Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 917175, 10 pages doi:10.1155/2009/917175 Research Article Fixed Point Theorems for Contractive Mappings in Complete G-Metric Spaces Zead Mustafa 1 and Brailey Sims 2 1 Department of Mathematics, The Hashemite University, P.O. Box 330127, Zarqa 13115, Jordan 2 School of Mathematical and Physical Sciences, The University of Newcastle, NSW 2308, Australia Correspondence should be addressed to Zead Mustafa, zmagablh@hu.edu.jo Received 31 December 2008; Accepted 7 April 2009 Recommended by H´ el` ene Frankowska We prove some fixed point results for mappings satisfying various contractive conditions on Complete G-metric Spaces. Also the Uniqueness of such fixed point are proved, as well as we showed these mappings are G-continuous on such fixed points. Copyright q 2009 Z. Mustafa and B. Sims. 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 Metric spaces are playing an increasing role in mathematics and the applied sciences. Over the past two decades the development of fixed point theory in metric spaces has attracted considerable attention due to numerous applications in areas such as variational and linear inequalities, optimization, and approximation theory. Different generalizations of the notion of a metric space have been proposed by Gahler 1, 2 and by Dhage 3, 4. However, HA et al. 5 have pointed out that the results obtained by Gahler for his 2 metrics are independent, rather than generalizations, of the corresponding results in metric spaces, while in 6the current authors have pointed out that Dhage’s notion of a D-metric space is fundamentally flawed and most of the results claimed by Dhage and others are invalid. In 2003 we introduced a more appropriate and robust notion of a generalized metric space as follows. Definition 1.1 see 7. Let X be a nonempty set, and let G : X × X × X → R be a function satisfying the following axioms: G 1 Gx, y, z 0 if x y z, G 2 0 <Gx, x, y, forall x, y ∈ X, with x / y,