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ISRN Applied Mathematics
Volume 2013 (2013), Article ID 935386, 6 pages
http://dx.doi.org/10.1155/2013/935386
Research Article

Common Fixed Point for Generalized Bose-Mukherjee-Type Fuzzy Mappings

Mathematics and Information Technology School, Jiangsu Second Normal University, Nanjing 210013, China

Received 17 May 2013; Accepted 14 July 2013

Academic Editors: M. Rojas-Medar and X. Xue

Copyright © 2013 Ming-liang Song and Zhong-qian Wang. 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 a common fixed point theorem for a pair of generalized Bose-Mukherjee-type fuzzy mappings in a complete metric space. An example is also provided to support the main result presented herein.

1. Introduction and Preliminaries

In many scientific and engineering applications, the fuzzy set concept plays an important role. The concept of fuzzy sets was introduced initially by Zadeh [1] in 1965. Since then, the study of fixed point theorems in fuzzy mathematics had been instigated by Weiss [2] and Butnariu [3]. Heilpern [4] introduced the concept of fuzzy contractive mappings and proved a fixed point theorem for these mappings in metric linear spaces. His result is a generalization of the fixed point theorem for point-to-set maps of Nadler [5]. Afterwards, several fixed point theorems for fuzzy contractive mappings have appeared in the literature (see [613]). Particularly, Vijayaraju and Marudai [6] studied a fixed point result for fuzzy (multivalued) mappings in a metric space . This result [6] is significant as it does not require the condition of approximate quantity for and linearity for . However, Azam and Arshad [7] pointed out that its proof [6] is incorrect and incomplete and presented the right version of this result. In fact, although there exist mistakes in the proof of Theorem 3.1 in [6], its conclusion is correct.

The aim of this work is to establish a common fixed point theorem for a pair of generalized Bose-Mukherjee-type fuzzy mappings in a complete metric space. Also, we give an example to show the validity of our result and by which indicate that our result improves and extends several known results in [6, 7, 14].

Let and be nonempty sets. A multivalued mapping from to , denoted by , is defined to be a function that assigns to each element of a nonempty subset of . Fixed points of the multivalued mapping will be the points such that .

Let be a metric space and let denote the set of all nonempty closed and bounded subsets of . For , define where .

A fuzzy set in is a function with domain and values in . If is a fuzzy set and , then the function value is called the grade of membership of in . The -level set of is denoted by and is defined as follows: Here, denotes the closure of the set . Let be the collection of all fuzzy sets in a metric space . For , means for each .

A mapping from to is called a fuzzy mapping if for each , (sometimes denoted by ) is a fuzzy set on and denotes the degree of membership of in . Let denote the set of all fuzzy sets on such that each of its -level is a nonempty closed bounded subset of .

The following lemmas are needed in the sequel.

Lemma 1 (Nadler [5]). Let be a metric space and ; then (1) for each , ,(2)for each , .

Lemma 2 (Nadler [5]). Let be a metric space and ; then for each and there exists an element such that .

2. Main Results

Lemma 3. Let ,,,, and be five nonnegative real numbers with ,, and either , or ,. Let ,; then .

Proof. If ,, then ,. Note that ; we have ,; that is, . Moreover, it is evident that , which further implies that That is, .
Similarly, if ,, then holds.

Theorem 4. Let be a complete metric space. Let be two generalized Bose-Mukherjee-type fuzzy mappings. Suppose that, for each , there exists such that and are nonempty closed bounded subsets of and for all , where ,,,, and are five nonnegative real numbers with , and either , or ,. Then there exists such that .

Proof. Let , and ,. By Lemma 3, we know that , and . Choosing , by hypotheses, there exists such that is nonempty closed bounded subset of . For convenience, we denote by . Let ; for this there exists such that is nonempty closed bounded subset of . Since , by Lemma 2, there exists such that Since , by the same argument, we can find and such that By induction, we produce a sequence of points of , such that For , applying (4), we obtain It implies that Note that ; we have Similarly, we have Using inductive method, for , by (11) and (12), we can obtain
Next, we prove that the sequence is a Cauchy sequence in . For , we have where . By the similar reasoning process, we can obtain Then, there exists with , for any , such that Since , it follows from Cauchy’s root test that is convergent and hence is a Cauchy sequence in . Since is a complete metric space, then there exists such that as . Then, by (4) and Lemma 1, we have Therefore, and hence as . Thus, .
Similarly, we can prove that . Hence, .
If ,, by the same argument, we can prove that the conclusion holds.

Corollary 5 (Vijayaraju and Marudai [6]). Let be a complete metric space. Let be two fuzzy mappings. Suppose that, for each , there exists such that and are nonempty closed bounded subsets of and for all , where ,,,, and are nonnegative real numbers with and either or . Then, there exists such that .

Proof. If ,, by , we can take . And let ,,,, and; then we have ,,,, and for all , which implies the conditions of Theorem 4 are satisfied. Similarly, we can prove that some cases of , or , or ,, respectively. Therefore, by Theorem 4, the corollary is proved.

Remark 6. Corollary 5 shows that, although there exist mistakes in the proof of Theorem 3.1 in [6], its conclusion is correct. Moreover, Corollary 5 also shows that Theorem 4 in [7] is not the right version of Theorem 3.1 in [6] but the special case of Theorem 3.1 in [6]. In addition, we give a correct proof of Theorem 3.1 in [6].

Corollary 7 (Azam and Arshad [7]). Let be a complete metric space. Let be two fuzzy mappings. Suppose that, for each , there exists such that and are nonempty closed bounded subsets of and for all , where ,,, and are nonnegative real numbers with . Then, there exists such that .

Theorem 8. Let be a complete metric space. Let be two generalized Bose-Mukherjee-type multivalued mappings. Suppose that, for all , where ,,,, and are five nonnegative real numbers with , and either , or ,. Then, there exists such that .

Proof. Let the fuzzy mappings be defined as and , where is the characteristic function on any subset of . Using the facts and for any , it is evident that and satisfy the condition of Theorem 4.

Similarly, as the proof of Corollary 5, from Theorem 8, we can obtain that common fixed point theorem for Bose-Mukherjee-type multivalued mappings in [14].

Corollary 9 (Bose and Mukherjee [14]). Let be a complete metric space. Let be two multivalued mappings. Suppose that, for all , where ,,,, and are nonnegative real numbers with and either or . Then, there exists such that .

Example 10. Let ; is an ordinary metric; then is a complete metric space. Define two fuzzy mappings as follows: Then, we have Now we take ,,,,; then we have , and ,.
Moreover, if and or, then, for all , If and , then, for all , If and , then, for all , If and , then, for all , Hence, the conditions of Theorem 4 are satisfied, and there exists such that . But, for any nonnegative real numbers ,,,, and with , we have for all . Thus, cannot satisfy the general contractive condition .

3. Conclusion

The aim of this work is to establish a common fixed point theorem for a pair of generalized Bose-Mukherjee-type fuzzy mappings in a complete metric space. Also, we give an example to show the validity of our result and by which indicate that our result improves and extends several known results in [6, 7, 14]. Moreover, we give a correct proof of Theorem 3.1 in [6] and point out the conclusion of Theorem 4.1 in [7] is the special case of Theorem 3.1 in [6]. Finally, we hope that this theory would provide a mathematical background to the ongoing work in the problems of scientific and engineering applications.

Acknowledgments

This work was supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant no. 13KJB110004) and Qing Lan Project of Jiangsu Province of China.

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