American Journal of Applied Mathematics
Volume 3, Issue 3-1, June 2015, Pages: 41-45

Fixed Point Theorems for Multivalued Contractive Mappings in Fuzzy Metric Spaces

Basit Ali1, *, Mujahid Abbas2

1Department of Mathematics, University of Management and Technology, Johar Town Lahore, Pakistan

2Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa

Email address:

(B. Ali)
(M. Abbas)

To cite this article:

Basit Ali, Mujahid Abbas. Fixed Point Theorems for Multivalued Contractive Mappings in Fuzzy Metric Spaces. American Journal of Applied Mathematics. Special Issue: Proceedings of the 1st UMT National Conference on Pure and Applied Mathematics (1st UNCPAM 2015). Vol. 3, No. 3-1, 2015, pp. 41-45. doi: 10.11648/j.ajam.s.2015030301.17


Abstract: In this paper, we introduce multivalued contractive mappings of Feng-Liu type in complete fuzzy metric spaces. We prove fixed point theorems for such mappings in the context of fuzzy metric spaces. We provide with an example to show that our results are more general than previously obtained results in the literature.

Keywords: Multivalued Mapping, Upper and Lower Semicontinuous, t-norm, Fuzzy Metric Space


1. Introduction

Due to its applications in mathematics and other related disciplines, Banach contraction principle has been generalized in many directions (for details one can see [1, 3, 4, 5, 8, 14, 19, 24, 25].

The concept of fuzzy sets was initiated by Zadeh [26] in 1965. Fuzzy metric spaces were introduced by Kramosil and Michalek [15]. Romaguera [21] introduced Hausdorff fuzzy metric on a set of nonempty closed and bounded subsets of a given fuzzy metric space. Fixed point theory in fuzzy metric spaces has been studied by a number of authors. For a wide survey we refer to ([6, 7, 11, 12, 16, 17, 18, 20, 23]) and the references therein.

Kiany et al. [14] proved fixed point theorems for multivalued fuzzy contraction maps in fuzzy metric spaces and obtained generalization of Banach contraction theorem in fuzzy metric spaces.

The aim of this paper is to obtain fixed point theorems for multivalued mapping in fuzzy metric spaces. As a results we extended the results given in ([8, 14] and reference therein) in fuzzy metric spaces.

Definition 1.1 [22] A binary operation  is called a   if for all  

       I.         is associative and commutative;

    II.         is continuous;

  III.       

  IV.         whenever  and

Definition 1.2 (compare [15]) A fuzzy metric space is a triple  such that  is a continuous t-norm and  is a fuzzy set in  such that for all

a.  

b.    if and only if  for all

c.  

d.    for all

e.    is left continuous.

The pair  (or simply,  if no confusion arises) is said to be a fuzzy metric on  It is well known and easy to see that for each   is a non-decreasing function on  Each fuzzy metric  on a set  induces a topology  on  which has a base the family of open balls

  

where

Observe that a sequence  converges to  (with respect to  if and only if  for all

It is also well known [13] that every fuzzy metric space  is metrizable, i.e., there exists a metric  on  whose induced topology agrees with  Conversely, if  is a metric space and we defin  by

 and

for all  then  (where  is minimum norm) is a fuzzy metric space and  is called the standard fuzzy metric of  [9]. Moreover, the topology  agrees with the topology induced by

A sequence  in a fuzzy metric space  is said to be a Cauchy sequence if for each  there exists  such that  for all  A fuzzy metric space  is said to be complete ([10]) if every Cauchy sequence converges. A subset  is said to be closed if for each convergent sequence  and  implies  A subset  is said to be compact if each sequence in  has a convergent subsequence. The set of all compact subsets of  will be denoted by .

Lemma 1.3 [11] For all   is nondecreasing.

Definition 1.4 Let  be a fuzzy metric space,  is said to be continuous on  if

whenever  is a sequence in  which converges to a point  that is,

Lemma 1.5 [11]  is a continuous function on

Kiany et al. [14] introduced the following Lemma in fuzzy metric spaces.

Lemma 1.6 ([14]) Let  be a fuzzy metric space satisfying

(1)

for every   and . Suppose  is sequences in  satisfying

for all  and  Then  is a Cauchy sequence.

Lemma 1.7 [21] Let  be a fuzzy metric space. Then, for each   and  there is  such that

Consistent with [21], we recall the notion of Hausdorff fuzzy metric induced by a fuzzy metric  as follows: For , and  define:

for all  where  Then  is called the Hausdorff fuzzy metric induced by the fuzzy metric  The triplet  is called Hausdorff fuzzy metric space.

Definition 1.8 A function  is called lower semicontinuous, if for any  and   implies A function  is called upper semicontinuous, if for any  and   implies  A multivalued mapping  (collection of all nonempty subsets of ) is called upper semicontinuous, if for any  and a neighborhood  of  there is a neighborhood  of  such that for any  we have  A multivalued mapping  (collection of all nonempty subsets of ) is called lower semicontinuous, if for any  and a neighborhood ,  there is a neighborhood  of  such that for any  we have

2. Fixed Point Theorems for Multivalued Contractive Mappings in Fuzzy Metric Spaces

In the following theorem we obtain fixed point for multivalued mapping satisfying a contractive condition.

Let  be a multivalued mapping. Define  for  For a positive constant  we define a set

(2)

Theorem 2.1 Let  be a complete fuzzy metric space and  be a multivalued mapping. If there exist a constant  such that for any  there is  satisfying

(3)

for  Assume that  satisfies (1) for some  in  then  has a fixed point provided  and  is upper semicontinuous.

Proof: Since  by Lemma 1.7  is nonempty for any  and . Let  be arbitrary, there exists  satisfying

and for  there exists  satisfying

Continuing this process, we obtain a sequence  in  such that  satisfying

(4)

On the other hand  gives

(5)

From (4) and (5) we obtain

That is

(6)

Let  then from (6)

(7)

for all  and  Pick the constant , such that  then  and  that is  we get

(8)

for all  and  Also, we have

(9)

Then by Lemma 6, we have

This shows that  is a Cauchy sequence in  Since  is complete, there exist  such that  From (4) and (5), it is clear that  is increasing and hence converges to  Since  is upper semicontinuous, so we have

This implies that  so  Hence by Lemma 7, we have

Kiany et al. [14] gave the following corollary.

Corollary 2.2 [14] Let  be a complete fuzzy metric. Suppose  be a multivalued mapping such that

(10)

for each   and  Furthermore, assume that  satisfies (1) for some  and  Then  has a fixed point.

Remark 2.3 Theorem 2.1 is a generalization of above corollary. Let  satisfies the conditions of above corollary and if  is upper semicontinuous, then from (10) for any   we get

Hence  satisfies all the conditions of Theorem 9, the existence of fixed point has been proved. Following example shows that Theorem 2.1 is an extension of Corollary 2.2.

Example 2.4 Let  

and

for . Then  is a complete metric space where  defined by  Let be defined as given in [8]

Since

which implies that  satisfies (1) Moreover

and

There does not exist any  such that (10) is satisfied. If it exists then

implies  a contradiction. On the other hand

is continuous. There exist  for any  such that

,

that is

So there exist  such that

Then the existence of fixed point follows from Theorem 2.1.

Acknowledgements

The authors are thankful to the anonymous reviewers for their useful suggestions that helped us to improve the presentation of the paper significantly.


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