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Zbl 1055.47052
Takahashi, W.; Toyoda, M.
Weak convergence theorems for nonexpansive mappings and monotone mappings.
(English)
[J] J. Optim. Theory Appl. 118, No. 2, 417-428 (2003). ISSN 0022-3239; ISSN 1573-2878/e

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.
MSC 2000:
*47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
55M20 Fixed points and coincidences (algebraic topology)
49J40 Variational methods including variational inequalities
47J20 Inequalities involving nonlinear operators

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

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