{"product_id":"generalisations-of-the-contraction-mapping-theorem-von-christopher-melsa","title":"Generalisations of the Contraction Mapping Theorem","description":"\u003cp\u003eVarious (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.\u003c\/p\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9783843384797\"\u003e\u003ch3\u003eA study on the Generalised Banach Contraction Conjecture\u003c\/h3\u003e\u003c\/div\u003e","brand":"Autorenwelt Shop","offers":[{"title":"Softcover - 9783843384797","offer_id":39497077456989,"sku":"9783843384797","price":49.0,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/779759e6-0e53-4e31-8278-7b3dd9c09163.jpg?v=1765262996","url":"https:\/\/shop.autorenwelt.de\/products\/generalisations-of-the-contraction-mapping-theorem-von-christopher-melsa","provider":"Autorenwelt Shop","version":"1.0","type":"link"}