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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 382094, 12 pages
http://dx.doi.org/10.1155/2012/382094
Research Article

Common Fixed Point Results Using Generalized Altering Distances on Orbitally Complete Ordered Metric Spaces

1Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud, Raipur 492101, India
2Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbia
3Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia

Received 9 December 2011; Accepted 21 March 2012

Academic Editor: Giuseppe Marino

Copyright © 2012 Hemant Kumar Nashine et al. 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 prove the existence of common fixed points for three relatively asymptotically regular mappings defined on an orbitally complete ordered metric space using orbital continuity of one of the involved maps. We furnish a suitable example to demonstrate the validity of the hypotheses of our results.

1. Introduction and Preliminaries

Browder and Petryshyn introduced the concept of asymptotic regularity of a self-map at a point in a metric space.

Definition 1.1 (see [1]). A self-map on a metric space is said to be asymptotically regular at a point if .

Recall that the set is called the orbit of the self-map at the point .

Definition 1.2 (see [2]). A metric space is said to be complete if every Cauchy sequence contained in (for some in ) converges in .

Here, it can be pointed out that every complete metric space is complete for any , but a complete metric space need not be complete.

Definition 1.3 (see [1]). A self-map defined on a metric space is said to be orbitally continuous at a point in if for any sequence (for some ), as implies as .

Clearly, every continuous self-mapping of a metric space is orbitally continuous, but not conversely.

Sastry et al. [3] extended the above concepts to two and three mappings and employed them to prove common fixed point results for commuting mappings. In what follows, we collect such definitions for three maps.

Definition 1.4 (see [3]). Let be three self-mappings defined on a metric space .

(1)If for a point , there exists a sequence in such that , , , then the set is called the orbit of at .(2)The space is said to be -orbitally complete at if every Cauchy sequence in converges in .(3)The map is said to be orbitally continuous at if it is continuous on .(4)The pair is said to be asymptotically regular (in short a.r.) with respect to at if there exists a sequence in such that , , , and as .

On the other side, Khan et al. [4] introduced the notion of an altering distance function, which is a control function that alters distance between two points in a metric space. This notion has been used by several authors to establish fixed point results in a number of subsequent works, some of which are noted in [59]. In [5], Choudhury introduced the concept of a generalized altering distance function in three variables which was further generalized by Rao et al. [10] to four variables and is defined as follows.

Definition 1.5 (see [10]). A function is said to be a generalized altering distance function if(i) is continuous,(ii) is nondecreasing in each variable,(iii). will denote the set of all functions satisfying conditions (i)–(iii).

Simple examples of generalized altering distance functions with four variables are

On the other hand, fixed point theory has developed rapidly in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [11] who presented its applications to matrix equations. Subsequently, Nieto and Rodríguez-López [12] extended this result for nondecreasing mappings and applied it to obtain a unique solution for a first order ordinary differential equation with periodic boundary conditions. Thereafter, several authors obtained many fixed point theorems in ordered metric spaces. For more details see [1320] and the references cited therein.

In this paper, an attempt has been made to derive some common fixed point theorems for three relatively asymptotically regular mappings defined on an orbitally complete ordered metric space, using orbital continuity of one of the involved maps and conditions involving a generalized altering distance function. The presented theorems generalize, extend, and improve some recent results given in [7, 14, 21, 22]. In the hypotheses, we have considered the space as not necessarily complete, the maps , and as not necessarily continuous and the range of and may not be contained in the range of .

2. Results

2.1. Notations and Definitions

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

If is a partially ordered set, then are called comparable if or holds. A subset of is said to be well ordered if every two elements of are comparable. If is such that, for , implies , then the mapping is said to be nondecreasing.

Definition 2.1. Let be a partially ordered set and . (1)[23] The pair is called weakly increasing if and for all .(2)[24] The pair is called partially weakly increasing if for all .(3)[24] The mapping is called a weak annihilator of if for all .(4)[24] The mapping is called dominating if for each .

Note that none of two weakly increasing mappings need to be nondecreasing. There exist some examples to illustrate this fact in [23]. Obviously, the pair is weakly increasing if and only if the ordered pairs and are partially weakly increasing. Following is an example of an ordered pair which is partially weakly increasing but not weakly increasing.

Example 2.2 (see [24]). Let be endowed with usual ordering.

(1)Let be defined by and . Clearly, is partially weakly increasing. But for implies that is not partially weakly increasing.(2)Let be defined by and . Obviously, for all . Thus is a weak annihilator of .(3)Let be defined by . Since for all , is a dominating map.

Definition 2.3 (see [25, 26]). Let be a metric space and . The mappings and are said to be compatible if , whenever is a sequence in such that for some .

Definition 2.4. Let be a nonempty set. Then is called an ordered metric space if(i) is a metric space,(ii) is a partially ordered set.

2.2. Main Results

The first main result is as follows.

Theorem 2.5. Let be an ordered metric space. Let be given mappings satisfying for all (for some ) such that and are comparable, where and   and are generalized altering distance functions (in ) and . We assume the following hypotheses:(i) is a.r. with respect to at ;(ii) is -orbitally complete at ;(iii) and are partially weakly increasing;(iv) and are dominating maps;(v) and are weak annihilators of ;(vi)for a nondecreasing sequence , for all and as imply that for all . Assume either(a) and are compatible; or is orbitally continuous at or(b) and are compatible; or is orbitally continuous at .Then and have a common fixed point. Moreover, the set of common fixed points of , and in is well ordered if and only if it is a singleton.

Proof. Since is a.r. with respect to at in , there exists a sequence in such that By the given assumptions, , and . Thus, for all , we have In view of (i), we have Now, we assert that is a Cauchy sequence in the metric space .
From (2.5), 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 integers , From (2.6) and using the triangular inequality, we get Letting in the above inequality and using (2.5), we obtain Again, the triangular inequality gives us Letting in the above inequality and using (2.5) and (2.8), we get Similarly, we have On the other hand, we have Then, from (2.5), (2.8), and the continuity of , we get by letting in (2.1) Now, using the considered contractive condition (2.1) for and , we have Then, from (2.5), (2.10), (2.11), and the continuity of and , we get by letting in the above inequality Now, combining (2.13) with the above inequality, we get which implies that , which is a contradiction since . Hence is a Cauchy sequence in . Since is -orbitally complete at , there exists some such that as .
Finally, we prove the existence of a common fixed point of the three mappings , and .
We have Suppose that (a) holds. Since are compatible, we have Also, . Now Assume that is orbitally continuous. Passing to the limit as , we obtain so , which implies that Now, and as , so by the assumption we have and (2.1) becomes Passing to the limit as in the above inequality and using (2.21), it follows that which holds unless , so Now, since and as implies that , from (2.1) Passing to the limit as , we have which gives that Therefore, , hence is a common fixed point of , and . The proof is similar when is orbitally continuous.
Similarly, the result follows when condition (b) holds.
Now, suppose that the set of common fixed points of , and in is well ordered. We claim that it cannot contain more than one point. Assume to the contrary that and but . By supposition, we can replace by and by in (2.1) to obtain a contradiction. Hence, . The converse is trivial.

Now, it is easy to state a corollary of Theorem 2.5 involving a contraction of integral type.

Corollary 2.6. Let , and satisfy the conditions of Theorem 2.5, except that condition (2.1) is replaced by the following: there exists a positive Lebesgue integrable function on such that for each and that Then, , and have a common fixed point. Moreover, the set of common fixed points of , and in is well ordered if and only if it is a singleton.

Remark 2.7. If we take for , then for all , and the contractive condition (2.1) becomes which corresponds to the contraction given by Theorem  2.1 in [24] by taking and . Hence, the result of Abbas et al. [24] is covered by Theorem 2.5 for three maps.
Other results could be derived for other choices of and .

As consequences of Theorem 2.5, we may state the following corollaries.

Corollary 2.8. Let be an ordered metric space. Let be given mappings satisfying for all such that and are comparable, where and are generalized altering distance functions (in ) and . We assume the following hypotheses:(i) is a.r. with respect to at ;(ii) is -orbitally complete at ;(iii) or is orbitally continuous at ;(iv) is partially weakly increasing;(v) is a dominating map;(vi) is a weak annihilator of ;(vii) and are compatible.Let for a nondecreasing sequence with   for all , as imply that for all .
Then and have a common fixed point. Moreover, the set of common fixed points of and in is well ordered if and only if it is a singleton.

Corollary 2.9. Let be an ordered metric space. Let be a mapping satisfying for all such that and are comparable, where and are generalized altering distance functions (in ) and . We assume the following hypotheses:(i) is a.r. at some point of ;(ii) is -orbitally complete at ;(iii) is a dominating map. Let for a nondecreasing sequence with   for all , as imply that for all .
Then has a fixed point. Moreover, the set of fixed points of in is well ordered if and only if it is a singleton.

We present an example showing the usage of our results.

Example 2.10. Let the set be equipped with the usual metric and the order defined by Consider the following self-mappings on : Take . Then it is easy to show that and all the conditions (i)–(vi) and (a)-(b) of Theorem 2.5 are fulfilled. Take and . Then contractive condition (2.1) takes the form for . Using substitution , , the last inequality reduces to and can be checked by discussion on possible values for . Hence, all the conditions of Theorem 2.5 are satisfied and have a common fixed point (which is 0).

Remark 2.11. It was shown by examples in [22] that (in similar situations)(1)if the contractive condition is satisfied just on , there might not exist a (common) fixed point;(2)under the given hypotheses (common), fixed point might not be unique in the whole space .

Acknowledgment

The authors thank the referees for their careful reading of the text and for suggestions that helped to improve the exposition of the paper. The second and third authors are thankful to the Ministry of Science and Technological Development of Serbia.

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