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Journal of Convex Analysis 26 (2019), No. 3, 911--924
Copyright Heldermann Verlag 2019

The Asymptotic Behaviour of Convex Combinations of Firmly Nonexpansive Mappings

Andrei Sipos
Dept. of Mathematics, Technische Universitšt, 64289 Darmstadt, Germany
and: Stoilow Inst. of Mathematics, Romanian Academy, 010702 Bucharest, Romania

We show that in the framework of CAT(0) spaces, any convex combination of two mappings which are firmly nonexpansive -- or which satisfy the more general property (P2) -- is asymptotically regular, conditional on its fixed point set being nonempty, and, in addition, also Δ-convergent to such a fixed point. These results are established by the construction and study of a convex combination metric on the Cartesian square of a CAT(0) space. We also derive a uniform rate of asymptotic regularity in the sense of proof mining. All these results are then interpreted in the special case of the mappings being projections onto closed, convex sets.

Keywords: CAT(0) spaces, firmly nonexpansive mappings, convex optimization, averaged projections, asymptotic regularity, proof mining.

MSC: 46N10, 47J25, 41A65, 03F10.

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