SOME CYCLIC FIXED POINT RESULTS FOR CONTRACTIVE MAPPINGS

In this paper, we obtained some new cyclic type fixed point theorems using C-class functions in the framework of metric spaces. Our results extend, generalize, improve and enrich some recent results in the existing literature.


INTRODUCTION AND PRELIMINARIES
Very recently, ( (Chandok et al., 2016), Theorem 14.) proved the following result: Theorem 1.1.((Chandok et al., 2016), Theorem 1).Let A and B be two closed subsets of complete metric spaces (X,d) such that and let be a mapping such that and .Assume that: (1.1) for all and , where , and . Then has a unique fixed point in .In this paper, using C-class functions, we prove some new cyclic type fixed point theorems.Our results extend, generalize, improve and enrich recently ones in existing literature.
First, we recall some notions and properties that will be needed throughout the paper.
Definition 1.1.((Chandok et al., 2016), Definition 5).One can say that is called −class function if it is continuous and satisfies the following axioms: (1) for all .
(1) (2) . Further, let denote the set of all monotone increasing continuous functions , with , as well as let denote the set of all continuous functions , with implies .In the sequel, let be a nonempty set, be its nonempty subsets.Recall that is said to be a cyclic representation of with respect to a mapping if: (1.2) We will use the following auxiliary result.Lemma 1.2.((Radenović, 2015b), Lemma 2.1).Let be a metric space, be a mapping and let be a cyclic representation of .Assume that: , where .If is not a Cauchy sequence, then there exist an and two sequences and of positive integers such that and the following sequences tend to as : where is chosen so that (mod ), for each .If we obtain very significant, important as well as useful result in the framework of metric space for standard fixed point results ( (Radenović et al., 2012), Lemma 2.1.).Hence, Lemma 1.2. is its generalization.

MAIN RESULTS
Our main result is the following improvement of Theorem 1.1.that is, Theorem 14 from (Chandok et al., 2016).It also generalizes several enough known results in existing literature.
Theorem 2.1.Let be a complete metric space, and let be a cyclic representation of with the respect to a mapping where the sets are closed.Assume that: (2.1) for all , where , and .Then, has a unique fixed point and .Moreover, each Picard sequence converges to z. Proof.First of all, if has a fixed point say , using (1.2), we have .(for more details see (Van Dung & Radenović, 2016), (Kadelburg et al., 2016), (Radenović et al., 2016), (Radenović, 2015a), (Radenović, 2016), (Radenović, 2015b) .Therefore, suppose that for each .Further, it can be proved in a standard way (e.g., as in the proof of (Chandok et al., 2016), Theorem 8) that is a decreasing sequence and converges to as .Further, we have to prove that is a Cauchy sequence.If it is not case, then, using Lemma 1.2, we have that there exist an and two sequences and of positive integers, with , such that the sequences (1.3) tend to when .Putting in (2.1), , we get: (2.4) Proof.Putting in Theorem 2.1.for all we obtain the result.□ Similarly as in (Kadelburg et al., 2016.), (Radenović et al., 2016), (Radenović 2015a), (Radenović 2016), (Radenović 2015b), one can easily prove that Theorem 2.1 and Corollary 2.1 are equivalent, that is, Theorem 2.1 holds if and only if Corollary 2.1 holds.
We now announce the following very significant as well as new result with new concept of C−class functions: Theorem 2.2.Theorem 2.1 and Corollary 2.1 are equivalent.
Finally, we have the following open question: Does the following Theorem holds?
Theorem 2.3.Let be a complete metric space, and let be a cyclic representation of with the respect to a mapping , where the sets are closed.Assume that for all there exists such that: (2.7) where and .Then, T has a unique fixed point and .Moreover, each Picard sequence converges to .

ACKNOWLEDGMENT
The authors are thankful to the anonymous learned referees for very careful reading and valuable suggestions.

CONFLICT OF INTERESTS
Authors declare that they have no any conflict of interest regarding the publication of this paper.