Abstract
The author extends two fixed point theorems (due to Gregori, Sapena, and Žikić, resp.) in fuzzy metric spaces to intuitionistic fuzzy metric spaces.
1. Introduction
In this paper, we pay our attention to the fixed point theory on intuitionistic fuzzy metric spaces. Since Zadeh [1] introduced the theory of fuzzy sets, many authors have studied the character of fuzzy metric spaces in different ways [2–5]. Among others, fixed point theorem was an important subject. Gregori and Sapena [6] investigated fixed point theorems for fuzzy contractive mappings defined on fuzzy metric spaces. Recently, Žikić [7] proved a fixed point theorem for mappings on fuzzy metric space which improved the result of Gregori and Sapena. As further development, Atanassov [8] introduced and studied the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets, and later there has been much progress in the study of intuitionistic fuzzy sets [9, 10]. Using the idea of intuitionistic sets, Park [11] defined the notion of intuitionistic fuzzy metric spaces with the help of continuous -norms and continuous -conorms as a generalization of fuzzy metric space. Recently, several authors studied the structure of intuitionistic fuzzy metric spaces and fixed point theorems for the mappings defined on intuitionistic fuzzy metric spaces. We refer the reader to [11–13] for further details. In this paper, we will prove the following two fixed point theorems.
The first theorem extends Gregori-Sapena's fixed point theorem [6] in fuzzy metric spaces to complete intuitionistic fuzzy metric spaces. As preparation, we recall the definition of -increasing sequence [6]. A sequence of positive real numbers is said to be an -increasing sequence if there exists such that , for all .
Theorem 1.1. Let be a complete intuitionistic fuzzy metric space such that for every -increasing sequence and arbitrary ,
hold.
Let and be a mapping satisfying and for all . Then has a unique fixed point.
The second theorem extends Žikić’s fixed point theorem [7] in fuzzy metric space to intuitionistic fuzzy metric space.
Theorem 1.2. Let be a complete intuitionistic fuzzy metric space such that for some and ,
hold.
Let and be a mapping satisfying and for all . Then has a unique fixed point.
2. Basic Notions and Preliminary Results
For the sake of completeness, in this section we will recall some definitions and preliminaries on intuitionistic fuzzy metric spaces.
Definition 2.1 (see [14]). Let be a nonempty fixed set. An intuitionistic fuzzy set is an object having the form where the functions and denote the degree of membership and the degree of nonmembership of each element to the set , respectively, and for each .
For developing intuitionistic fuzzy topological spaces, in [10], Çoker introduced the intuitionistic fuzzy sets and in as follows.
Definition 2.2 (see [10]). and .
By Definition 2.2, Çoker defined the notion of intuitionistic fuzzy topological spaces.
Definition 2.3 (see [10]). An intuitionistic fuzzy topology on a nonempty set is a family of intuitionistic fuzzy sets in satisfying the following axioms:
(T1); (T2) for any ; (T3) for any arbitrary family .
In this case, the pair is called an intuitionistic fuzzy topological space.
Definition 2.4 (see [15]). A binary operation is a continuous -norm (triangular norm) if satisfies the following conditions: is associative and commutative; is continuous; for all ; whenever and , and .
By this definition, it is easy to see that . According to condition (a), the following product is well defined: , and we will denote it by .
Definition 2.5 (see [15]). A binary operation is a continuous -conorm (triangular conorm) if satisfies the following conditions: (e) is associative and commutative; (f) is continuous; (g) for all ; (h) whenever and , and .
By this definition, it is easy to see that . According to condition (e), the following product is well defined: , and we also denote this product by .
Remark 2.6. The origin of concepts of -norms and related -conorms was in the theory of statistical metric spaces in the work of Menger [5]. These concepts are known as the axiomatic skeletons that we use for characterizing fuzzy intersections and unions, respectively. Basic examples of -norms are and , and basic examples of -conorms are and .
Definition 2.7 (see [13]). A 5-tuple is said to be an intuitionistic fuzzy metric space if is an arbitrary set, is a continuous -norm, is a continuous -conorm, and are fuzzy sets on satisfying the following conditions: (IFm 1); (IFm 2); (IFm 3) for all if and only if ; (IFm 4); (IFm 5) for all ; (IFm 6) is left continuous;(IFm 7) for all ; (IFm 8); (IFm 9) for all if and only if ;(IFm 10); (IFm 11) for all ; (IFm 12) is right continuous; (IFm 13) for all .
We denote by the intuitionistic fuzzy metric on . In intuitionistic fuzzy metric space , it is easy to see is nondecreasing and is nonincreasing for all . We also note that the successive product with respect to is in the sense of and the successive product with respect to is in the sense of throughout this paper.
Definition 2.8. Let be an intuitionistic fuzzy metric space. Then (I) a sequence in is Cauchy sequence if and only if for each and , (II) a sequence in is convergent to if and only if for each ,
Definition 2.9. An intuitionistic fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent.
3. Proof of Theorem 1.1
In this section, we will give the proof of Theorem 1.1 of the present paper.
Proof. Select an arbitrary point . Let . We have
By induction it follows that and .
Let . For , without loss of generality, we suppose ; if we choose , satisfying , then we have
In particular, since , taking , one achieves
We define . It is preliminary to show that , as , so is an -increasing sequence, and hence we get
The combination of (3.3), (3.4), and (3.5) implies and for . Hence is a Cauchy sequence. Since is complete, there is such that . We claim is a fixed point of . In fact, it is easy to see
Thus and , and we obtain . In the sequel, we show the uniqueness of the fixed point. We assume for some . We have
Thus we get and , and hence . The proof is complete.
4. Proof of Theorem 1.2
In this section, we will give the proof of Theorem 1.2 by three lemmas.
Lemma 4.1. For any monotonely nondecreasing function , the following implication holds: for all , where the infinite product is in the sense of .
Proof
Case 1 (). For , , and since is nondecreasing, hold. And hence . So implication (4.1) holds.Case 2 (). If , it follows
Then we have . And . Thus it follows that for . Since is nondecreasing, it is easy to show for .
For an arbitrary , there exists such that , and we can repeat the above process -times to get .
Lemma 4.2. For any monotonely nonincreasing function , the following implication holds: for all , where the infinite product is in the sense of .
Proof. One can take a similar procedure as in the proof of Lemma 4.1 to complete the proof of this lemma. For simplicity, we omit the detailed argument. We refer the reader to [7] for further details.
Lemma 4.3. We define . Then is a Cauchy sequence.
Proof. We assume and for . Then () is nondecreasing (nonincreasing) mapping from into . Taking , by Lemmas 4.1 and 4.2, we have
Since , , for any there exists such that . For the above , if and ,
hold.
And according to (4.4), we have and for . So is Cauchy sequence.
Since is complete, there exists some such that . One can prove is the unique fixed point of by repeating the same process as in the proof of Theorem 1.1. Thus, we complete the proof of Theorem 1.2.
Acknowledgments
The author is grateful to Professor Nazim I. Mahmudov and the anonymous referees for their helpful comments and constructive suggestions. This work was supported partly by the National Natural Science Foundation of China (nos. 11026105 and 11026106), the Scientific Research Fund of Zhejiang Province Education Department (no. Y201119292), and the Foundation for Development of Mathematics Subject of HDU (ZX100204004-10).