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Báo cáo hóa học: " Research Article A Continuation Method for Weakly Contractive Mappings under the Interior Condition"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Continuation Method for Weakly Contractive Mappings under the Interior Condition | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 809315 8 pages doi 10.1155 2009 809315 Research Article A Continuation Method for Weakly Contractive Mappings under the Interior Condition David Ariza-Ruiz and Antonio Jimenez-Melado Departamento de Analisis Matemdtico Facultad de Ciencias Universidad de Mdlaga 29071 Malaga Spain Correspondence should be addressed to Antonio Jimenez-Melado melado@uma.es Received 29 July 2009 Accepted 8 October 2009 Recommended by Marlene Frigon Recently Frigon proved that for weakly contractive maps the property of having a fixed point is invariant by a certain class of homotopies obtaining as a consequence a Leray-Schauder alternative for this class of maps in a Banach space. We prove here that the Leray-Schauder condition in the aforementioned result can be replaced by a modification of it the interior condition. We also show that our arguments work for a certain class of generalized contractions thus complementing a result of Agarwal and O Regan. Copyright 2009 D. Ariza-Ruiz and A. Jimenez-Melado. 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. 1. Introduction Suppose that X is a Banach space that U c X is an open bounded subset of X containing the origin and that f U X is a mapping. It is well known that if f satisfies the Leray-Schauder condition defined as f x Ỷ kx for x e dU x 1 L-S and f is a strict set-contraction or more generally condensing then f has a fixed point in U see e.g. 1 or 2 . The first continuation method in the setting of a complete metric space for contractive maps comes from the hands of Granas 3 in 1994 who gave a homotopy result for contractive maps for more information on this topic see e.g. 4 5 or 6 . On the other hand it has been recently shown in 7 that for condensing mappings the condition L-S can be