Geodesic
The Asymptotic Behaviour of Convex Combinations of Firmly Nonexpansive Mappings
Journal of convex analysis, Tome 26 (2019) no. 3, pp. 911-924
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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.
Classification : 46N10, 47J25, 41A65, 03F10
Mots-clés : CAT(0) spaces, firmly nonexpansive mappings, convex optimization, averaged projections, asymptotic regularity, proof mining 
@article{JCA_2019_26_3_JCA_2019_26_3_a12,
     author = {A. Sipos},
     title = {The {Asymptotic} {Behaviour} of {Convex} {Combinations} of {Firmly} {Nonexpansive} {Mappings}},
     journal = {Journal of convex analysis},
     pages = {911--924},
     year = {2019},
     volume = {26},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JCA_2019_26_3_JCA_2019_26_3_a12/}
}
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A. Sipos. The Asymptotic Behaviour of Convex Combinations of Firmly Nonexpansive Mappings. Journal of convex analysis, Tome 26 (2019) no. 3, pp. 911-924. http://geodesic.mathdoc.fr/item/JCA_2019_26_3_JCA_2019_26_3_a12/