Proceedings of the American Mathematical Society

Fenchel duality, Fitzpatrick functions and the extension of firmly nonexpansive mappings

Author(s): Heinz H. Bauschke
Journal: Proc. Amer. Math. Soc. 135 (2007), 135-139.
MSC (2000): Primary 46C05, 47H09; Secondary 52A41, 90C25
Posted: August 16, 2006
Retrieve article in: PDF

Abstract: Recently, S. Reich and S. Simons provided a novel proof of the Kirszbraun-Valentine extension theorem using Fenchel duality and Fitzpatrick functions. In the same spirit, we provide a new proof of an extension result for firmly nonexpansive mappings with an optimally localized range.

1.: H. H. Bauschke and J. M. Borwein, ``On projection algorithms for solving convex feasibility problems,'' SIAM Review, vol. 38, pp. 367-426, 1996. MR 1409591 (98f:90045)
2.: H. H. Bauschke and P. L. Combettes, ``A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces,'' Mathematics of Operations Research, vol. 26, pp. 248-264, 2001. MR 1895827 (2003f:65101)
3.: H. H. Bauschke, P. L. Combettes, and S. Reich, ``The asymptotic behavior of the composition of two resolvents,'' Nonlinear Analysis: Theory, Methods, and Applications, vol. 56, pp. 283-301, 2005. MR 2101879 (2006d:47088)
4.: J. M. Borwein, ``Maximal monotonicity via convex analysis," to appear in Journal of Convex Analysis.
5.: J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis, Springer-Verlag, 2005. MR 2144010
6.: H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, 1973. MR 0348562 (50:1060)
7.: R. E. Bruck and S. Reich, ``Nonexpansive projections and resolvents of accretive operators in Banach spaces,'' Houston Journal of Mathematics, vol. 3, pp. 459-470, 1977. MR 0470761 (57:10507)
8.: Y. Censor and S. Reich, ``Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization,'' Optimization, vol. 37, pp. 323-339, 1996. MR 1402641 (98j:47161)
9.: P. L. Combettes, ``Construction d'un point fixe commun à une famille de contractions fermes,'' Comptes Rendus des Séances de l'Académie des Sciences, Série I, Mathématique, vol. 320, pp. 1385-1390, 1995. MR 1338291 (96c:47087)
10.: P. L. Combettes, ``Solving monotone inclusions via compositions of nonexpansive averaged operators,'' Optimization, vol. 53, pp. 475-504, 2004. MR 2115266 (2005i:47088)
11.: H. Debrunner and P. Flor, ``Ein Erweiterungssatz für monotone Mengen,'' Archiv der Mathematik, vol. 15, pp. 445-447, 1964. MR 0170189 (30:428)
12.: S. Fitzpatrick, ``Representing monotone operators by convex functions," Workshop/Miniconference on Functional Analysis and Optimization (Canberra 1988), Proceedings of the Centre for Mathematical Analysis, Australian National University vol. 20, Canberra, Australia, pp. 59-65, 1988. MR 1009594 (90i:47054)
13.: K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990. MR 1074005 (92c:47070)
14.: K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, 1984. MR 0744194 (86d:58012)
15.: W. Kaczor, ``Fixed points of asymptotically regular nonexpansive mappings on nonconvex sets,'' Abstract and Applied Analysis, vol. 2003, pp. 83-91, 2003. MR 1960139 (2004a:47066)
16.: M. D. Kirszbraun, ``Über die zusammenziehende und Lipschitzsche Transformationen," Fundamenta Mathematicae, vol. 22, pp. 77-108, 1934.
17.: K. C. Kiwiel and B. $\L$ opuch, ``Surrogate projection methods for finding fixed points of firmly nonexpansive mappings,'' SIAM Journal on Optimization, vol. 7, pp. 1084-1102, 1997. MR 1479616 (98j:90049)
18.: G. J. Minty, `Monotone (nonlinear) operators in Hilbert space,'' Duke Mathematical Journal, vol. 29, pp. 341-346, 1962. MR 0169064 (29:6319)
19.: J.-J. Moreau, ``Proximité et dualité dans un espace hilbertien,'' Bulletin de la Société Mathématique de France, vol. 93, pp. 273-299, 1965. MR 0201952 (34:1829)
20.: S. Park, ``Generalized Kirszbraun-Minty type inequalities,'' in Fixed point theory and applications, pp. 197-203, Nova Science Publishers, 2002. MR 2083504 (2005d:49017)
21.: S. Reich, ``Extension problems for accretive sets in Banach spaces,'' Journal of Functional Analysis, vol. 26, pp. 378-395, 1977. MR 0477893 (57:17393)
22.: S. Reich and I. Shafrir, ``The asymptotic behavior of firmly nonexpansive mappings,'' Proceedings of the American Mathematical Society, vol. 101, pp. 245-250, 1987. MR 0902536 (88i:47030)
23.: S. Reich and S. Simons, ``Fenchel duality, Fitzpatrick functions and the Kirszbraun-Valentine extension theorem,'' Proceedings of the American Mathematical Society, vol. 133, pp. 2657-2660, 2005. MR 2146211 (2006d:46025)
24.: S. Simons and C. Zalinescu, ``Fenchel duality, Fitzpatrick functions and maximal monotonicity,'' Journal of Nonlinear and Convex Analysis, vol. 6, pp. 1-22, 2005. MR 2138099 (2005k:49102)
25.: P. Tseng, ``On the convergence of the products of firmly nonexpansive mappings,'' SIAM Journal on Optimization, vol. 2, pp. 425-434, 1992. MR 1172499 (93f:90161)
26.: F. A. Valentine, ``On the extension of a vector function so as to preserve a Lipschitz condition," Bulletin of the American Mathematical Society, vol. 49, pp. 100-108, 1943. MR 0008251 (4:269d)
27.: C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing, 2002. MR 1921556 (2003k:49003)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46C05, 47H09, 52A41, 90C25

Heinz H. Bauschke
Affiliation: Department of Mathematics, Irving K. Barber School, University of British Columbia Okanagan, Kelowna, British Columbia, Canada V1V 1V7
Email: heinz.bauschke@ubc.ca

DOI: 10.1090/S0002-9939-06-08770-3
PII: S 0002-9939(06)08770-3
Keywords: Fenchel duality, firmly nonexpansive mapping, Fitzpatrick function, Kirszbraun-Valentine theorem
Received by editor(s): July 24, 2005
Posted: August 16, 2006
Communicated by: Jonathan M. Borwein
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS