Weak convergence theorems for nonexpansive mappings and monotone mappings. (English) Zbl 1055.47052

Takahashi, W.; Toyoda, M.

J. Optim. Theory Appl. 118, No. 2, 417-428 (2003).

Let \(K\) be a closed convex subset of a real Hilbert space \(H\), \(A:K\rightarrow H\) be inverse strongly monotone, and \(S:K\rightarrow K\) be nonexpansive. Assuming that the set of solutions of the variational inequality for \(A\) and the set of fixed points of \(S\) have a nonempty intersection, the authors introduce an iteration process that is shown to generate a sequence converging weakly to an element of this intersection. This is the main result of the paper, which is then applied to obtain a sequence converging to a common fixed point of a nonexpansive map and a strictly pseudocontractive map.

Reviewer: Nicolas Hadjisavvas (Hermoupolis)

MSC:

47H10	Fixed-point theorems
55M20	Fixed points and coincidences in algebraic topology
49J40	Variational inequalities
47J20	Variational and other types of inequalities involving nonlinear operators (general)

Keywords:

fixed points; nonexpansive mappings; variational inequalities; inverse strongly-monotone mappings

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