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International Journal of Analysis
Volume 2013 (2013), Article ID 763261, 10 pages
http://dx.doi.org/10.1155/2013/763261
Research Article

Employing Common Limit Range Property to Prove Unified Metrical Common Fixed Point Theorems

1Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
2Near Nehru Training Centre, H. No. 274, Nai Basti B-14, Bijnor, Uttar Pradesh 246 701, India

Received 29 January 2013; Accepted 4 April 2013

Academic Editor: Chuanxi Qian

Copyright © 2013 Mohammad Imdad and Sunny Chauhan. 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 purpose of this paper is to emphasize the role of “common limit range property” to ascertain the existence of common fixed point in metric spaces satisfying an implicit function essentially due to the paper of Ali and Imdad (2008). As an application to our main result, we derive a fixed point theorem for four finite families of self-mappings which can be utilized to derive common fixed point theorems involving any finite number of mappings. Our results improve and extend a host of previously known results including the ones contained in the paper of Ali and Imdad (2008). We also furnish some illustrative examples to support our main results.

1. Introduction and Preliminaries

The well-known Banach fixed point theorem is often referred to as Banach contraction principle which appeared in its explicit form in the thesis of Banach in 1922 [1]. Owing to its simplicity and usefulness, it became a very popular and powerful tool in solving existence problems in pure and applied sciences which include biology, medicine, physics, and computer science (see [2, 3]). This natural theorem asserts that every contraction mapping defined on a complete metric space has a unique fixed point and that fixed point can be explicitly obtained as limit of repeated iteration of the mapping at any point of the underlying space. Evidently, every contraction mapping is a continuous but not conversely.

A result on the existence and uniqueness of common fixed point in metric spaces generally involves conditions on commutativity, continuity, and contraction along with a suitable condition on the containment of range of one mapping into the range of other. Hence, one is always required to improve one or more of these conditions to prove a new fixed point theorem. In the recent past, several authors have contributed to the vigorous development of metric fixed point theory (e.g., [418]).

In metric fixed point theory, recently implicit relations are utilized to cover several contraction conditions in one go rather than proving a separate theorem for each contraction condition. The first ever attempt to coin an implicit relation can be traced back to Popa [19]. In 2008, Ali and Imdad [20] introduced a new class of implicit functions which covers several classes of contraction conditions such as Ćirić quasi-contractions, generalized contractions, -type contractions, rational inequalities, and among others. They (i.e., Ali and Imdad [20]) proved some fixed point theorems for weakly compatible mappings satisfying common property (E.A). Thereafter, many researchers utilized various implicit relations to prove a number of fixed point theorems in different settings (e.g., [2127]).

In this paper, utilizing the implicit function of Ali and Imdad [20], we prove some common fixed point theorems for two pairs of weakly compatible mappings employing common limit range property. In process, many known results (especially the ones contained in Ali and Imdad [20]) are enriched and improved. Some related results are also derived besides furnishing illustrative examples.

Definition 1. Let be two self-mappings of a metric space . Then the pair is said to be (1)commuting if , for all , (2)weakly commuting (cf. [28]) if , for all , (3)compatible (cf. [29]) if for each sequence in such that , (4)noncompatible (cf. [30]) if there exists a sequence in such that but is either nonzero or nonexistent, (5)weakly compatible (cf. [31]) if they commute at their coincidence points, that is, whenever , for some .
For more details on systematic comparisons and illustrations of earlier described notions, we refer to Murthy [32].

Definition 2 (cf. [33]). A pair of self-mappings of a metric space is said to satisfy the property (E.A) if there exists a sequence in such that for some .

Hence any pair of compatible as well as noncompatible self-mappings of a metric space satisfy the property (E.A) but a pair of mappings satisfying the property (E.A) need not be noncompatible (see [34, Example 1]).

In 2005, Liu et al. [35] defined the notion of common property (E.A) for hybrid pairs of mappings which contains the property (E.A).

Definition 3. Two pairs and of self-mappings of a metric space are said to satisfy the common property (E.A), if there exist two sequences and in such that for some .

Motivated by the observation that the majority of the fixed point results always require the closedness of the underlying subspaces for the existence of common fixed point under the property (E.A) and common property (E.A), Sintunavarat and Kumam [36] coined the idea of “common limit range property” (also see [37]). Recently, Imdad et al. [38] extended the notion of common limit range property to two pairs of self-mappings which relaxes the requirements on closedness of the underlying subspaces.

Definition 4 (cf. [36]). A pair of self-mappings of a metric space is said to satisfy the common limit range property with respect to , denoted by , if there exists a sequence in such that where .

Thus, one can infer that a pair satisfying the property (E.A) along with closedness of the subspace always enjoys the property with respect to the mapping (see [38, Examples 2.16-2.17]).

Definition 5. Two pairs and of self-mappings of a metric space are said to satisfy the common limit range property with respect to mappings and , denoted by , if there exist two sequences and in such that where .

Definition 6 (cf. [39]). Two families of self-mappings and are said to be pairwise commuting if (1) for all , (2) for all , (3) for all and .

2. Implicit Functions

In an attempt to cover Ćirić’s quasi-contractions, Ali and Imdad [20] introduced a new class of implicit functions which includes a variety of contraction classes such as Ćirić quasi-contractions, generalized contractions, -type contractions, rational inequalities, and among others.

Let be the family of lower semicontinuous functions satisfying the following conditions:, for all ,, for all ,, for all .

The following examples furnished in Ali and Imdad’s [20] establishes the utility of the preceeding definition.

Example 7. Define as

Example 8. Define as

Example 9. Define as

Example 10. Define as where and .

Example 11. Define as where and .

Example 12. Define as where and .

Example 13. Define as where and .

Example 14. Define as where is an upper semicontinuous function such that and for all .

Example 15. Define as where is an upper semicontinuous function such that , , for each .

Example 16. Define as where is an upper semicontinuous function such that , , for each .

Example 17. Define as where and .

Example 18. Define as where and .

Example 19. Define as where and .

Example 20. Define as where .

Example 21. Define as where .

Example 22. Define as

Example 23. Define as where .

Example 24. Define as where and .

Example 25. Define as

Example 26. Define as where and .

Apart from earlier stated definitions, still there are many contractive definitions which meet the requirements , and but due to paucity of the space we have not opted to include more examples.

3. Main Results

Firstly, we observe the following.

Lemma 27. Let , and be self-mappings of a metric space . Suppose the following: (1)the pair satisfies the property (or satisfies the property), (2)or , (3)or is a closed subset of , (4)there exists such that for all
Then the pairs and share the property.

Proof. If the pair enjoys the property with respect to mapping , then there exists a sequence in such that where . Since , for each sequence there exists a sequence in such that . Therefore, due to closedness of , where . Thus in all, we have , and as . Now, we show that as . On using inequality (25) with , , we have
Let on contrary that as . Then, on taking limit as in the preceeding inequality, we get or which is a contradiction to . Hence, which shows that the pairs and share the property. This concludes the proof.

Remark 28. In general, the converse of Lemma 27 is not true (see [38, Example 3.5]).

Now, we state and prove our main result for two pairs of weakly compatible mappings satisfying the property.

Theorem 29. Let , and be self-mappings of a metric space satisfying the inequality (25) of Lemma 27. If the pairs and share the property, then and have a coincidence point each. Moreover, and have a unique common fixed point provided that both pairs and are weakly compatible.

Proof. Since the pairs and satisfy the property, there exist two sequences and in such that where . Since , there exists a point such that . We show that . If not, then using inequality (25) with , , we get which on making reduces to or a contradiction to . Hence, . Therefore, is a coincidence point of the pair .
Also , there exists a point such that . We assert that . If not, then using inequality (25) with , , we get which reduces to or a contradiction to . Hence, , which shows that is a coincidence point of the pair .
Since the pair is weakly compatible and , hence . Now, we show that is a common fixed point of the pair . Suppose that , using inequality (25) with , , we have or a contradiction to ; we get which shows that is a common fixed point of the pair .
Also the pair is weakly compatible and , then . Suppose that , then using inequality (25) with , , we have or a contradiction to . Therefore, which shows that is a common fixed point of the pair and in all is a common fixed point of both pairs and . Uniqueness of common fixed point is an easy consequence of the inequality (25) in view of condition . This concludes the proof.

Remark 30. Theorem 29 improves the corresponding results contained in Ali and Imdad’s [20, Theorem 3.1] as closedness of the underlying subspaces is not required.

Now, we present an example which demonstrates the validity of the hypotheses and degree of generality of our main result over comparable ones from the existing literature.

Example 31. Consider equipped with the usual metric. Define four self-mappings , and by
Also, define implicit function as or for all and . By a routine calculation, one can verify the inequality (44) with . If we choose two sequences as , (or , ), then the pairs and satisfy the property:
Also, and . Thus all the conditions of Theorem 29 are satisfied and 5 is a unique common fixed point of the pairs and which also remains a point of coincidence as well. Here, one may notice that all the involved mappings are discontinuous at their unique common fixed point 5.
Notice that the subspaces and are not closed subspaces of ; therefore, Theorem 29 of Ali and Imdad [20] cannot be used in the context of this example which establishes the genuineness of our extension.

Theorem 32. Let , and be self-mappings of a metric space satisfying all the hypotheses of Lemma 27. Then , and have a unique common fixed point provided that both pairs and are weakly compatible.

Proof. In view of Lemma 27, the pairs and share the property so that there exist two sequences and in such that where . The rest of the proof can be completed on the lines of the proof of Theorem 29. This completes the proof.

The following example demonstrates the utility of Theorem 32 over Theorem 29.

Example 33. In the setting of Example 31, replace the self-mappings and by the following besides retaining the rest (i.e., and ):
Then, like earlier example, the pairs and satisfy the property. Consider the implicit function as which implies for all and . By a routine calculation, one can verify inequality (49) with . Also, and . The pairs and commute at 5 which is also their common coincidence point as well. Thus all the conditions of Theorems 32 are satisfied and 5 is a unique common fixed point of the involved mappings , and .
Here, it can be pointed out that Theorem 29 is not applicable to this example as both and are closed subsets of which demonstrates the situational utility of Theorem 32 over Theorem 29.

Corollary 34. The conclusions of Lemma 27, Theorems 29, and 32 remain true if inequality (25) is replaced by one of the following contraction conditions. For all , where .
Consider where .
Consider where .
Consider where and .
Consider where and .
Consider where and .
Consider where and .
Consider where is an upper semicontinuous function such that and for all .
Consider where is an upper semicontinuous function such that , for each .
Consider where is an upper semicontinuous function such that , for each .
In the following contraction conditions, one denotes and : where and , where and , where and , where , where where , where and , where and .

Proof. The proof of each inequality (50)–(67) easily follows from Theorem 29 in view of Examples 726.

Remark 35. Corollary 34 improves and generalizes a multitude of well-known results especially those contained in [12, 14, 29, 30, 35, 4042].
By choosing , and suitably, we can deduce corollaries involving two as well as three self-mappings. For the sake of naturality, we only derive the following corollary involving a pair of self-mappings.

Corollary 36. Let and be self-mappings of a metric space . Suppose the following: (1)the pair satisfies the property, (2)there exists such that for all .
Then has a coincidence point. Moreover, if the pair is weakly compatible, then the pair has a unique common fixed point in .

As an application of Theorem 29, we have the following result involving four finite families of self-mappings.

Theorem 37. Let , , , and be four finite families of self-mappings of a metric space with , , and satisfying the condition (25). Suppose that the pairs and enjoy the property, then and have a point of coincidence each.
Moreover, , and have a unique common fixed point if the families and commute pairwise wherein , , , and .

Proof. The proof of this theorem can be completed on the lines of Theorem 2.2 of Imdad et al. [10, Theorem 2.2].

Remark 38. (1) A result similar to Theorem 37 can be outlined in respect of Theorem 29.
(2) Theorem 37 improves and extends the result of Ćirić et al. [5] and Pathak et al. [25].
Now, we indicate that Theorem 37 can be utilized to derive common fixed point theorems for any finite number of mappings. As a sample, for five mappings, we can derive the following by setting one family of two members while the remaining three of single members.

Corollary 39. Let , and be self-mappings of a metric space . Suppose the following: (1)the pairs and share the property, (2)there exists such that for all .
Then and have a coincidence point each. Moreover, , and have a unique common fixed point provided that both pairs and commute pairwise, that is, , , , .

Similarly, we can derive a common fixed point theorem for six mappings by setting two families of two members while the rest two of single members.

Corollary 40. Let , and be self-mappings of a metric space . Suppose that (1)the pairs and share the property, (2)there exists such that for all .
Then and have a coincidence point each. Moreover, , and have a unique common fixed point provided that both pairs and commute pairwise, that is, , , , , , and .

By setting , , and in Theorem 37, we deduce the following.

Corollary 41. Let , and be self-mappings of a metric space . Suppose the following: (1)the pairs and share the property, (2)there exists such that for all where are fixed positive integers.
Then , and have a unique common fixed point provided that and .

Remark 42. Corollary 41 is a slight but partial generalization of Theorem 29 as the commutativity requirements (i.e., and ) in this corollary are relatively stronger as compared to weak compatibility in Theorem 29.

Remark 43. Results similar to Corollary 41 can be derived from Theorem 29 and Corollary 34.

Remark 44. It may be noticed that the earlier proved results, namely, Theorems 2937 (also Corollaries 3641) remain valid in symmetric space whenever is continuous.

Acknowledgment

The authors wish to dedicate this work to Professor Mohammad Saeed Khan for his crucial role in their initial development in general and his fruitful observations on the first draft of this paper in particular.

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