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 Quasicontraction Mappings in Modular Spaces without Δ2 -Condition | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008 Article ID 916187 6 pages doi 2008 916187 Research Article Quasicontraction Mappings in Modular Spaces without A2-Condition M. A. Khamsi Department of Mathematical Science The University of Texas at El Paso El Paso TX 79968 USA Correspondence should be addressed to M. A. Khamsi mohamed@ Received 22 May 2008 Accepted 1 July 2008 Recommended by William A. Kirk As a generalization to Banach contraction principle Ciric introduced the concept of quasicontraction mappings. In this paper we investigate these kinds of mappings in modular function spaces without the A2-condition. In particular we prove the existence of fixed points and discuss their uniqueness. Copyright 2008 M. A. Khamsi. 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 Let M d be a metric space. A mapping T M M is said to be quasicontraction if there exists k 1 such that d T x T y kmax d x y d x T x d y T y d x T y d y T x for any x y M. In 1974 Ciric 1 introduced these mappings and proved an existence fixed point result very similar to the original Banach contraction fixed point theorem. Recently the authors 2 tried to extend their ideas to modular spaces. Though their conclusions are very similar to Ciric s results proved in metric spaces they were unable to escape the A2-condition. They also asked whether Ciric s results may be proved in the modular setting without the very restrictive A2-condition. In this work we give a proof in the affirmative. Recall that modular spaces were initiated by Nakano in 1950 3 in connection with the theory of order spaces and redefined and generalized by Luxemburg 4-13 and Orlicz in 1959. These spaces were developed following the successful theory of Orlicz spaces which replaces the particular
