Fuzzy fixed point theorems in Hausdorff fuzzy metric spaces
© Phiangsungnoen et al.; licensee Springer. 2014
Received: 18 November 2013
Accepted: 29 April 2014
Published: 22 May 2014
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.
In 1965, Zadeh  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 . Later on, George and Veeramani  modified the notion of fuzzy metric spaces due to Kramosil and Michalek  and studied a Hausdorff topology of fuzzy metric spaces. Recently,Gregori et al. gave many interesting examples of fuzzy metrics in the sense of Georgeand Veeramani  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  introduced the Hausdorff fuzzy metric on a collection of nonempty compactsubsets of a given fuzzy metric spaces. Recently, Kiany and Amini-Harandi  proved fixed point and endpoint theorems for multivalued contractionmappings in fuzzy metric spaces.
On the other hand, Heilpern  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 . 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.
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 . If A is a fuzzy set and, then the function-value is called the grade of membership of x inA. 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 is said to be a fuzzy mapping.
where denotes the closure of B in X.
Definition 2.2 Let X be an arbitrary nonempty set, T befuzzy mapping from X into and . If there exists such that , 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 ()
A binary operation is a continuous t-norm if it satisfies thefollowing conditions:
(T1) ∗ is associative and commutative,
(T2) ∗ is continuous,
(T3) for all ,
(T4) whenever and for all .
Examples of a continuous t-norm are Lukasievicz t-norm, that is,, product t-norm, that is, and minimum t-norm, that is,.
The concept of fuzzy metric space is defined by George and Veeramani  as follows.
Definition 2.4 ()
Let X be an arbitrary nonempty set, ∗ be a continuous t-norm,and M be a fuzzy set on . The 3-tuple is called a fuzzy metric space if satisfying thefollowing conditions, for each and ,
(M2) if and only if ,
(M5) is continuous.
Remark 2.5 It is worth pointing out that (for all ) provided (see ).
A subset is called open if for each , there exist and such that . 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 ).
Example 2.6 ()
for all and . Then 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..
Example 2.7 ()
for all and . Then is a fuzzy metric space on X where ∗is the product t-norm.
Example 2.8 ()
for all and . Then is a fuzzy metric space on X where ∗is the product t-norm.
Example 2.9 ()
for all and . Then is a fuzzy metric space on X where ∗is a Lukasievicz t-norm.
Definition 2.10 ()
A sequence in X is said to be convergent to a point if for all .
A sequence in X is called a Cauchy sequence if, for each and , there exists such that for each .
A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.
A fuzzy metric space in which every sequence has a convergent subsequence is said to be compact.
Lemma 2.11 ()
Letbe a fuzzy metric space. For all, is non-decreasing function.
If is a fuzzy metric space, then the mapping Mis continuous on , that is, if are sequences such that , and verifies then .
Lemma 2.12 ()
Ifbe a fuzzy metric space, then M is a continuous function on.
In 2004, Rodriguez-López and Romaguera  introduced the notion for Hausdorff fuzzy metric of a given fuzzy metricspace on , where denotes the set of its nonempty compact subsets.
Definition 2.13 ()
for all and .
Lemma 2.14 ()
Letbe a fuzzy metric space. Then the 3-tupleis a fuzzy metric space.
Lemma 2.15 ()
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.
for alland. Thenis a Cauchy sequence.
Proof It follows proof similar to the proof of Lemma 1 of Kiany andAmini-Harandi . Then, in order to avoid repetition, the details areomitted. □
Now we are ready to prove our main result.
for alland, then T has an α-fuzzy fixed point.
for all . From Lemma 3.1, we get is a Cauchy sequence. Since is a complete fuzzy metric space, there exists such that , which means , for each .
that is, . It follows from being a compact subset of X and that . 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.
The function H is called the Hausdorff metric. Further, it is well knownthat is a metric spaces.
for all, where. Then T has an α-fuzzy fixed point.
for each and each . Therefore, the conclusion follows fromTheorem 3.2. □
Next, we give an example to support the validity of our results.
for all , where . Therefore all conditions of Theorem 3.2 holdand thus we can claim the existence of a point such that , that is, we have an α-fuzzy fixedpoint of T. Thus is an α-fuzzy fixed point ofT.
or for all . 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.
Then there existssuch that.
Now condition (7) becomes condition (5). Therefore, Corollary 3.3 can beapplied to obtain such that . This implies that the multivalued mapping Ghas a fixed point. This completes the proof. □
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.
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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