TY - JOUR
T1 - Asymptotic behavior of compositions of under-relaxed nonexpansive operators
AU - Baillon, Jean Bernard
AU - Combettes, Patrick L.
AU - Cominetti, Roberto
N1 - Funding Information:
2010 Mathematics Subject Classification. 47H09, 47H10, 47N10, 65K15. Key words and phrases. Cyclic projections, De Pierro’s conjecture, fixed point, nonexpansive operator, projection operator, under-relaxed cycles. The research of P. L. Combettes was supported in part by the European Union under the 7th Framework Programme “FP7-PEOPLE-2010-ITN”, grant agreement number 264735-SADCO. The research of R. Cominetti was supported by Fondecyt 1130564 and Núcleo Milenio Información y Coordinación en Redes ICM/FIC P10-024F.
Publisher Copyright:
© American Institute of Mathematical Sciences.
PY - 2014
Y1 - 2014
N2 - In general there exists no relationship between the fixed point sets of the composition and of the average of a family of nonexpansive operators in Hilbert spaces. In this paper, we establish an asymptotic principle connecting the cycles generated by under-relaxed compositions of nonexpansive operators to the fixed points of the average of these operators. In the special case when the operators are projectors onto closed convex sets, we prove a conjecture by De Pierro which has so far been established only for projections onto affine subspaces.
AB - In general there exists no relationship between the fixed point sets of the composition and of the average of a family of nonexpansive operators in Hilbert spaces. In this paper, we establish an asymptotic principle connecting the cycles generated by under-relaxed compositions of nonexpansive operators to the fixed points of the average of these operators. In the special case when the operators are projectors onto closed convex sets, we prove a conjecture by De Pierro which has so far been established only for projections onto affine subspaces.
UR - http://www.scopus.com/inward/record.url?scp=84920984357&partnerID=8YFLogxK
U2 - 10.3934/jdg.2014.1.331
DO - 10.3934/jdg.2014.1.331
M3 - Article
AN - SCOPUS:84920984357
SN - 2164-6074
VL - 1
SP - 331
EP - 346
JO - Journal of Dynamics and Games
JF - Journal of Dynamics and Games
IS - 3
ER -