Putting several existing ideas together, in this paper we define the concept of cyclic coupled Kannan type contraction. We establish a strong coupled fixed point theorem for such mappings. The theorem is supported with an illustrative example. 1. Introduction and Mathematical Preliminaries In this paper, we establish a strong coupled fixed point result by using cyclic coupled Kannan type contractions. The following are two of several reasons why Kannan type mappings feature prominently in metric fixed point theory. They are a class of contractive mappings which are different from Banach contraction and have unique fixed 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 fixed 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 fixed points for Kannan type mappings implies that the corresponding metric space is complete. The same is not true with the Banach contractions. In fact, there is an example of an incomplete metric space where every contraction has a fixed 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 for some (see [1, 2]). Let and be two nonempty subsets of a set . A mapping is cyclic (with respect to and ) if and . The fixed point theory of cyclic contractive mappings has a recent origin. Kirk et al. [11] in 2003 initiated this line of research. This work has been followed by works like those in [12–15]. Cyclic contractive mappings are mappings of which the contraction condition is only satisfied between any two points and with and . The above notion of cyclic mapping is extended to the cases of mappings from to in the following definition. 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 fixed point problems have a large share in the recent development of the fixed point theory. Some examples of these works are in [16–22] and references therein. The definition of the coupled fixed point is the following. Definition 2
References
[1]
R. Kannan, “Some results on fixed points,” Bulletin of the Calcutta Mathematical Society, vol. 60, pp. 71–76, 1968.
[2]
R. Kannan, “Some results on fixed points II,” The American Mathematical Monthly, vol. 76, pp. 405–408, 1969.
[3]
N. Shioji, T. Suzuki, and W. Takahashi, “Contractive mappings, Kannan mappings and metric completeness,” Proceedings of the American Mathematical Society, vol. 126, no. 10, pp. 3117–3124, 1998.
[4]
P. V. Subrahmanyam, “Completeness and fixed-points,” Monatshefte für Mathematik, vol. 80, no. 4, pp. 325–330, 1975.
[5]
E. H. Connell, “Properties of fixed point spaces,” Proceedings of the American Mathematical Society, vol. 10, pp. 974–979, 1959.
[6]
D. Ariza-Ruiz and A. Jimnez-Melado, “A continuation method for weakly kannan maps,” Fixed Point Theory and Applications, vol. 2010, Article ID 321594, 12 pages, 2010.
[7]
B. S. Choudhury and K. Das, “Fixed points of generalized Kannan type mappings in generalized Menger spaces,” Korean Mathematical Society. Communications, vol. 24, no. 4, pp. 529–537, 2009.
[8]
B. S. Choudhury and A. Kundu, “A Kannan-like contraction in partially ordered spaces,” Demonstratio Mathematica, vol. 46, no. 2, pp. 327–334, 2013.
[9]
B. Damjanovi？ and D. Dragan, “Multivalued generalizations of the Kannan fixed point theorem,” Filomat, vol. 25, no. 1, pp. 125–131, 2011.
[10]
S. Moradi and D. Alimohammadi, “New extensions of Kannan fixed-point theorem on complete metric and generalized metric spaces,” International Journal of Mathematical Analysis, vol. 5, no. 45–48, pp. 2313–2320, 2011.
[11]
W. A. Kirk, P. S. Srinivasan, and P. Veeramani, “Fixed points for mappings satisfying cyclical contractive conditions,” Fixed Point Theory, vol. 4, no. 1, pp. 79–89, 2003.
[12]
B. S. Choudhury, K. Das, and S. K. Bhandari, “A generalized cyclic -contraction principle in Menger spaces using a control function,” International Journal of Applied Mathematics, vol. 24, no. 5, pp. 663–673, 2011.
[13]
E. Karap？nar, “Fixed point theory for cyclic weak -contraction,” Applied Mathematics Letters, vol. 24, no. 6, pp. 822–825, 2011.
[14]
I. A. Rus, “Cyclic representations and fixed points,” Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity, vol. 3, pp. 171–178, 2005.
[15]
M. P？curar and I. A. Rus, “Fixed point theory for cyclic -contractions,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 72, no. 3-4, pp. 1181–1187, 2010.
[16]
V. Berinde, “Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 74, no. 18, pp. 7347–7355, 2011.
[17]
V. Berinde, “Coupled fixed point theorems for -contractive mixed monotone mappings in partially ordered metric spaces,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 75, no. 6, pp. 3218–3228, 2012.
[18]
B. S. Choudhury and A. Kundu, “A coupled coincidence point result in partially ordered metric spaces for compatible mappings,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 73, no. 8, pp. 2524–2531, 2010.
[19]
B. S. Choudhury and P. Maity, “Coupled fixed point results in generalized metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 73–79, 2011.
[20]
T. Gnana Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 65, no. 7, pp. 1379–1393, 2006.
[21]
V. Lakshmikantham and L. ？iri？, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 70, no. 12, pp. 4341–4349, 2009.
[22]
B. Samet and C. Vetro, “Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 74, no. 12, pp. 4260–4268, 2011.
