Open Access

Fuzzy fixed point theorems in Hausdorff fuzzy metric spaces

Journal of Inequalities and Applications20142014:201

https://doi.org/10.1186/1029-242X-2014-201

Received: 18 November 2013

Accepted: 29 April 2014

Published: 22 May 2014

Abstract

In this paper, we introduce the concept of fuzzy mappings in Hausdorff fuzzymetric spaces (in the sense of George and Veeramani (Fuzzy Sets Syst.64:395-399, 1994)). We establish the existence of α-fuzzyfixed point theorems for fuzzy mappings in Hausdorff fuzzy metric spaces, whichcan be utilized to derive fixed point theorems for multivalued mappings. We alsogive an illustrative example to support our main result.

Keywords

α-fuzzy fixed point fuzzy mappings fuzzy metric spaces Hausdorff fuzzy metric spaces

1 Introduction

In 1965, Zadeh [1] introduced and studied the concept of a fuzzy set in his seminal paper.Afterward, several researches have extensively developed the concept of fuzzy set,which also include interesting applications of this theory in different fields suchas mathematical programming, modeling theory, control theory, neural network theory,stability theory, engineering sciences, medical sciences, color image processing,etc. The concept of fuzzy metric spaces was introduced initially byKramosil and Michalek [2]. Later on, George and Veeramani [3] modified the notion of fuzzy metric spaces due to Kramosil and Michalek [2] and studied a Hausdorff topology of fuzzy metric spaces. Recently,Gregori et al.[4] gave many interesting examples of fuzzy metrics in the sense of Georgeand Veeramani [3] and have also applied these fuzzy metrics to color image processing.Several researchers proved the fixed point theorems in fuzzy metric spaces such asin [520] and the references therein. In 2004, López and Romaguera [21] introduced the Hausdorff fuzzy metric on a collection of nonempty compactsubsets of a given fuzzy metric spaces. Recently, Kiany and Amini-Harandi [22] proved fixed point and endpoint theorems for multivalued contractionmappings in fuzzy metric spaces.

On the other hand, Heilpern [23] first introduced the concept of fuzzy contraction mappings and proved afixed point theorem for fuzzy contraction mappings in a complete metric linearspaces, which seems to be the first to establish a fuzzy analog of Nadler’scontraction principle [24]. His work opened an avenue for further development of fixed point in thisdirection. Many researchers used different assumptions on various kinds of fuzzymappings and proved several fuzzy fixed point theorems (see [2535]) and references therein.

To the best of our knowledge, there is no discussion so far concerning the fuzzyfixed point theorems for fuzzy mappings in Hausdorff fuzzy metric spaces. The objectof this paper is to study the role of some type of fuzzy mappings to ascertain theexistence of fuzzy fixed point in Hausdorff fuzzy metric spaces. We also presentsome relation of multivalued mappings and fuzzy mappings.

2 Preliminaries

Firstly, we recall some definitions and properties of an α-fuzzy fixedpoint.

Let X be an arbitrary nonempty set. A fuzzy set in X is a functionwith domain X and values in [ 0 , 1 ] . If A is a fuzzy set and x X , then the function-value A ( x ) is called the grade of membership of x inA. F ( X ) stands for the collection of all fuzzy sets inX unless and until stated otherwise.

Definition 2.1 Let X and Y be two arbitrary nonempty sets. Amapping T from the set X into F ( Y ) is said to be a fuzzy mapping.

If X is endowed with a topology, for α [ 0 , 1 ] , the α-level set of A isdenoted by [ A ] α and is defined as follows:
[ A ] α = { x X : A ( x ) α } ; α ( 0 , 1 ] ,
and
[ A ] 0 = { x X : A ( x ) > 0 } ¯ ,

where B ¯ denotes the closure of B in X.

Definition 2.2 Let X be an arbitrary nonempty set, T befuzzy mapping from X into F ( X ) and z X . If there exists α [ 0 , 1 ] such that z [ T z ] α , then a point z is called anα-fuzzy fixed point of T.

The following notations as regards t-norm and fuzzy metric space will beused in the sequel.

Definition 2.3 ([36])

A binary operation : [ 0 , 1 ] 2 [ 0 , 1 ] is a continuous t-norm if it satisfies thefollowing conditions:

(T1) is associative and commutative,

(T2) is continuous,

(T3) a 1 = a for all a [ 0 , 1 ] ,

(T4) a b c d whenever a c and b d for all a , b , c , d [ 0 , 1 ] .

Examples of a continuous t-norm are Lukasievicz t-norm, that is, a L b = max { a + b 1 , 0 } , product t-norm, that is, a P b = a b and minimum t-norm, that is, a M b = min { a , b } .

The concept of fuzzy metric space is defined by George and Veeramani [3] as follows.

Definition 2.4 ([3])

Let X be an arbitrary nonempty set, be a continuous t-norm,and M be a fuzzy set on X 2 × ( 0 , ) . The 3-tuple ( X , M , ) is called a fuzzy metric space if satisfying thefollowing conditions, for each x , y , z X and t , s > 0 ,

(M1) M ( x , y , t ) > 0 ,

(M2) M ( x , y , t ) = 1 if and only if x = y ,

(M3) M ( x , y , t ) = M ( y , x , t ) ,

(M4) M ( x , y , t ) M ( y , z , s ) M ( x , z , t + s ) ,

(M5) M ( x , y , ) : ( 0 , ) [ 0 , 1 ] is continuous.

Remark 2.5 It is worth pointing out that 0 < M ( x , y , t ) < 1 (for all t > 0 ) provided x y (see [37]).

Let ( X , M , ) be a fuzzy metric space. For t > 0 , the open ball B ( x , r , t ) with a center x X and a radius 0 < r < 1 is defined by
B ( x , r , t ) = { y X : M ( x , y , t ) > 1 r } .

A subset A X is called open if for each x A , there exist t > 0 and 0 < r < 1 such that B ( x , r , t ) A . Let τ denote the family of all opensubsets of X. Then τ is a topology on X, called thetopology induced by the fuzzy metric M. This topology is metrizable (see [38]).

Example 2.6 ([3])

Let ( X , d ) be a metric space. Define a b = a b (or a b = min { a , b } ) for all a , b [ 0 , 1 ] , and define M : X 2 × ( 0 , ) [ 0 , 1 ] as
M ( x , y , t ) = t t + d ( x , y )

for all x , y X and t > 0 . Then ( X , M , ) is a fuzzy metric space. We call this fuzzy metricinduced by the metric d the standard fuzzy metric.

Now we give some examples of fuzzy metric space due to Gregori et al.[4].

Example 2.7 ([4])

Let X be a nonempty set, f : X R + be a one-one function and g : R + [ 0 , ) be an increasing continuous function. For fixed α , β > 0 , define M : X 2 × ( 0 , ) [ 0 , 1 ] as
M ( x , y , t ) = ( ( min { f ( x ) , f ( y ) } ) α + g ( t ) ( max { f ( x ) , f ( y ) } ) α + g ( t ) ) β

for all x , y X and t > 0 . Then ( X , M , ) is a fuzzy metric space on X where is the product t-norm.

Example 2.8 ([4])

Let ( X , d ) be a metric space and g : R + [ 0 , ) be an increasing continuous function. Define M : X 2 × ( 0 , ) [ 0 , 1 ] as
M ( x , y , t ) = e ( d ( x , y ) g ( t ) )

for all x , y X and t > 0 . Then ( X , M , ) is a fuzzy metric space on X where is the product t-norm.

Example 2.9 ([4])

Let ( X , d ) be a bounded metric space with d ( x , y ) < k (for all x , y X , where k is fixed constant in ( 0 , ) ) and g : R + ( k , ) be an increasing continuous function. Define afunction M : X 2 × ( 0 , ) [ 0 , 1 ] as
M ( x , y , t ) = 1 d ( x , y ) g ( t )

for all x , y X and t > 0 . Then ( X , M , ) is a fuzzy metric space on X where is a Lukasievicz t-norm.

Definition 2.10 ([3])

Let ( X , M , ) be a fuzzy metric space.
  1. (1)

    A sequence { x n } in X is said to be convergent to a point x X if lim n M ( x n , x , t ) = 1 for all t > 0 .

     
  2. (2)

    A sequence { x n } in X is called a Cauchy sequence if, for each 0 < ϵ < 1 and t > 0 , there exists n 0 N such that M ( x n , x m , t ) > 1 ϵ for each n , m n 0 .

     
  3. (3)

    A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.

     
  4. (4)

    A fuzzy metric space in which every sequence has a convergent subsequence is said to be compact.

     

Lemma 2.11 ([6])

Let ( X , M , ) be a fuzzy metric space. For all x , y X , M ( x , y , ) is non-decreasing function.

If ( X , M , ) is a fuzzy metric space, then the mapping Mis continuous on X 2 × ( 0 , ) , that is, if { x n } , { y n } X are sequences such that { x n } M x X , { y n } M y X and { t n } ( 0 , ) verifies { t n } t ( 0 , ) then { M ( x n , y n , t n ) } M ( x , y , t ) .

Lemma 2.12 ([21])

If ( X , M , ) be a fuzzy metric space, then M is a continuous function on X 2 × ( 0 , ) .

In 2004, Rodriguez-López and Romaguera [21] introduced the notion for Hausdorff fuzzy metric of a given fuzzy metricspace ( X , M , ) on K M ( X ) , where K M ( X ) denotes the set of its nonempty compact subsets.

Definition 2.13 ([21])

Let ( X , M , ) be a fuzzy metric space. The Hausdorff fuzzy metric H M : ( K M ( X ) ) 2 × ( 0 , ) is defined by
H M ( A , B , t ) : = min { inf x A ( sup y B M ( x , y , t ) ) , inf y B ( sup x A M ( x , y , t ) ) }

for all A , B K M ( X ) and t > 0 .

Lemma 2.14 ([21])

Let ( X , M , ) be a fuzzy metric space. Then the 3-tuple ( K M ( X ) , H M , ) is a fuzzy metric space.

Lemma 2.15 ([21])

Let ( X , M , ) be a fuzzy metric space and t > 0 be fixed. If A and B are nonempty compact subsets of X and x A , then there exists a point y B such that
sup b B M ( x , b , t ) = M ( x , y , t ) .

3 Main result

In this section, we establish the existence theorem of fuzzy fixed point forα-fuzzy mapping in Hausdorff fuzzy metric and reduce our result tometric space. The following lemma is essential in proving our main result.

Lemma 3.1 Let ( X , M , ) be a fuzzy metric space and { x n } is a sequence in X such that for all n N ,
M ( x n + 1 , x n + 2 , k t ) M ( x n , x n + 1 , t ) ,
where 0 < k < 1 . Suppose that
lim n i = n M ( x 1 , x 2 , t h i ) = 1
(1)

for all t > 0 and h > 1 . Then { x n } is a Cauchy sequence.

Proof It follows proof similar to the proof of Lemma 1 of Kiany andAmini-Harandi [22]. Then, in order to avoid repetition, the details areomitted. □

Now we are ready to prove our main result.

Theorem 3.2 Let ( X , M , ) be a complete fuzzy metric space and α : X ( 0 , 1 ] be a mapping such that [ T x ] α ( x ) is a nonempty compact subset of X for all x X . Suppose that T : X F ( X ) is a fuzzy mapping such that
H M ( [ T x ] α ( x ) , [ T y ] α ( y ) , k t ) M ( x , y , t )
(2)
for all t > 0 , where k ( 0 , 1 ) . If there exist x 0 X and x 1 [ T x 0 ] α ( x 0 ) such that
lim n i = n M ( x 0 , x 1 , t h i ) = 1
(3)

for all t > 0 and h > 1 , then T has an α-fuzzy fixed point.

Proof We start from x 0 X and x 1 [ T x 0 ] α ( x 0 ) under the hypothesis. From the assumption, we have [ T x 1 ] α ( x 1 ) is a nonempty compact subset of X. If [ T x 0 ] α ( x 0 ) = [ T x 1 ] α ( x 1 ) , then x 1 [ T x 1 ] α ( x 1 ) and so x 1 is an α-fuzzy fixed point of Tand the proof is finished. Therefore, we may assume that [ T x 0 ] α ( x 0 ) [ T x 1 ] α ( x 1 ) . Since x 1 [ T x 0 ] α ( x 0 ) and [ T x 1 ] α ( x 1 ) is a nonempty compact subset of X then byLemma 2.15 and condition (2), there exists x 2 [ T x 1 ] α ( x 1 ) satisfying
M ( x 1 , x 2 , k t ) = sup x 2 [ T x 1 ] α ( x 1 ) M ( x 1 , x 2 , k t ) H M ( [ T x 0 ] α ( x 0 ) , [ T x 1 ] α ( x 1 ) , k t ) M ( x 0 , x 1 , t ) .
If [ T x 1 ] α ( x 1 ) = [ T x 2 ] α ( x 2 ) , then x 2 [ T x 2 ] α ( x 2 ) . This implies that x 2 is an α-fuzzy fixed point of Tand then the proof is finished. Therefore, we may assume that [ T x 1 ] α ( x 1 ) [ T x 2 ] α ( x 2 ) . Since x 2 [ T x 1 ] α ( x 1 ) and [ T x 2 ] α ( x 2 ) is a nonempty compact subset of X, by usingLemma 2.15 and condition (2), there exists x 3 [ T x 2 ] α ( x 2 ) satisfying
M ( x 2 , x 3 , k t ) = sup x 3 [ T x 2 ] α ( x 2 ) M ( x 2 , x 3 , k t ) H M ( [ T x 1 ] α ( x 1 ) , [ T x 2 ] α ( x 2 ) , k t ) M ( x 1 , x 2 , t ) .
By induction, we can construct the sequence { x n } in X such that x n [ T x n 1 ] α ( x n 1 ) and
M ( x n , x n + 1 , k t ) M ( x n 1 , x n , t )

for all n N . From Lemma 3.1, we get { x n } is a Cauchy sequence. Since ( X , M , ) is a complete fuzzy metric space, there exists x X such that lim n x n = x , which means lim n M ( x n , x , t ) = 1 , for each t > 0 .

Now we claim that x [ T x ] α ( x ) . Since
H M ( [ T x n ] α ( x n ) , [ T x ] α ( x ) , k t ) M ( x n , x , t )
and lim n M ( x n , x , t ) = 1 , then for each t > 0 , we get
lim n H M ( [ T x n ] α ( x n ) , [ T x ] α ( x ) , k t ) = 1 .
(4)
This implies that
lim n sup x [ T x ] α ( x ) M ( x n + 1 , x , t ) = 1
and thus there exists a sequence { x n } in [ T x ] α ( x ) such that
lim n M ( x n , x n , t ) = 1 ,
for each t > 0 . For each n N , we have
M ( x n , x , t ) M ( x n , x n , t 2 ) M ( x n , x , t 2 ) .
Since lim n M ( x n , x n , t 2 ) = 1 and lim n M ( x n , x , t 2 ) = 1 , we get
lim n M ( x n , x , t ) = 1 ,

that is, lim n x n = x . It follows from [ T x ] α ( x ) being a compact subset of X and x n [ T x ] α ( x ) that x [ T x ] α ( x ) . Therefore, x is an α-fuzzyfixed point of T. This completes the proof. □

Next, we apply Theorem 3.2 to α-fuzzy fixed point theorems inmetric space. Before we study the following results, we give the followingnotation.

Let ( X , d ) be a metric space and K ( X ) denote the collection of all nonempty compact subsetsof X. For A , B K ( X ) , we denote
H ( A , B ) : = max { sup x A ( inf y B d ( x , y ) ) , sup y B ( inf x A d ( x , y ) ) } .

The function H is called the Hausdorff metric. Further, it is well knownthat ( K ( X ) , H ) is a metric spaces.

Corollary 3.3 Let ( X , d ) be a complete metric space and α : X ( 0 , 1 ] be a mapping such that [ T x ] α ( x ) is a nonempty compact subset of X for all x X . Suppose that T : X F ( X ) be a fuzzy mapping such that
H ( [ T x ] α ( x ) , [ T y ] α ( y ) ) k d ( x , y )
(5)

for all t > 0 , where k ( 0 , 1 ) . Then T has an α-fuzzy fixed point.

Proof Let ( X , M , ) be standard fuzzy metric space induced by the metricd with a b = a b . Now we show that the conditions of Theorem 3.2are satisfied. Since ( X , d ) is a complete metric space then ( X , M , ) is complete. It is easy to see that ( X , M , ) satisfies (3). From Proposition 3 in [21], for each nonempty compact subset of X, we have
H M ( A , B , t ) = t t + H ( A , B ) .
(6)
By the above equality, we have
H M ( [ T x ] α ( x ) , [ T y ] α ( y ) , k t ) = k t k t + H ( [ T x ] α ( x ) , [ T y ] α ( y ) ) k t k ( t + d ( x , y ) ) t t + d ( x , y ) = M ( x , y , t ) ,

for each t > 0 and each x , y X . Therefore, the conclusion follows fromTheorem 3.2. □

Next, we give an example to support the validity of our results.

Example 3.4 Let X = { 1 , 2 , 3 } and define metric d : X × X R by
d ( x , y ) = { 0 , x = y , 1 2 , x y  and  x , y { 1 , 3 } , 1 , x y  and  x , y { 1 , 2 } , 6 11 , x y  and  x , y { 2 , 3 } .
It is easy to see that ( X , d ) is a complete metric space. Denote a b = a b (or a b = min { a , b } ) for all a , b [ 0 , 1 ] and
M ( x , y , t ) = t t + d ( x , y )
for all x , y X and t > 0 . Then we find that ( X , M , ) is a complete fuzzy metric space. Define the fuzzymapping T : X F ( X ) by
( T 1 ) ( t ) = ( T 2 ) ( t ) { 3 4 , t = 1 , 0 , t = 2 , 3 , ( T 3 ) ( t ) = { 0 , t = 1 , 3 , 3 4 , t = 2 .
Define α : X ( 0 , 1 ] by α ( x ) = 3 4 for all x X . Now we obtain
[ T x ] 3 4 = { { 1 } , x = 1 , 2 { 2 } , x = 3 .
For x , y X , we get
H ( [ T x ] 3 4 , [ T y ] 3 4 ) = { 0 , x = y , H ( { 1 } , { 1 } ) = 0 , x y  and  x = 1 , y = 2 , H ( { 1 } , { 2 } ) = 1 , x y  and  x { 1 , 2 } , y = 3 , H ( { 2 } , { 1 } ) = 1 , x y  and  x = 3 , y { 1 , 2 } .
By a simple calculation, we get
H M ( [ T x ] α ( x ) , [ T y ] α ( y ) , k t ) M ( x , y , t )

for all t > 0 , where k = 1 4 . Therefore all conditions of Theorem 3.2 holdand thus we can claim the existence of a point z X such that z [ T z ] α ( z ) , that is, we have an α-fuzzy fixedpoint of T. Thus z = 1 is an α-fuzzy fixed point ofT.

Remark 3.5 From Example 3.4, we have
H ( [ T 1 ] 3 4 , [ T 3 ] 3 4 ) = H ( { 1 } , { 2 } ) = 1 > α 2 = α d ( 1 , 3 )

or H ( [ T 2 ] 3 4 , [ T 3 ] 3 4 ) = H ( { 1 } , { 2 } ) = 1 > 6 α 11 = α d ( 2 , 3 ) for all α [ 0 , 1 ) . Therefore, Corollary 3.3 is not applicable toclaim the existence of an α-fuzzy fixed point of T.

Here, we study some relations of multivalued mappings and fuzzy mappings. Indeed, weindicate that Corollary 3.3 can be utilized to derive a fixed point for amultivalued mapping.

Corollary 3.6 Let ( X , d ) be a complete metric space and G : X K ( X ) be multivalued mapping such that for all x , y X , we have
H ( G x , G y ) k d ( x , y ) , where  k ( 0 , 1 ) .
(7)

Then there exists ν X such that ν G ν .

Proof Let α : X ( 0 , 1 ] be an arbitrary mapping and T : X F ( X ) be defined by
( T x ) ( t ) = { α ( x ) , t G x , 0 , t G x .
By a routine calculation, we obtain
[ T x ] α ( x ) = { t : ( T x ) ( t ) α ( x ) } = G x .

Now condition (7) becomes condition (5). Therefore, Corollary 3.3 can beapplied to obtain ν X such that ν [ T ν ] α ( ν ) = G ν . This implies that the multivalued mapping Ghas a fixed point. This completes the proof. □

4 Conclusions

In the present work we introduced a new concept of fuzzy mappings in the Hausdorfffuzzy metric space on compact sets, which is a partial generalization of fuzzycontractive mappings in the sense of George and Veeramani. Also, we derived theexistence of α-fuzzy fixed point theorems for fuzzy mappings in theHausdorff fuzzy metric space. Moreover, we reduced our result from fuzzy mappings inHausdorff fuzzy metric spaces to fuzzy mappings in metric space.

Finally, we showed some relation of multivalued mappings and fuzzy mappings, whichcan be utilized to derive fixed point for multivalued mappings.

Declarations

Acknowledgements

The authors were supported by the Higher Education Research Promotion andNational Research University Project of Thailand, Office of the Higher EducationCommission (NRU2557). Moreover, the third author is grateful to Department ofMathematics, Faculty of Science, King Mongkut’s University of technologyThonburi (KMUTT) for providing the opportunity for him to attend theInternational Conference Anatolian Communications in Nonlinear Analysis(ANCNA2013) which was held in July 2013 in Bolu, Turkey. We are also grateful toProfessor Dr. Erdal Karapinar for the kind hospitality.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT)
(2)
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center

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© Phiangsungnoen et al.; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permitsunrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly credited.