Abstract and Applied Analysis

Volume 2013 (2013), Article ID 879084, 7 pages

http://dx.doi.org/10.1155/2013/879084

Research Article

Common Fixed Points for Weak ψ-Contractive Mappings in Ordered Metric Spaces with Applications

Sumit Chandok¹ and Simona Dinu²

¹Department of Mathematics, Khalsa College of Engineering & Technology (Punjab Technical University), Ranjit Avenue, Amritsar 143001, India

²Faculty of Applied Sciences, University Politehnica of Bucharest, 313 Splaiul Independentei, 060042 Bucharest, Romania

Received 7 July 2013; Accepted 29 August 2013

Academic Editor: Lishan Liu

Copyright © 2013 Sumit Chandok and Simona Dinu. 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 obtain some new common fixed point theorems satisfying a weak contractive condition in the framework of partially ordered metric spaces. The main result generalizes and extends some known results given by some authors in the literature.

1. Introduction and Preliminaries

Fixed point and common fixed point theorems for different types of nonlinear contractive mappings have been investigated extensively by various researchers (see [1–41]). Fixed point problems involving weak contractions and the mappings satisfying weak contractive type inequalities have been studied by many authors (see [10–20] and references cited therein).

Recently, many researchers have obtained fixed point, common fixed point, coupled fixed point, and coupled common fixed point results in partially ordered metric spaces (see [3, 6–8, 10–12, 29, 30, 32, 36]) and other spaces (see [5, 15, 31, 35, 38, 40, 41]).

Let be a partially ordered set and two self-mappings on . A pair of self-mappings of is said to be weakly increasing [4] if and for any . An ordered pair is said to be partially weakly increasing if for all .

Note that a pair is weakly increasing if and only if the ordered pairs and are partially weakly increasing.

Example 1 (see [3]). Let be endowed with usual ordering and two mappings given by and . Clearly, the pair is partially weakly increasing. But for any implies that the pair is not partially weakly increasing.

Let be a partially ordered set. A mapping is called a weak annihilator of a mapping if for all .

Example 2 (see [3]). Let be endowed with usual ordering and be two mappings given by and . It is clear that for implies that is a weak annihilator of .

Let be a partially ordered set. A mapping is called a dominating if for any .

Example 3 (see [3]). Let be endowed with usual ordering and a mapping defined by , since for implies that is a dominating mapping.

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

Let be a nonempty subset of a metric space . Let and be mappings from a metric space into itself. A point is a common fixed (resp., coincidence) point of and if . The set of fixed points (resp., coincidence points) of and is denoted by (resp., ).

In 1986, Jungck [24] introduced the more generalized commuting mappings in metric spaces, called compatible mappings, which also are more general than the concept of weakly commuting mappings (that is, the mappings are said to be weakly commuting if for all ) introduced by Sessa [34] as follows.

Definition 4. Let and be mappings from a metric space into itself. The mappings and are said to be compatible if

whenever is a sequence in such that for some .

In general, commuting mappings are weakly commuting and weakly commuting mappings are compatible, but the converses are not necessarily true and some examples can be found in [24–26].

In [27], Jungck and Rhoades introduced the concept of weakly compatible mappings and proved some common fixed point theorems for these mappings.

Definition 5. The mappings and are said to be weakly compatible if they commute at coincidence points of and .

In Djoudi and Nisse [21], we can find an example to show that there exists weakly compatible mappings which are not compatible mappings in metric spaces.

Let denote the set of all functions such that

(a)is continuous;(b) is strictly increasing in all the variables;(c)for all ,

It is easy to verify that the following functions are from the class , see [18]:

Definition 6 (see [18]). Let be a partially ordered set and suppose that there exists a metric in such that is a metric space. The mapping is said to be a - contractive mapping, if

for .

Recently, Chen introduced -contractive mappings. The purpose of this paper is to extend the results of Chen for four mappings, in the framework of ordered metric spaces.

2. Main Results

Now, we give the main results in this paper.

Theorem 7. Let be a partially ordered set, and suppose that there exists a metric on such that is a complete metric space. Suppose that , , , and are self-mappings on , the pairs and are partially weakly increasing with and , and the dominating mappings and are weak annihilators of and , respectively. Further, suppose that for any two comparable elements , and ,

holds. If, for a nondecreasing sequence with for all , implies that and either

(a) and are compatible, or is continuous, and are weakly compatible or(b) and are compatible, or is continuous, and are weakly compatible,

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

Proof. Let be an arbitrary point. Since and , we can construct the sequences and in such that

for each . By assumptions, we have

for each . Thus, for each , we have . Without loss of generality, we assume that for each .

Now, we claim that for all , we have

Suppose to the contrary that ) for some . Since and are comparable, from (5), we have

which is a contradiction. Hence for each .

Similarly, we can prove that for each .

Therefore, we can conclude that (8) holds.

Let us denote . Then, from (8), is a nonincreasing sequence and bounded below. Thus, it must converge to some . If , then by the above inequalities, we have . Taking the limit, as , we have , which is a contradiction. Hence,

Now, we show that is a Cauchy sequence.

Suppose that is not a Cauchy sequence. Then, there exists for which we can find two sequences of natural numbers and with such that

From (11), it follows that

Letting and using (10), we have

Again,

Letting in the above inequalities and using (10) and (13), we have

Again,

Letting in the above inequalities and using (10) and (15), we have

Similarly, we have

Also, again from (10), (15), and the inequality

it follows that

Now, we have

Letting , we get

which is a contradiction. Thus is a Cauchy sequence. Since is a complete metric space, there exists such that . Therefore, we have

Assume that is continuous. Since and are compatible, we have

Also, . Now, we have

Letting , we get

which implies that .

Now, it follows that and , . From (5), we have

Letting , we get

which implies that . Since , there exists such that . Suppose that . Since implies , from (5), we obtain

which implies that . Since and are weakly compatible, . Thus, is a coincidence point of and .

Now, and implies . Thus, from (5), we obtain

Letting , we get

which implies that . Therefore, we have .

If is continuous, then, following the similar arguments, also we get the result.

Similarly, the result follows when (b) holds.

Now, suppose that the set of common fixed points of , , , and is well ordered.

We claim that common fixed points of , , , and are unique.

Assume that and , but . Then, from (5), we have

This implies that , and hence .

Conversely, if , , , and have only one common fixed point, then the set of common fixed point of , , , and being singleton is well ordered. This completes the proof.

Example 8. Consider with usual ordering and

Then is a complete partially ordered metric space.

Let , and be self-mappings on defined as

Define function by the formula

Note that , and satisfy all the conditions given in Theorem 7. Moreover, is a common fixed point of , and .

If , then we have the following result.

Corollary 9. Let be a partially ordered set, and suppose that there exists a metric on such that is a complete metric space. Suppose that , , and are self-mappings on , the pairs and are partially weakly increasing with and , and the dominating mapping is a weak annihilator of and . Further, suppose that there exists the function such that, for any two comparable elements ,

holds. If, for a nondecreasing sequence with for all , implies that and either

(a) are compatible, or is continuous, and are weakly compatible or(b) are compatible, or is continuous, and are weakly compatible,

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

Corollary 10. Let be a partially ordered set, and suppose that there exists a metric on such that is a complete metric space. Suppose that , , and are self-mappings on , the pairs and are partially weakly increasing with and , and the dominating mappings and are weak annihilators of . Further, suppose that there exists the function such that, for any two comparable elements ,

holds. If, for a nondecreasing sequence with for all , implies that and either

(a) are compatible, or is continuous, and are weakly compatible or(b) are compatible, or is continuous, and are weakly compatible,

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

Corollary 11. Let be a partially ordered set, and suppose that there exists a metric on such that is a complete metric space. Suppose that and are self-mappings on , the pair is partially weakly increasing with , and the dominating mapping is a weak annihilator of . Further, suppose that there exists the function such that, for any two comparable elements ,

holds. If, for a nondecreasing sequence with for all , implies that and, further, are compatible, or is continuous, and are weakly compatible, then and have a common fixed point in . 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 in .

3. Applications

The aim of the section is to apply our new results to mappings involving contractions of integral type. For this purpose, denote by the set of functions satisfying the following hypotheses:

(h1) is a Lebesgue-integrable mapping on each compact of ;(h2)for any , we have .

Corollary 12. Let be a partially ordered set, and suppose that there exists a metric on such that is a complete metric space. Suppose that , , , and are self-mappings on , the pairs and are partially weakly increasing with and , and the dominating mappings and are weak annihilators of and , respectively. Further, suppose that there exists the function such that, for any two comparable elements ,

holds, where . If, for a nondecreasing sequence with for all , implies that and either

(a) are compatible, or is continuous, and are weakly compatible or(b) are compatible, or is continuous, and are weakly compatible,

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

Corollary 13. Let be a partially ordered set, and suppose that there exists a metric on such that is a complete metric space. Suppose that , , and are self-mappings on , the pairs and are partially weakly increasing with and , and the dominating mappings and are weak annihilators of . Further, suppose that there exists the function such that, for any two comparable elements ,

holds, where . If, for a nondecreasing sequence with for all , implies that and either

(a) are compatible, or is continuous, and are weakly compatible or(b) are compatible, or is continuous, and are weakly compatible,

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

Corollary 14. Let be a partially ordered set, and suppose that there exists a metric on such that is a complete metric space. Suppose that and are self-mappings on , the pair is a partially weakly increasing with , and the dominating mapping is a weak annihilator of . Further, suppose that there exists the function such that, for any two comparable elements ,

holds, where . If, for a nondecreasing sequence with for all , implies that and, further, are compatible, or is continuous, and are weakly compatible, then and have a common fixed point in . 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 in .

Conflict of Interests

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

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