Generalisations of the Contraction Mapping Theorem: A study on the Generalised Banach Contraction Conjecture - Softcover

Melsa, Christopher

Softcover

ISBN 10: 3843384797 ISBN 13: 9783843384797

Publisher: LAP LAMBERT Academic Publishing, 2011

Synopsis

Various (putative) generalisations of the Contraction Mapping Theorem in metric space theory in the literature are considered. In particular, generalisations due to Kannan and Suzuki are studied, as is a result of Merryfield, Rothschild and Stein with interesting links to Ramsey theory. The main subsequent business is a question of Stein about when there is a finite family of maps of the metric space, at least one of which contracts any particular pair of points: does some composition of members of the family have to have a unique fixed point? The answer is shown to be 'yes' in the special case of two commuting continuous maps of the metric space, but to be 'no' in general, by the counterexample of Austin based on combinatorics on words. Thus in complete generality the Generalised Banach Contraction Conjecture is proven false. Relevant background on metric spaces is also developed.

About this title

About the Author

Obtained First Class Honours Degree in Mathematics at Essex University. A particular interest was discovered in the field of Metric Spaces and my passion for research gave way to two years studying towards a Masters on the topic of the Contraction Mapping Theorem. The aim is to work towards gaining a PhD in the wider field of topology.

Edition details

• Publisher: LAP LAMBERT Academic Publishing
• Publication date: 2011
• Language: English
• ISBN 10: 3843384797
• ISBN 13: 9783843384797
• Binding: Paperback
• Number of pages: 120