VISCOSITY APPROXIMATION METHODS OF RANDOM FIXED POINT SOLUTIONS AND RANDOM VARIATIONAL INEQUALITIES IN HILBERT SPACES

In this paper, we construct random iterative processes for nonexpansive random operators and study necessary conditions for these processes. It is shown that these random iterative processes converge to random fixed points of nonexpansive random operators and solve some random variational inequalities. We also proved that an implicit random iterative process converges to the random fixed point and solves these random variational inequalities. Our results can be viewed as a refinement and improvement of the previously known results for variational inequality theory and also give generalization stochastic version of some results of Xu [23].