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 [25]. 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 𝑡-norms and continuous 𝑡-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 [1113] 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 𝑠-increasing sequence [6]. A sequence {𝑡𝑛} of positive real numbers is said to be an 𝑠-increasing sequence if there exists 𝑚0 such that 𝑡𝑚+1𝑡𝑚+1, for all 𝑚𝑚0.

Theorem 1.1. Let (𝑋,𝑀,𝑁,,) be a complete intuitionistic fuzzy metric space such that for every 𝑠-increasing sequence {𝑡𝑛} and arbitrary 𝑥,𝑦𝑋, lim𝑛𝑖=𝑛𝑀𝑥,𝑦,𝑡𝑖=1,lim𝑛𝑖=𝑛𝑁𝑥,𝑦,𝑡𝑖=0(1.1) hold.
Let 𝑘(0,1) and 𝑇𝑋𝑋 be a mapping satisfying 𝑀(𝑇𝑥,𝑇𝑦,𝑘𝑡)𝑀(𝑥,𝑦,𝑡) and 𝑁(𝑇𝑥,𝑇𝑦,𝑘𝑡)𝑁(𝑥,𝑦,𝑡) for all 𝑥,𝑦𝑋. Then 𝑇 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 (𝑋,𝑀,𝑁,,) be a complete intuitionistic fuzzy metric space such that for some 𝜎0(0,1) and 𝑥0𝑋, lim𝑛𝑖=𝑛𝑀𝑥0,𝑇𝑥0,1𝜎𝑖0=1,lim𝑛𝑖=𝑛𝑁𝑥0,𝑇𝑥0,1𝜎𝑖0=0(1.2) hold.
Let 𝑘(0,1) and 𝑇𝑋𝑋 be a mapping satisfying 𝑀(𝑇𝑥,𝑇𝑦,𝑘𝑡)𝑀(𝑥,𝑦,𝑡) and 𝑁(𝑇𝑥,𝑇𝑦,𝑘𝑡)𝑁(𝑥,𝑦,𝑡) for all 𝑥,𝑦𝑋. Then 𝑇 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 𝑋 be a nonempty fixed set. An intuitionistic fuzzy set 𝐴 is an object having the form 𝐴=𝑥,𝜇𝐴(𝑥),𝜈𝐴(𝑥)𝑥𝑋,(2.1) where the functions 𝜇𝐴𝑋[0,1] and 𝜈𝐴𝑋[0,1] denote the degree of membership and the degree of nonmembership of each element 𝑥𝑋 to the set 𝐴, respectively, and 0𝜇𝐴(𝑥)+𝜈𝐴(𝑥)1 for each 𝑥𝑋.

For developing intuitionistic fuzzy topological spaces, in [10], Çoker introduced the intuitionistic fuzzy sets 0 and 1 in 𝑋 as follows.

Definition 2.2 (see [10]). 0={𝑥,0,1𝑥𝑋} and 1={𝑥,1,0𝑥𝑋}.

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 𝑋 is a family 𝜏 of intuitionistic fuzzy sets in 𝑋 satisfying the following axioms:
(T1)0,1𝜏; (T2)𝐺1𝐺2𝜏 for any 𝐺1,𝐺2𝜏; (T3)𝐺𝑖𝜏 for any arbitrary family {𝐺𝑖𝑖𝐽}𝜏.
In this case, the pair (𝑋,𝜏) is called an intuitionistic fuzzy topological space.

Definition 2.4 (see [15]). A binary operation [0,1]×[0,1][0,1] is a continuous 𝑡-norm (triangular norm) if satisfies the following conditions: (a) is associative and commutative;(b) is continuous; (c)𝑎1=𝑎 for all 𝑎[0,1]; (d)𝑎𝑏𝑐𝑑 whenever 𝑎𝑐 and 𝑏𝑑, and 𝑎,𝑏,𝑐,𝑑[0,1].
By this definition, it is easy to see that 11=1. According to condition (a), the following product is well defined: 𝑀(𝑥1,𝑦1,𝑡1)𝑀(𝑥2,𝑦2,𝑡2)𝑀(𝑥𝑛,𝑦𝑛,𝑡𝑛), and we will denote it by 𝑖=𝑛𝑖=1𝑀(𝑥𝑖,𝑦𝑖,𝑡𝑖).

Definition 2.5 (see [15]). A binary operation [0,1]×[0,1][0,1] is a continuous 𝑡-conorm (triangular conorm) if satisfies the following conditions: (e) is associative and commutative; (f) is continuous; (g)𝑎0=𝑎 for all 𝑎[0,1]; (h)𝑎𝑏𝑐𝑑 whenever 𝑎𝑐 and 𝑏𝑑, and 𝑎,𝑏,𝑐,𝑑[0,1].
By this definition, it is easy to see that 00=0. According to condition (e), the following product is well defined: 𝑁(𝑥1,𝑦1,𝑡1)𝑁(𝑥2,𝑦2,𝑡2)𝑁(𝑥𝑛,𝑦𝑛,𝑡𝑛), and we also denote this product by 𝑖=𝑛𝑖=1𝑁(𝑥𝑖,𝑦𝑖,𝑡𝑖).

Remark 2.6. The origin of concepts of 𝑡-norms and related 𝑡-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 𝑡-norms are 𝑎𝑏=𝑎𝑏 and 𝑎𝑏=min{𝑎,𝑏}, and basic examples of 𝑡-conorms are 𝑎𝑏=max{𝑎,𝑏} and 𝑎𝑏=min{1,𝑎+𝑏}.

Definition 2.7 (see [13]). A 5-tuple (𝑋,𝑀,𝑁,,) is said to be an intuitionistic fuzzy metric space if 𝑋 is an arbitrary set, is a continuous 𝑡-norm, is a continuous 𝑡-conorm, and 𝑀,𝑁 are fuzzy sets on 𝑋×𝑋×[0,) satisfying the following conditions: (IFm 1)𝑀(𝑥,𝑦,𝑡)+𝑁(𝑥,𝑦,𝑡)1; (IFm 2)𝑀(𝑥,𝑦,0)=0; (IFm 3)𝑀(𝑥,𝑦,𝑡)=1 for all 𝑡>0 if and only if 𝑥=𝑦; (IFm 4)𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡); (IFm 5)𝑀(𝑥,𝑦,𝑡)𝑀(𝑦,𝑧,𝑠)𝑀(𝑥,𝑧,𝑡+𝑠) for all 𝑥,𝑦,𝑧𝑋,𝑠,𝑡>0; (IFm 6)𝑀(𝑥,𝑦,)[0,)[0,1] is left continuous;(IFm 7)lim𝑡𝑀(𝑥,𝑦,𝑡)=1 for all 𝑥,𝑦𝑋; (IFm 8)𝑁(𝑥,𝑦,0)=1; (IFm 9)𝑁(𝑥,𝑦,𝑡)=0 for all 𝑡>0 if and only if 𝑥=𝑦;(IFm 10)𝑁(𝑥,𝑦,𝑡)=𝑁(𝑦,𝑥,𝑡); (IFm 11)𝑁(𝑥,𝑦,𝑡)𝑁(𝑦,𝑧,𝑠)𝑁(𝑥,𝑧,𝑡+𝑠) for all 𝑥,𝑦,𝑧𝑋,𝑠,𝑡>0; (IFm 12)𝑁(𝑥,𝑦,)[0,)[0,1] is right continuous; (IFm 13)lim𝑡𝑁(𝑥,𝑦,𝑡)=0 for all 𝑥,𝑦𝑋.
We denote by (𝑀,𝑁) the intuitionistic fuzzy metric on 𝑋. In intuitionistic fuzzy metric space 𝑋, it is easy to see 𝑀(𝑥,𝑦,) is nondecreasing and 𝑁(𝑥,𝑦,) is nonincreasing for all 𝑥,𝑦𝑋. We also note that the successive product with respect to 𝑀(𝑥,𝑦,𝑡) is in the sense of and the successive product with respect to 𝑁(𝑥,𝑦,𝑡) is in the sense of throughout this paper.

Definition 2.8. Let (𝑋,𝑀,𝑁,,) be an intuitionistic fuzzy metric space. Then (I) a sequence {𝑥𝑛} in 𝑋 is Cauchy sequence if and only if for each 𝑡>0 and 𝑝>0, lim𝑛𝑀𝑥𝑛,𝑥𝑛+𝑝,𝑡=1,lim𝑛𝑁𝑥𝑛,𝑥𝑛+𝑝,𝑡=0,(2.2)(II) a sequence {𝑥𝑛} in 𝑋 is convergent to 𝑥𝑋 if and only if for each 𝑡>0, lim𝑛𝑀𝑥𝑛,𝑥,𝑡=1,lim𝑛𝑁𝑥𝑛,𝑥,𝑡=0.(2.3)

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 𝑥𝑋. Let 𝑥𝑛=𝑇𝑛(𝑥),𝑛. We have 𝑀𝑥1,𝑥2,𝑡=𝑀𝑇(𝑥),𝑇2𝑡(𝑥),𝑡𝑀𝑥,𝑇(𝑥),𝑘=𝑀𝑥,𝑥1,𝑡𝑘,𝑁𝑥1,𝑥2𝑇,𝑡=𝑁(𝑥),𝑇2𝑡(𝑥),𝑡𝑁𝑥,𝑇(𝑥),𝑘=𝑁𝑥,𝑥1,𝑡𝑘.(3.1) By induction it follows that 𝑀(𝑥𝑛,𝑥𝑛+1,𝑡)𝑀(𝑥,𝑥1,𝑡/𝑘𝑛) and 𝑁(𝑥𝑛,𝑥𝑛+1,𝑡)𝑁(𝑥,𝑥1,𝑡/𝑘𝑛).
Let 𝑡>0. For 𝑚,𝑛, without loss of generality, we suppose 𝑛<𝑚; if we choose 𝑠𝑖>0,𝑖=𝑛,,𝑚1, satisfying 𝑠𝑛+𝑠𝑛+1++𝑠𝑚11, then we have 𝑀𝑥𝑛,𝑥𝑚𝑥,𝑡𝑀𝑛,𝑥𝑛+1,𝑠𝑛𝑡𝑥𝑀𝑚1,𝑥𝑚,𝑠𝑚1𝑡𝑀𝑥,𝑥1,𝑠𝑛𝑡𝑘𝑛𝑀𝑥,𝑥1,𝑠𝑚1𝑡𝑘𝑚1,𝑁𝑥𝑛,𝑥𝑚𝑥,𝑡𝑁𝑛,𝑥𝑛+1,𝑠𝑛𝑡𝑥𝑁𝑚1,𝑥𝑚,𝑠𝑚1𝑡𝑁𝑥,𝑥1,𝑠𝑛𝑡𝑘𝑛𝑁𝑥,𝑥1,𝑠𝑚1𝑡𝑘𝑚1.(3.2) In particular, since 𝑛=11/𝑛(𝑛+1)=1, taking 𝑠𝑖=1/𝑖(𝑖+1),𝑖=𝑛,,𝑚1, one achieves 𝑀𝑥𝑛,𝑥𝑚,𝑡𝑀𝑥,𝑥1,𝑡𝑛(𝑛+1)𝑘𝑛𝑀𝑥,𝑥1,𝑡(𝑚1)𝑚𝑘𝑚1𝑁𝑥,(3.3)𝑛,𝑥𝑚,𝑡𝑁𝑥,𝑥1,𝑡𝑛(𝑛+1)𝑘𝑛𝑁𝑥,𝑥1,𝑡(𝑚1)𝑚𝑘𝑚1.(3.4) We define 𝑡𝑛=𝑡/𝑛(𝑛+1)𝑘𝑛. It is preliminary to show that (𝑡𝑛+1𝑡𝑛), as 𝑛, so {𝑡𝑛} is an 𝑠-increasing sequence, and hence we get lim𝑚𝑛=𝑚𝑀𝑥,𝑥1,𝑡𝑛(𝑛+1)𝑘𝑛=1,lim𝑚𝑛=m𝑁𝑥,𝑥1,𝑡𝑛(𝑛+1)𝑘𝑛=0.(3.5) The combination of (3.3), (3.4), and (3.5) implies lim𝑛𝑀(𝑥𝑛,𝑥𝑚,𝑡)=1 and lim𝑛𝑁(𝑥𝑛,𝑥𝑚,𝑡)=0 for 𝑚>𝑛. Hence {𝑥𝑛} is a Cauchy sequence. Since 𝑋 is complete, there is 𝑦𝑋 such that lim𝑛𝑥𝑛=𝑦. We claim 𝑦 is a fixed point of 𝑇. In fact, it is easy to see 𝑀(𝑇(𝑦),𝑦,𝑡)lim𝑛𝑀𝑥𝑇(𝑦),𝑇𝑛,𝑡2lim𝑛𝑀𝑥𝑛+1𝑡,𝑦,2lim𝑛𝑀𝑦,𝑥𝑛,𝑡2𝑘lim𝑛𝑀𝑥𝑛+1𝑡,𝑦,2=11,𝑁(𝑇(𝑦),𝑦,𝑡)lim𝑛𝑁𝑥𝑇(𝑦),𝑇𝑛,𝑡2lim𝑛𝑁𝑥𝑛+1𝑡,𝑦,2lim𝑛𝑁𝑦,𝑥𝑛,𝑡2𝑘lim𝑛𝑁𝑥𝑛+1𝑡,𝑦,2=00.(3.6) Thus 𝑀(𝑇(𝑦),𝑦,𝑡)=1 and 𝑁(𝑇(𝑦),𝑦,𝑡)=0, and we obtain 𝑇(𝑦)=𝑦. In the sequel, we show the uniqueness of the fixed point. We assume 𝑇(𝑧)=𝑧 for some 𝑧𝑋. We have 𝑡1𝑀(𝑦,𝑧,𝑡)=𝑀(𝑇𝑦,𝑇𝑧,𝑡)𝑀𝑦,𝑧,𝑘𝑡=𝑀𝑇(𝑦),𝑇(𝑧),𝑘𝑡𝑀𝑦,𝑧,𝑘2𝑡=𝑀𝑇(𝑦),𝑇(𝑧),𝑘2lim𝑛𝑀𝑡𝑦,𝑧,𝑘𝑛𝑡=1,0𝑁(𝑦,𝑧,𝑡)=𝑁(𝑇𝑦,𝑇𝑧,𝑡)𝑁𝑦,𝑧,𝑘𝑇𝑡=𝑁(𝑦),𝑇(𝑧),𝑘𝑡𝑁𝑦,𝑧,𝑘2𝑇𝑡=𝑁(𝑦),𝑇(𝑧),𝑘2lim𝑛𝑁𝑡𝑦,𝑧,𝑘𝑛=0.(3.7) Thus we get 𝑀(𝑦,𝑧,𝑡)=1 and 𝑁(𝑦,𝑧,𝑡)=0, and hence 𝑦=𝑧. 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 𝐹(0,)[0,1], the following implication holds: lim𝑛𝑖=𝑛𝐹𝜎𝑖0=0lim𝑛𝑖=𝑛𝐹𝜎𝑖=0(4.1) for all 𝜎(0,1), where the infinite product is in the sense of .

Proof
Case 1 (𝜎<𝜎0). For 𝑖, 𝜎𝑖<𝜎𝑖0, and since 𝐹 is nondecreasing, 𝐹(𝜎𝑖)𝐹(𝜎𝑖0) hold. And hence 𝑖=𝑛𝐹(𝜎𝑖)𝑖=𝑛𝐹(𝜎𝑖0),𝑛. So implication (4.1) holds.Case 2 (𝜎𝜎0). If 𝜎=𝜎0, it follows 𝑖=2𝑚𝐹𝜎𝑖=𝑖=𝑚𝐹𝜎2𝑖𝑖=𝑚𝐹𝜎2𝑖+1𝑖=𝑚𝐹𝜎𝑖0𝑖=𝑚𝐹𝜎𝑖0.(4.2) Then we have lim𝑚𝑖=2𝑚𝐹(𝜎𝑖)00=0. And lim𝑚𝑖=2𝑚+1𝐹(𝜎𝑖)lim𝑚𝑖=2𝑚+2𝐹(𝜎𝑖)=0. Thus it follows that lim𝑚𝑖=𝑚𝐹(𝜎𝑖)=0 for 𝜎=𝜎0. Since 𝐹 is nondecreasing, it is easy to show lim𝑚𝑖=𝑚𝐹(𝜎𝑖)=0 for 𝜎<𝜎0.
For an arbitrary 𝜎>𝜎0, there exists 𝑚 such that 𝜎<𝜎[(1/2)𝑚]0, and we can repeat the above process 𝑚-times to get lim𝑚𝑖=𝑚𝐹(𝜎𝑖)=0.

Lemma 4.2. For any monotonely nonincreasing function 𝐺(0,)[0,1], the following implication holds: lim𝑛𝑖=𝑛𝐺𝜎𝑖0=1lim𝑛𝑖=𝑛𝐺𝜎𝑖=1(4.3) for all 𝜎(0,1), 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 𝑥𝑛=𝑇𝑛(𝑥0)(𝑛). Then {𝑥𝑛} is a Cauchy sequence.

Proof. We assume 𝐹(𝑥)=𝑁(𝑥0,𝑇(𝑥0),1/𝑥) and 𝐺(𝑥)=𝑀(𝑥0,𝑇(𝑥0),1/𝑥) for 𝑥>0. Then 𝐹(𝑥) (𝐺(𝑥)) is nondecreasing (nonincreasing) mapping from (0,) into [0,1]. Taking 1>𝜎>𝑘, by Lemmas 4.1 and 4.2, we have lim𝑛𝑖=𝑛𝑀𝑥0𝑥,𝑇0,1(𝑘/𝜎)𝑖=1,lim𝑛𝑖=𝑛𝑁𝑥0𝑥,𝑇0,1(𝑘/𝜎)𝑖=0.(4.4) Since 𝜎<1, 𝑛=1𝜎𝑛<, for any 𝜀0>0 there exists 𝑛0 such that 𝑛=𝑛0𝜎𝑛<𝜀0. For the above 𝜀0>0, if 𝑚>𝑛>𝑛0 and 𝑡>𝜀0, 𝑀𝑥𝑛,𝑥𝑚𝑥,𝑡𝑀𝑛,𝑥𝑚,𝜀0𝑚1𝑖=𝑛𝑀𝑥𝑖,𝑥i1,𝜎𝑖𝑚1𝑖=𝑛𝑀𝑥0,𝑇𝑥0,𝜎𝑖𝑘𝑖=𝑚1𝑖=𝑛𝑀𝑥0,𝑇𝑥0,1(𝑘/𝜎)𝑖,𝑁𝑥𝑛,𝑥𝑚𝑥,𝑡𝑁𝑛,𝑥𝑚,𝜀0𝑚1𝑖=𝑛𝑁𝑥𝑖,𝑥𝑖1,𝜎𝑖𝑚1𝑖=𝑛𝑁𝑥0,𝑇𝑥0,𝜎𝑖𝑘𝑖=𝑚1𝑖=𝑛𝑁𝑥0,𝑇𝑥0,1(𝑘/𝜎)𝑖(4.5) hold.
And according to (4.4), we have lim𝑛𝑀(𝑥𝑛,𝑥𝑚,𝑡)=1 and lim𝑛𝑁(𝑥𝑛,𝑥𝑚,𝑡)=0 for 𝑚>𝑛. So {𝑥𝑛} is Cauchy sequence.
Since 𝑋 is complete, there exists some 𝑦𝑋 such that lim𝑛𝑥𝑛=𝑦. One can prove 𝑦 is the unique fixed point of 𝑇 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).