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Journal of Operators
Volume 2013 (2013), Article ID 186910, 11 pages
http://dx.doi.org/10.1155/2013/186910
Research Article

Unified Metrical Common Fixed Point Theorems in 2-Metric Spaces via an Implicit Relation

1Department of Mathematics, R.H. Government Postgraduate College, Kashipur 244 713, U.S. Nagar, Uttarakhand, India
2Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
3Università degli Studi di Palermo, Dipartimento di Matematica e Informatica, Via Archirafi 34, 90123 Palermo, Italy

Received 29 March 2013; Accepted 22 May 2013

Academic Editor: Lingju Kong

Copyright © 2013 Sunny Chauhan 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 some common fixed point theorems for two pairs of weakly compatible mappings in 2-metric spaces via an implicit relation. As an application to our main result, we derive Bryant's type generalized 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. Moreover, we study the existence of solutions of a nonlinear integral equation.

1. Introduction and Preliminaries

In 1963, Gähler [1] initiated the concept of 2-metric space as a natural generalization of a metric space. The topology induced by 2-metric space is called 2-metric topology which is generated by the set of all open spheres with two centers (see [2, 3]). In this course of development, Iséki [4] studied the fixed point theorems in 2-metric spaces. For more references on the recent development of common fixed point theory in 2-metric spaces, we refer readers to [519].

In metric fixed point theory, implicit relations are often 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 function can be traced back to Popa [20]. Recently, Popa et al. [21] proved some interesting fixed point results for weakly compatible mappings in 2-metric spaces satisfying an implicit relation.

In this paper, utilizing the implicit function due to Popa et al. [21], 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 Popa et al. [21, 22]) are enriched and improved. Some related results are also derived. Finally, we study the existence of solutions of a nonlinear integral equation using the presented results.

A 2-metric space is a set equipped with a real valued function on which satisfies the following conditions: For distinct points , there exists a point such that . if at least two of , , are equal., for all ., for all .

The function is called a 2-metric on the set whereas the pair stands for 2-metric space. Geometrically a 2-metric represents the area of a triangle with vertices , , and .

It has been known since Gähler [1] that a 2-metric is a nonnegative continuous function in any one of its three arguments but it does not need that to be continuous in two arguments. A 2-metric is said to be continuous if it is continuous in all of its arguments. Throughout this paper stands for a continuous 2-metric.

Definition 1. A sequence in a 2-metric space is said to be (1) convergent to a point , denoted by , if for all ;(2) Cauchy sequence if for all .

A 2-metric space is said to be complete if every Cauchy sequence in is convergent.

In 1986, Jungck [23] introduced the notion of compatible mappings and utilized the same (as a tool) to improve commutativity conditions due to Sessa [24] in common fixed point theorems. However, the study of common fixed points of noncompatible mappings is also equally interesting which has been initiated by Pant [25, 26]. Jungck [27] introduced the notion of weakly compatible mappings in ordinary metric spaces and proved common fixed point theorems under minimal commutativity requirement. In recent years, using this idea several general common fixed point theorems have been proved in metric spaces.

Definition 2. Let be two self-mappings of a 2-metric space . Then the pair is said to be (1) compatible [28] if for all , whenever is a sequence in such that , for some ;(2) noncompatible [22] if there exists a sequence in such that for some but for at least one is either nonzero or nonexistent;(3) weakly compatible [29] 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 readers to [28, 29].

Inspired by the work of Aamri and El Moutawakil [30], Popa et al. [22] studied the notion of property (E.A) in the settings of 2-metric spaces.

Definition 3. A pair of self-mappings of a 2-metric space is said to satisfy the property (E.A) if there exists a sequence in such that for all .

In 2005, Liu et al. [31] defined the notion of common property (E.A) for hybrid pairs of mappings which contains the property (E.A).

Definition 4. Two pairs and of self-mappings of a 2-metric space are said to satisfy the common property (E.A), if there exist two sequences and in such that for all and some .

Notice that the recent results, contained in Popa et al. [22] proved for weakly compatible mappings under the property (E.A), always require the completeness of the underlying subspace for the existence of common fixed point. In 2011, Sintunavarat and Kumam [32] introduced the notion of “common limit range property” which relaxes the requirement on completeness (or closedness) of the underlying subspaces (also see [3335]). Since then, Imdad et al. [36, 37] extended the notion of common limit range property to two pairs of self-mappings and proved common fixed point theorems in Menger and metric spaces, respectively.

Now we define the notion of common limit range property in 2-metric spaces as follows.

Definition 5. A pair of self-mappings of a 2-metric space is said to satisfy the common limit range property with respect to mapping , denoted by , if there exists a sequence in such that where and for all .

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 [36, Examples 2.16-2.17]).

Definition 6. Two pairs and of self-mappings of a 2-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 and for all .

Definition 7 (see [38]). 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

Let be the set of all lower semicontinuous functions satisfying the following conditions: implies , implies , implies .

The following examples are furnished in Popa et al. [21] establishing the utility of the preceding definition. Another examples can be found in Pathak et al. [39].

Example 8. Define as where .

Example 9. Define as where , , , and .

Example 10. Define as where , , , and .

Example 11. Define as where .

Example 12. Define as where , and .

Example 13. Define as where with at least one nonzero and .

Example 14. Define as where such that .

Example 15. Define as where .

Example 16. Define as where and .

Example 17. Define as where and .

Example 18. Define as where is an increasing upper semicontinuous function with and for each .

Example 19. Define as where is upper semicontinuous and non-decreasing function in each coordinate variable such that for each and with .

Example 20. Define as where is upper semicontinuous and non-decreasing function in each coordinate variable such that for each and with .

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

We begin with the following observation.

Lemma 21. Let , , , and be self-mappings of a 2-metric space . Suppose that (1) the pair satisfies the property (or satisfies the property), (2) (or ), (3) (or ) is a closed subset of , (4) converges for every sequence in whenever converges (or converges for every sequence in whenever converges), (5) there exists such that (for all and ) Then the pairs and share the property.

Proof. Since the pair satisfies the property with respect to mapping , there exists a sequence in such that where . As , for each sequence there exists a sequence in such that . Therefore, due to closedness of , where . Thus in all, we have , , and as . By (4), the sequence converges, and in all, we need to show that as . On using inequality (18) with , , we have Let on contrary as . Then, taking limit as , we get or yielding thereby for all (due to ). Hence , which shows that the pairs and share the property. This concludes the proof.

Remark 22. In general, the converse of Lemma 21 is not true (see [36, Example ]).

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

Theorem 23. Let , , , and be self-mappings of a metric space satisfying the inequality (18) (of Lemma 21). If the pairs and share the property, then and have a coincidence point each. Moreover, , , , and have a unique common fixed point provided both the pairs and are weakly compatible.

Proof. Since the pairs and share the property, there exist two sequences and in such that where . Since , there exists a point such that . We show that . On using inequality (18) with and , we get which on making , reduces to or implying thereby for all (due to ). Hence which shows that is a coincidence point of the pair .
As , there exists a point such that . We assert that . On using inequality (18) with , , we get which reduces to or yielding thereby for all (due to ). Hence , which shows that is a coincidence point of the pair .
Since the pair is weakly compatible and , hence . Now, we assert that is a common fixed point of the pair . On using inequality (18) with , , we have or implying thereby for all (due to ). Hence which shows that is a common fixed point of the pair .
Also the pair is weakly compatible and , then . On using inequality (18) with and , we have or yielding thereby for all (due to ). Therefore, which shows that is a common fixed point of the pair . Hence is a common fixed point of both the pairs and .
To prove the uniqueness, let be another common fixed point of , , , and . On using inequality (18) with and , we have or or implying thereby for all (due to ). Hence . This concludes the proof.

Remark 24. Theorem 23 improves the corresponding result contained in Popa et al. [21, Theorem 4.1] as completeness (or 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 25. Let be a metric space wherein equipped with usual metric . Define a 2-metric on by , for all . Also the self-mappings , , , and are defined by ,
Then we have and . Consider the implicit function given by where . By a routine calculation, one can verify the following inequality for all , and : Now, if we choose two sequences as , (or , ), then the pairs and satisfy the property. In fact where . Thus all the conditions of Theorem 23 are satisfied and 1 is a unique common fixed point of the pairs and which also remains a point of coincidence as well.
Notice that in the context of this example, and are not complete (or closed) subsets of ; therefore, Theorem 4.1 of Popa et al. [21] cannot be used in the context of this example which establishes the genuineness of our extension.

Theorem 26. Let , , , and be self-mappings of a 2-metric space satisfying all the hypotheses of Lemma 21. Then , , , and have a unique common fixed point provided both the pairs and are weakly compatible.

Proof. In view of Lemma 21, the pairs and enjoy 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 23; therefore, we omit the details.

The following example demonstrates the utility of Theorem 26 over Theorem 23.

Example 27. In the setting of Example 25, replace the self-mappings and by the following, besides retaining the rest:
Then, like earlier example, the pairs and enjoy the property. Consider the implicit function given by which yields the inequality for all and . Clearly, inequality (45) holds true. Also, , and the pairs and commute at 1 which is also their common coincidence point as well. Thus all the conditions of Theorem 26 are satisfied and 1 is a unique common fixed point of the involved mappings , , , and .

Here, it can be pointed out that Theorem 23 is not applicable to this example as both , are complete subspaces of ; this demonstrates the situational utility of Theorem 26 over Theorem 23.

Corollary 28. The conclusions of Lemma 21, Theorems 23 and 26 remain true if inequality (18) is replaced by one of the following contractive conditions, for all , where ; where , , and ; where , , and ; where ; where , and ; where with at least one nonzero and ; where such that ; where ; where and ; where and ; where is an increasing upper semicontinuous function with and for each ; where is an upper semicontinuous and non-decreasing function in each coordinate variable such that for each and with ; where is an upper semicontinuous and non-decreasing function in each coordinate variable such that for each and with .

Proof. The proof of each of inequalities (46)–(58) easily follows from Theorem 23 in view of Examples 820.

Remark 29. Corollary 28 improves and generalizes a multitude of well-known results especially those contained in [812, 15, 17, 21, 22, 28, 4044] and others whereas some of these present 2-metric space version of certain existing results of literature (e.g., Chugh and Kumar [45], Ali and Imdad [46], Jeong and Rhoades [47], Hardy and Rogers [48], Lal et al. [49], and others) besides yielding some results which are seeming new to the literature.

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 30. Let and be self-mappings of a 2-metric space . Suppose that (1) the pair satisfies the property, (2)  there exists such that for all and .
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 23, we state a Bryant’s [50] type generalized common fixed point theorem involving four finite families of self-mappings.

Theorem 31. Let , , , and be four finite families of self-mappings of a 2-metric space with , , , and satisfying condition (18). 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 the corresponding theorem of Imdad et al. [38].

Remark 32. Note that (1) a result similar to Theorem 31 can be outlined in respect of Theorem 23; (2) Theorem 31 improves and extends the corresponding results contained in Popa et al. [22].

Now, we indicate that Theorem 31 can be utilized to derive common fixed point theorems for any finite number of mappings. As a sample, we derive the following theorem for five mappings by setting one family of two members while the rest are three, of single members.

Corollary 33. Let , , , , and be self-mappings of a 2-metric space . Suppose that (1) the pairs and share the property, (2) there exists such that for all and .
Then and have a coincidence point each. Moreover, , , , , and have a unique common fixed point provided both pairs and commute pairwise; that is, , , , .

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

Corollary 34. Let , , , , , and be self-mappings of a 2-metric space . Suppose that (1) the pairs and share the property, (2) there exists such that for all and .
Then and have a coincidence point each. Moreover, , , , , , and have a unique common fixed point provided both of the pairs and commute pairwise; that is, , , , , , and .

By setting , , , and in Theorem 31, we deduce the following.

Corollary 35. Let , , , and be self-mappings of a 2-metric space . Suppose that (1) the pairs and share the property, (2) there exists such that for all , and where , , , are fixed positive integers.
Then , , , and have a unique common fixed point provided and .

Remark 36. Corollary 35 is a slight but partial generalization of Theorem 23 as the commutativity requirements (i.e., and ) in this corollary are relatively stronger as compared to weak compatibility in Theorem 23.

Remark 37. Results similar to Corollary 35 can be derived in respect of Theorem 23 and Corollary 28.

4. Application

Inspired by Pathak et al. [51], we study the existence of solutions of a nonlinear integral equation using the results proved in Section 3. Let be the set of continuous real valued functions defined on endowed with the metric given by Clearly, is a complete metric space. Now, consider the integral equation: for all , where with are known, .

Now, we formulate our result.

Theorem 38. Assume that the following hypotheses hold: (a) and ; (b) for all there exists such that , for all ; (c) for all , there exists such that , for all ; (d) ; (e) , for all ;(f) , for all ; (g) there exists a sequence in such that .
Then, the integral equation (64) has a unique solution in .

Proof. We consider the operators defined by It is easy to show that is a solution to (64) if and only if is a coincidence point of and . To establish the existence of such a point, we will use our Corollary 30. Then, we have to check that all the hypotheses of Corollary 30 are satisfied.
Firstly, suppose that is the 2-metric on given by for all . Since is a complete metric space, then also is complete. Next, we show that . In fact, by using hypothesis (f), for all we get Therefore, from hypothesis and containment of ranges, we deduce that the pair satisfies the property. Now, by using hypotheses (a) and (b), for all and , we have Similarly, we get This implies easily Therefore, combining opportunely (67) and (69), we obtain for all . Moreover, using hypothesis (e) and after routine calculations (omitted) we obtain for all . Then, from (70), (71), and hypothesis (d), we conclude that condition (53) is satisfied with , , and .
Now, applying Corollary 30, where inequality (59) is replaced by inequality (53), we obtain the existence of a solution to (64).
On the lines of the proof of Theorem  4.1 in [51], one can show that hypothesis (f) also implies that the pair is weakly compatible. Thus, from Corollary 30 we obtain the uniqueness of the solution to (64).

Acknowledgment

The authors are grateful to all anonymous referees for their fruitful comments on the paper.

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