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Journal of Mathematics
Volume 2013 (2013), Article ID 617532, 6 pages
http://dx.doi.org/10.1155/2013/617532
Research Article

Common Fixed Point Theorems of Multivalued Maps in Fuzzy Ultrametric Spaces

Mathematics Department, Al-Lith University College, Umm Al-Qura University, P.O. Box 112, Al-Lith, Makkah Al-Mukarramah 21961, Saudi Arabia

Received 15 April 2013; Revised 31 August 2013; Accepted 14 September 2013

Academic Editor: Feng Feng

Copyright © 2013 A. F. Sayed. 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

In the setting of fuzzy ultrametric spaces, we study common fixed point theorems of multivalued maps. Our results unify, extend, and generalize some related common fixed point theorems of the literature for both ultrametric spaces (Wang and Song (2013), Gajić (2002) and (2001)) and fuzzy metric spaces (Vijayaraju and Sajath (2011)).

1. Introduction

In 1965, Zadeh [1] introduced the theory of fuzzy sets. Many authors introduced the notion of fuzzy metric space in different ways. George and Veeramani [2] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [3] and defined Hausdorff topology in fuzzy metric space. Several authors [410] studied and developed the concept in different directions and proved fixed point theorems in fuzzy metric spaces. Vijayaraju and Sajath [11] extended some previous results and proved some common fixed points theorems for hybrid pair of single and multivalued maps under hybrid contractive conditions. Wang and Song [12] established some results on coincidence and common fixed point for two pairs of multivalued and single-valued maps in ultrametric spaces. In 2009, Savchenko and Zarichnyi [13] introduced the concept of fuzzy ultrametric space. Sedghi and Shobe [14] proved common fixed point theorems for self-maps satisfying contractive conditions on spherically complete fuzzy ultrametric spaces. In this paper, in the setting of fuzzy ultrametric spaces, we study common fixed point theorems of multivalued maps. Our results unify, extend, and generalize some related common fixed point theorems of the literature for both ultrametric spaces [12, 15, 16] and fuzzy metric spaces [11].

2. Preliminaries and Notations

Definition 1 (see [1]). Letbe any nonempty set. A fuzzy setinis a function with domainand values in .

Definition 2 (see [17]). A binary operation is called a continuous triangular norm (shortly t-norm) if it satisfies the following conditions:(1) is associative and commutative,(2) is continuous,(3) for all ,(4) whenever and for all .

Definition 3 (see [2]). The 3-tuple (X, M, *) is called a fuzzy metric space ifis an arbitrary (nonempty) set, is a continuous t-norm, and M is a fuzzy set on satisfying the following conditions, for all and each and :(1),(2) if and only if ,(3),(4),(5) is continuous.
is called a fuzzy metric on. The functions denote the degree of nearness between x and y with respect to, respectively.
Let (,, *) be a fuzzy metric space. For , the open ball with center and radius is defined by
A subset is called open if, for each , there existand such that . Let denote the family of all open subsets of. Then is a topology on induced by the fuzzy metric. This topology is Hausdorff and first countable.

Definition 4 (see [18]). Let () be a metric space. If the metricsatisfies strong triangle inequality thenis called an ultrametric onand the pair () is called an ultrametric space.

Definition 5 (see [18]). An ultrametric space () is said to be spherically complete if every shrinking collection of balls inhas a nonempty intersection.

Definition 6 (see [14]). Let (,, *) be a fuzzy metric space. If the fuzzy metricsatisfies strong triangle inequality thenis called a fuzzy ultrametric onand the 3-tuple (,, *) is called a fuzzy ultrametric space.

Definition 7 (see [14]). A fuzzy ultrametric space (,, *) is said to be spherically complete if every shrinking collection of balls inhas a nonempty intersection.

Remark 8 (see [14]). (i) Letbe an ultrametric onand for all . For each , define for all ,, and . Then fuzzy metricis also a fuzzy ultrametric.
(ii) Let an ultrametric space () be spherically complete and for all . For each , define for all ,, and . Then fuzzy metric space (,, *) is also spherically complete.

3. Main Results

Let (,, *) be a fuzzy ultrametric space. Let denote the set of all nonempty closed bounded subsets of. For every and , we define where .

Remark 9 (see [8]). It is obvious that, for all and , we have(i), whenever ,(ii) if and only if .
It is easy to prove that is a Hausdorff ultrametric space.

Definition 10. Let (,, *) be a fuzzy ultrametric space. An element is said to be a coincidence point of a multivalued map and a single-valued map if . We denote the set of coincidence points of a multivalued mapand single-valued map.

Definition 11. Let (,, *) be a fuzzy ultrametric space, a multivalued map, and if is a single-valued map. andare said to be coincidentally computing at if .

Definition 12. Let (,, *) be a fuzzy ultrametric space. An element is said to be a common fixed point of two multivalued maps and a single-valued map if

Theorem 13. Let (,, *) be a fuzzy ultrametric space. Let be a pair of multivalued maps and a pair of single-valued maps satisfying the following conditions:(a) is spherically complete; (b),,  for all , with , and for ; (c),,,,,; (d)If ,, then there exist pointsandin, such that

Proof. Let denote the closed ball with centered and radius Let be the collection of all the spheres for all . Then the relation is a partial order on .
Consider a totally ordered subfamily of . Since is spherically complete, we have
Let , where and . Then . Hence If , then . Assume that . Let, for all , Sinceis a nonempty closed bounded set, then there exists an element , such that is a nonempty closed bounded set; then there exists an element such that
From the two conditions , and (13), we now have
So ; we have just proved that , for every . Thus is an upper bound in for the family , and hence by Zorn’s lemma, there is a maximal element in , say . There exists an element such that .
Suppose
Since and are nonempty closed bounded sets, then there exist and such that
From the two conditions (b), (c) and (19), we have
From the two conditions (b), (c) and (19)–(22), we have
From (21) and (23), we have
From (22) and (3.7), we have
If then from (24), we have .
Hence . It is a contradiction to the maximality of in , since .
If then from (25), we have .
Hence . It is a contradiction to the maximality of in , since ,
So In addition, .
Using the two conditions (b), (c) and (28), we obtain
Hence . Then the proof is completed.

Theorem 14. Let (,*) be a fuzzy ultrametric space. Let be a pair of multivalued maps and let be a single-valued map satisfying the following conditions:(a) is spherically complete;(b),  for all , with , and for ;(c),,;(d)If ,.
Then ,, and have a coincidence point in . Moreover, if and , and are coincidentally commuting at and , then ,, and have a common fixed point in.

Proof. If in Theorem 13, we obtain that there exist points andinsuch that
As , and are coincidentally commuting at and .
Writing , then . Then we have
Now, since also and are coincidentally commuting at and , so we obtain
Thus, we have proved that ; that is, is a common fixed point of ,, and .

The proof of the following corollary is easy, so it is omitted.

Corollary 15. Let (,*) be a spherically complete fuzzy ultrametric space. Let be a pair of multivalued maps satisfying the following conditions:(a), for all , with , and for ;(b).
Then, there exists a point in such that and .

Now we give an example to illustrate Theorem 13.

Example 16. Let (,*) be a fuzzy ultrametric space in which , for all , and for all .
Define the maps ,,, on as follows:
Define where is a function from satisfying the following conditions:) is increasing in and decreasing in () implies that , for all and .
Then for any , the following inequality is satisfied for all , with , and for all since the LHS of the inequality is .
Clearly all conditions in Theorem 13 are satisfied, so is the unique common fixed point of ,,, and .

4. Conclusion

In this paper, we get coincidence point theorems and common fixed point theorems for two pairs of multivalued and single-valued maps satisfying different contractive conditions on spherically complete fuzzy ultrametric space, which are generalized results for both ultrametric spaces [12, 15, 16] and fuzzy metric spaces [11].

Acknowledgment

The author is thankful to the referees for their valuable suggestions in preparing this paper.

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