A fixed point theorem for cyclic generalized contractions in metric spaces

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 [13]), 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 [8] 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 [8])

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

where

Remark 1.2 (see [8])

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 [6] obtained an extension of Banach’s fixed point theorem for mappings satisfying cyclical contractive conditions. Some generalizations of the results given in [6], using the setting of so-called fixed point structures, are presented in I. A. Rus [12]. In [10], Păcurar and Rus established a fixed point theorem for cyclic φ-contractions and they further discussed fixed point theory in metric spaces. In [3], 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.

We need first to recall a known concept.

Definition 2.1 ([3])

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:

(i) , where are nonempty sets for each ;

(ii) .

Let be a complete metric space. Selecting an arbitrary , we denote

Our main result is as follows.

Theorem 2.2Letbe a complete metric space, mbe a positive integer, be closed nonempty subsets ofX, , be an increasing function vanishing with continuity at zero, andbe an operator. Assume that:

1. is a cyclic representation ofYwith respect tof;

2. For every, , and (, where), we have

where, andare fixed constants.

Then, we have the following conclusions:

(i) fis a Picard operator, i.e., fhas a unique fixed pointand the Picard iteration sequenceconverges to, for any initial point;

(ii) 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

Consequently,

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 [8], 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 .

(ii) Since is a fixed point and , we obtain that

By (2.4), it follows that

 □

In view of Remark 1.1, we immediately obtain the following corollary.

Corollary 2.3 (Kirk, Srinivasan, Veeramani [2], Theorem 1.3])

Letbe a complete metric space, mbe a positive integer, be closed nonempty subsets ofX, andbe an operator. Assume that:

(i) is a cyclic representation ofYwith respect tof;

(ii) there existssuch that, for any, , where, we have

Thenfhas 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 2.4Letbe 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:

1. is a cyclic representation ofYwith respect tof.

2. There existssuch thatis closed.

3. For everyand each, we have

where, andare 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 [8] 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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