Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 562130, 9 pages doi:10.1155/2008/562130 Research Article On Coincidence and Fixed-Point Theorems in Symmetric Spaces Seong-Hoon Cho, 1 Gwang-Yeon Lee, 1 and Jong-Sook Bae 2 1 Department of Mathematics, Hanseo University, Chungnam 356-706, South Korea 2 Department of Mathematics, Moyngji University, Youngin 449-800, South Korea Correspondence should be addressed to Seong-Hoon Cho, shcho@hanseo.ac.kr Received 28 August 2007; Revised 4 February 2008; Accepted 5 March 2008 Recommended by Lech Gorniewicz We give an axiom C.C in symmetric spaces and investigate the relationships between C.C and axioms W3, W4, and H.E. We give some results on coinsidence and fixed-point theorems in symmetric spaces, and also, we give some examples for the results of Imdad et al. 2006. Copyright q 2008 Seong-Hoon Cho et al. 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 In 1, the author introduced the notion of compatible mappings in metric spaces and proved some fixed-point theorems. This concept of compatible mappings was frequently used to show the existence of common fixed points. However, the study of the existence of common fixed points for noncompatible mappings is, also, very interesting. In 2, the author initially proved some common fixed-point theorems for noncompatible mappings. In 3, the authors gave a notion E-A which generalizes the concept of noncompatible mappings in metric spaces, and they proved some common fixed-point theorems for noncompatible mappings under strict contractive conditions. In 4, the authors proved some common fixed-point theorems for strict contractive noncompatible mappings in metric spaces. Recently, in 5 the authors extended the results of 3, 4 to symmetricsemimetric spaces under tight conditions. In 6, the author gave a common fixed-point theorem for noncompatible self-mappings in a symmetric spaces under a contractive condition of integral type. In this paper, we give some common fixed-point theorems in symmetricsemimetric spaces and give counterexamples for the results of Imdad et al. 5. In order to obtain common fixed-point theorems in symmetric spaces, some axioms are needed. In 5, the authors assumed axiom W3, and in 6 the author assumed axioms W3, W4, and H.E; see Section 2 for definitions.