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

Abstract

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 (Ćirić [5]) if there exists such that Ćirić [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 contraction with respect to and if is cyclic with respect to and and, for some , satisfies the following condition: where and .

Definition 3 (see [7]). Let be a nonempty set. An element is called a coupled fixed point of a mapping if and . A point is called a strong coupled fixed point if .

Theorem 4 (see [7]). Let and be two nonempty closed subsets of a complete metric space , a cyclic coupled Kannan contraction with respect to and with . Then has a strong coupled fixed point in .

The aim of this paper is to obtain a generalization of Theorem 4 in the context of Ciric-type contractions.

2. Main Results

Following Definition 2 we formulate the following definition.

Definition 5. Let and be two nonempty subsets of a metric space . A mapping is called a cyclic coupled Ciric-type mapping with respect to and if is cyclic with respect to and and, for some constants , satisfies the following condition: where and and is given by

Examples of coupled cyclic Ciric-type mappings include coupled cyclic Kannan-type mappings studied in [7] since the later imply the former. We now prove the main theorem of this work.

Theorem 6. Let and be two nonempty closed subsets of a complete metric space , a cyclic coupled Kannan contraction with respect to and with . Then has a strong coupled fixed point in .

Proof. Let and ; we put , , , , and so forth. Then we obtain the following as estimates for respective displacements: where .
Similarly, using the fact that we obtain
Next, we will use (9), (11), and (12) to obtain the following estimates for the displacements and : Similarly,
Now, to estimate and we proceed as follows: given that ,  ,   , and  , we have the following: Inductively, we assume and ; . Also, we assume and ; . Then It follows from (19) and (20) that Therefore, using (19)–(23) we conclude that for all we have and . Also, using the fact that we conclude that for all implying that if and then and . To prove that the sequences and are convergent it suffices to verify that they are Cauchy sequences.
To this end, we observe that if then for some so that Therefore, the sequences and are Cauchy sequences and since and are closed sets with we submit that the strong cyclic coupled fixed point of so derived satisfies .

3. Conclusion

Further studies include proving our main result and Theorem 4 without the restriction . This can be studied as a best proximity point problem.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.