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Hussain, N.; Parvaneh, V.; Hoseini Ghoncheh, S. J.

doi:https://doi.org/10.1155/2014/941086

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Research Article | Open Access

Volume 2014 | Article ID 941086 | https://doi.org/10.1155/2014/941086

Generalized Contractive Mappings and Weakly -Admissible Pairs in -Metric Spaces

N. Hussain,¹V. Parvaneh,²and S. J. Hoseini Ghoncheh³

Academic Editor: Erdal Karapinar

Received26 Jun 2014

Accepted22 Jul 2014

Published17 Aug 2014

Abstract

The aim of this paper is to present some coincidence and common fixed point results for generalized (, )-contractive mappings using partially weakly --admissibility in the setup of -metric space. As an application of our results, periodic points of weakly contractive mappings are obtained. We also derive certain new coincidence point and common fixed point theorems in partially ordered -metric spaces. Moreover, some examples are provided here to illustrate the usability of the obtained results.

1. Introduction and Mathematical Preliminaries

The concept of a generalized metric space, or a -metric space, was introduced by Mustafa and Sims [1]. In recent years, many authors have obtained different fixed point theorems for mappings satisfying various contractive conditions on -metric spaces. For a survey of fixed point theory, its applications, different contractive conditions, and related topics in -metric spaces we refer the reader to [1–33] and the references mentioned therein.

Recall that very recently Samet et al. [33] and Jleli and Samet [22] proved that several results in G-metric spaces can be deduced from the usual one. Later on, Agarwal and Karapnar [23] and Asadi et al. [25] suggested some new contraction mapping type to fail the approaches in [22, 33].

Definition 1 (-metric space [1]). Let be a nonempty set and let be a function satisfying the following properties:(G1) if ;(G2), for all with ;(G3), for all with ;(G4) (symmetry in all three variables);(G5), for all (rectangle inequality).
Then, the function is called a -metric on and the pair is called a -metric space.

Definition 2 (see [1]). Let be a -metric space and let be a sequence of points of . A point is said to be the limit of the sequence if . In this case, one says that the sequence is -convergent to . Thus, if in a -metric space , then, for any , there exists a positive integer such that , for all , .

Definition 3 (see [1]). Let be a -metric space. A sequence is called -Cauchy if for every , there is a positive integer such that , for all , , ; that is, if , as , , .

Lemma 4 (see [1]). Let be a -metric space. Then, the following are equivalent:(1) is -convergent to ;(2), as ;(3), as ;(4), as , .

Lemma 5 (see [34]). If is a -metric space, then is a -Cauchy sequence if and only if for every , there exists a positive integer such that , for all .

Definition 6 (see [1]). A -metric space is said to be -complete (or complete -metric space) if every -Cauchy sequence in is -convergent in .

Proposition 7 (see [1]). Let be a -metric space. Then for each it follows that(1)if then ,(2),(3),(4).

Definition 8 (see [1]). Let and be two -metric spaces. Then a function is -continuous at a point if and only if it is -sequentially continuous at ; that is, whenever is -convergent to , is -convergent to .

The concept of an altering distance function was introduced by Khan et al. [35] as follows.

Definition 9. The function is called an altering distance function, if the following properties are satisfied.(1) is continuous and nondecreasing.(2) if and only if .

Samet et al. [36] defined the notion of -admissible mappings in the framework of metric spaces as follows.

Definition 10. Let be a self-mapping on and let be a function. We say that is an -admissible mapping if For more details on -admissible mappings we refer the reader to [37–39].

Definition 11 (see [40]). Let be a -metric space and let be a self-mapping on and let be a function. We say that is a --admissible mapping if

Definition 12. Let be an arbitrary set, , and . A mapping is called an -dominating map on if or for each in .

Example 13. Let . Let be defined by and let be defined by . Then, for all . That is, . Thus, is an -dominating map.

Definition 14. Let be a -metric space. We say that is -regular if and only if the following hypothesis holds.
For any sequence in with such that as , it follows that or or , for all .

Definition 15. Let be a set and let be given mappings. We say that the pair is partially weakly --admissible if and only if for all .

Let be a nonempty set and a given mapping. For every , let .

Definition 16. Let be a set and let be given mappings. We say that the pair is partially weakly --admissible with respect to if and only if for all , , where .

If , we say that is partially weakly --admissible with respect to .

If (the identity mapping on , then the previous definition reduces to the partially weakly --admissible pair.

Following is an example of mappings , , , , , and for which ordered pairs , , and are partially weakly --admissible with respect to , , and , respectively.

Example 17. Let . We define functions by Also, let .

Jungck in [41] introduced the following definition.

Definition 18 (see [42]). Let be a -metric space and let . The pair is said to be compatible if and only if , whenever is a sequence in such that for some .

The aim of this paper is to prove some coincidence and common fixed point theorems for nonlinear weakly -contractive mappings , , and which are partially weakly -admissible with respect to , , and , respectively, in a -metric space.

2. Main Results

Let be a metric space and let be six self-mappings. In the rest of this paper, unless otherwise stated, for all , let From now on, let be a function having the following property: For example, one can take by .

Our first result is the following.

Theorem 19. Let be a -complete -metric space. Let be six mappings such that , , and . Suppose that, for every three elements , , and with , one has where are altering distance functions. Let , , , , , and be continuous, the pairs , , and compatible, and the pairs , , and partially weakly -admissible with respect to , , and , respectively. Then, the pairs , , and have a coincidence point in . Moreover, if , then is a coincidence point of , , , , , and .

Proof. Let be an arbitrary point. Since , we can choose such that . Since , we can choose such that . Also, as , we can choose such that .
Continuing this process, we can construct a sequence defined by for all .
Now, since , , and and , , and are partially weakly -admissible with respect to , , and , respectively, we obtain that
Continuing this process, from (5), we get for all .
Define . Suppose , for some . Then, . In the case that , then gives . Indeed, If then
Thus, which implies that .
Analogously, for other values of , we can get this result.
Similarly, if , then gives . Also, if , then implies that . Consequently, is a coincidence point of the pairs , , and . Indeed, let . Then, we know that .
So,
This means that , , and .
On the other hand, the pairs , , and are compatible. So, they are weakly compatible. Hence, , , and or, equivalently, , , and .
Now, since , we have , , and .
In the other cases, when (), similarly, one can show that () is a coincidence point of the pairs , , and .
So, suppose that for each ; that is, for each .
We complete the proof in three steps as follows.
Step 1. We will prove that
Since , using (6), we obtain that
Since is a nondecreasing function, we get that
If , then (17) becomes
If then, from (G3) and (G4) in Definition 1, and then (17) will be
If then, again from (G3) and (G4), and then (17) becomes
If then, again from (G3) and (G4), and then (17) becomes
Finally, if then and then (17) becomes
Similarly it can be shown that
Hence, we conclude that is a nondecreasing sequence of nonnegative real numbers. Thus, there is an such that
Reviewing the above argument, from (17), we have
In general, we can show that
Letting in (34), we get that
Letting and using (6), (35), and the continuity of and , we get , and hence . This gives us that from our assumptions about . Also, from Definition 1, part (G3), we have
Step 2. We will show that is a -Cauchy sequence in . So, we will show that, for every , there exists such that, for all , .
Suppose the above statement is false. Then, there exists for which we can find subsequences and of such that satisfying that and is the smallest number such that (38) holds; that is, From rectangle inequality, Hence, in (40), if , using (36) and (39), we have
Also, Hence, in (42), if , using (36) and (38), we have
On the other hand, Hence, in (44), if , from (43), we have
Also, Hence, in (46), if , using (36) and (38), we have
In a similar way, we have Therefore, from (48) by taking limit when , using (36) and (45), we get that Further, we can obtain that
Also, or, equivalently,
Also, Hence, in (53), if , using (36) and (38), we have Since , putting , , and in (6), for all , we have where
If = , from (54) and (41), if in (55), we have which is a contradiction to (43).
If from (37), (45), (49), and (54), if in (55), we have which is a contradiction to (47).
If from (37), (50), and (54), if in (55), we have which is a contradiction to (52).
If from (37) and (54), if in (55), we have
If from (37) and (54), if in (55), we have
Hence, (63) and (65) yield that which is a contradiction. Consequently, is a -Cauchy sequence.
Step 3. We will show that , , , , , and have a coincidence point.
Since is a -Cauchy sequence in the complete -metric space , there exists such that Hence,
As is compatible, so
Moreover, from , , and the continuity of and , we obtain
By the rectangle inequality, we have
Taking limit as in (70), using (68) and (69), we obtain which implies that ; that is, is a coincidence point of and .
Similarly, we can obtain that and .
Now, let . By (6), we have where
Let ; that is, or .
If , from (72), we have hence, , a contradiction.
If and , then so, from (72), we have that is, hence, , a contradiction to .
In the other cases, by a similar manner, we can show that .

In the following theorem, we will omit the compatibility and continuity assumptions.

Theorem 20. Let be an -regular -metric space and six mappings such that , , and and , , and are -complete subsets of . Suppose that, for elements , , and with , we have where are altering distance functions. Then, the pairs , , and have a coincidence point in provided that the pairs , , and are weakly compatible and the pairs , , and are partially weakly -admissible with respect to , , and , respectively. Moreover, if , then is a coincidence point of , , , , , and .

Proof. Following the proof of Theorem 19, there exists such that Since is -complete and , therefore , so there exists such that and Similarly, there exists such that and Now we prove that is a coincidence point of and .
As as , -regularity of implies that . Therefore, from (6), we have where
Let .
Letting in (82), from the continuity of and , we get so .
As and are weakly compatible, we have . Thus is a coincidence point of and .
Similarly, in other cases for , it can be shown that is a coincidence point of the pairs and .
The rest of the proof is similar to the proof of Theorem 19.

Assume that

Taking (the identity mapping on ) in the previous theorems, we obtain the following common fixed point result.

Corollary 21. Let be a -complete -metric space. Let be three mappings. Suppose that, for every three elements , , and with , we have where are altering distance functions. Let the pairs , , and be partially weakly -admissible. Then, the triple has a common fixed point in provided that (a) , , and are continuous or (b) is -regular.

Assume that

Taking in Theorems 19 and 20, we obtain the following coincidence point result.

Corollary 22. Let be a -metric space. Let be two mappings such that . Suppose that, for every three elements , , and with , we have where are altering distance functions. Let the pair be compatible and is partially weakly -admissible w.r.t. . Then, has a coincidence point in provided that (a) and are continuous and is a -complete -metric space or (b) is -regular and is -complete.

Theorem 23. Under the hypotheses of Corollary 22, and have a common fixed point in if is an -dominating map. Moreover, and have one and only one common fixed point if or , where and are common fixed points of and .

Proof. Corollary 22 guarantees that there is a such that . Since and are weakly compatible (since the pair is compatible), we have . Let . Therefore, we have
Since is an -dominating map,
If , then is a common fixed point of and . If , then, from (90) , from (88), we have where
Let . Then, from (91),
Therefore, . So, . Now, since and , we have .
Suppose that or , where and are common fixed points of and . We claim that common fixed point of and is unique. Assume on the contrary that and and . Without any loss of generality, we may assume that . Using (88), we obtain
Let . Then we have
Therefore, , a contradiction.
In the other cases the proof will be done in a similar way.

Example 24. Let , on given by , for all , and given by . Define self-maps , , , , , and on by To prove that is partially weakly -admissible with respect to , let and ; that is, . By the definition of and , we have . So, and hence Therefore, .
To prove that is partially weakly -admissible with respect to , let and ; that is, . By the definition of and , we have . So, Therefore, .
To prove that is partially weakly -admissible with respect to , let and ; that is, . By the definition of and , we have . So, Therefore, .
Furthermore, .
Define as and , for all , where .
Using the mean value theorem for all , , and with we have Thus, (6) is true for . Therefore, all the conditions of Theorem 19 are satisfied. Moreover, is a coincidence point of all six maps.

3. Periodic Point Results

Let be the fixed point set of .

Clearly, a fixed point of is also a fixed point of , for every ; that is, . However, the converse is false. For example, the mapping , defined by , has the unique fixed point , but every is a fixed point of . If , for every , then is said to have property . For more details, we refer the reader to [5, 42–45] and the references mentioned therein.

Assume that

Taking (the identity mapping on ) in Corollary 22, we obtain the following fixed point result.

Corollary 25. Let be a -complete -metric space. Let be a mapping such that is partially weakly -admissible and, for every such that , where is an altering distance function. Then, has a fixed point in provided that (a) is continuous or (b) is -regular.

Theorem 26. Let and be as in Corollary 25. Then has property if is an -dominating map.

Proof. From Corollary 25, . Let for some . We will show that . We have , as is -dominating. Using (6), we obtain that where
If , then, from (103), we have
Starting from and repeating the above process, we get which from our assumptions about implies that for all . Now, taking , we have .
Now, let
So, we have that is,
Repeating the above process, we get
From the above inequalities, we have
Therefore, which from our assumptions about implies that for all . Now, taking , we have .
In other three cases, the proof will be done in a similar way.

4. Results in Ordered -Metric Spaces

Fixed point theorems for monotone operators in ordered metric spaces are widely investigated and have found various applications in differential and integral equations (see [46–48], and references therein). As an application of our results, we derive some new coincidence point and common fixed point theorems for partially weakly increasing contractions which generalize many results in the literature.

Definition 27 (see [49]). Let be a partially ordered -metric space. We say that is regular if and only if the following hypothesis holds.
For any nondecreasing sequence in such that as , it follows that for all .

Definition 28 (see [49]). Let be a partially ordered set and given mappings such that and . We say that and are weakly increasing with respect to if and only if for all , , for all , and , for all .

If , we say that is weakly increasing with respect to .

Definition 29 (see [49]). Let be a partially ordered set and and two self-maps on . An ordered pair is said to be partially weakly increasing with respect to if , for all .

If (the identity mapping on , then the previous definition reduces to the weakly increasing mapping [50] (also see [51, 52]).

Note that a pair is weakly increasing with respect to if and only if ordered pairs and are partially weakly increasing with respect to it.

Let be a partially ordered set and let

Theorem 30. Let be a partially ordered -complete -metric space. Let be six mappings such that , , and . Suppose that, for every three elements , one has where are altering distance functions. Let , , , , , and be continuous, the pairs , , and compatible, and the pairs , , and partially weakly increasing with respect to , , and , respectively. Then, the pairs , , and have a coincidence point in . Moreover, if , then is a coincidence point of , , , , , and .

Theorem 31. Let be a regular partially ordered -metric space, six mappings such that , , and , and , , and -complete subsets of . Suppose that, for elements , one has where are altering distance functions. Then, the pairs , , and have a coincidence point in provided that the pairs , , and are weakly compatible and the pairs , , and are partially weakly increasing with respect to , , and , respectively. Moreover, if , then is a coincidence point of , , , , , and .

Conflict of Interests

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

Acknowledgment

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

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Copyright © 2014 N. 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.

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