NDLI: A Dichotomy Theorem on Fixed Points of Several Nonexpansive Mappings

Please wait, while we are loading the content...

A Dichotomy Theorem on Fixed Points of Several Nonexpansive Mappings

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Feder, Toms
Copyright Year	2006
Abstract	The problem of finding a fixed point of a nonexpansive mapping on a hypercube is that it has a polynomial time algorithm. In fact, it is known that one can find a 2-satisfiability characterization of the set of all fixed points in polynomial time. This implies that the problem of finding a vertex that is a common fixed point of several given nonexpansive mappings on a hypercube is that it has a polynomial time algorithm.We consider the problem of finding a vertex that is a common fixed point of several given nonexpansive mappings on a more general Cartesian product of graphs. For a single nonexpansive mapping, a known polynomial time algorithm finds a fixed point and a 2-satisfiability-like characterization of all fixed points. We introduce graphs with a farthest point property (also called apiculate graphs in [H. J. Bandelt and V. Chepoi, The Algebra of Metric Betweenness: Subdirect Representations, Retracts, and Axiomatics, manuscript]), and show that finding a common fixed point of several nonexpansive mappings on Cartesian products of such graphs involves using a polynomial time algorithm. We generalize this result to any family of graphs having a majority function.By contrast, the smallest graph (in the sense of having the fewest vertices, and the fewest edges of those having the fewest vertices) without the farthest point property is K2,3 , and finding a vertex that is a fixed point of two given nonexpansive mappings (retractions) on a Cartesian product of graphs isomorphic to K2,3 is NP-complete. More generally, we exhibit an infinite family of graphs without the farthest point property giving NP-completeness. We show that for any family of graphs not having a majority function, the existence of a common fixed point of two nonexpansive mappings on Cartesian products of such graphs is NP-complete. This proves a dichotomy for the problem based on the existence of a majority function; a similar dichotomy is obtained for the special case of nonexpansive mappings that are retractions. Finally we characterize the families of chordal graphs corresponding to both dichotomies.
Starting Page	291
Ending Page	301
Page Count	11
File Format	PDF
ISSN	08954801
DOI	10.1137/S0895480103427734
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	2
Volume Number	20
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-08-01
Access Restriction	Subscribed
Subject Keyword	fixed points retractions product graphs Graph theory apiculate graphs nonexpansive mappings
Content Type	Text
Resource Type	Article
Subject	Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

An Iteration Process for Nonexpansive Mappings with Applications to Fixed Point Theory in Product Spaces

Approximating Fixed Points of Nonexpansive Mappings

Common fixed points of G-nonexpansive mappings on Banach spaces with a graph

Fixed Points of Nonexpansive Mappings

On Fixed Points of Nonexpansive Mappings in Nonconvex Sets

On Fixed Points of Nonexpansive Set-Valued Mappings

Construction of Fixed Points for Asymptotically Nonexpansive Mappings

Approximation to Fixed Points of Generalized Nonexpansive Mappings

A Fixed Point Theorem for Asymptotically Nonexpansive Mappings

A Dichotomy Theorem on Fixed Points of Several Nonexpansive Mappings

Similar Documents

An Iteration Process for Nonexpansive Mappings with Applications to Fixed Point Theory in Product Spaces

Approximating Fixed Points of Nonexpansive Mappings

Common fixed points of G-nonexpansive mappings on Banach spaces with a graph

Fixed Points of Nonexpansive Mappings

On Fixed Points of Nonexpansive Mappings in Nonconvex Sets

On Fixed Points of Nonexpansive Set-Valued Mappings

Construction of Fixed Points for Asymptotically Nonexpansive Mappings

Approximation to Fixed Points of Generalized Nonexpansive Mappings

A Fixed Point Theorem for Asymptotically Nonexpansive Mappings

A Dichotomy Theorem on Fixed Points of Several Nonexpansive Mappings