Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 340631, 12 pages doi: Research Article Measures of Noncircularity and Fixed Points of Contractive Multifunctions Isabel Marrero Departamento de An´ lisis Matem´ tico, Universidad de La Laguna, 38271 La Laguna (Tenerife), Spain a a Correspondence should be addressed to Isabel Marrero, imarrero@ Received 24 October 2010; Accepted 8 December 2010 Academic Editor: N. J. Huang Copyright q 2010 Isabel Marrero. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In. | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 340631 12 pages doi 2010 340631 Research Article Measures of Noncircularity and Fixed Points of Contractive Multifunctions Isabel Marrero Departamento de Analisis Matemdtico Universidad de La Laguna 38271 La Laguna Tenerife Spain Correspondence should be addressed to Isabel Marrero imarrero@ Received 24 October 2010 Accepted 8 December 2010 Academic Editor N. J. Huang Copyright 2010 Isabel Marrero. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. In analogy to the Eisenfeld-Lakshmikantham measure of nonconvexity and the Hausdorff measure of noncompactness we introduce two mutually equivalent measures of noncircularity for Banach spaces satisfying a Cantor type property and apply them to establish a fixed point theorem of Darbo type for multifunctions. Namely we prove that every multifunction with closed values defined on a closed set and contractive with respect to any one of these measures has the origin as a fixed point. 1. Introduction Let X II II be a Banach space over the field K e R C . In what follows we write BX x e X xH 1 for the closed unit ball of X. Denote by 2X the collection of all subsets of X and consider b X c e 2X 0 C bounded . For c D e b X define their nonsymmetric Hausdorff distance by h C D sup inf c - d ceC deD 2 Fixed Point Theory and Applications and their symmetric Hausdorff distance or Hausdorff-Pompeiu distance by H C D max h C D h D C . This H is a pseudometric on b X since ry J-T 7 r7 J-T c ni J Ti7 1 H C D H C D H C D H C D . where A denotes the closure of A e 2X. Around 1955 Darbo 1 ensured the existence of fixed points for so-called condensing operators on Banach spaces a result which generalizes both Schauder fixed point theorem and Banach contractive .
