In this paper, we extend a recent result of V. Pata (J. Fixed Point Theory Appl. 10:299-305, 2011) in the frame of a cyclic representation of a complete metric space.
One of the fundamental result in fixed point theory is the Banach contraction principle. It has various non-trivial applications in many branches of pure and applied sciences (see, for instance, [2,7,14] and references cited therein).
In terms of Picard operator theory (see ), Banach contraction principle asserts that if f is a contraction and is complete, then f is a Picard operator. This result has been extended to other important classes of maps. Recently, Pata  proved that if is a complete metric space and is an operator such that there exists fixed constants , and such that, for every and every ,
Remark 1.1 (see )
The condition (1.2) is weaker than the contraction condition (1.1). In fact, if
Remark 1.2 (see )
Kirk, Srinivasan and Veeramani  obtained an extension of Banach’s fixed point theorem for mappings satisfying cyclical contractive conditions. Some generalizations of the results given in , using the setting of so-called fixed point structures, are presented in I. A. Rus . In , Păcurar and Rus established a fixed point theorem for cyclic φ-contractions and they further discussed fixed point theory in metric spaces. In , Karapinar proved a fixed point theorem for cyclic weak φ-contraction mappings. Some other recent results concerning this topic are given in [1,4,5,9,11].
In the present paper, we obtain a fixed point theorem for a generalized contraction in the sense of the assumption (1.2), defined on a cyclic representation of a complete metric space.
2 Main results
We need first to recall a known concept.
Definition 2.1 ()
Our main result is as follows.
Then, we have the following conclusions:
(ii) the following estimates hold:
So, we have
the relation (2.3) becomes
On the other hand, the sequence has an infinite number of terms in each , for every . Since is complete, in each , we can construct a subsequence of which converges to y. Since each is closed for , we get that . Then and we can consider the restriction
which satisfies the conditions of Theorem 1 in , since is also closed and complete. From this result, it follows that g has a unique fixed point, say .
We claim now that for any initial value , we get the same limit point . Indeed, for , by repeating the above process, the corresponding iterative sequence yields that g has a unique fixed point, say . Since , , we have , for all and, hence, and are well defined. We can write (2.1) in the form
If equality occurs, the relation
By (2.4), it follows that
In view of Remark 1.1, we immediately obtain the following corollary.
Corollary 2.3 (Kirk, Srinivasan, Veeramani , Theorem 1.3])
Finally, we will prove a periodic point theorem. For this purpose, notice first that if f satisfies (1.2) with constants α, β, γ and function ψ, and if for each , then its m-iterate also satisfies the condition (1.2) with constants α, β, mγ and function ψ. Indeed, let us suppose that f satisfies (1.2) with constants α, β, γ. Then, for every , we have
Theorem 2.4Letbe a complete metric space, mbe a positive integer, be nonempty subsets ofX, , be an increasing function vanishing with continuity at zero andbe an operator such thatfor each. Assume that:
Proof Notice that, by the above considerations, is a self mapping on and it satisfies the condition (1.2) with constants α, β, mγ and function ψ. Thus, by Theorem 1 in  we get the conclusion. □
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
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