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The resolvent average : an expansive analysis of firmly nonexpansive mappings and maximally monotone operators

University of British Columbia

Monotone operators and firmly nonexpansive mappings are essential to modern optimization and fixed point theory. Minty first discovered the link between these two classes of operators; every resolvent of a monotone operator is firmly nonexpansive and every firmly nonexpansive mapping is a resolvent of a monotone operator. This thesis provides an in-depth study of the relationship between firmly nonexpansive mappings and maximally monotone operators. First, corresponding properties between maximally monotone operators and their resolvents are collected. Then a new method of averaging monotone operators is presented, called the resolvent average, which is based on the convex combination of the resolvents of monotone operators. Several new results are given concerning the asymptotic regularity of compositions and convex combinations of firmly nonexpansive mappings. Finally, the resolvent average is studied with respect to which properties the average inherits from the averaged operators.

Text

2014-12-18

Attribution-NonCommercial-NoDerivs 2.5 Canada

http://hdl.handle.net/2429/51593

Mathematics

University of British Columbia

UBCO

http://creativecommons.org/licenses/by-nc-nd/2.5/ca/