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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 736367, 23 pages
http://dx.doi.org/10.1155/2012/736367
Research Article

Generalized Altering Distances and Common Fixed Points in Ordered Metric Spaces

1Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud, Raipur 492101, India
2Institut Superiéur d'Informatique et des Technologies de Communication de Hammam Sousse, Route GP1-4011, Hammam Sousse, Tunisia

Received 27 March 2012; Revised 6 June 2012; Accepted 6 June 2012

Academic Editor: Teodor Bulboaca

Copyright © 2012 Hemant Kumar Nashine and Hassen Aydi. 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

Coincidence point and common fixed point results with the concept of generalized altering distance functions in complete ordered metric spaces are derived. These results generalize the existing fixed point results in the literature. To illustrate our results and to distinguish them from the existing ones, we equip the paper with examples. As an application, we study the existence of a common solution to a system of integral equations.

1. Introduction and Preliminaries

The study of common fixed points of mappings satisfying certain contractive conditions has been researched extensively by many mathematicians since fixed point theory plays a major role in mathematics and applied sciences (see [140] and others). A new category of contractive fixed point problems was addressed by Khan et al. [1]. In this work, they introduced the notion of an altering distance function, which is a control function that alters distance between two points in a metric space.

Definition 1.1 (see [1]). A function is called an altering distance function if and only if(i)is continuous,(ii) is nondecreasing,(iii).
Khan et al. [1] proved the following result.

Theorem 1.2 (see [1]). Let be a complete metric space, an altering distance function, and a self-mapping which satisfies the following inequality: for all and for some . Then has a unique fixed point.

Putting in Theorem 1.2, we retrieve immediately the Banach contraction principle.

In 1997, Alber and Guerre-Delabriere [2] introduced the concept of weak contractions in Hilbert spaces.

This concept was extended to metric spaces by Rhoades in [3].

Definition 1.3. A mapping , where is a metric space, is said to be weakly contractive if and only if where is an altering distance function.

Theorem 1.4 (see [3]). Let be a complete metric space and a weakly contractive map. Then, admits a unique fixed point.

Weak inequalities of the above type have been used to establish fixed point results in a number of subsequent works, some of which are noted in [47, 20]. In [5], Choudhury introduced the concept of a generalized altering distance function for three variables.

Definition 1.5 (see [5]). A function is said to be a generalized altering distance function if and only if(i) is continuous,(ii) is nondecreasing in all the three variables,(iii).

In [5], Choudhury proved the following common fixed point theorem using altering distances for three variables.

Theorem 1.6 (see [5]). Let be a complete metric space and two self mappings such that the following inequality is satisfied: for all , where and are generalized altering distance functions and . Then and have a common fixed point.

In [31], Rao et al. generalized Theorem 1.6 for four mappings satisfying a generalized contractive condition and used the following generalized altering distance function for four variables.

Definition 1.7 (see [31]). A function is said to be a generalized altering distance function if and only if(i) is continuous,(ii) is nondecreasing in all the three variables,(iii).
Let denote the set of all functions satisfying (i)–(iii) in Definition 1.7.

Recently, there have been so many exciting developments in the field of existence of fixed point in partially ordered sets (see [819, 2136] and the references cited therein). The first result in this direction was given by Turinici [36], where he extended the Banach contraction principle in partially ordered sets. Ran and Reurings [33] presented some applications of Turinici’s theorem to matrix equations. The obtained result by Turinici was further extended and refined in [2832]. Subsequently, Harjani and Sadarangani [19] generalized their own results [18] by considering pair of altering functions . Nashine and Altun [21] and Nashine and Altun [22] generalized the results of Harjani and Sadarangani [18, 19]. Also, Nashine and Altun [22] and Shatanawi and Samet [41] worked for a pair of weakly increasing mappings with respect to a third mapping . In another paper, Nashine et al. [24] prove coincidence point and common fixed point results for mappings satisfying a contractive inequality which involves two generalized altering distance functions for three variables in ordered complete metric spaces. As application, they studied the existence of a common solution to a system of integral equations.

The aim of this paper is to generalize the results of Nashine et al. [24] in the sense of four variables. We obtain coincidence point and common fixed point theorems in complete ordered metric spaces for mappings satisfying a contractive condition which involves two generalized altering distance functions in four variables. Presented theorems are ordered version of Theorem  2.1 of Rao et al. [31] for three mappings. In addition, an application to the study of the existence of a common solution to a system of integral equations is given.

2. Main Results

First we introduce some notations and definitions that will be used later.

2.1. Notations and Definitions

The following definition was introduced by Jungck in [37].

Definition 2.1 (see [37]). Let be a metric space and . If , for some , then is called a coincidence point of and , and is called a point of coincidence of 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 a given mapping. For every , we denote by the subset of defined by

Definition 2.2 (see [23]). 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

Remark 2.3. If is the identity mapping ( for all ), then and are weakly increasing with respect to whichimplies that and are weakly increasing mappings. Note that the notion of weakly increasing mappings was introduced in [9] (also see [17, 38]).

Example 2.4. Let be endowed with the partial order given by Define the mappings by We will show that the mappings and are weakly increasing with respect to .
Let such that . By the definition of , we have . On the other hand, and . Thus, we have for all .
Let such that . By the definitions of and , we have . Then we have . On the other hand, . Then, for all .
Thus, we proved that and are weakly increasing with respect to .

Definition 2.5. Let be a nonempty set. Then is called an ordered metric space if and only if:(i) is a metric space,(ii) is a partially ordered set.

2.2. Results

Our first result is the following.

Theorem 2.6. Let be an ordered complete metric space. Let be given mappings satisfying for every pair such that and are comparable, where and are generalized altering distance functions (in ) and . One assumes the following hypotheses:(i), , and are continuous,(ii), ,(iii) and are weakly increasing with respect to ,(iv) the pairs and are compatible.
Then, , , and have a coincidence point, that is, there exists such that .

Proof. Let be an arbitrary point. Since , there exists such that . Since , there exists such that . Continuing this process, we can construct a sequence in defined by We claim that To this aim, we will use the increasing property of the mappings and with respect to . From (2.6), we have Since , then , and we get Again, Since , we get Hence, by induction, (2.7) holds.
Without loss of the generality, we can assume that
Now, we will prove our result on three steps.
Step  1. We claim that Putting and , from (2.7) and the considered contraction (2.5), we have Suppose, for some , that Using (2.15) and a triangular inequality, we have Using this and (2.15) together with a property of the generalized altering function , we get Hence, we obtain This implies that which yields that Hence, we obtain a contradiction with (2.12). We deduce that Similarly, putting and , from (2.7) and the considered contraction (2.5), we have Suppose, for some , that Then, by a triangular inequality, we have Hence, from this, (2.22), and (2.23), we obtain This implies that which leads to Hence, we obtain a contradiction with (2.12). We deduce that Combining (2.21) and (2.28), we obtain Then, is a nonincreasing sequence of positive real numbers. This implies that there exists such that By (2.14), we have Letting in (2.31) and using the continuities of and , we obtain which implies that , so . Hence Hence, (2.13) is proved.
Step  2. We claim that is a Cauchy sequence.
From (2.13), it will be sufficient to prove that is a Cauchy sequence. We proceed by negation and suppose that is not a Cauchy sequence. Then, there exists for which we can find two sequences of positive integers and such that, for all positive integer , From (2.34) and using a triangular inequality, we get Letting in the previous inequality and using (2.13), we obtain Again, a triangular inequality gives us Letting in the above inequality and using (2.13) and (2.36), we get On the other hand, we have Then, from (2.13), (2.36), and the continuity of , we get by letting in the above inequality Now, using the considered contractive condition (2.5) for and , we have Then, from (2.13), (2.38), and the continuities of and , we get by letting in the above inequality Now, combining (2.40) with the previous inequality, we get which implies that , that is a contradiction since . We deduce that is a Cauchy sequence.
Step  3. We claimexistence of a coincidence point.
Since is a Cauchy sequence in the complete metric space , there exists such that From (2.44) and the continuity of , we get By the triangular inequality, we have On the other hand, we have Since and are compatible mappings, this implies that Now, from the continuity of and (2.44), we have Combining (2.45), (2.48), and (2.49) and letting in (2.46), we obtain that is, Again, by a triangular inequality, we have On the other hand, we have Since and are compatible mappings, this implies that Now, from the continuity of and (2.44), we have Combining (2.45), (2.54), and (2.55) and letting in (2.52), we obtain that is, Finally, from (2.51) and (2.57), we have that is, is a coincidence point of , , and . This makes end to the proof.

In the next theorem, we omit the continuity hypotheses satisfied by , , and .

Definition 2.7. Let be a partially ordered metric space. We say that is regular if and only if the following hypothesis holds: if is a nondecreasing sequence in with respect to such that as , then for all .

Now, our second result is the following.

Theorem 2.8. Let be an ordered complete metric space. Let be given mappings satisfying for every pair such that and are comparable, where and are generalized altering distance functions and . We assume the following hypotheses:(i) is regular,(ii) and are weakly increasing with respect to ,(iii) is a complete subspace of ,(iv), .
Then, , , and have a coincidence point.

Proof. Following the proof of Theorem 2.6, we have is a Cauchy sequence in . Since is a complete, there exists , such that Since is a nondecreasing sequence and is regular, it follows from (2.60) that for all . Hence, we can apply the contractive condition (2.5). Then, for and , we obtain Letting in the above inequality and using (2.13), (2.60), and the properties of and , we obtain This implies that , which gives us that , that is, Similarly, for and , we obtain Letting in the above inequality, we get This implies that , and then, Now, combining (2.63) and (2.66), we obtain Hence, is a coincidence point of , , and . This makes end to the proof.

Now, we give a sufficient condition that assures the uniqueness of the common coincidence point of and .

Theorem 2.9. Under the hypotheses of Theorem 2.6 (resp., Theorem 2.8) and suppose that is a totally ordered set and is one-to-one mapping, then one obtains a unique common coincidence point of and .

Proof. Following the proof of Theorem 2.6 (resp., Theorem 2.8), the set of common coincidence points of and is nonempty. Let be two common coincidence points of and , that is, This implies that , and then, and are comparable with respect to . Then, we can apply the contractive condition (2.5). We have This implies that , which gives us that , that is, . From (2.68), we get . Since is one-to-one, we have . This makes end to the proof.

Now, it is easy to state a corollary of Theorem 2.6 or Theorem 2.8 involving contractions of integral type.

Corollary 2.10. Let and satisfy the conditions of Theorem 2.6 or Theorem 2.8, except that condition (2.5) is replaced by the following: there exists a positive Lebesgue integrable function on such that for each and that Then, , , and have a coincidence point.

If is the identity mapping, we can deduce easily the following common fixed point results.

The next result is an immediate consequence of Theorem 2.6.

Corollary 2.11. Let be an ordered complete metric space. Let be given mappings satisfying for every pair such that and are comparable, where and are generalized altering distance functions and . One assumes the following hypotheses:(i) and are continuous,(ii) and are weakly increasing.
Then, and have a common fixed point, that is, there exists such that .

The following result is an immediate consequence of Theorem 2.8.

Corollary 2.12. Let be an ordered complete metric space. Let be given mappings satisfying for every pair such that and are comparable, where and are generalized altering distance functions and . One assume the following hypotheses:(i) is regular,(ii) and are weakly increasing.
Then, and have a common fixed point.

Now, we give some examples to support our results.

Example 2.13. Let be endowed with the usual metric for all , and . Consider the mappings We define the functions by Clearly, for all . Now, we will check that all the hypotheses required by Theorem 2.8 are satisfied.
(i) is regular.
Let be a nondecreasing sequence in with respect to such that as . We have for all .(a)If , then . From the definition of , we have . By induction, we get for all and . Then, for all .(b)If , then . From the definition of , we have . By induction, we get for all and . Then, for all .(c)If , then . From the definition of , we have . By induction, we get for all . Suppose that there exists such that . From the definition of , we get for all . Thus, we have and for all . Now, suppose that for all . In this case, we get and for all .
Thus, we proved that, in all cases, we have for all . Then, is regular.
(ii) and are weakly increasing.
Since , we have to check that for all .
For , we have For , we have For , we have Thus, we proved that and are weakly increasing.
On the other hand, it is very easy to show that (2.5) is satisfied for all such that .
Now, all the hypotheses of Theorem 2.8 are satisfied. Then , , and have a coincidence point .
Note that inequality (2.5) is not satisfied for and . Indeed, Then, Theorem  2.1 of Rao et al. [31] cannot be applied (for three maps) in this case.

Example 2.14. Let with the Euclidean distance . is, obviously, a complete metric space. Moreover, we consider the order in given by . Notice that the elements in are only comparable to themselves, so is regular. Also we consider given by It is easy that, for all , , and , so the pair is weakly increasing.
Define by Then, for all .
As the elements in are only comparable to themselves, condition (2.5) appearing in Theorem 2.8 is, obviously, satisfied. Now, all the hypotheses of Theorem 2.8 are satisfied. and are the coincidence points of the mappings , , and .
On the other hand, the inequality (2.5) is not satisfied for and . Indeed, Then, again Theorem  2.1 of Rao et al. [31] cannot be applied (for three maps) in this case.

A number of fixed point results may be obtained by assuming different forms for the functions and . In particular, fixed point results under various contractive conditions may be derived from the above theorems. For example, if we consider where and , we obtain the following results.

The next result is an immediate consequence of Corollaries 2.11 and 2.12.

Corollary 2.15. Let be an ordered complete metric space. Let be given mappings satisfying for every pair such that and are comparable, where and . One assumes the following hypotheses:(i) and are continuous or is regular,(ii) and are weakly increasing.
Then, and have a common fixed point, that is, there exists such that .

Remark 2.16. Other fixed point results may also be obtained under specific choices of and .

3. Application

Consider the integral equations: where .

The purpose of this section is to give an existence theorem for common solution of (3.1) using Corollary 2.15. This application is inspired by [9].

Previously, we consider the space () of continuous functions defined on . Obviously, this space with the metric given by is a complete metric space. can also be equipped with the partial order given by Moreover, in [28], it is proved that is regular.

Now, we will prove the following result.

Theorem 3.1. Suppose that the following hypotheses hold:
(i) and are continuous,
(ii) for all ,
(iii) there exist such that where
and , for every and and .
Then, the integral equations (3.1) have a solution .

Proof. Define by Now, we will prove that and are weakly increasing. From (ii), for all , we have Similarly, Then, we have and for all . This implies that and are weakly increasing.
Now, for all such that , by (iii), we have Hence Then for all such that .
This implies that, for all