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Mathematical Problems in Engineering
Volume 2013, Article ID 504609, 7 pages
http://dx.doi.org/10.1155/2013/504609
Research Article

-Metric Space: A Generalization

1Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran
2Department of Mathematics, Atilim University, İncek, 06836 Ankara, Turkey
3Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia

Received 27 May 2013; Accepted 25 July 2013

Academic Editor: Bashir Ahmad

Copyright © 2013 Farshid Khojasteh et al. 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 introduce the notion of -metric as a generalization of a metric by replacing the triangle inequality with a more generalized inequality. We investigate the topology of the spaces induced by a -metric and present some essential properties of it. Further, we give characterization of well-known fixed point theorems, such as the Banach and Caristi types in the context of such spaces.

1. Introduction

Celebrated Banach contraction mapping principle [1] can be considered as a revolution in fixed point theory and hence in nonlinear functional analysis. The statement of this well-known principle is simple, but the consequences are so strong: every contraction mapping in a complete metric space has a unique fixed point. Since fixed point theory has been applied to different sciences and also distinct branches of mathematics, this pioneer result of Banach has been generalized, extended, and improved in various ways in several abstract spaces. After that, many authors stated many types, generalizations, and applications of fixed point theory until now (see also [2, 3]).

In 1976, Caristi [4] defined an order relation in a metric space by using a functional under certain conditions and proved a fixed point theorem for such an ordered metric space. Denote and as the sets of all real and natural numbers, respectively.

The order relation is defined as follows.

Lemma 1. Let be a metric space and a functional. Define the relation “” on by Then, is a partial order relation on introduced by , and is called an ordered metric space introduced by . Apparently, if , then .

Caristi’s fixed point theorem states that a mapping has a fixed point provided that is a complete metric space and there exists a lower semicontinuous map such that This general fixed point theorem has found many applications in nonlinear analysis.

Many authors generalized Caristi’s fixed point theorem and stated many types of it in complete metric spaces (see [58]). In particular, in 2010, Amini-Harandi [6] extended Caristi’s fixed point and Takahashi’s minimization theorems in complete metric space via the extension of partial ordered relation which is introduced in Lemma 1 and introduced some applications of such results.

One of the interesting generalizations of the notion of a metric is the concept of a fuzzy metric, given by Kramosil and Michálek [9], and Grabiec [10], independently. Later, George and Veeramani [11] investigated fuzzy metric structure and observed some important topological properties of such spaces. Furthermore, the authors [911] announced existence and uniqueness of a fixed point of certain mappings in the framework of such spaces.

Definition 2 (see [11]). A binary operation is a continuous -norm if it is a topological monoid with unit 1 such that whenever and ().

Definition 3 (see [9, 10]). The 3-tuple is said to be a fuzzy metric space if is an arbitrary set, is a continuous -norm, and is a fuzzy set satisfying the following conditions: (1), (2) for all if and only if , (3), (4), (5) is left continuous, where and .

In this paper, inspired from the definition of fuzzy metric spaces, we will introduce -metric as an extension of metric spaces which is obtained by replacing the triangle inequality with a more generalized inequality. We also investigate the topology of the -metric space and observe some fundamental properties of it. Furthermore, we give the characterization of the Banach and Caristi type fixed point theorems in the context of -metric space.

First, we give the following definition.

Definition 4. Let be a continuous mapping with respect to each variable. Let . A mapping is called an -action if and only if it satisfies the following conditions: (I) and for all ,(II)(III)for each and for each , there exists such that , (IV), for all .

We denote by the set of all -actions.

Example 5. The following functions are examples of -action: () , where , () , where , () , () , () , () () .

Example 5 shows that the category of -actions is uncountable. Next, we derive some lemmas which play a crucial role in our main results.

Lemma 6. Let Then, there exists a correspondence between and . In other words, is an infinite set.

Proof. For each , define , where and . It is previous that . Now, define is well-defined and injective function which completes the proof.

Lemma 7. Let be a -action. For each and , there exist and a function such that . Then, one derives the following. (). () and . () is continuous with respect to the first variable. ()If , then .

Proof. By (III) of Definition 4, for each and for each , there exists such that . Now, define . Let and be two values such that and . If , then or . If , then and this is a contradiction. For , we have the same argument. Thus, is well defined.
On the other hand, , , and are straightforward from (III) of Definition 4. Also, holds since if and is a sequence in such that , then . Thus, If or , then by (II) of Definition 4 and (6), we conclude a contradiction. So, is continuous with respect to the first variable.

The following definition arises from Lemma 7.

Definition 8. The function , mentioned in Lemma 7, is called -inverse action of . One says that is regular if satisfies , for each .

The set of all regular -inverse actions will be denoted by .

Example 9. Let and . It is evident that and . Furthermore, , satisfy all the conditions of Lemma 7. Note that , are regular.

Lemma 10. If , and , then implies that .

Proof. Suppose, on the contrary, that . Then, we have a contradiction.

2. Main Results

In this section, we introduce the notion -metric and discuss the induced topology generated by on a nonempty subset . In particular, we will show that is Hausdorff and first countable. Further, we derive that is a metrizable topological space.

Definition 11. Let be a nonempty set. A mapping is called a -metric on with respect to -action if satisfies the following:() if and only if , (), for all , (), for all . A pair is called a -metric space.

Remark 12. If is a -metric space, , and , then is a metric space.
Conversely, for , , we have that there exists metric space which is not -metric space. For example, if and defined by is a metric, but is not a metric.

Example 13. Let and defined by For , the function forms a metric, and hence, the pair is a -metric space.

Remark 14. Notice that , in Example 13, is not a metric on , since we have .

Definition 15. Let be a -metric space. An open ball at a center with a radius is defined as follows:

Lemma 16. Every open ball is an open set.

Proof. We show that, for each and and for each , there exists such that By (III) of Definition 4, we can choose such that . Now, if , then we have It means that and (11) is proved.

Lemma 17. If is a -metric space, then the collection of open sets forms a topology, denoted by . A pair is called topological space induced by a -metric.

Lemma 18. The set is a local base at , and the above topology is first countable.

Proof. For each and , we can find such that . Thus, . This means that is a local base at and the above topology is first countable.

Theorem 19. A topological space is Hausdorff.

Proof. Let , be two distinct points of . Suppose that is arbitrary. By Definition 4, we conclude that . Therefore, there exist such that . It is clear that . For if there exists , then a contradiction.

Theorem 20. Let be a -metric space and the topology induced by the -metric. Then, for a sequence in , if and only if as .

Proof. Suppose that . Then, for each , there exists such that , for all . Thus, ; that is, as . The converse is verified easily.

Theorem 21. Let be a -metric space and , .Then,

Proof. For each , there exists such that, for all , Thus, by the continuity of with respect to each variable, we have Therefore, Thus, .

Lemma 22. Let be a -metric space. Let be a sequence in and . Then, is unique.

Proof. Suppose that and . We show that . For each , there exists such that and . By the continuity of , we have Hence, we have .

Definition 23. Let be a -metric space. Then, for a sequence in , one says that is a Cauchy sequence if for each , there exists such that, for all , .

Definition 24. Let be a -metric space. One says that is complete -metric space if every Cauchy sequence is convergent in .

Lemma 25 (see [12]). A Hausdorff topological space is metrizable if and only if it admits a compatible uniformity with a countable base.

In the following theorems we apply the previous lemma and the concept of uniformity (see [12] for more information) to prove the metrizability of a topological space .

Theorem 26. Let be a -metric space. Then, is a metrizable topological space.

Proof. For each , define We will prove that is a base for a uniformity on whose induced topology coincides with .
We first note that for each , On the other hand, for each , there is, by the continuity of , an such that Then, : indeed, let and . Thus, Therefore, . Hence, is a base for a uniformity on . Since for each and each , . We deduce from Lemma 25 that is a metrizable topological space.

Let us recall that a metrizable topological space is said to be completely metrizable if it admits a complete metric [13].

Theorem 27. Let be a complete -metric space. Then, is completely metrizable.

Proof. It follows from the proof of Theorem 26 that is a base for a uniformity on compatible with , where , for every . Then, there exists a metric on whose induced uniformity coincides with . We want to show that the metric is complete. Indeed, given a Cauchy sequence in , we will prove that is a Cauchy sequence in . To this end, fix . Choose such that . Then, there exists such that for every . Consequently, for each , . We have shown that is a Cauchy sequence in the complete -metric space and so is convergent with respect to . Thus, is a complete metric space.

3. Two Fixed Point Theorems

In this section, we introduce two fixed point theorems in -metric spaces. First, we introduce the Banach fixed point and Caristi’s fixed point theorems in such spaces.

3.1. Banach Fixed Point Theorem

Theorem 28. Let be a complete -metric space and a mapping that satisfies the following: for each , where . Then, has a unique fixed point.

Proof. Let and . We divide our proof into four steps.
Step  1. We claim that . Indeed, we have Thus, we have .
Step  2. We assert that the sequence is bounded. Suppose, on the contrary, that is an unbounded sequence. Thus, there exists subsequence such that and for each , is minimal in the sense that the relation does not hold and holds for all . Hence, by using the triangle inequality, we derive that By taking the limit from two sides of (27) and using (II) of Definition 4, we derive that Also, we have which implies that Since , we have , a contradiction. Thus, the sequence is bounded.
Step  3. We will show that is a Cauchy sequence. Let with Since is a bounded sequence, therefore, ; that is, is a Cauchy sequence. Thus, there exists such that . Further, we derive that It means that ; that is, .
Step  4. In the last step, we prove that the is the unique fixed point of . Suppose, on the contrary, that , are two distinct fixed points of . So, we get that is a contradiction. This completes the proof.

3.2. Caristi-Type Fixed Point Theorem

Definition 29. Suppose that be a complete -metric space and the class of all maps which satisfies the following conditions: ()there exists such that is bounded below and lower semicontinuous, and is upper semicontinuous for each , (), for each , (), for each .

Lemma 30. By Definition 29, one has for each .

Proof. By Lemma 10, we obtain the desired result.

Example 31. Let thus, . Now, let be a lower bounded, lower semicontinuous function and Then, satisfies all conditions of Definition 29.

Example 32. Let ; thus, . Now, let be a lower bounded, lower semicontinuous function and Then, satisfies all conditions of Definition 29. Also, , and is regular.

From now to end, we assume that is regular (see Definition 8).

Lemma 33. Let be a complete -metric space and . Let be -subadditive; that is, , for each , nondecreasing continuous map such that . Define the order on by for any . Then, is a partial order set which has minimal elements.

Proof. At first, we show that is a partial ordered set. For each , we have . Thus, . If and , then and . Thus, we give It means that . Finally, if and , then and . Thus, we give It means that . Thus, is a partial ordered set.
To show that has minimal elements, we show that any decreasing chain has a lower bound. Indeed, let be a decreasing chain; then we have by definition of , we have . Thus, is decreasing net of reals which is bounded below. Let be an increasing sequence of elements from such that Then, for each , we infer that By taking limit from two sides of (42), the regularity of , and continuity of , we give Then, our assumption on implies that is a Cauchy sequence and therefore converges to some . Since is continuous and is upper semicontinuous, then we have This shows that for all , which means that is lower bound for . In order to see that is also a lower bound for , let be such that for all . Then, for each , we have Hence, for all , which implies that Thus, from (46), we get , which implies that . Therefore, for any , there exists such that ; that is, is a lower bound of . Zorn’s lemma will therefore imply that has minimal elements.

Theorem 34. Let be a complete -metric space and . Let be as in Lemma 33. Let be a map satisfying the following: for any . Then, has a fixed point.

Proof. By Lemma 33, has a minimal element say . Thus, . It means that .

Corollary 35. Let be a complete -metric space and . Let be as in Lemma 33. Let be a multivalued mapping satisfying the following: Then, has an endpoint; that is, there exists such that .

In Corollary 35, we can introduce many types of Caristi’s fixed point theorem as follows.

If we set as in Example 32, then (48) has the following form:

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