Abstract and Applied Analysis

Volume 2012 (2012), Article ID 379212, 21 pages

doi:10.1155/2012/379212

Research Article

Common Fixed Point Theorems for Six Mappings in Generalized Metric Spaces

Feng Gu

Institute of Applied Mathematics and Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

Received 23 September 2012; Accepted 6 October 2012

Academic Editor: RuDong Chen

Copyright © 2012 Feng Gu. 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.

By using the weakly commutative and weakly compatible conditions of self-mapping pairs, we prove some new common fixed point theorems for six self-mappings in the framework of generalized metric spaces. An example is provided to support our result. The results presented in this paper generalize the well-known comparable results in the literature due to Abbas, Nazir, Saadati, Mustafa, and Sims.

1. Introduction and Preliminaries

The study of fixed points of mappings satisfying certain conditions has been at the center of vigorous research activity. In 2006, Mustafa and Sims [1] introduced a new structure of generalized metric spaces, which are called -metric spaces as follows.

Definition 1.1 (see [1]). Let be a nonempty set and let be a function satisfying the following properties:

if ;, for all with ; for all with ;, symmetry in all three variables; for all (rectangle inequality).

Then the function is called a generalized metric, or, more specifically, a -metric on , and the pair is called a -metric space.

Since then the fixed point theory in -metric spaces has been studied and developed by authors (see [2–43]). Fixed point problems have also been considered in partially ordered -metric spaces (see [44–56]).

The purpose of this paper is to use the concept of weakly commuting mappings and weakly compatible mappings to discuss some new common fixed point problem for six self-mappings in -metric spaces. The results presented in this paper extend and improve the corresponding results of Abbas et al. [2] and Mustafa and Sims [3].

We now recall some of the basic concepts and results in -metric spaces.

Proposition 1.2 (see [1]). Let be a -metric space, then the function is jointly continuous in three of its variables.

Definition 1.3 (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 , and we say that the sequence is -convergent to or  -convergent to , that is, for any , there exists such that for all (throughout this paper we mean by the set of all natural numbers).

Proposition 1.4 (see [1]). Let be a -metric space, then the following are equivalent:

(i) is -convergent to ;(ii) as ;(iii) as ;(iv) as .

Definition 1.5 (see [1]). Let be a -metric space. A sequence is called -cauchy if for every , there is such that for all , that is as .

Proposition 1.6 (see [1]). Let be a -metric space, then the following are equivalent:

(i)the sequence is -cauchy;(ii)for every , there is such that for all .

Definition 1.7 (see [1]). A -metric space is -complete if every -cauchy sequence in is -convergent in .

Definition 1.8 (see [1]). Let and be -metric spaces, and let be a function. Then is said to be -continuous at a point if and only if for every , there is such that , and implies . A function is -continuous at if and only if it is -continuous at all .

Proposition 1.9 (see [1]). Let and be -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 .

Definition 1.10 (see [4]). Two self-mappings and of a -metric space are said to be weakly commuting if for all in .

Definition 1.11 (see [4]). Let and be two self-mappings from a -metric space into itself. Then the mappings and are said to be weakly compatible if whenever .

Proposition 1.12 (see [1]). Let be a -metric space. Then, for all in , it follows that

(i)if , then ;(ii);(iii);(iv);(v); (vi).

2. Common Fixed Point Theorems

Theorem 2.1. Let be a complete -metric space, and let , , , , , and be six mappings of into itself satisfying the following conditions:

, , ;,

or

where . Then one of the pairs , , and has a coincidence point in . Further, if one of the following conditions is satisfied, then the mappings , , , , , and have a unique common fixed point in . Either or is -continuous, the pair is weakly commutative, the pairs and are weakly compatible; Either or is -continuous, the pair is weakly commutative, the pairs and are weakly compatible; Either or is -continuous, the pair is weakly commutative, the pairs and are weakly compatible.

Proof. Suppose that mappings , , , , , and satisfy condition (2.1).

Let in be arbitrary point, since , , , there exist the sequences and in such that

for .

If for some , with , then is a coincidence point of the pair ; if for some , with , then is a coincidence point of the pair ; if for some , with , then is a coincidence point of the pair . Without loss of generality, we can assume that for all .

Now we prove that is a -cauchy sequence in .

Actually, using the condition (2.1) and (), we have

which further implies that

Thus

where Obviously .

Similarly it can be shown that

It follows from (2.6) and (2.7) that, for all ,

Therefore, for all , , by and we have

Hence is a -cauchy sequence in , since is complete -metric space, there exists a point such that .

Since the sequences , , and are all subsequences of , then they all converge to , that is,

Now we prove that is a common fixed point of , , , , , and under the condition (a).

First, we suppose that is continuous, the pair is weakly commutative, the pairs and are weakly compatible.

Step  1. We prove that .

By (2.10) and weakly commutativity of mapping pair , we have

Since is continuous, then , . By (2.11), we know that .

From the condition (2.1), we get

Letting and using the Proposition 1.12(iii), we have

Hence, and , since .

Again by using condition (2.1), we have

Letting , we have

From the Proposition 1.12(iii), we get

Hence, and , since .

So we have .

Step  2. We prove that .

Since and , there is a point such that . Again by using condition (2.1), we have

Letting , using and the Proposition 1.12(iii), we obtain

Hence, and so , since .

Since the pair is weakly compatible, we have

Again by using condition (2.1), we have

Letting , using , and the Proposition 1.12(iii), we have

Hence, and so , since .

So we have .

Step  3. We prove that .

Since and , there is a point such that . Again by using condition (2.1), we have

Using , and the Proposition 1.12(iii), we obtain

Hence, =  0 and so , since .

Since the pair is weakly compatible, we have

Again by using condition (2.1), we have

Using , , and the Proposition 1.12(iii), we have

Hence, =  0 and so , since .

Therefore, is the common fixed point of , , , , , and when is continuous and the pair is weakly commutative, the pairs and are weakly compatible.

Next, we suppose that is continuous, the pair is weakly commutative, the pairs and are weakly compatible.

Step  1. We prove that .

By (2.10) and weak commutativity of mapping pair , we have

Since is continuous, then , . By (2.10), we know that .

From the condition (2.1), we have

Letting and noting the Proposition 1.12(iii), we have

Hence, =  0 and so , since .

Step  2. We prove that .

Since and , there is a point such that . Again by using condition (2.1), we have

Letting and using and the Proposition 1.12(iii), we have

Hence =  0 and so , since .

Since the pair is weakly compatible, we have

Again by using condition (2.1), we have

Letting and using , and the Proposition 1.12(iii), we have

Hence, and so , since .

So we have .

Step  3. We prove that .

Since and , there is a point such that . Again by using condition (2.1), we have

Letting and using and the Proposition 1.12(iii), we obtain

Hence, and so , since .

Since the pair is weakly compatible, we have

Again by using condition (2.1), we have

Letting and using and the Proposition 1.12(iii), we have

Hence, and so , since .

Step  4. We prove that .

Since and , there is a point such that . Again by using condition (2.1), we have

Using , , and the Proposition 1.12(iii), we obtain

Hence and , since .

Since the pair is weakly compatible, we have

Therefore, is the common fixed point of , , , , , and when is continuous and the pair is weakly commutative, the pairs and are weakly compatible.

Similarly, we can prove the result that is a common fixed point of , , , , , and when under the condition of (b) or (c).

Finally, we prove uniqueness of common fixed point .

Let and be two common fixed points of , , , , , and , by using condition (2.1), we have

Hence, and so , since Thus common fixed point is unique.

The proof using (2.2) is similar. This completes the proof.

Now we introduce an example to support Theorem 2.1.

Example 2.2. Let , and let be a -metric space defined by = for all in . Let , , , , , and be self-mappings defined by

Note that is -continuous in , and , , , , and are not -continuous in .

Clearly we can get , ,  and .

Actually, since , , , , = ,  and , so we know , , and .

By the definition of the mappings of and , for all , = , so we can get the pair is weakly commuting.

By the definition of the mappings of and , only for , , at this time = = = , so , so we can obtain that the pair is weakly compatible. Similarly, we can show that the pair is also weakly compatible.

Now we proof that the mappings , , , , , and are satisfying the condition (2.1) of Theorem 2.1 with . Let

Case 1. If , then

Thus, we have

Case 2. If , then

Therefore, we get

Case 3. If , then

Hence, we have

Case 4. If , then

So we get

Case 5. If , then

If , then

If , then

And so we have

for all . Hence we have

Case 6. If , then

Thus, we have

Case 7. If , then

If , then

If , then

And so we have

for all . Hence we have

Case 8. If , then

Then in all the above cases, the mappings , , , , , and are satisfying the condition (2.1) of Theorem 2.1 with so that all the conditions of Theorem 2.1 are satisfied. Moreover, is the unique common fixed point for all of the mappings , , , , , and .

In Theorem 2.1, if we take ( is identity mapping, the same as below), then we have the following corollary.

Corollary 2.3 (see [2, Theorem ]). Let be a complete -metric space, and let , , and be three mappings of into itself satisfying the following conditions:

or

, where Then , , and have a unique common fixed point in .

Also, if we take and in Theorem 2.1, then we get the following.

Corollary 2.4 (see [3, Theorem  2.4]). Let be a complete -metric space, and let be a mapping of into itself satisfying the following conditions:

or

, where Then, has a unique fixed point in .

Remark 2.5. Theorem 2.1 and Corollaries 2.3 and 2.4 generalize and extend the corresponding results of Abbas and Rhoades [5] and Mustafa et al. [6].

Remark 2.6. In Theorem 2.1, if we take: (1) ; (2) ; (3) and ; (4) , , several new results can be obtained.

Theorem 2.7. Let be a complete -metric space, and let , , , , , and be six mappings of into itself satisfying the following conditions:

(i), , ;(ii)the pairs , , and are commutative mappings;(iii),

or

where , then , , , , , and have a unique common fixed point in .

Proof. Suppose that mappings , , , , , and satisfy the condition (2.71). Since , so that . Similarly, we can show that and . From the the Theorem 2.1, we see that , , , , , and have a unique common fixed point .

Since , so that

note that and the Proposition 1.12(iii), we obtain

Since , hence and so .

By the same argument, we can prove that and . Thus, we have = = = = = = so that , , , , , and have a common fixed point in . Let be any other common fixed point of , , , , , and , then by using condition (2.71), we have

Hence, and so , since . Thus, common fixed point is unique.

The proof using (2.72) is similar. This completes the proof.

In Theorem 2.7, if we take , then we have the following corollary.

Corollary 2.8 (see [2, Corollary  2.5]). Let be a complete -metric space, and let , , and be three mappings of into itself satisfying the following conditions:

or

, where , ; then , , and have a unique common fixed point in .

Also, if we take and in Theorem 2.7, then we get the following.

Corollary 2.9 (see [3, Corollary  2.5]). Let be a complete -metric space, and let be a mapping of into itself satisfying the following conditions:

or

, where ; then has a unique fixed point in .

Remark 2.10. In Theorem 2.7, if we take: (1) ; (2) and ; (3) , , several new results can be obtained.

Corollary 2.11. Let be a complete -metric space, and let , , , , , and be six mappings of into itself satisfying the following conditions:

(i), , ;(ii),

 or

 where Then one of the pairs , , and has a coincidence point in . Further, if one of the following conditions is satisfied, then the mappings , , , , , and have a unique common fixed point in . (a) Either or is -continuous, the pair is weakly commutative, the pairs and are weakly compatible; (b) Either or is -continuous, the pair is weakly commutative, the pairs and are weakly compatible; (c) Either or is -continuous, the pair is weakly commutative, the pairs and are weakly compatible.

Proof. Suppose that mappings , , , , , and satisfy the condition (2.80). For , let

Then

So, if

then . Taking in Theorem 2.1, the conclusion of Corollary 2.11 can be obtained from Theorem 2.1 immediately.

The proof using (2.81) is similar. This completes the proof.

Corollary 2.12. Let be a complete -metric space, and let , , , , , and be six mappings of into itself satisfying the following conditions:

(i), , ;(ii)the pairs , and are commutative mappings;(iii),

or

where , ; then , , , , , and have a unique common fixed point in .

Proof. The proof follows from Corollary 2.11, and from an argument similar to that used in Theorem 2.7.

Remark 2.13. In Corollaries 2.11 and 2.12, if we take: (1) ; (2) ; (3) ; (4) and ; (5) , , several new results can be obtained.

Acknowledgment

This work was supported by the National Natural Science Foundation of China Grant (nos. 11071169, 11271105) and the Natural Science Foundation of Zhejiang Province (nos. Y6110287, LY12A01030).

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