A Triple Fixed Point Theorem for Multimap in a Hausdorff Fuzzy Metric Space
We obtain two triple fixed point theorems for a multimap in a Hausdorff fuzzy metric space.
1. Introduction and Preliminaries
The concept of fuzzy sets was introduced by Zadeh  in 1965 as a mathematical tool to represent vagueness in every day life. Since then, it was developed extensively by many authors, which also include interesting applications of this theory in diverse areas. To use this concept in topology and analysis, several researchers have defined fuzzy metric spaces in several ways (e.g., [2–4]). George and Veeramani  have modified the concept of fuzzy metric space introduced by Kramosil and Michálek  and also have succeeded in inducing a Hausdroff topology on such fuzzy metric space which is often used in current research these days. Later Grabiec  proved the contraction principle in the setting of fuzzy metric spaces introduced in . Fuzzy metric spaces have many applications, for example, the various concepts of fuzzy topology have already been found in vital applications in quantum particle physics particularly in connection with both string and theory which were studied and formulated by El Naschie  and also most recently Gregori et al.  have furnished several interesting examples of fuzzy metrics in the sense of George and Veeramani  and have also utilized such fuzzy metrics to color image processing. For fixed point theorems in fuzzy metric spaces some of the interesting references are in [2, 5, 8–16].
The theory of set valued maps has applications in control theory, convex optimization, differential inclusions, and economics. The study of fixed points for multivalued contraction mappings using the Hausdorff metric was initiated by Nadler . In 2004, Rodríguez-López and Romaguera  introduced Hausdorff’s fuzzy metric on the set of the nonempty compact subsets of a given fuzzy metric space. Later several authors proved some fixed point theorems for multivalued maps in fuzzy metric spaces (e.g., [19–22]). The existence of fixed points for various multivalued contractive mappings has been studied by many authors under different conditions. For details, we refer the reader to [3, 17, 23–29] and the references therein. In 2006, Gnana Bhaskar and Lakshmikantham  introduced the notion of a coupled fixed point in partially ordered metric spaces, also discussed some problems of the uniqueness of a coupled fixed point, and applied their results to the problems of the existence and uniqueness of a solution for the periodic boundary value problems. In 2011, Samet and Vetro  extended the coupled fixed point theorems for a multivalued mapping and later several authors, namely, Hussain and Alotaibi , Aydi et al. , and Abbas et al.  proved coupled coincidence point theorems in partially ordered metric spaces. Borcut  observed that the coupled fixed points technique cannot solve a system with the following form: and hence Berinde and Borcut  introduced the concept of triple fixed points and obtained a tripled fixed point theorem for a single valued map in partially ordered metric spaces. Moreover, these results could be used to study the existence of solutions of periodic boundary value problem involving .
In this paper, we obtain a triple fixed point theorem for a multimap in a Hausdorff fuzzy metric space and using it, we obtain a common triple fixed point for a multi- and single valued maps.
In the sequel, we need the following.
Definition 1 (see ). A binary operation is a continuous -norm if it satisfies the following conditions: (1) is associative and commutative, (2) is continuous, (3) for all ,(4) whenever and , for each .
Two typical examples of continuous -norm are and .
Definition 2 (see ). A 3-tuple is called a fuzzy metric space if is an arbitrary (nonempty) set, is a continuous -norm, and is a fuzzy set on , satisfying the following conditions for each and each and , (1), (2) if and only if , (3), (4), (5) is continuous.
Let be a fuzzy metric space. For , the open ball with centre and radius is defined by .
A subset is called open if for each , there exist and such that . Let denote the family of all open subsets of . Then is called the topology on induced by the fuzzy metric . This topology is Hausdorff and first countable. A subset of is said to be F-bounded if there exist and such that for all .
Grabiec  obtained the following important lemma.
Lemma 3 (see ). Let be a fuzzy metric space. Then is nondecreasing with respect to , for all in .
Rodríguez-López and Romaguera  defined the continuity of fuzzy metric and obtained the following lemma relating to the continuity of .
Definition 4. Let be a fuzzy metric space. is said to be continuous on if whenever a sequence in converges to a point , that is, whenever
Lemma 5 (see ). Let be a fuzzy metric space. Then is a continuous function on .
From now onwards, will denote the following condition: In 1994, Mishra et al.  proved the following lemma relating to Cauchy sequences in fuzzy metric spaces.
Lemma 6 (see ). Let be a sequence in fuzzy metric space () satisfying . If there exists a positive number such that then is a Cauchy sequence in .
Definition 7 ([18, Definition 2.2]). Let be a nonempty subset of a fuzzy metric space . For and , define .
Throughout the paper, let denote the class of all nonempty compact subsets of .
Lemma 8 ([18, Lemma 1]). Let be a fuzzy metric space. Then for each and , there exists such that .
Definition 9 (see ). Let be a fuzzy metric space. For each and , set The -tuple is called a Hausdorff fuzzy metric space.
Lemma 10 (see ). Let be a nonempty set and be a mapping. Then there exists a subset such that and is one one.
Now, we give the following definitions for a hybrid pair of mappings (see also ).
Definition 11. Let be a nonempty set, (collection of all nonempty subsets of ) and . (i)The point is called a tripled fixed of if (ii)The point is called a tripled coincident point of and if (iii)The point is called a tripled common fixed point of and if
Definition 12. Let be a multivalued map and be a self-map on . The Hybrid pair is called -compatible if whenever is a tripled coincidence point of and .
2. Main Result
First we prove a slightly different result from Lemma 6 which is essential in proving our main result.
Now, we are ready to prove our first main result.
Proof. Let .
Choose , .
Since is compact valued, by Lemma 8, there exists such that Since is compact valued, by Lemma 8, there exists such that Since is compact valued, by Lemma 8, there exists such that Thus, Continuing in this way, we can obtain sequences , and in such that , and such that Hence, by Lemma 13, and are Cauchy sequences in .
Since is complete, there exist such that , and . Consider Letting , we get Similarly, we can show that Since , , and , from (20) and (21), we have Hence there exist sequences , and such that Now, for each , we have Letting , we obtain Similarly, we can show that Since , , and are compact, we have , , and .
Thus, is a tripled fixed point of .
Using the above theorem, we now prove a tripled coincidence and common fixed point theorem for a hybrid pair of multivalued and single-valued mapping.
Theorem 15. Let be a complete fuzzy metric space satisfying condition and and be a mappings satisfying for all , and . Further assume that , then and have a tripled coincidence point. Moreover, and have a tripled common fixed point if one of the following conditions holds. (a)The pair is -compatible and there exist such that , , and , whenever is a tripled coincidence point of and and is continuous at . (b)There exist such that , and , whenever is a tripled coincidence point of and and is continuous at .
Proof. By Lemma 10, There exists such that is one to one and .
Now, define by for all . Since is one-one on , is well defined.
Now, Hence satisfies (13) and all the conditions of Theorem 14.
By Theorem 14, has a tripled fixed point . Thus, Since , there exist such that , , and . So from (29), we have This implies that is a tripled coincidence point of and .
Now, following as in  except from the inequalities satisfied by we can show that and have a tripled common fixed point.
The authors are thankful to the referee for his valuable suggestions.
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