Research Article Cyclic Coupled Fixed Point Result Using Kannan Type Contractions Binayak S. Choudhury and Pranati Maity Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah, West Bengal 711103, India Correspondence should be addressed to Pranati Maity; pranati.math@gmail.com Received 17 January 2014; Revised 9 June 2014; Accepted 9 June 2014; Published 2 July 2014 Academic Editor: Aref Jeribi Copyright © 2014 B. S. Choudhury and P. Maity. Tis 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. Putting several existing ideas together, in this paper we defne the concept of cyclic coupled Kannan type contraction. We establish a strong coupled fxed point theorem for such mappings. Te theorem is supported with an illustrative example. 1. Introduction and Mathematical Preliminaries In this paper, we establish a strong coupled fxed point result by using cyclic coupled Kannan type contractions. Te following are two of several reasons why Kannan type mappings feature prominently in metric fxed point theory. Tey are a class of contractive mappings which are diferent from Banach contraction and have unique fxed points in complete metric spaces. Unlike the Banach condition, they may be discontinuous functions. Following their appearance in [1, 2], many persons created contractive conditions not requiring continuity of the mappings and established fxed points results of such mappings. Today, this line of research has a vast literature. Another reason for the importance of the Kannan type mapping is that it characterizes completeness which the Banach contraction does not. It has been shown in [3, 4], the necessary existence of fxed points for Kannan type mappings implies that the corresponding metric space is complete. Te same is not true with the Banach contractions. In fact, there is an example of an incomplete metric space where every contraction has a fxed point [5]. Kannan type mappings, its generalizations, and extensions in various spaces have been considered in a large number of works some of which are in [6–10] and in references therein. A mapping :→, where (,) is a metric space, is called a Kannan type mapping if (,)≤[(,)+(,)] (1) for some 0<<1/2 (see [1, 2]). Let  and  be two nonempty subsets of a set .A mapping :→ is cyclic (with respect to  and ) if ()⊆ and ()⊆. Te fxed point theory of cyclic contractive mappings has a recent origin. Kirk et al. [11] in 2003 initiated this line of research. Tis work has been followed by works like those in [12–15]. Cyclic contractive mappings are map- pings of which the contraction condition is only satisfed between any two points  and  with ∈ and ∈. Te above notion of cyclic mapping is extended to the cases of mappings from × to  in the following defnition. Definition 1. Let  and  be two nonempty subsets of a given set . We call any function :×→ such that (,) ∈  if ∈ and ∈ and (,) ∈  if ∈ and ∈ a cyclic mapping with respect to  and . Coupled fxed point problems have a large share in the recent development of the fxed point theory. Some examples of these works are in [16–22] and references therein. Te defnition of the coupled fxed point is the following. Definition 2 (coupled fixed point [20]). An element (,) ∈ ×, where  is any nonempty set, is called a coupled fxed point of the mapping :×→ if (,) =  and (,)=. Hindawi Publishing Corporation Journal of Operators Volume 2014, Article ID 876749, 5 pages http://dx.doi.org/10.1155/2014/876749