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Fuzzy fixed point theorems in Hausdorff fuzzy metric spaces
Journal of Inequalities and Applications volume 2014, Article number: 201 (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.64:395399, 1994)). We establish the existence of αfuzzyfixed 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.
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 [5–20] 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 AminiHarandi [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 [25–35]) 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 αfuzzy 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 andx\in X, then the functionvalue A(x) is called the grade of membership of x inA. \mathcal{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 \mathcal{F}(Y) is said to be a fuzzy mapping.
If X is endowed with a topology, for \alpha \in [0,1], the αlevel set of A isdenoted by {[A]}_{\alpha} and is defined as follows:
and
where \overline{B} denotes the closure of B in X.
Definition 2.2 Let X be an arbitrary nonempty set, T befuzzy mapping from X into \mathcal{F}(X) and z\in X. If there exists \alpha \in [0,1] such that z\in {[Tz]}_{\alpha}, then a point z is called anαfuzzy fixed point of T.
The following notations as regards tnorm and fuzzy metric space will beused in the sequel.
Definition 2.3 ([36])
A binary operation \ast :{[0,1]}^{2}\to [0,1] is a continuous tnorm if it satisfies thefollowing conditions:
(T1) ∗ is associative and commutative,
(T2) ∗ is continuous,
(T3) a\ast 1=a for all a\in [0,1],
(T4) a\ast b\le c\ast d whenever a\le c and b\le d for all a,b,c,d\in [0,1].
Examples of a continuous tnorm are Lukasievicz tnorm, that is,a{\ast}_{L}b=max\{a+b1,0\}, product tnorm, that is,a{\ast}_{P}b=ab and minimum tnorm, that is,a{\ast}_{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 tnorm,and M be a fuzzy set on {X}^{2}\times (0,\mathrm{\infty}). The 3tuple (X,M,\ast ) is called a fuzzy metric space if satisfying thefollowing conditions, for each x,y,z\in 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)\ast M(y,z,s)\le M(x,z,t+s),
(M5) M(x,y,\cdot ):(0,\mathrm{\infty})\to [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\ne y (see [37]).
Let (X,M,\ast ) be a fuzzy metric space. For t>0, the open ball B(x,r,t) with a center x\in X and a radius 0<r<1 is defined by
A subset A\subset X is called open if for each x\in A, there exist t>0 and 0<r<1 such that B(x,r,t)\subset 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\ast b=ab (or a\ast b=min\{a,b\}) for all a,b\in [0,1], and define M:{X}^{2}\times (0,\mathrm{\infty})\to [0,1] as
for all x,y\in X and t>0. Then (X,M,\ast ) 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\to {\mathbb{R}}^{+} be a oneone function and g:{\mathbb{R}}^{+}\to [0,\mathrm{\infty}) be an increasing continuous function. For fixed\alpha ,\beta >0, define M:{X}^{2}\times (0,\mathrm{\infty})\to [0,1] as
for all x,y\in X and t>0. Then (X,M,\ast ) is a fuzzy metric space on X where ∗is the product tnorm.
Example 2.8 ([4])
Let (X,d) be a metric space and g:{\mathbb{R}}^{+}\to [0,\mathrm{\infty}) be an increasing continuous function. DefineM:{X}^{2}\times (0,\mathrm{\infty})\to [0,1] as
for all x,y\in X and t>0. Then (X,M,\ast ) is a fuzzy metric space on X where ∗is the product tnorm.
Example 2.9 ([4])
Let (X,d) be a bounded metric space withd(x,y)<k (for all x,y\in X, where k is fixed constant in(0,\mathrm{\infty})) and g:{\mathbb{R}}^{+}\to (k,\mathrm{\infty}) be an increasing continuous function. Define afunction M:{X}^{2}\times (0,\mathrm{\infty})\to [0,1] as
for all x,y\in X and t>0. Then (X,M,\ast ) is a fuzzy metric space on X where ∗is a Lukasievicz tnorm.
Definition 2.10 ([3])
Let (X,M,\ast ) be a fuzzy metric space.

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

(2)
A sequence \{{x}_{n}\} in X is called a Cauchy sequence if, for each 0<\u03f5<1 and t>0, there exists {n}_{0}\in \mathbb{N} such that M({x}_{n},{x}_{m},t)>1\u03f5 for each n,m\ge {n}_{0}.

(3)
A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.

(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,\ast )be a fuzzy metric space. For allx,y\in X, M(x,y,\cdot )is nondecreasing function.
If (X,M,\ast ) is a fuzzy metric space, then the mapping Mis continuous on {X}^{2}\times (0,\mathrm{\infty}), that is, if \{{x}_{n}\},\{{y}_{n}\}\subseteq X are sequences such that \{{x}_{n}\}\stackrel{M}{\to}x\in X, \{{y}_{n}\}\stackrel{M}{\to}y\in X and \{{t}_{n}\}\subset (0,\mathrm{\infty}) verifies \{{t}_{n}\}\to t\in (0,\mathrm{\infty}) then \{M({x}_{n},{y}_{n},{t}_{n})\}\to M(x,y,t).
Lemma 2.12 ([21])
If(X,M,\ast )be a fuzzy metric space, then M is a continuous function on{X}^{2}\times (0,\mathrm{\infty}).
In 2004, RodriguezLópez and Romaguera [21] introduced the notion for Hausdorff fuzzy metric of a given fuzzy metricspace (X,M,\ast ) on {\mathcal{K}}_{M}(X), where {\mathcal{K}}_{M}(X) denotes the set of its nonempty compact subsets.
Definition 2.13 ([21])
Let (X,M,\ast ) be a fuzzy metric space. The Hausdorff fuzzy metric{H}_{M}:{({\mathcal{K}}_{M}(X))}^{2}\times (0,\mathrm{\infty}) is defined by
for all A,B\in {\mathcal{K}}_{M}(X) and t>0.
Lemma 2.14 ([21])
Let(X,M,\ast )be a fuzzy metric space. Then the 3tuple({\mathcal{K}}_{M}(X),{H}_{M},\ast )is a fuzzy metric space.
Lemma 2.15 ([21])
Let(X,M,\ast )be a fuzzy metric space andt>0be fixed. If A and B are nonempty compact subsets of X andx\in A, then there exists a pointy\in Bsuch that
3 Main result
In this section, we establish the existence theorem of fuzzy fixed point forαfuzzy 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,\ast )be a fuzzy metric space and\{{x}_{n}\}is a sequence in X such that for alln\in \mathbb{N},
where0<k<1. Suppose that
for allt>0andh>1. Then\{{x}_{n}\}is a Cauchy sequence.
Proof It follows proof similar to the proof of Lemma 1 of Kiany andAminiHarandi [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,\ast )be a complete fuzzy metric space and\alpha :X\to (0,1]be a mapping such that{[Tx]}_{\alpha (x)}is a nonempty compact subset of X for allx\in X. Suppose thatT:X\to \mathcal{F}(X)is a fuzzy mapping such that
for allt>0, wherek\in (0,1). If there exist{x}_{0}\in Xand{x}_{1}\in {[T{x}_{0}]}_{\alpha ({x}_{0})}such that
for allt>0andh>1, then T has an αfuzzy fixed point.
Proof We start from {x}_{0}\in X and {x}_{1}\in {[T{x}_{0}]}_{\alpha ({x}_{0})} under the hypothesis. From the assumption, we have{[T{x}_{1}]}_{\alpha ({x}_{1})} is a nonempty compact subset of X. If{[T{x}_{0}]}_{\alpha ({x}_{0})}={[T{x}_{1}]}_{\alpha ({x}_{1})}, then {x}_{1}\in {[T{x}_{1}]}_{\alpha ({x}_{1})} and so {x}_{1} is an αfuzzy fixed point of Tand the proof is finished. Therefore, we may assume that {[T{x}_{0}]}_{\alpha ({x}_{0})}\ne {[T{x}_{1}]}_{\alpha ({x}_{1})}. Since {x}_{1}\in {[T{x}_{0}]}_{\alpha ({x}_{0})} and {[T{x}_{1}]}_{\alpha ({x}_{1})} is a nonempty compact subset of X then byLemma 2.15 and condition (2), there exists {x}_{2}\in {[T{x}_{1}]}_{\alpha ({x}_{1})} satisfying
If {[T{x}_{1}]}_{\alpha ({x}_{1})}={[T{x}_{2}]}_{\alpha ({x}_{2})}, then {x}_{2}\in {[T{x}_{2}]}_{\alpha ({x}_{2})}. This implies that {x}_{2} is an αfuzzy fixed point of Tand then the proof is finished. Therefore, we may assume that{[T{x}_{1}]}_{\alpha ({x}_{1})}\ne {[T{x}_{2}]}_{\alpha ({x}_{2})}. Since {x}_{2}\in {[T{x}_{1}]}_{\alpha ({x}_{1})} and {[T{x}_{2}]}_{\alpha ({x}_{2})} is a nonempty compact subset of X, by usingLemma 2.15 and condition (2), there exists {x}_{3}\in {[T{x}_{2}]}_{\alpha ({x}_{2})} satisfying
By induction, we can construct the sequence \{{x}_{n}\} in X such that {x}_{n}\in {[T{x}_{n1}]}_{\alpha ({x}_{n1})} and
for all n\in \mathbb{N}. From Lemma 3.1, we get \{{x}_{n}\} is a Cauchy sequence. Since (X,M,\ast ) is a complete fuzzy metric space, there existsx\in X such that {lim}_{n\to \mathrm{\infty}}{x}_{n}=x, which means {lim}_{n\to \mathrm{\infty}}M({x}_{n},x,t)=1, for each t>0.
Now we claim that x\in {[Tx]}_{\alpha (x)}. Since
and {lim}_{n\to \mathrm{\infty}}M({x}_{n},x,t)=1, then for each t>0, we get
This implies that
and thus there exists a sequence \{{x}_{n}^{\prime}\} in {[Tx]}_{\alpha (x)} such that
for each t>0. For each n\in \mathbb{N}, we have
Since {lim}_{n\to \mathrm{\infty}}M({x}_{n}^{\prime},{x}_{n},\frac{t}{2})=1 and {lim}_{n\to \mathrm{\infty}}M({x}_{n},x,\frac{t}{2})=1, we get
that is, {lim}_{n\to \mathrm{\infty}}{x}_{n}^{\prime}=x. It follows from {[Tx]}_{\alpha (x)} being a compact subset of X and{x}_{n}^{\prime}\in {[Tx]}_{\alpha (x)} that x\in {[Tx]}_{\alpha (x)}. Therefore, x is an αfuzzyfixed point of T. This completes the proof. □
Next, we apply Theorem 3.2 to αfuzzy fixed point theorems inmetric space. Before we study the following results, we give the followingnotation.
Let (X,d) be a metric space and \mathcal{K}(X) denote the collection of all nonempty compact subsetsof X. For A,B\in \mathcal{K}(X), we denote
The function H is called the Hausdorff metric. Further, it is well knownthat (\mathcal{K}(X),H) is a metric spaces.
Corollary 3.3 Let(X,d)be a complete metric space and\alpha :X\to (0,1]be a mapping such that{[Tx]}_{\alpha (x)}is a nonempty compact subset of X for allx\in X. Suppose thatT:X\to \mathcal{F}(X)be a fuzzy mapping such that
for allt>0, wherek\in (0,1). Then T has an αfuzzy fixed point.
Proof Let (X,M,\ast ) be standard fuzzy metric space induced by the metricd with a\ast b=ab. Now we show that the conditions of Theorem 3.2are satisfied. Since (X,d) is a complete metric space then(X,M,\ast ) is complete. It is easy to see that(X,M,\ast ) satisfies (3). From Proposition 3 in [21], for each nonempty compact subset of X, we have
By the above equality, we have
for each t>0 and each x,y\in 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\times X\to \mathbb{R} by
It is easy to see that (X,d) is a complete metric space. Denotea\ast b=ab (or a\ast b=min\{a,b\}) for all a,b\in [0,1] and
for all x,y\in X and t>0. Then we find that (X,M,\ast ) is a complete fuzzy metric space. Define the fuzzymapping T:X\to \mathcal{F}(X) by
Define \alpha :X\to (0,1] by \alpha (x)=\frac{3}{4} for all x\in X. Now we obtain
For x,y\in X, we get
By a simple calculation, we get
for all t>0, where k=\frac{1}{4}. Therefore all conditions of Theorem 3.2 holdand thus we can claim the existence of a point z\in X such that z\in {[Tz]}_{\alpha (z)}, that is, we have an αfuzzy fixedpoint of T. Thus z=1 is an αfuzzy fixed point ofT.
Remark 3.5 From Example 3.4, we have
or H({[T2]}_{\frac{3}{4}},{[T3]}_{\frac{3}{4}})=H(\{1\},\{2\})=1>\frac{6\alpha}{11}=\alpha d(2,3) for all \alpha \in [0,1). Therefore, Corollary 3.3 is not applicable toclaim the existence of an αfuzzy 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 andG:X\to \mathcal{K}(X)be multivalued mapping such that for allx,y\in X, we have
Then there exists\nu \in Xsuch that\nu \in G\nu.
Proof Let \alpha :X\to (0,1] be an arbitrary mapping and T:X\to \mathcal{F}(X) be defined by
By a routine calculation, we obtain
Now condition (7) becomes condition (5). Therefore, Corollary 3.3 can beapplied to obtain \nu \in X such that \nu \in {[T\nu ]}_{\alpha (\nu )}=G\nu. 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 αfuzzy 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.
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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.
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Phiangsungnoen, S., Sintunavarat, W. & Kumam, P. Fuzzy fixed point theorems in Hausdorff fuzzy metric spaces. J Inequal Appl 2014, 201 (2014). https://doi.org/10.1186/1029242X2014201
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DOI: https://doi.org/10.1186/1029242X2014201