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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 701910, 18 pages
http://dx.doi.org/10.1155/2012/701910
Research Article

On Generalized Weakly -Contractive Mappings in Partially Ordered -Metric Spaces

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

Received 3 August 2012; Accepted 6 September 2012

Academic Editor: Naseer Shahzad

Copyright © 2012 A. Razani and V. Parvaneh. 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

The aim of this paper is to present some coincidence and common fixed point results for generalized weakly -contractive mappings in the setup of partially ordered -metric space. We also provide an example to illustrate the results presented herein. As an application of our results, periodic points of weakly -contractive mappings are obtained.

1. Introduction and Mathematical Preliminaries

The concept of a generalized metric space, or a -metric space, was introduced by Mustafa et al. [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, comparison of different contractive conditions, and related topics in -metric spaces we refer the reader to [114] and the references mentioned therein.

Definition 1.1 (-metric space [1]). Let be a nonempty set and be a function satisfying the following properties:(G1) if and only 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 1.2 (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 and 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 1.3 (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 1.4 (see [1]). Let be a -metric space. Then, the following are equivalent:(1) is -convergent to .(2), as .(3), as .(4), as .

Lemma 1.5 (see [15]). 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 1.6 (see [1]). A -metric space is said to be -complete (or complete -metric space) if every -Cauchy sequence in is convergent in .

Definition 1.7 (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. [16] as follows.

Definition 1.8. The function is called an altering distance function, if the following properties are satisfied.(1) is continuous and nondecreasing.(2) if and only if .

In [5], Aydi et al. established some common fixed point results for two self-mappings and on a generalized metric space . They presented the following definitions.

Definition 1.9 (see [5]). Let be a -metric space and be two mappings. We say that is a generalized weakly -contraction mapping of type with respect to if for all , the following inequality holds: where(1) is an altering distance function;(2) is a continuous function with if and only if .

Definition 1.10 (see [5]). Let be a -metric space and be given mappings. We say that is a generalized weakly -contraction mapping of type with respect to if for all , the following inequality holds: where(1) is an altering distance function;(2) is a continuous function with if and only if .
Note that the concept of a generalized weakly -contraction is the extension of the concept of weakly C-contraction which has been defined by Choudhury in [17]. For more details on weakly C-contractive mappings we refer the reader to [18, 19].

Definition 1.11 (see [20]). Let be a partially ordered set. A mapping is called a dominating map on if for each in .

Example 1.12 (see [20]). Let be endowed with the usual ordering. Let be defined by . Then, for all . Thus, is a dominating map.

Example 1.13 (see [20]). Let be endowed with the usual ordering. Let be defined by for and for , for any . Then, for all ; that is, is a dominating map.

A subset of a partially ordered set is said to be well ordered if every two elements of be comparable [20].

The following definition is Definition 2.5 of [21], but in the setup of partially ordered -metric spaces.

Definition 1.14. 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 .

Jungck in [22] introduced the following definition.

Definition 1.15 (see [22]). Let be a metric space and . The pair is said to be compatible if and only if , whenever is a sequence in such that for some .

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

Definition 1.16 (see [21]). Let be a partially ordered set and are given mappings such that and . We say that and are weakly increasing with respect to if and only if for all , we have
If , we say that is weakly increasing with respect to .

If (the identity mapping on , then the above definition reduces to the weakly increasing mapping [23] (also see [21, 24]).

Definition 1.17. Let be a -metric space and . The pair is said to be compatible if and only if , whenever is a sequence in such that for some .

Note that the concept of compatibility in a -metric space has been defined by Kumar in [25] (Definition 2.1). In the above definition we only modify his definition, using the fact that , for all .

The aim of this paper is to prove some coincidence and common fixed point theorems for nonlinear weakly -contractive mappings in partially ordered -metric spaces.

2. Main Results

From now, we assume

Our first result is the following.

Theorem 2.1. Let be a partially ordered complete -metric space. Let be two mappings such that ; is weakly increasing with respect to and for every such that , where is an altering distance function and . Then and have a coincidence point in provided that and are continuous and the pair is compatible.

Proof. Let be an arbitrary point. Since , we can construct a sequence defined by: , for all .
Now, since and , as is weakly increasing with respect to , we obtain
Continuing this process, we get:
We complete the proof in three steps.
Step I. We will prove that .
Since , using (2.2) we obtain that
Since is a nondecreasing function, from (2.5), we have
Hence, we conclude that is a nondecreasing sequence of nonnegative real numbers. Thus, there is an such that
Letting in (2.6), we get that that is,
Again, from (2.5) we have
Letting and using (2.7), (2.9), and the continuities of and , we get , and hence . This gives us that from our assumptions about .
Step II. We will show that is a -Cauchy sequences 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 and where is the smallest index with this property, that is,
From rectangle inequality,
Making in (2.15), from (2.11), (2.13), and (2.14) we conclude that
Again, from rectangle inequality,
Hence in (2.17), if , using (2.11), and (2.16), we have
On the other hand, and
Hence in (2.19) and (2.20), if , from (2.11), (2.16) and (2.18) we have
In a similar way, we have and therefore, from (2.22) by taking limit when , using (2.11) and (2.18), we get that
Also,
So, from (2.11), (2.23), and (2.24), we have
Finally,
Hence in (2.26), if and using (2.11) and (2.25), we have
Since , putting , , and in (2.2), for all , we have
Now, if in (2.28), from (2.11), (2.21), (2.25), and (2.27), we have
Hence, which is a contradiction. Consequently, is -Cauchy.
Step III. We will show that and have a coincidence point.
Since is a -Cauchy sequence in the complete -metric space , there exists such that
From (2.30) and the continuity of , we get
By the rectangle inequality, we have
From (2.30), as , we have
Since the pair is compatible, this implies that
Now, from the continuity of and (2.30), we have
Combining (2.31), (2.32), and (2.34) and letting in (2.35), we obtain which implies that , that is, is a coincidence point of and .

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

Theorem 2.2. Let be a partially ordered -metric space. Let be two mappings such that ; is weakly increasing with respect to and for every such that , where is an altering distance function and . Then, and have a coincidence point in if is regular and is a -complete subset of .

Proof. Following the proof of Theorem 2.1, there exists such that
Since is -complete and , we have and hence there exists such that and
Now, we will prove that is a coincidence point of and .
We know that is a nondecreasing sequence in . Regularity of yields that . So, from (2.2) we have
Letting in (2.40), from the continuity of and , we get
As , we have
Hence, . So, and hence, . This means that and have a coincidence point.

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

Corollary 2.3. Let be a partially ordered complete -metric space. Let be a mapping such that , for all and for every such that , where is an altering distance function and . Then, has a fixed point in provided that one of the following two conditions is satisfied:(a) is continuous, or,(b) is regular.

Taking , where , in the above corollary, we obtain the following result.

Corollary 2.4. Let be a partially ordered complete -metric space. Let be a mapping such that , for all and for every such that , where . Then, has a fixed point in if one of the following two conditions is satisfied:(a) is continuous, or,(b) is regular.

Theorem 2.5. Under the hypotheses of Theorem 2.1, and have a common fixed point in if is a nondecreasing dominating map.
Moreover, the set of common fixed points of and is well ordered if and only if and have one and only one common fixed point.

Proof. Following the proof of the Theorem 2.1 we obtain that the sequence is -convergent to and . Since and are weakly compatible (since the pair is compatible), we have . Let . Therefore, we have As is a nondecreasing dominating map,
If , then is a common fixed point. If , then, since from (2.46) , from (2.2) we have
Therefore, . So, . Now, since and , we have . This completes the proof.
Suppose that the set of common fixed points of and is well ordered. We claim that common fixed point of and is unique. Assume on contrary that, and , and . Without any loss of generality, we may assume that . Using (2.2), we obtain
Therefore, , a contradiction. Conversely, if and have only one common fixed point then, clearly, the set of common fixed points of and is well ordered.

Theorem 2.6. Under the hypotheses of Theorem 2.2, and have a common fixed point in provided that and are weakly compatible and is a nondecreasing dominating map.
Moreover, the set of common fixed points of and is well ordered if and only if and have one and only one common fixed point.

Proof. The proof is done as in Theorem 2.5.

Following arguments similar to those given in the proof of Theorems 2.1 and 2.2, we have the following results for a generalized weakly -contractive mapping of type .

Theorem 2.7. Let be a partially ordered complete -metric space. Let be two mappings such that is weakly increasing with respect to and for every such that , where is an altering distance function and . Then and have a coincidence point in provided that and are continuous and the pair is compatible.
Moreover, and have a common fixed point in if is a nondecreasing dominating map.
Also, the set of common fixed points of and is well ordered if and only if and have one and only one common fixed point.

Theorem 2.8. Let be a partially ordered -metric space. Let be two mappings such that is weakly increasing with respect to and for every such that , where is an altering distance function and . Then and have a coincidence point in provided that is regular and is a -complete subset of .
Moreover, and have a common fixed point in if and are weakly compatible and is a nondecreasing dominating map.
Also, the set of common fixed points of and is well ordered if and only if and have one and only one common fixed point.

The following corollary is an immediate consequence of the above theorems.

Corollary 2.9. Let be a partially ordered complete -metric space. Let be a mapping such that , for all and for every such that , where is an altering distance function and . Then has a fixed point in provided that one of the following two conditions is satisfied:(a) is continuous, or,(b) is regular.

Example 2.10. Let be endowed with the usual order in and on be given as
Define as for all .
Define by and by .
Let . Now, we have
There are other 3 cases as follows:(1) and .(2) and .(3).
By a careful calculation for the remained cases above, we see that all the conditions of Theorems 2.1 and 2.5 are satisfied. Moreover, is the unique common fixed point of and .

Denote by the set of all functions verifying the following conditions:(I) is a positive Lebesgue integrable mapping on each compact subset of .(II)for all , .

Other consequences of the main theorems are the following results for mappings satisfying a contraction of integral type.

Corollary 2.11. Replace the contractive condition (2.2) of Theorem 2.1 by the following condition.
There exists a such that
Then, and have a coincidence point, if the other conditions of Theorem 2.1 are satisfied.

Proof. Consider the function . Then, (2.55) becomes
Taking and and applying Theorem 2.1, we obtain the proof (it is easy to verify that is an altering distance function and ).

Similar to [21], let be fixed. Let be a family of functions which belong to . For all , we define We have the following result.

Corollary 2.12. Replace the inequality (2.2) of Theorem 2.1 by the following condition:
Then, and have a coincidence point if the other conditions of Theorem 2.1 are satisfied.

Proof. Consider and .

3. Periodic Point Results

Let , 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 [6, 2628] and the references mentioned therein.

Theorem 3.1. Let and be as in Corollary 2.3. If is a dominating map on , then has property .

Proof. From Corollary 2.3, . Let for some . We will show that . Since is dominating on , we have , which implies that , as is nondecreasing. Using (2.2), we obtain that 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 .

Analogously, we have the following theorem.

Theorem 3.2. Let and be as in Corollary 2.12. If is a dominating map on , then has property .

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