`Abstract and Applied AnalysisVolume 2012 (2012), Article ID 630457, 13 pageshttp://dx.doi.org/10.1155/2012/630457`
Research Article

Common Fixed Point Theorems of Altman Integral Type Mappings in -Metric Spaces

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

Received 5 August 2012; Accepted 18 October 2012

Copyright © 2012 Feng Gu and Hongqing Ye. 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 concept of -weakly commuting self-mapping pairs in -metric space. Using this concept, we establish a new common fixed point theorem of Altman integral type for six self-mappings in the framework of complete -metric space. An example is provided to support our result. The results obtained in this paper differ from the recent relative results in the literature.

1. Introduction and Preliminaries

Metric fixed point theory is an important mathematical discipline because of its applications in areas as variational and linear inequalities, optimization theory. Many results have been obtained by many authors considering different contractive conditions for self-mappings in metric space. In 1975, Altman [1] proved a fixed point theorem for a mapping which satisfies the condition , where is an increasing function satisfying the following conditions:(i);(ii) is a decreasing function;(iii) for some positive number , there holds .

Remark 1.1. By condition (i) and that is increasing, we know that and .

Gu and Deng [2], Liu [3], Zhang [4], and Li and Gu [5] discussed common fixed point theorems for Altman type mappings in metric space. In 2006, a new structure of generalized metric space was introduced by Mustafa and Sims [6] as an appropriate notion of generalized metric space called -metric space. Abbas and Rhoades [7] initiated the study of common fixed point in generalized metric space. Recently, many fixed point and common fixed point theorems for certain contractive conditions have been established in -metric spaces, and for more details, one can refer to [842]. Coupled fixed point problems have also been considered in partially ordered -metric spaces (see [4356]). However, no one has discussed the common fixed point theorems for two or three pairs combining Altman type mappings recently.

Inspired by that, the purpose of this paper is to study common fixed point problem of Altman integral type for six self-mappings in -metric space. We introduce a new concept of -weakly commuting self-mapping pairs in -metric space, and a new common fixed point theorem for six self-mappings has been established through this concept. The results obtained in this paper differ from the recent relative results in the literature.

Throughout the paper, we mean by the set of all natural numbers.

Definition 1.2 (see [6]). Let be a nonempty set, and let be a function satisfying the following axioms: 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.

Definition 1.3 (see [6]). Let be a -metric space and let be a sequence of points in , a point in is said to be the limit of the sequence if , and one says that sequence is -convergent to .

Thus, if in a -metric space , then for any , there exists such that , for all .

Proposition 1.4 (see [6]). Let be a -metric space, then the followings are equivalent:(1) is -convergent to , (2) as , (3) as , (4) as .

Definition 1.5 (see [6]). Let be a -metric space. A sequence is called -Cauchy sequence if, for each there exists a positive integer such that for all ; that is, as .

Definition 1.6 (see [6]). A -metric space is said to be -complete, if every -Cauchy sequence in is -convergent in .

Proposition 1.7 (see [6]). Let be a -metric space. Then the followings are equivalent.(1) The sequence is -Cauchy;(2) For every , there exists such that , for all .

Proposition 1.8 (see [6]). Let be a -metric space. Then the function is jointly continuous in all three of its variables.

Definition 1.9 (see [6]). Let and be -metric space, and be a function. Then is said to be -continuous at a point if and only if for every , there is such that , and imply . A function is -continuous at if and only if it is -continuous at all .

Proposition 1.10 (see [6]). Let and be -metric space. Then is -continuous at if and only if it is -sequentially continuous at , that is, whenever is -convergent to , is -convergent to .

Proposition 1.11 (see [6]). Let be a -metric space. Then, for any in it follows that:(i) if , then ,(ii), (iii), (iv), (v), (vi).

Definition 1.12 (see [8]). Self-mappings and of a -metric space are said to be compatible if and , whenever is a sequence in such that , for some .

In 2010, Manro et al. [9] introduced the concept of weakly commuting mappings, -weakly commuting mappings into -metric space as follows.

Definition 1.13 (see [9]). A pair of self-mappings of a -metric space is said to be weakly commuting if

Definition 1.14 (see [9]). A pair of self-mappings of a -metric space is said to be -weakly commuting, if there exists some positive real number such that

Remark 1.15. If , then -weakly commuting mappings are weakly commuting.

Now we introduce the new concept of -weakly commuting mappings as follow.

Definition 1.16. A pair of self-mappings of a -metric space is said to be -weakly commuting, if there exists a continuous function , such that

Remark 1.17. Commuting mappings are weakly commuting mappings, but the reverse is not true. For example: let , , for all , define , , through a straightforward calculation, we have: , , , but , hence, , but .

Remark 1.18. Weakly commuting mappings are -weakly commuting mappings, but the reverse is not true. For example: let , define , for all , , , then , , , , , when , we get that and are -weakly commuting mappings, but not weakly commuting mappings.

Remark 1.19. -weakly commuting mappings are -weakly commuting mappings but the reverse is not true. For example: let , , for all , , thus, we have , , , . Let , then For any given , since , there exists such that , so we get . Therefore, and are -weakly commuting mappings, but not -weakly commuting mappings.

Lemma 1.20. Let be Lebesgue integrable, and , for all , let , then is an increasing function in .

Definition 1.21. Let and be self-mappings of a set . If for some in , then is called a coincidence point of and , and is called s point of coincidence of and .

2. Main Results

In this paper, we denote the function satisfying , for all .

Theorem 2.1. Let be a complete -metric space and let , , , , , and be six mappings of into itself. If there exists an increasing function satisfying the conditions (i)~(iii) and the following conditions:(iv), (v), for all ,where is a Lebesgue integrable function which is summable nonnegative such that Then,(a) one of the pairs , , and has a coincidence point in ,(b) if , , and are three pairs of continuous -weakly commuting mappings, then the mappings , , , , , and have a unique common fixed point in .

Proof. Let be an arbitrary point in , from the condition (iv), there exist such that By induction, there exist two sequences , in , such that
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 .
On the other hand, if there exists such that , then for any . This implies that is a -Cauchy sequence.
In fact, if there exists such that , then applying the contractive condition (v) with , , and , and the property of , we get
From Lemma 1.20 and the property of , we have Which implies that . So we find for any . This implies that is a -Cauchy sequence. The same conclusion holds if , or for some .
Without loss of generality, we can assume that for all and .
Now we prove that is a -Cauchy sequence in .
Let , then we have for all . Actually, from the condition (v), (2.3) and the property of , we have By Lemma 1.20 and the property of , we have
Again, using condition (v), (2.3) and the property of , we get From Lemma 1.20 and the property of we have
Similarly, we can get From Lemma 1.20 and property of , we have Combining (2.8), (2.10), and (2.12), we know that the (2.6) holds. This implies that is a nonnegative sequence which is strictly decreasing, hence, is convergent and , for all .
For any , , by combining , , and (2.6), we have From the convergence of the sequence and the condition (iii) we assure that Thus, is a -Cauchy sequence in , since is a complete -metric space, there exists such that , hence Since are -weakly commuting mappings, thus we have On taking at both sides, noting that and are continuous mappings, we have Which gives that . Similarly, we can get .
By using condition (v) and the property of , we get Thus, by Lemma 1.20, and the property of , noting that , , , we have Which implies that By Remark 1.1, we have , therefore, . So, immediately, we can have . Setting Since are -weakly commuting mappings, we have Which gives that . By the same argument, we can get , , So we have , , . Again, by condition (v), we have By the Lemma 1.20 and the property of , we have Which implies that . Thus, by the property of , we have , hence, . Similarly, we can prove that , , so we get , which means that is a common fixed point of , , , , , and .
Now, we will show the common fixed point of , , , , and is unique. Actually, assume is another common fixed point of , , , , , and , then by condition (v), we have By Lemma 1.20 and the property of , we have It is a contradiction, unless , that is, , , , , , and have a unique common fixed point in . This completes the proof of Theorem 2.1.

Remark 2.2. If we take: (1) ; (2) ; (3) ( is identity mapping); (4) and ; (5) , ; (6) in Theorem 2.1, then several new results can be obtained.

Corollary 2.3. Let be a complete -metric space and let , , , , , and be six mappings of into itself. If there exists an increasing function satisfying the conditions (i)~(iii) and the following conditions:(iv), , ,(v), .Then,(a) one of the pairs , , and has a coincidence point in .(b) if , , and are three pairs of continuous -weakly commuting mappings, then the mappings , , , , , and have a unique common fixed point in .

Proof. Taking in Theorem 2.1, the conclusion of Corollary 2.3 can be obtained from Theorem 2.1 immediately. This completes the proof of Corollary 2.3.

Now we give an example to support Corollary 2.3.

Example 2.4. Let , , for all . Let be defined by , , , , , . Clearly, we can get , , . Through calculation, we have Thus, we have Now we choose , and , then we have and satisfies (i)~(iii). Thus, we have

On the other hand, let for all , we have for all . Which means that , , and are three pairs of continuous -weakly commuting mappings in . So that all the conditions of Corollary 2.3 are satisfied. Moreover, is the unique common fixed point for all of the mappings , , , , , and .

Corollary 2.5. Let be a complete -metric space, are three self-mappings in , and function satisfies conditions (i)~(iii) and the following condition: Then , , and have a unique common fixed point in .

Proof. Taking in Corollary 2.3, where is an identity mapping. Then the conclusion of Corollary 2.5 can be obtained from Corollary 2.3 immediately. This completes the proof of Corollary 2.5.

Corollary 2.6. Let be a complete -metric space, is a self-mapping in , and function satisfies conditions (i)~(iii) and the following condition: Then has a unique fixed point in .

Proof. Taking in Corollary 2.5, the conclusion of Corollary 2.6 can be obtained from Corollary 2.5 immediately. This completes the proof of Corollary 2.6.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11071169, 11271105), the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030), and the Physical Experiment Center in Hangzhou Normal University.

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