Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 761086, 16 pages doi:10.1155/2009/761086 Research Article Assad-Kirk-Type Fixed Point Theorems for a Pair of Nonself Mappings on Cone Metric Spaces S. Jankovi´ c, 1 Z. Kadelburg, 2 S. Radenovi´ c, 3 and B. E. Rhoades 4 1 Mathematical Institute SANU, Knez Mihailova 36, 11001 Beograd, Serbia 2 Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbia 3 Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia 4 Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA Correspondence should be addressed to S. Radenovi´ c, radens@beotel.yu Received 7 February 2009; Accepted 27 April 2009 Recommended by William A. Kirk New fixed point results for a pair of non-self mappings defined on a closed subset of a metrically convex cone metric space which is not necessarily normal are obtained. By adapting Assad-Kirk’s method the existence of a unique common fixed point for a pair of non-self mappings is proved, using only the assumption that the cone interior is nonempty. Examples show that the obtained results are proper extensions of the existing ones. Copyright q 2009 S. Jankovi´ c 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 and Preliminaries Cone metric spaces were introduced by Huang and Zhang in 1, where they investigated the convergence in cone metric spaces, introduced the notion of their completeness, and proved some fixed point theorems for contractive mappings on these spaces. Recently, in 2–4, some common fixed point theorems have been proved for maps on cone metric spaces. However, in 1–3, the authors usually obtain their results for normal cones. In this paper we do not impose the normality condition for the cones. We need the following definitions and results, consistent with 1, in the sequel. Let E be a real Banach space. A subset P of E is a cone if i P is closed, nonempty and P /  {0}; ii a, b ∈ R, a,b ≥ 0, and x, y ∈ P imply ax  by ∈ P ; iii P ∩ -P  {0}. Given a cone P ⊂ E, we define the partial ordering ≤ with respect to P by x ≤ y if and only if y - x ∈ P . We write x<y to indicate that x ≤ y but x /  y, while x ≪ y stands for y - x ∈ int P the interior of P .