{"title":"On a Notion of Averaged Mappings in \\(\\operatorname{CAT}(0)\\) Spaces","authors":"A. Bërdëllima","doi":"10.1134/S0016266322010038","DOIUrl":null,"url":null,"abstract":"<p> We introduce a notion of averaged mappings in the broader class of <span>\\(\\operatorname{CAT}(0)\\)</span> spaces. We call these mappings <span>\\(\\alpha\\)</span>-firmly nonexpansive and develop basic calculus rules for ones that are quasi-<span>\\(\\alpha\\)</span>-firmly nonexpansive and have a common fixed point. We show that the iterates <span>\\(x_n:=Tx_{n-1}\\)</span> of a nonexpansive mapping <span>\\(T\\)</span> converge weakly to an element in <span>\\(\\operatorname{Fix} T\\)</span> whenever <span>\\(T\\)</span> is quasi-<span>\\(\\alpha\\)</span>-firmly nonexpansive. Moreover, <span>\\(P_{\\operatorname{Fix} T}x_n\\)</span> converge strongly to this weak limit. Our theory is illustrated with two classical examples of cyclic and averaged projections. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"56 1","pages":"27 - 36"},"PeriodicalIF":0.6000,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Functional Analysis and Its Applications","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1134/S0016266322010038","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
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.
期刊介绍:
Functional Analysis and Its Applications publishes current problems of functional analysis, including representation theory, theory of abstract and functional spaces, theory of operators, spectral theory, theory of operator equations, and the theory of normed rings. The journal also covers the most important applications of functional analysis in mathematics, mechanics, and theoretical physics.