Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings

In this paper we introduce an iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The iterative process is based on two well-known methods: hybrid and extragradient. We obtain a strong convergence theorem for three sequences generated by this process. Based on this result, we also construct an iterative process for finding a common fixed point of two mappings, such that one of these mappings is nonexpansive and the other is taken from the more general class of Lipschitz pseudocontractive mappings.