantom{\rule{4pt}{0ex}}\langle Tu,v-u\rangle +\varphi \left(v\right)-\varphi \left(u\right)\ge 0,\phantom{\rule{1.em}{0ex}}$ for all

$\phantom{\rule{4pt}{0ex}}v\in H$ with a Hilbert space

$H$, a nonlinear operator

$T:H\to H$ and a lower semicontinuous function

$\varphi :H\to R\cup \{+\infty \}\xb7$ The resolvent operator technique is used to study a number of splitting methods for solving the mixed variational inequalities. The methods are based on the fact that a solution

$u\in H$ of a variational inequality satisfies the relation

$u={J}_{\varphi}[u-\rho Tu]$, where

${J}_{\varphi}={(I+\rho \partial \varphi )}^{-1}$ is the resolvent operator. The resolvent operator method requires the assumption that

$T$ must be strongly monotone for the convergence. In order to overcome this difficulty the author suggests the double resolvent formula

$\phantom{\rule{4pt}{0ex}}u={J}_{\varphi}\u3014u-\rho T{J}_{\varphi}[u-\rho Tu]\u3015\xb7$ Several splitting methods for solving the iteration method connected with fixed point algorithms are analyzed.