Fixed Point Theory and Applications 
Volume 2006 (2006), Article ID 79075, 8 pages
doi:10.1155/FPTA/2006/79075

Diametrically contractive maps and fixed points

Marco Baronti,1 Emanuele Casini,2 and Pier Luigi Papini3

 1Sezione Metodi e Modelli Matematici Dipartimento di Ingegneria della Produzione, Termoenergetica e Metodi e Modelli Matematici, Università degli Studi di Genova, Piazzale Kennedy, Genova 16129, Italy
2Dipartimento di Fisica e Matematica, Università dell'Insubria, Via Valleggio 11, Como 22100, Italy
3Dipartimento di Matematica, Università degli Studi di Bologna, Piazza Porta S. Donato 5, Bologna 40126, Italy
Received 13 January 2006; Revised 3 May 2006; Accepted 3 May 2006

Abstract

Contractive maps have nice properties concerning fixed points; a big amount of literature has been devoted to fixed points of nonexpansive maps. The class of shrinking (or strictly contractive) maps is slightly less popular: few specific results on them (not applicable to all nonexpansive maps) appear in the literature and some interesting problems remain open. As an attempt to fill this gap, a condition half way between shrinking and contractive maps has been studied recently. Here we continue the study of the latter notion, solving some open problems concerning these maps.