Báo cáo hóa học: " Research Article Fixed Point Theory for Contractive Mappings Satisfying Φ-Maps in G-Metric Spaces"

Research Article Fixed Point Theory for Contractive Mappings Satisfying Φ-Maps in G-Metric Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 181650 9 pages doi 2010 181650 Research Article Fixed Point Theory for Contractive Mappings Satisfying O-Maps in G-Metric Spaces W. Shatanawi Department of Mathematics Hashemite University . Box 150459 Zarqa 13115 Jordan Correspondence should be addressed to W. Shatanawi swasfi@ Received 23 March 2010 Revised 13 May 2010 Accepted 1 June 2010 Academic Editor Brailey Sims Copyright 2010 W. Shatanawi. 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. We prove some fixed point results for self-mapping T X X in a complete G-metric space X under some contractive conditions related to a nondecreasing map ị 0 x 0 x with limn .y_i t 0 for all t e 0 x . Also we prove the uniqueness of such fixed point aswell as studying the G-continuity of such fixed point. 1. Introduction The fixed point theorems in metric spaces are playing a major role to construct methods in mathematics to solve problems in applied mathematics and sciences. So the attraction of metric spaces to a large numbers of mathematicians is understandable. Some generalizations of the notion of a metric space have been proposed by some authors. In 2006 Mustafa in collaboration with Sims introduced a new notion of generalized metric space called G-metric space 1 . In fact Mustafa et al. studied many fixed point results for a self-mapping in G-metric space under certain conditions see 1-5 . In the present work we study some fixed point results for self-mapping in a complete G-metric space X under some contractive conditions related to a nondecreasing map ị 0 x 0 x with limn f t 0 for all t e 0 x . 2. Basic Concepts In this section we present the necessary definitions and theorems in G-metric spaces. Definition see 1 . Let X be a nonempty set and .