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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 962784, 7 pages
http://dx.doi.org/10.1155/2014/962784
Research Article

Discussion on Contractions on Generalized Metric Spaces

1Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey
2Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia

Received 22 December 2013; Accepted 10 January 2014; Published 20 February 2014

Academic Editor: Chi-Ming Chen

Copyright © 2014 Erdal Karapınar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We discuss the existence and uniqueness of fixed points of contractive mappings in complete generalized metric spaces, introduced by Branciari. Our results generalize and improve several results in the literature.

1. Introduction and Preliminaries

Recently, Branciari [1] introduced the notion of a generalized metric, also known as rectangular metric, by replacing the triangle inequality with a more general inequality, namely, quadrilateral inequality. Since quadrilateral inequality is weaker than triangle inequality, each metric space is a generalized metric space. As it is expected, the converse of this statement is false [1]. By using this generalized metric, Branciari successfully defined an open ball and hence a topology. On the other hand, the topology of this metric fails to provide some useful topological properties:()generalized metric needs not to be continuous;()a convergent sequence in generalized metric space needs not to be Cauchy;()generalized metric space needs not to be Hausdorff, and hence the uniqueness of limits cannot be guaranteed.

It is quite natural to ask whether the existing fixed point results are still valid in the setting of generalized metric spaces. The first answer to this question was given by Branciari [1] by proving an analog of the well-known Banach contraction principle. Although the statement is true, in the proof, Branciari [1] used the continuity of generalized metric which cannot be guaranteed. Samet [2] gives an example for a generalized metric which is not continuous. Later, the proof of Branciari [1] corrected by several authors; see, for example, [35]. The challenging nature of the topology of generalized metric has attracted attention and hence various fixed points results for different type contraction mappings on generalized metric spaces have been investigated (see, e.g., [213] and the references therein).

Very recently, Samet et al. [14] suggested a very interesting class of mappings, contraction mappings, to investigate the existence and uniqueness of fixed point. Several well-known fixed point theorems, including the Banach mapping principle, were concluded as consequences of the main result of this interesting paper. The techniques used in this paper have been studied and improved by a number of authors; see, for example, [1519] and the references therein.

In this paper, we investigate the existence and uniqueness of fixed point of contraction mappings in the setting of generalized metric spaces by carrying the problems mentioned above. Notice that in the literature there are distinct notions that are called “generalized metric.” In the sequel, when we mention “generalized metric,” we correspond to the notion of “generalized metric” defined by Branciari [1].

We now recollect some fundamental definitions, notations, and basic results that will be used throughout this paper.

Let be the family of functions satisfying the following conditions:(i) is nondecreasing;(ii)there exist and and a convergent series of nonnegative terms such that for and any .

In the literature such functions are called either Bianchini-Grandolfi gauge functions (see, e.g., [2022]) or -comparison functions (see, e.g., [23]).

Lemma 1 (see, e.g., [23]). If , then the following hold:(i) converges to as for all ;(ii), for any ;(iii) is continuous at ;(iv)the series converges for any .

In what follows, we recall the notion of generalized metric spaces.

Definition 2 (see [1]). Let be a nonempty set and let satisfy the following conditions for all and all distinct each of which is different from and : Then the map is called generalized metric and abbreviated as GMS. Here, the pair is called generalized metric space.

In the above definition, if satisfies only (GMS1) and (GMS2), then it is called semimetric (see, e.g., [6]).

The concepts of convergence, Cauchy sequence, and completeness in a GMS are defined below.

Definition 3. A sequence in a GMS is GMS convergent to a limit if and only if as .
A sequence in a GMS is GMS Cauchy if and only if for every there exists positive integer such that for all .
A GMS is called complete if every GMS Cauchy sequence in is GMS convergent.

Let be nonempty set and let be an endomorphism, . A point in is called (i) a fixed point of if , (ii) a preperiodic point of if there are distinct natural and such that , and (iii) a periodic point of if there exists a natural number so that , where is the th iteration of . It is evident that all periodic points are preperiodic.

The following assumption was suggested by Wilson [6] to replace the triangle inequality with the weakened condition.()For each pair of (distinct) points , there is a number such that, for every ,

Proposition 4 (see [4]). In a semimetric space, assumption (W) is equivalent to the assertion that limits are unique.

Proposition 5 (see [4]). Suppose that is a Cauchy sequence in a GMS with , where . Then for all . In particular, the sequence does not converge to if .

Definition 6 (see [14]). For a nonempty set , let and be mappings. One says that self-mapping on is -admissible if, for all , one has

Example 7. Let . We let and be mappings that are defined by and respectively. Then is -admissible.

Example 8. For we define and by and respectively. Then is -admissible.

Some interesting examples of such mappings were given in [14, 18].

The notion of contractive mapping is defined in the following way.

Definition 9 (see [14]). Let be a metric space and let be a given mapping. One says that is an contractive mapping if there exist two functions and such that

It is obvious that any contractive mapping, that is, a mapping satisfying Banach contraction, is an contractive mapping with for all and , .

2. Main Results

We present our main results in this section. First we give the analog of the notion of contractive mapping, in the context of generalized metric space as follows.

Definition 10. Let be a generalized metric space and let be a given mapping. One says that is an contractive mapping if there exist two functions and such that

Now, we state the following fixed point theorem.

Theorem 11. Let be a complete generalized metric space and let be an contractive mapping. Suppose that (i) is admissible;(ii)there exists such that and ;(iii) is continuous.
Then there exists . such that .

Proof. Let be an arbitrary point such that and . Notice that the existence of such a point is guaranteed by assumption (ii) of the theorem. We construct an iterative sequence in by for all . If we have for some , then is a fixed point of . Hence, for the rest of the proof, we presume that Since is admissible, we have Recursively, we obtain that Analogously, we derive that Iteratively, we get that Regarding (6) and (9), we deduce that for all .
Inductively, we derive that It is evident from Lemma 1 that
Regarding (6) and (11), we deduce that for all .
By utilizing inequality (15), we derive that Owing to Lemma 1, we find that
Let for some with . Without loss of generality, assume that . Thus, . Consider now Due to (ii) of Lemma 1, inequality (18) turns into which is a contradiction. Hence, has no periodic point.
In what follows, we will prove that the sequence is Cauchy. For this purpose, it is sufficient to examine two cases. Case (I): suppose that and is odd. Let , . Then, by using the quadrilateral inequality together with (16), we find
Case (II): let and let be even. Let , . Then, by applying the quadrilateral inequality together with (16) and (17), we find By combining the expressions (20) and (21) we conclude that is a Cauchy sequence in . Since is complete, there exists such that Since is continuous, we obtain from (22) that From (22) and (23) we get immediately that . Taking Proposition 5 into account, we conclude that is a fixed point of ; that is, .

Theorem 12. Let be a complete generalized metric space and let be an contractive mapping. Suppose that (i) is admissible;(ii)there exists such that and ;(iii)if is a sequence in such that for all and as , then for all . Then there exists a such that .

Proof. Following the lines in the proof of Theorem 11, we know that the sequence defined by for all converges for some . From (9) and condition (iii), there exists a subsequence of such that for all . Applying (6), for all , we get that
Letting in the above equality, we find that By Proposition 5, we obtain that is a fixed point of ; that is, .

For the uniqueness, we need an additional condition.(U)For all , we have , where denotes the set of fixed points of .

Theorem 13. Adding condition (U) to the hypotheses of Theorem 11 (resp., Theorem 12), one obtains that is the unique fixed point of .

Proof. In what follows we will show that is a unique fixed point of . We will use the reductio ad absurdum. Let be another fixed point of with . It is evident that .
Now, due to (6), we have which is a contradiction. Hence, .

As an alternative condition for the uniqueness of a fixed point of a contractive mapping, we will consider the following hypothesis.(H)For all , there exists such that and .

Theorem 14. Adding conditions (H) and (W) to the hypotheses of Theorem 11 (resp., Theorem 12), one obtains that is the unique fixed point of .

Proof. Suppose that is another fixed point of . From , there exists such that Since is -admissible, from (27), we have Define the sequence in by for all and . From (28), for all , we have
Iteratively, by using inequality (29), we get that for all . Letting in the above inequality, we obtain Similarly, one can show that Regarding together with (31) and (32), it follows that . Thus we proved that is the unique fixed point of .

Corollary 15. Adding condition (H) to the hypotheses of Theorem 11 (resp., Theorem 12) and assuming that is Hausdorff, one obtains that is the unique fixed point of .

The proof is clear and hence it is omitted. Indeed, Hausdorffness implies the uniqueness of the limit. Thus, the theorem above yields the conclusions.

Example 16. Let where , , and . Define the generalized metric on as follows: It is clear that does not satisfy triangle inequality on . Indeed, Notice that (GMS3) holds, so is a generalized metric.
Let be defined as Define
It is clear that is an contractive mapping with for all . Moreover, there exists such that . In fact, for , we have
Notice also that is -admissible mapping. To show this assume that with . It yields that . Owing to the definition of the mapping , we have
Thus, the mapping is -admissible. It is clear that the mapping is not continuous. On the other hand, if is a sequence in such that for all , then . Recall that the sequence is defined iteratively, by for each integer , with an initial (arbitrary) point . In this case, the initial point lies in either or . If , then the sequence is a constant sequence and hence tends to . Thus, for all implies that . If , then the sequence is a constant sequence and hence tends to . Thus, for all implies that .
Then satisfies the conditions of Theorem 12 and has a (unique) fixed point on ; that is, .

3. Consequences

Now, we will show that many existing results in the literature can be deduced easily from our Theorems 11 and 12.

Corollary 17. Let be a complete generalized metric space and let be a given mapping. Suppose that there exists a function such that for all . Then has a unique fixed point.

Proof. Let be the mapping defined by , for all . Then is an -contraction mapping. It is evident that all conditions of Theorem 11 are satisfied. Hence, has a unique fixed point.

The following fixed point theorems follow immediately from Corollary 17 by taking , where .

Corollary 18 (Branciari [1]). Let be a complete generalized metric space and let be a given mapping. Suppose that there exists a constant such that for all . Then has a unique fixed point.

Remark 19. Our results improve and correct the results of Branciari [1] in which the analog of Banach fixed point theorem was proved. In the literature, to correct the proof of Branciari [1], some authors assume some superfluous conditions such as Hausdorffness of the induced topology of generalized metric space and continuity of generalized metric function. Inspired by the interesting papers of [3, 4] we prove the analog of Banach fixed point theorem in the context of generalized metric space without any further condition.

The notion of transitivity of mapping was introduced in [24, 25] as follows.

Definition 20 (see [24, 25]). Let . One says that is -transitive (on ) if for all .
In particular, we say that is transitive if it is 1-transitive; that is,

As consequences of Definition 20, we obtain the following remarks.

Remark 21 (see [24, 25]). Any function is 0-transitive.
If is -transitive, then it is -transitive for all .
If is transitive, then it is -transitive for all .
If is -transitive, then it is not necessarily transitive for all .

Corollary 22. Let be a complete generalized metric space and let be an contractive mapping. Suppose that (i) is admissible;(ii)there exists such that and is transitive;(iii)either(a) is continuous,or(b)if is a sequence in such that for all and as , then for all .
Then there exists a such that .

Proof. Regarding assumption (ii) of the theorem, there exists such that . Consequently, we have , by (i). Since is transitive, we derive that . Hence all conditions of Theorem 11 (and, resp., Theorem 12) are satisfied.

The following are evident from Theorems 13 and 14.

Corollary 23. Adding condition (U) to the hypotheses of Corollary 22, one concludes that is the unique fixed point of .

Corollary 24. Adding conditions (H) and (W) to the hypotheses of Corollary 22, one obtains that is the unique fixed point of .

Conflict of Interests

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

Acknowledgment

The author thanks are due to anonymous referees for their remarkable comments, suggestion, and ideas that help to improve this paper.

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