Regularization of Fixed-Point Problems and Staircase Iteration

Abstract

Let C be a nonempty closed convex subset of the Hilbert space H and P be a nonexpansive mapping from C into C. The Tikhonov regularization method is extended to the fixed point problem for P. This method generates a family of strongly contractive mappings P r from H into H by composition of P with the projector onto C and with the resolvent of a given maximal and strongly monotone operator R on H with positive parameter r. If the fixed point set 5 of P is nonempty (for instance C bounded), then, as r tends to zero, u r converges to u ⋆ in S the unique solution to the variational inequality defined by R and the closed convex subset S. Moreover, the iteration method suitably combined, by a staircase technique, with approximation of P by a sequence of nonexpansive mappings P n and with regularization generates a sequence that converges strongly to u ⋆. Applications to some variational problems are consideredKeywordsApproximationfixed pointiterationmonotonenonexpansiveregularizationstaircase iterationvariational inequalitywell-posed