Existence of Strong Coupled Fixed Points for Cyclic Coupled Ciric-Type Mappings

In this short communication the concept of cyclic coupled Kannan-type contractions is generalized using a certain class of Ciric-type mappings. 1. Introduction and Preliminaries The Banach contraction condition in a metric space given by , has so many significant generalizations which include the class of generalized contractions defined by Ciric [1] as follows. A mapping is called a generalized contraction if and only if there exist nonnegative numbers .？？, , and such that is called contractive if It is worth mentioning that the contractive condition (2) restricts applications only to the class of continuous operators while the contractive conditions (1) accommodate discontinuous operators as well. The search for contractive conditions that do not require continuity of operators culminated in 1969 with the appearance of the Kannan [2] contractive condition below: The Chatterjea [3] contractive condition which followed is independent of both the contractive condition (2) and the Kannan condition (3) which in turn is independent of (2). Consequently, unlike condition (2) the Kannan condition (3) does not generalize the well-known Banach condition above. In a first attempt, the three contractive conditions were combined by Zamfirescu [4] in one theorem to generalize and extend the Banach fixed point theorem. Following Zamfirescu Ciric unified contractive conditions mentioned above by introducing the larger and unifying class of operators called quasi-contractions. is called a quasi-contraction (？iri？ [5]) if there exists such that ？iri？ [5] observed that the class of quasi-contractions contains the class of generalized contractions as a proper subclass. Rhoades [6] noted that the Zamfirescu result is generalized by the Ciric contractive condition (4). There have been numerous generalizations and extensions of the Banach fixed point theorem in literature and they are, basically, modifications of those mentioned above. Very recently Choudhury and Maity [7] introduced the concept of cyclic coupled Kannan-type contractions and established a strong cyclic coupled fixed point result below. We recall the following definition. Let and be two nonempty subsets of a given set . A mapping , such that if and and if and , is called a cyclic mapping with respect to and . Definition 1 (see [8]). Let be a metric space and nonempty subsets. is called a cyclic (or 2-cyclic) -contraction if and and the following condition is satisfied: for all . Definition 2 (see [7]). Let and be two nonempty subsets of a metric space . A mapping is called a cyclic coupled Kannan-type