Geodesic
On a notion of averaged mappings in $\operatorname{CAT}(0)$ spaces
Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 1, pp. 37-50.

Voir la notice de l'article provenant de la source Math-Net.Ru

We introduce a notion of averaged mappings in the broader class of $\operatorname{CAT}(0)$ spaces. We call these mappings $\alpha$-firmly nonexpansive and develop basic calculus rules for ones that are quasi-$\alpha$-firmly nonexpansive and have a common fixed point. We show that the iterates $x_n:=Tx_{n-1}$ of a nonexpansive mapping $T$ converge weakly to an element in $\operatorname{Fix} T$ whenever $T$ is quasi-$\alpha$-firmly nonexpansive. Moreover, $P_{\operatorname{Fix} T}x_n$ converge strongly to this weak limit. Our theory is illustrated with two classical examples of cyclic and averaged projections.
Keywords: averaged mapping, firmly nonexpansive mapping, $\operatorname{CAT}(0)$ space. 
@article{FAA_2022_56_1_a2,
     author = {A. B\"erd\"ellima},
     title = {On a notion of averaged mappings in $\operatorname{CAT}(0)$ spaces},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {37--50},
     publisher = {mathdoc},
     volume = {56},
     number = {1},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2022_56_1_a2/}
}
TY  - JOUR
AU  - A. Bërdëllima
TI  - On a notion of averaged mappings in $\operatorname{CAT}(0)$ spaces
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2022
SP  - 37
EP  - 50
VL  - 56
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2022_56_1_a2/
LA  - ru
ID  - FAA_2022_56_1_a2
ER  - 
%0 Journal Article
%A A. Bërdëllima
%T On a notion of averaged mappings in $\operatorname{CAT}(0)$ spaces
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2022
%P 37-50
%V 56
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2022_56_1_a2/
%G ru
%F FAA_2022_56_1_a2
A. Bërdëllima. On a notion of averaged mappings in $\operatorname{CAT}(0)$ spaces. Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 1, pp. 37-50. http://geodesic.mathdoc.fr/item/FAA_2022_56_1_a2/

[1] A. D. Aleksandrov, “Odna teorema o treugolnikakh v metricheskom prostranstve i nekotorye ee prilozheniya”, Sbornik statei. Posvyaschaetsya akademiku Ivanu Matveevichu Vinogradovu k ego 60-letiyu, Trudy MIAN SSSR, 38, Izd-vo AN SSSR, M., 1951, 5–23

[2] D. Ariza-Ruiz, G. López-Acedo, A. Nicolae, “The asymptotic behavior of the composition of firmly nonexpansive mappings”, J. Optim. Theory Appl., 167:2 (2015), 409–429 | DOI | MR | Zbl

[3] W. Ballmann, Lectures on Spaces of Nonpositive Curvature, Birkhäuser-Verlag, Basel, 1995 | MR | Zbl

[4] H. H. Bauschke, J. M. Borwein, “On projection algorithms for solving convex feasibility problems”, SIAM Review, 38:3 (1996), 367–426 | DOI | MR | Zbl

[5] H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, Berlin, 2011 ; 2nd ed., 2017 | MR | Zbl

[6] H. H. Bauschke, Projection algorithms and monotone operators, PhD, Simon Fraser University, Buranby, 1996 | MR

[7] A. Bërdëllima, Investigations in Hadamard Spaces, Georg-August-Universität Göttingen, Göttingen, 2020

[8] I. D. Berg, I. G. Nikolaev, “Quasilinearization and curvature of Aleksandrov spaces”, Geom. Dedicata, 133 (2008), 195–218 | DOI | MR | Zbl

[9] M. Baćak, Convex Analysis and Optimization in Hadamard Spaces, De Gruyter Series in Nonlinear Analysis and Applications, 22, De Gruyter, Berlin, 2014 | MR | Zbl

[10] P. L. Combettes, I. Yamada, “Compositions and convex combinations of averaged nonexpansive operators”, J. Math. Anal. Appl., 425:1 (2015), 55–70 | DOI | MR | Zbl

[11] M. Gromov, “Hyperbolic groups”, Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, Springer-Verlag, New York, 1987, 75–263 | MR

[12] M. Gromov, Metric Structures for Riemannian and non-Riemannian Spaces, Progress in Mathematics, 152, Birkhäuser, Boston, 1999 | MR | Zbl

[13] R. E. Bruck Jr., “Nonexpansive projections on subsets of Banach spaces”, Pacific J. Math., 47 (1973), 341–355 | DOI | MR | Zbl

[14] M. A. Krasnoselskii, “Dva zamechaniya o metode posledovatelnykh priblizhenii”, UMN, 10:1(63) (1955), 123–127 | MR | Zbl

[15] T.-C. Lim, “Remarks on some fixed point theorems”, Proc. Amer. Math. Soc., 60 (1976), 179–182 | DOI | MR

[16] D. R. Luke, N. H. Thao, M. K. Tam, “Quantitative convergence analysis of iterated expansive, set-valued mappings”, Math. Oper. Res., 43:4 (2018), 1143–1176 | DOI | MR | Zbl

[17] A. Lytchak, A. Petrunin, Cyclic projections in Hadamard spaces, arXiv: 2104.09026

[18] M. R. Bridson, A. Haefliger, Metric Spaces of Nonpositive Curvature, A Series of Comprehensive Studies in Mathematics, 319, Birkhäuser Boston Inc., Boston, 1999 | MR

[19] W. R. Mann, “Mean value methods in iteration”, Proc. Amer. Math. Soc., 4 (1953), 506–510 | DOI | MR | Zbl