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Journal of Applied Mathematics
Volume 2011, Article ID 410214, 12 pages
http://dx.doi.org/10.1155/2011/410214
Research Article

Semicompatibility and Fixed Point Theorems for Reciprocally Continuous Maps in a Fuzzy Metric Space

1School of Studies in Mathematics, Vikram University, Madhya Pradesh, Ujjain 456010, India
2Department of Mathematics, Star Academy of Technology and Management, Madhya Pradesh, Indore 453331, India

Received 6 September 2010; Accepted 17 February 2011

Academic Editor: Yongkun Li

Copyright © 2011 V. H. Badshah and Varsha Joshi. 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 aim of this paper is to prove a common fixed point theorem for six mappings on fuzzy metric space using notion of semicompatibility and reciprocal continuity of maps satisfying an implicit relation. We proposed to reanalysis the theorems of Imdad et al. (2002), Popa (2001), Popa (2002) and Singh and Jain (2005).

1. Introduction

The fuzzy theory has become an area of active research for the last forty years. It has a wide range of applications in the field of science and engineering, for example, population dynamics, computer programming, nonlinear dynamical systems, medicine and so forth. The concept of fuzzy sets was introduced initially by Zadeh [1] in 1965. Since then, to use this concept in topology and analysis, many authors have expansively developed the theory of fuzzy sets and application. Following the concept of fuzzy sets, fuzzy metric spaces have been introduced by Kramosil and Michalek [2], and George and Veeramani [3] modified the notion of fuzzy metric spaces with the help of continuous -norms. Later, many authors, for example, [46] proved common fixed point theorems in fuzzy metric spaces. Grabiec’s [7] followed Kramosil and Michalek and obtained the fuzzy version of Banach contraction principle. Vasuki [8] obtained the fuzzy version of common fixed point theorem which had extra conditions. Pant [9] introduced the notion of reciprocal continuity of mappings in metric spaces. Pant and Jha [10] proved an analogue of the result given by Balasubramaniam et al. [11]. Popa [12] proved theorem for weakly compatible noncontinuous mapping using implicit relation. Recently Singh and Jain [13], and Singh and Chauhan [14] have introduced semicompatible, compatible, and weak compatible maps in fuzzy metric space.

The purpose of this paper is to prove a common fixed point theorem in fuzzy metric spaces using weak compatibility, semicompatibility, an implicit relation, and reciprocal continuity. Here, we generalize the result of [15] by(1)increasing the number of self maps from four to six,(2)using the notion of reciprocal continuity.

2. Preliminaries

Definition 2.1. A binary operation is called a continuous -norm if is an abelian topological monoid with unit 1 such that whenever and for all and .

Examples of -norm are and .

Definition 2.2. The 3-tuple is called a fuzzy metric space if is an arbitrary set, is a continuous -norm, and is a fuzzy set in satisfying the following conditions for all and :  ,  , for all if and only if ,  ,  ,   is left continuous.
Note that can be thought of as the degree of nearness between and with respect to .

Remark 2.3. Every metric space induces a fuzzy metric space , where and for all , , for all , , which is called the fuzzy metric space induced by the metric .

Lemma 2.4. For all , let be a nondecreasing function.

Definition 2.5. Let be a fuzzy metric space.(a)The sequence in is said to be convergent to a point if (b)The sequence in is said to be a Cauchy sequence in if (c)The space is said to be complete if every Cauchy sequence in it converges to a point of it.

Remark 2.6. Since is continuous, it follows from (FMS-4) that the limit of a sequence in a fuzzy metric space is unique.
In this paper, is considered to be the fuzzy metric space with condition
  

Lemma 2.7. Let be a sequence in a fuzzy metric space with the condition (FMS-6). If there exists a number such that for all , then is a Cauchy sequence in .

Lemma 2.8. Let and be two self-maps on a complete fuzzy metric space such that for some , for all and , Then, and have a unique common fixed point in .

Definition 2.9. Let and be mappings from a fuzzy metric space into itself. Then, the mappings are said to be weak compatible if they commute at their coincidence points, that is, implies that .

Definition 2.10. Let and be mappings from a fuzzy metric space into itself. Then, the mappings are said to be compatible if whenever is a sequence in such that

Remark 2.11. Compatibility implies weak compatibility. The converse is not true.

Definition 2.12. Let and be mappings from a fuzzy metric space into itself. Then the mappings are said to be semicompatible if whenever is a sequence in such that

Remark 2.13. Semicompatibility implies weak compatibility. The converse is not true.

Proposition 2.14. Let and be self-maps on a fuzzy metric space . If is continuous, then is semicompatible if and only if is compatible.

Remark 2.15. Semicompatibility of the pair does not imply the semicompatibility of as seen in Example 2.16.

Example 2.16. Let and let be the fuzzy metric space with , for all , . Define self-map as follows: Let be the identity map on and . Then, and. Thus, . Hence, is not semicompatible. Again, as is commuting, it is compatible. Further, for any sequence in such that and , we have . Hence, is always semicompatible.

Definition 2.17. Let and be mappings from a fuzzy metric space into itself. Then, the mappings are said to be reciprocally continuous if whenever is a sequence in such that

If and are both continuous, then they are obviously reciprocally continuous, but the converse is not true.

2.1. A Class of Implicit Relations

Let be the set of all real continuous functions

, nondecreasing in first argument and satisfying the following conditions:(i)for , or implies that ;(ii) implies that .

3. Main Result

Theorem 3.1. Let , and be self-mappings of a complete fuzzy metric space such that(a) and ;(b)the pair is semicompatible and is weak compatible;(c)the pair is reciprocally continuous.

For some , there exists such that for all and , Then , and have a unique common fixed point. Furthermore, if the pairs , and are commuting mapping then, , and have a unique common fixed point.

Proof. Let be an arbitrary point in . Since and , there exist such that and . Inductively, we construct the sequences and in such that for . Now putting in (3.1) , , we obtain that is, Using (i), we get Analogously, putting , in (3.2), we have Using (i), we get Thus, from (3.6) and (3.8), for and , we have Hence, by Lemma 2.7, is a Cauchy sequence in , which is complete. Therefore, converges to . The sequences , , , and , being subsequences of , also converge to , that is, The reciprocal continuity of the pair gives The semicompatibility of the pair gives From the uniqueness of the limit in a fuzzy metric space, we obtain that
Step 1. By putting , in (3.1), we obtain Letting tends to infinity and using (3.10) and (3.13), we get As is nondecreasing in first argument, we have Using (ii), we have for all , which gives , that is,
Step 2. As , there exists such that .
Putting , in (3.1), we obtain that Letting tends to infinity and using (3.10), we get Using (i), we have for all , which gives .
Thus, . Therefore, . Since is weak compatible, we get , that is,

Step 3. By putting , in (3.1) and using (3.17) and (3.20), we obtain that is, .
As is nondecreasing in first argument, we have Using (ii), we have for all , which gives .
Thus, Therefore, , that is, is a common fixed point of , and .
Uniqueness. Let be another common fixed point of , and . Then, .
By putting and in (3.1), we get that is, .
As is nondecreasing in first argument, we have Using (ii), we have for all , which gives , that is, . Therefore, is the unique common fixed point of the self-maps , and .
Finally, we need to show that is also a common fixed point of , and . For this, let be the unique common fixed point of both the pairs and . Then, by using commutativity of the pair , , and , we obtain which shows that and are common fixed point of , yielding thereby in the view of uniqueness of the common fixed point of the pair . Similarly, using the commutativity of , , , it can be shown that Now, we need to show that also remains a common fixed point of both the pairs and . For this, put and in (3.1), and using (3.27) and (3.28), we get that is, As is nondecreasing in first argument, we have Using (ii), we obtain which gives , that is, . Similarly, it can be shown that . Thus, is the unique common fixed point of , and . This completes the proof of our theorem.

Since semicompatibility implies weak compatibility, we have the following.

Corollary 3.2. Let , and be self-maps of a complete fuzzy metric space satisfying the conditions (a), (3.1), and (3.2) of the above theorem, and the pairs and are semicompatible and one of the pair, or , is reciprocally continuous. Then, , and have a unique common fixed point. Furthermore, if the pairs , and are commuting mapping, then , and have a unique common fixed point.

If we take  =  = identity map in Theorem 3.1, then we have the following.

Corollary 3.3. Let , and be self-mappings of a complete fuzzy metric space such that(a) and ;(b), are semicompatible commuting pair of maps;(c)the pair or is reciprocally continuous.

For some , there exists such that for all and , Then, , and have a unique common fixed point.

This theorem proves that the theorem of Singh holds even when the pairs are reciprocally continuous.

On taking  = identity mapping in Theorem 3.1, we have the following result for 5 self-maps.

Corollary 3.4. Let , and be self-mappings of a complete fuzzy metric space such that(a),(b) is weak compatible.

For some , there exists such that for all and , Then, , and have a unique common fixed point. Furthermore, if the pairs , , , and are commuting mapping then , and have a unique common fixed point.

If we take  identity mapping in Theorem 3.1, then the conditions (a), (b), and (c) are satisfied trivially, and we get the following result.

Corollary 3.5. Let and be self-mappings of a complete fuzzy metric space such that for some , there exists such that for all and , Then, and have a unique common fixed point in .

Theorem 3.6. Let , and be self-maps on a complete fuzzy metric space satisfying conditions (a), (c), (3.1), and (3.2) of Theorem 3.1, and is compatible, and is weak compatible. Then, , and have a unique common fixed point. Furthermore, if the pairs , , , , , and are commuting mapping then , and have a unique common fixed point.

Proof. As in the proof of Theorem 3.1, the sequence converges to , and (3.10) is satisfied.
The reciprocal continuity of the pair gives The compatibility of the pair gives From the uniqueness of the limit in a fuzzy metric space, we obtain that .
Step 1. By putting , in (3.1), we obtain Letting tends to infinity and using (3.10) and (3.13), we get As is nondecreasing in first argument, we have Using (ii), we have for all , which gives , that is,
Step 2. As , there exists such that .
Putting , in (3.1), we obtain that Letting tends to infinity and using (3.10), we get Using (i), we have for all , which gives . Thus, . Therefore, . Since is weak compatible, we get  , that is,

Step 3. By putting , in (3.1) and using (3.17) and (3.20), we obtain that is, .
As is nondecreasing in first argument, we have Using (ii), we have for all , which gives . Thus,
Therefore, , that is, is a common fixed point of , and .
The rest of the proof is the same as in Theorem 3.1.

Since compatibility implies weak compatibility, we have the following.

Corollary 3.7. Let , and be self-maps on a complete fuzzy metric space satisfying conditions (a), (3.1), and (3.2) of Theorem 3.1, and the pairs and are compatible, and one of the pair, or , is reciprocally continuous. Then, , and have a unique common fixed point in . Furthermore, if the pairs , , , , , and are commuting mapping, then , and have a unique common fixed point.

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