%0 Journal Article
%T A Coupled Fixed Point Theorem in Fuzzy Metric Space Satisfying -Contractive Condition
%A B. D. Pant
%A Sunny Chauhan
%A Jelena Vujakovi£¿
%A Muhammad Alamgir Khan
%A Calogero Vetro
%J Advances in Fuzzy Systems
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/826596
%X The intent of this paper is to prove a coupled fixed point theorem for two pairs of compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) mappings, satisfying -contractive conditions in a fuzzy metric space. We also furnish some illustrative examples to support our results. 1. Introduction The evolution of fuzzy mathematics commenced with the introduction of the notion of fuzzy sets by Zadeh [1], where the concept of uncertainty was introduced in the theory of sets, in a nonprobabilistic manner. Fuzzy set theory has applications in applied sciences such as mathematical programming, model theory, engineering sciences, image processing, and control theory. In 1975, Kramosil and Michalek [2] introduced the concept of fuzzy metric space as a generalization of the statistical (probabilistic) metric space. Afterwards, Grabiec [3] defined the completeness of the fuzzy metric space and extended the Banach contraction principle to fuzzy metric spaces. Since then, many authors contributed to the development of this theory, also in relation to fixed point theory (e.g., [4¨C9]). Mishra et al. [10] extended the notion of compatible mappings (introduced by Jungck [11] in metric spaces) to fuzzy metric spaces and proved common fixed point theorems in presence of continuity of at least one of the mappings, completeness of the underlying space, and containment of the ranges amongst involved mappings. Further, Singh and Jain [12] weakened the notion of compatibility by using the notion of weakly compatible, mappings in fuzzy metric spaces and showed that every pair of compatible mappings is weakly compatible but converse is not true. Inspired by Bouhadjera and Godet-Thobie [13, 14], Gopal and Imdad [15] extended the notions of subcompatibility and subsequential continuity to fuzzy metric spaces and proved fixed point theorems using these notions together due to Imdad et al. [16]. In recent past, several authors proved various fixed point theorems employing more general contractive conditions (e.g., [17¨C26]). On the other hand, Bhaskar and Lakshmikantham [27] and Lakshmikantham and £¿iri£¿ [28] gave some coupled fixed point theorems in partially ordered metric spaces (see also [29¨C31]). In 2010, Sedghi et al. [32] proved common coupled fixed point theorems in fuzzy metric spaces for commuting mappings. Motivated by the results of [33], Hu [34] proved a coupled fixed point theorem for compatible mappings satisfying -contractive conditions in fuzzy metric spaces with continuous t-norm of H-type and generalized the result of Sedghi et
%U http://www.hindawi.com/journals/afs/2013/826596/