On ๐›ผ-Firmly Nonexpansive Operators in r-Uniformly Convex Spaces

dc.contributor.authorBรซrdรซllima, Arian
dc.contributor.authorSteidl, Gabriele
dc.date.accessioned2021-12-09T19:29:50Z
dc.date.available2021-12-09T19:29:50Z
dc.date.issued2021-08-02
dc.description.abstractWe introduce the class of ๐›ผ-firmly nonexpansive and quasi ๐›ผ-firmly nonexpansive operators on r-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where ๐›ผ-firmly nonexpansive operators coincide with so-called ๐›ผ-averaged operators. For our more general setting, we show that ๐›ผ-averaged operators form a subset of ๐›ผ-firmly nonexpansive operators. We develop some basic calculus rules for (quasi) ๐›ผ-firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) ๐›ผ-firmly nonexpansive. Moreover, we will see that quasi ๐›ผ-firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browderโ€™s demiclosedness principle, we prove for r-uniformly convex Banach spaces that the weak cluster points of the iterates ๐‘ฅ๐‘›+1:=๐‘‡๐‘ฅ๐‘› belong to the fixed point set Fix๐‘‡ whenever the operator T is nonexpansive and quasi ๐›ผ-firmly. If additionally the space has a Frรฉchet differentiable norm or satisfies Opialโ€™s property, then these iterates converge weakly to some element in Fix๐‘‡. Further, the projections ๐‘ƒFix๐‘‡๐‘ฅ๐‘› converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in ๐ฟ๐‘, ๐‘โˆˆ(1,โˆž)โˆ–{2} spaces on probability measure spaces.en
dc.description.sponsorshipDFG, 390685689, EXC 2046: MATH+: Berlin Mathematics Research Centeren
dc.description.sponsorshipTU Berlin, Open-Access-Mittel โ€“ 2021en
dc.identifier.eissn1420-9012
dc.identifier.issn1422-6383
dc.identifier.urihttps://depositonce.tu-berlin.de/handle/11303/14015
dc.identifier.urihttp://dx.doi.org/10.14279/depositonce-12788
dc.language.isoenen
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/en
dc.subject.ddc510 Mathematikde
dc.subject.otheraveraged operatorsen
dc.subject.otherBanach spacesen
dc.subject.othercontractive projectionsen
dc.subject.otherfirmly nonexpansive operatorsen
dc.subject.otherfixed point theoryen
dc.subject.otheruniformly convex spacesen
dc.titleOn ๐›ผ-Firmly Nonexpansive Operators in r-Uniformly Convex Spacesen
dc.typeArticleen
dc.type.versionpublishedVersionen
dcterms.bibliographicCitation.articlenumber172en
dcterms.bibliographicCitation.doi10.1007/s00025-021-01481-8en
dcterms.bibliographicCitation.issue4en
dcterms.bibliographicCitation.journaltitleResults in Mathematicsen
dcterms.bibliographicCitation.originalpublishernameSpringer Natureen
dcterms.bibliographicCitation.originalpublisherplaceBerlin ; Heidelbergen
dcterms.bibliographicCitation.volume76en
tub.accessrights.dnbfreeen
tub.affiliationFak. 2 Mathematik und Naturwissenschaften::Inst. Mathematik::FG Angewandte Mathematikde
tub.affiliation.facultyFak. 2 Mathematik und Naturwissenschaften
tub.affiliation.groupFG Angewandte Mathematik
tub.affiliation.instituteInst. Mathematik
tub.publisher.universityorinstitutionTechnische Universitรคt Berlinen

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