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A fixed point theorem for cyclic generalized contractions in metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 122 (2012)
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).
Let be a metric space and be an operator. We say that f is a contraction if there exists such that, for all ,
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 ,
(where is an increasing function vanishing with continuity at zero and , with arbitrary ), then f has a unique fixed point in X.
Remark 1.1 (see )
The condition (1.2) is weaker than the contraction condition (1.1). In fact, if
then it can be verified that, for every , we have
Remark 1.2 (see )
The function defined as
has a unique fixed point , but fails to be a contraction on any neighborhood both of 1 and of ∞.
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 ()
Let X be a nonempty set, m be a positive integer and an operator. Then, we say that is a cyclic representation of X with respect to f if:
, where are nonempty sets for each ;
Let be a complete metric space. Selecting an arbitrary , we denote
Our main result is as follows.
Theorem 2.2 Let be a complete metric space, m be a positive integer, be closed nonempty subsets of X, , be an increasing function vanishing with continuity at zero, and be an operator. Assume that:
is a cyclic representation of Y with respect to f;
For every , , and (, where ), we have(2.1)
where , and are fixed constants.
Then, we have the following conclusions:
f is a Picard operator, i.e., f has a unique fixed point and the Picard iteration sequence converges to , for any initial point ;
the following estimates hold:
Proof (i) For convenience of notation, if , define where and . Let . Starting from , let be the Picard iteration defined by the sequence
and set . Assume for all n. By (2.1), we have
First, we prove that the sequence is bounded. By (2.2) we get that
Since and , from (2.1), we obtain that
where , , and for some . Thus,
If there is a subsequence , the choice leads to the contradiction
Therefore, the sequence is bounded.
From (2.2) we obtain that the sequence is nonincreasing and then it is convergent to the real number
Now we show that . Assume that . Let and . By (2.1), we have
for some . Letting , we obtain
which implies . This leads to a contradiction, therefore
For , suppose there exists j, , such that , i.e., . Now, let p be fixed, and let
So, we have
Since , and lie in different sets and , for some . Then by (2.1) we have
where . Choosing for each n
the relation (2.3) becomes
Since , it follows that
This shows that is a Cauchy sequence in the complete metric space and, thus, it is convergent to a point . The case similar.
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
for some . Suppose that . Then we have
If equality occurs, the relation
is valid for every , which implies . Thus, is the unique fixed point of f for any initial value .
To prove that the Picard iteration converges to , let us consider . Then there exists such that . As it follows that as well. By the continuity of f, we obtain
Letting , it follows that , i.e., the Picard iteration converges to the unique fixed point of f for any initial point .
Since is a fixed point and , we obtain that(2.4)
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])
Let be a complete metric space, m be a positive integer, be closed nonempty subsets of X, and be an operator. Assume that:
is a cyclic representation of Y with respect to f;
there exists such that, for any , , where , we have
Then f has a unique fixed point .
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
Thus, we immediately get that, for with , we have
Notice also that if is a cyclic representation of X with respect to f, then each () is an invariant set with respect to . Using these two remarks, we get the following periodic point theorem.
Theorem 2.4 Let be a complete metric space, m be a positive integer, be nonempty subsets of X, , be an increasing function vanishing with continuity at zero and be an operator such that for each . Assume that:
is a cyclic representation of Y with respect to f.
There exists such that is closed.
For every and each , we have
where , and are fixed constants.
Then, has a fixed point.
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. □
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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.
An erratum to this article is available at http://dx.doi.org/10.1186/1687-1812-2013-39.
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Alghamdi, M.A., Petruşel, A. & Shahzad, N. A fixed point theorem for cyclic generalized contractions in metric spaces. Fixed Point Theory Appl 2012, 122 (2012). https://doi.org/10.1186/1687-1812-2012-122
- Point Theorem
- Fixed Point Theorem
- Nonempty Subset
- Fixed Point Theory
- Unique Fixed Point