Abstract and Applied Analysis

Volume 2014 (2014), Article ID 180698, 14 pages

http://dx.doi.org/10.1155/2014/180698

Research Article

A Unification of -Metric, Partial Metric, and -Metric Spaces

Nawab Hussain,¹ Jamal Rezaei Roshan,² Vahid Parvaneh,³ and Abdul Latif¹

¹Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²Department of Mathematics, Islamic Azad University, Qaemshahr Branch, Qaemshahr, Iran

³Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran

Received 9 October 2013; Accepted 18 January 2014; Published 5 March 2014

Academic Editor: Mohamed Amine Khamsi

Copyright © 2014 Nawab Hussain 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.

Using the concepts of -metric, partial metric, and -metric spaces, we define a new concept of generalized partial -metric space. Topological and structural properties of the new space are investigated and certain fixed point theorems for contractive mappings in such spaces are obtained. Some examples are provided here to illustrate the usability of the obtained results.

1. Introduction and Mathematical Preliminaries

The concept of a -metric space was introduced by Czerwik in [1, 2]. After that, several interesting results about the existence of fixed point for single-valued and multivalued operators in (ordered) -metric spaces have been obtained (see, e.g., [3–13]).

Definition 1 (see [1]). Let be a (nonempty) set and a given real number. A function is a -metric on if, for all , the following conditions hold:

) if and only if ,(),().

In this case, the pair is called a -metric space.

The concept of a generalized metric space, or a -metric space, was introduced by Mustafa and Sims [14].

Definition 2 (see [14]). Let be a nonempty set and a function satisfying the following properties:

() if ;(), for all with ;(), for all with ;(), where is any permutation of (symmetry in all the three variables);(), for all (rectangle inequality).

Then, the function is called a -metric on and the pair is called a -metric space.

Aghajani et al. in [15] introduced the class of generalized -metric spaces (-metric spaces) and then they presented some basic properties of -metric spaces.

The following is their definition of -metric spaces.

Definition 3 (see [15]). Let be a nonempty set and a given real number. Suppose that a mapping satisfies

() if ,() for all with ,() for all with ,(), where is a permutation of (symmetry),() for all (rectangle inequality).

Then is called a generalized -metric and the pair is called a generalized -metric space or a -metric space.

On the other hand, Matthews [16] has introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. In partial metric spaces, self-distance of an arbitrary point need not to be equal to zero.

Definition 4 (see [16]). A partial metric on a nonempty set is a mapping such that, for all :

() if and only if ,(), (), ().

In this case, is called a partial metric space.

For a survey of fixed point theory, its applications, and comparison of different contractive conditions and related results in both partial metric spaces and -metric spaces we refer the reader to, for example, [17–27] and references mentioned therein.

Recently, Zand and Nezhad [28] have introduced a new generalized metric space (-metric spaces) as a generalization of both partial metric spaces and -metric spaces.

We will use the following definition of a -metric space.

Definition 5 (see [29]). Let be a nonempty set. Suppose that a mapping satisfies

() if ;() for all with ;(), where is any permutation of , and (symmetry in all the three variables);() for all (rectangle inequality).

Then is called a -metric and is called a -metric space.

As a generalization and unification of partial metric and -metric spaces, Shukla [30] presented the concept of a partial -metric space as follows.

Definition 6 (see [30]). A partial -metric on a nonempty set is a mapping such that, for all :

() if and only if ,(),(),().

A partial -metric space is a pair such that is a nonempty set and is a partial -metric on . The number is called the coefficient of .

In a partial -metric space , if and , then , but the converse may not be true. It is clear that every partial metric space is a partial -metric space with coefficient and every -metric space is a partial -metric space with the same coefficient and zero self-distance. However, the converse of these facts needs not to be hold.

Example 7 (see [30]). Let , a constant, and defined by

Then is a partial -metric space with the coefficient , but it is neither a -metric nor a partial metric space.

Note that in a partial -metric space the limit of a convergent sequence may not be unique (see [30, Example 2]).

In [31], Mustafa et al. introduced a modified version of ordered partial -metric spaces in order to obtain that each partial -metric generates a -metric .

Definition 8 (see [31]). Let be a (nonempty) set and a given real number. A function is a partial -metric if, for all , the following conditions are satisfied:

(),(),(),().

The pair is called a partial -metric space.

Since , from , we have

Hence, a partial -metric in the sense of Definition 8 is also a partial -metric in the sense of Definition 6.

The following example shows that a partial -metric on (in the sense of Definition 8) is neither a partial metric nor a -metric on .

Example 9 (see [31]). Let be a metric space and , where and are real numbers. Then is a partial -metric with .

Proposition 10 (see [31]). Every partial -metric defines a -metric , where

for all .

Hence, the importance of our definition of partial -metric is that by using it we can define a dependent -metric which we call the -metric associated with .

Now, we present some definitions and propositions in a partial -metric space.

Definition 11 (see [31]). Let be a partial -metric space. Then, for an and an , the -ball with center and radius is

Lemma 12 (see [31]). Let be a partial -metric space. Then,

(A)if , then ;(B)if , then .

Proposition 13 (see [31]). Let be a partial -metric space, , and . If then there exists a such that .

Thus, from the above proposition the family of all open -balls

is a base of a -topology on which we call the -metric topology.

The topological space is but need not be .

The following lemma shows the relationship between the concepts of -convergence, -Cauchyness, and -completeness in two spaces and .

Lemma 14 (see [31]). (1) A sequence is a -Cauchy sequence in a partial -metric space if and only if it is a -Cauchy sequence in the -metric space .

(2) A partial -metric space is -complete if and only if the -metric space is -complete. Moreover, if and only if

Now, we introduce the concept of generalized partial -metric space, a -metric space, as a generalization of both partial -metric space and -metric space.

Definition 15. Let be a nonempty set. Suppose that the mapping satisfies the following conditions:

() if ;() for all with ;(), where is any permutation of , or (symmetry in all three variables);()  +    +    +     +    for all (rectangle inequality).

Then is called a -metric and is called a -metric space.

Since , so from we have the following inequality:

The -metric space is called symmetric if

holds for all . Otherwise, is an asymmetric -metric.

Now we present some examples of -metric space.

Example 16. Let and let be given by , where . Obviously, is a symmetric -metric space which is not a -metric space. Indeed, if , then . It is easy to see that are satisfied. Now we show that holds. For each , we have

so

Thus,

which implies the required inequality

Example 17. Let . Let

Define by

It is easy to see that is an asymmetric -metric space with coefficient .

Proposition 18. Every -metric space defines a -metric space , where

for all .

Proof. Let . Then we have

With straightforward calculations, we have the following proposition.

Proposition 19. Let be a -metric space. Then for each it follows that

(1);(2)  +    −    +  ;(3)  +    + ;(4)  +    −    +  , .

Lemma 20. Let be a -metric space. Then,

(A)if , then ;(B)if , then .

Proof. If , then from we have , so from we obtain (A). To prove (B), on the contrary, if , then from (A) we have , a contradiction.

Definition 21. Let be a -metric space. Then for an and an , the -ball with center and radius is

By motivation of Proposition 4 in [31], we provide the following proposition.

Proposition 22. Let be a -metric space, , and . If , then there exists a such that .

Proof. Let ; if , then we choose . Suppose that ; then, by Lemma 20, we have . Now, we consider two cases.

Case 1. If , then for we choose . If , then we consider the set

By Archmedean property, is a nonempty set; then by the well ordering principle, has a least element . Since , we have and we choose . Let ; by property we have

Hence, .

Case 2. If , then, from property , we have , and. for , we consider the set

Similarly, by the well ordering principle there exists an element such that , and we choose . One can easily obtain that .

For , we consider the set

and by the well ordering principle there exists an element such that and we choose . Let . By property we have

Hence, .

Thus, from the above proposition the family of all open -balls

is a base of a -topology on which we call the -metric topology.

The topological space is , but need not be .

Definition 23. Let be a -metric space. Let be a sequence in .

(1)A point is said to be a limit of the sequence , denoted by , if .(2) is said to be a -Cauchy sequence, if exists (and is finite).(3) is said to be -complete if every -Cauchy sequence in is -convergent to an .

Using the above definitions, one can easily prove the following proposition.

Proposition 24. Let be a -metric space. Then, for any sequence in X and a point , the following statements are equivalent:

(1)  is  -convergent to  .(2), as  .(3), as  .

Based on Lemma 2.2 of [27], we prove the following essential lemma.

Lemma 25. (1) A sequence is a -Cauchy sequence in a -metric space if and only if it is a -Cauchy sequence in the -metric space .

(2) A -metric space is -complete if and only if the -metric space is -complete. Moreover, if and only if

Proof. First, we show that every -Cauchy sequence in is a -Cauchy sequence in . Let be a -Cauchy sequence in . Then, there exists such that, for arbitrary , there is with

for all . Hence,

for all . Hence, we conclude that is a -Cauchy sequence in .

Next, we prove that -completeness of implies -completeness of . Indeed, if is a -Cauchy sequence in , then it is also a -Cauchy sequence in . Since the -metric space is -complete we deduce that there exists such that . Hence,

therefore; .

On the other hand,

Also, from ,

Hence, we obtain that is a -convergent sequence in .

Now, we prove that every -Cauchy sequence in is a -Cauchy sequence in . Let . Then, there exists such that for all . Since

then

Consequently, the sequence is bounded in and so there exists such that a subsequence is convergent to ; that is,

Now, we prove that is a Cauchy sequence in . Since is a -Cauchy sequence in , for given , there exists such that , for all . Thus, for all ,

Therefore, .

On the other hand,

for all . Hence, , and consequently is a -Cauchy sequence in .

Conversely, let be a -Cauchy sequence in . Then, is a -Cauchy sequence in and so it is -convergent to a point with

Then, for given , there exists such that

Therefore,

whenever . Therefore, is -complete.

Finally, let . So

On the other hand,

Definition 26. Let and be two generalized partial -metric spaces and let be a mapping. Then is said to be -continuous at a point if, for a given , there exists such that and imply that . The mapping is -continuous on if it is -continuous at all . For simplicity, we say that is continuous.

From the above definition, with straightforward calculations, we have the following proposition.

Proposition 27. Let and be two generalized partial -metric spaces. Then a mapping is -continuous at a point if and only if it is sequentially continuous at ; that is, whenever is -convergent to , is -convergent to .

Definition 28. A triple is called an ordered generalized partial -metric space if is a partially ordered set and is a generalized partial -metric on .

We will need the following simple lemma of the -convergent sequences in the proof of our main results.

Lemma 29. Let be a -metric space and suppose that , and are -convergent to , , and , respectively. Then we have

In particular, if are constant, then

Proof. Using the rectangle inequality, we obtain

Taking the lower limit as in the first inequality and the upper limit as in the second inequality we obtain the desired result.

If , then

Let denote the class of all real functions satisfying the condition

In order to generalize the Banach contraction principle, Geraghty proved the following result.

Theorem 30 (see [32]). Let be a complete metric space and let be a self-map. Suppose that there exists such that

holds for all . Then f has a unique fixed point and for each the Picard sequence converges to z.

In [33], some fixed point theorems for mappings satisfying Geraghty-type contractive conditions are proved in various generalized metric spaces.

As in [33], we will consider the class of functions , where if and has the property

Theorem 31 (see [33]). Let and let be a complete metric type space. Suppose that a mapping satisfies the condition

for all and some . Then has a unique fixed point and for each converges to in .

The aim of this paper is to present certain new fixed point theorems for hybrid rational Geraghty-type and -contractive mappings in partially ordered -metric spaces. Our results improve and generalize many comparable results in literature. Some examples are established to prove the generality of our results.

2. Main Results

Recall that denotes the class of all functions satisfying the following condition:

Theorem 32. Let be a partially ordered set and suppose that there exists a generalized partial -metric on such that is a -complete -metric space and let be an increasing mapping with respect to with for some . Suppose that

for all comparable elements , where

If is continuous, then has a fixed point.

Proof. Put . Since and is an increasing function we obtain by induction that

Step 1. We will show that . Since for each , then by (49) we have

because

Therefore, is decreasing. Then there exists such that . Letting in (52) we have

Since , we deduce that ; that is,

Step 2. Now, we prove that the sequence is a -Cauchy sequence. By rectangular inequality and (49), we have

Letting in the above inequality and applying (55) we have

Here,

Letting in the above inequality we get

Hence, from (57) and (59), we obtain

and so we get

Since we deduce that

Consequently, is a -Cauchy sequence in . Thus, from Lemma 25, is a -Cauchy sequence in the -metric space . Since is -complete, then, from Lemma 25, is a -complete -metric space. Therefore, the sequence -converges to some ; that is, . Again, from Lemma 25 and (62),

Step 3. Now, we show that is a fixed point of . Suppose to the contrary; that is, ; then, from Lemma 20, we have .

Using the rectangular inequality, we get

Letting and using the continuity of and (63), we get

Note that, from (49), we have

where by (65)

Hence, as for all , we have . Thus, by (65) we obtain that . But then, using (66), we get that , which is a contradiction. Hence, we have . Thus, is a fixed point of .

Now we replace the continuity of in Theorem 32 by the regularity of the space to get the required conclusion.

Theorem 33. Under the same hypotheses of Theorem 32, instead of the continuity assumption of , assume that, whenever is a nondecreasing sequence in such that , one has for all . Then has a fixed point.

Proof. Repeating the proof of Theorem 32, we construct an increasing sequence in such that . Using the assumption on we have . Now, we show that . By Lemma 29 and (63)

where

Therefore, we deduce that . Hence, we have .

If in the above theorems we assume , where , we obtain the following corollary.

Corollary 34. Let be a partially ordered set and suppose that there exists a -metric on such that is a -complete -metric space, and let be an increasing mapping with for some . Suppose that

for all comparable elements , where and

If is continuous or for any nondecreasing sequence in such that one has for all , then has a fixed point.

Corollary 35. Let be a partially ordered set and suppose that there exists a -metric space on such that is a -complete -metric space, and let be an increasing mapping with respect to such that there exists an element with . Suppose that

for all comparable elements , where and .

If is continuous or for any nondecreasing sequence in such that one has for all , then has a fixed point.

Proof. Since

taking , all the conditions of Corollary 34 hold and hence has a fixed point.

Let be the family of all continuous and nondecreasing functions such that

for all . It is known that, if , then and for all .

Theorem 36. Let be a partially ordered set and suppose that there exists a generalized partial -metric on such that is a -complete -metric space, and let be an increasing mapping with for some . Suppose that

where

for all comparable elements . If is continuous, then has a fixed point.

Proof. Since and is an increasing function we obtain by induction that

Putting , we have

If there exists such that then and so we have nothing to prove. Hence, for all , we assume that .

Step 1. We will prove that

Using condition (75), we obtain

Here,

Hence,

By induction, we get that

As , we conclude that

Step 2. We will prove that is a -Cauchy sequence.

Suppose to the contrary that is not a -Cauchy sequence. Then there exists for which we can find two subsequences and of such that is the smallest index for which

This means that

From (85) and using the rectangle inequality, we get

Taking the upper limit as , we get

From the definition of we have

and if , by (84) and (86), we have

Now, from (75) we have

Again, if by (88) we obtain

which is a contradiction. Consequently, is a -Cauchy sequence in . Thus, from Lemma 25 we have proved that is a -Cauchy sequence in the -metric space . Since is -complete, then, from Lemma 25, is a -complete -metric space. Therefore, the sequence -converges to some ; that is, . Again, from Lemma 25 and (62),

Step 3. Now we show that is a fixed point of . Suppose to the contrary, that ; then, from Lemma 20, we have .

Using the rectangle inequality, we get

Letting and using the continuity of , we get

Note that, from (75), we have

where

Hence, as is nondecreasing, we have . Thus, by (95) we obtain that

Equation (96) yields that . This is impossible, according to our assumptions on . Hence, we have . Thus, is a fixed point of .

Theorem 37. Under the hypotheses of Theorem 36, instead of the continuity assumption of , assume that, whenever is a nondecreasing sequence in such that , one has for all . Then has a fixed point.

Proof. Following the proof of Theorem 36, we construct an increasing sequence in such that . Using the given assumption on we have . Now, we show that . By (75) we have

where

Letting in the above, from (93), we get

Again, taking the upper limit as in (99) and using Lemma 29 and (101) we get

So we get . That is, .

Corollary 38. Let be a partially ordered set and suppose that there exists a generalized partial -metric on such that is a -complete -metric space, and let be an increasing mapping with for some . Suppose that

where and

for all comparable elements . If is continuous or, for any nondecreasing sequence in such that , we have for all , then has a fixed point.

We conclude this section by presenting some examples that illustrate our results.

Example 39. Let be equipped with the usual order and -metric function given by with . Consider the mapping defined by and the function given by , and . It is easy to see that is an increasing function on and . We show that is -continuous on . By Proposition 27 it is sufficient to show that is sequentially continuous on . Let be a sequence in such that , so we have , equally , and hence . On the other hand we have

So is sequentially continuous on .

For all comparable elements and the fact that is an increasing function on we have

Hence, satisfies all the assumptions of Theorem 32 and thus it has a fixed point (which is .

Example 40. Let be equipped with the usual order and -metric function given by with . Consider the mapping defined by and the function given by . It is easy to see that is increasing function and . Now we show that is -continuous function on .

Let be a sequence in such that , so we have , equally , and hence . On the other hand we have

So is -sequentially continuous on .

For all comparable elements , we have

Hence, satisfies all the assumptions of Theorem 36 and thus it has a fixed point (which is .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first and fourth authors acknowledge with thanks DSR, KAU, for financial support.

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