Hindawi Publishing Corporation International Journal of Analysis Volume 2013, Article ID 852727, 6 pages http://dx.doi.org/10.1155/2013/852727 Research Article On a Theorem of Khan in a Generalized Metric Space Jamshaid Ahmad, 1 Muhammad Arshad, 2 and Calogero Vetro 3 1 Department of Mathematics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad 44000, Pakistan 2 Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan 3 Dipartimento di Matematica e Informatica, Universit` a degli Studi di Palermo, Via Archirafi, 34, 90123 Palermo, Italy Correspondence should be addressed to Calogero Vetro; cvetro@math.unipa.it Received 25 August 2012; Revised 20 January 2013; Accepted 23 January 2013 Academic Editor: Ying Hu Copyright © 2013 Jamshaid Ahmad et al. his 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. Existence and uniqueness of ixed points are established for a mapping satisfying a contractive condition involving a rational expression on a generalized metric space. Several particular cases and applications as well as some illustrative examples are given. 1. Introduction and Preliminaries In the last decades, several attempts have been made in order to generalize the concept of metric space, and many impor- tant results have been reported. For instance, see quasi metric spaces [1], generalized quasi metric spaces [2], pseudometric spaces ([3], Chapter 2), approach spaces [4], -spaces [5], inframetric spaces [6], and -metric spaces [7]. Sometimes, as in [8, 9], even the very notion of generalized metric spaces (or even gms [10]) is used, but it has a diferent meaning. In 2000, Branciari [11] introduced the notion of generalized metric space where the triangle inequality of a metric space is replaced by a rectangular inequality involving four terms instead of three. He also extended the Banach’s contraction principle in such spaces. In 2008, Azam and Arshad [12] obtained suicient conditions for existence of unique ixed point of Kannan type mappings deined on generalized met- ric spaces. Recently, Samet [13] and Sarma et al. [14] showed that some propositions in [11] are not true. Moreover, in [14], a rigorous and nice proof of the Banach’s contraction principle is presented, by assuming that the generalized metric space is Hausdorf. Aterwards, many authors studied various existence theorems of ixed points in such spaces. For more details about ixed point theory in generalized metric spaces, we refer the reader to [13, 15–24]. On the other hand, in [25] Khan proved the following ixed point theorem. heorem 1. Let (,) be a complete metric space and let  be a self-mapping on  that satisfies the following contractive condition: (,)≤ (,) (,)+(,)(,) (,)+(,) (1) for all ,∈ and for some ∈[0,1). hen  has a unique fixed point in . Remark 2. In (1) if the denominator vanishes, then  =  and  =  and consequently also the numerator vanishes. Moreover, we have (,) = (,), and so the contrac- tive condition is not well deined. he aim of this paper is to give a version of heorem 1 in the setting of generalized metric spaces. he following deinitions will be needed in the sequel. Definition 3 (see [11]). Let  be a nonempty set and let : × → [0,+∞) be a mapping such that for all ,∈ and for all distinct points , V ∈, each of them diferent from  and , one has (gm1) (,)=0 if and only if =, (gm2) (,)=(,), (gm3) (,)≤(,)+(, V)+(V,) (the rectangular inequality).