On Fixed Point Theorems in Intuitionistic Fuzzy Metric Spaces

The author extends two fixed point theorems (due to Gregori, Sapena, and Žikic, resp.) in fuzzy metric spaces to intuitionistic fuzzy metric spaces.


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 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 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 s-increasing sequence 6 .A sequence {t n } of positive real numbers is said to be an s-increasing sequence if there exists m 0 ∈ N such that t m 1 ≤ t m 1 , for all m ≥ m 0 .Theorem 1.1.Let X, M, N, * , ♦ be a complete intuitionistic fuzzy metric space such that for every s-increasing sequence {t n } and arbitrary x, y ∈ X, Let k ∈ 0, 1 and T : X → X be a mapping satisfying M Tx, Ty, kt ≥ M x, y, t and N Tx, Ty, kt ≤ N x, y, t for all x, y ∈ X.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 σ 0 ∈ 0, 1 and Let k ∈ 0, 1 and T : X → X be a mapping satisfying M Tx, Ty, kt ≥ M x, y, t and N Tx, Ty, kt ≤ N x, y, t for all x, y ∈ X.Then T has a unique fixed point.

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 where the functions μ A : X → 0, 1 and ν A : X → 0, 1 denote the degree of membership and the degree of nonmembership of each element x ∈ X to the set A, respectively, and 0 ≤ μ A x ν A x ≤ 1 for each x ∈ X.
For developing intuitionistic fuzzy topological spaces, in 10 , C ¸oker introduced the intuitionistic fuzzy sets 0 ∼ and 1 ∼ in X as follows.
By Definition 2.2, C ¸oker 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: In this case, the pair X, τ is called an intuitionistic fuzzy topological space.By this definition, it is easy to see that 1 * 1 1.According to condition a , the following product is well defined:  By this definition, it is easy to see that 0♦0 0. According to condition e , the following product is well defined: 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, y ∈ X.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 {x n } in X is Cauchy sequence if and only if for each t > 0 and p > 0, Definition 2.9.An intuitionistic fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent.

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 x ∈ X.Let x n T n x , n ∈ N. We have

3.1
By induction it follows that M x n , x n 1 , t ≥ M x, x 1 , t/k n and N x n , x n 1 , t ≤ N x, x 1 , t/k n .
Let t > 0. For m, n ∈ N, without loss of generality, we suppose n < m; if we choose

3.2
In particular, since 3.4 We define t n t/n n 1 k n .It is preliminary to show that t n 1 − t n → ∞, as n → ∞, so {t n } is an s-increasing sequence, and hence we get The combination of 3.3 , 3.4 , and 3.5 implies lim n → ∞ M x n , x m , t 1 and lim n → ∞ N x n , x m , t 0 for m > n.Hence {x n } is a Cauchy sequence.Since X is complete, there is y ∈ X such that lim n → ∞ x n y.We claim y is a fixed point of T .In fact, it is easy to see

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 z ∈ X.We have

3.7
Thus we get M y, z, t 1 and N y, z, t 0, and hence y z.The proof is complete.

Proof of Theorem 1.2
In this section, we will give the proof of Theorem 1.2 by three lemmas.

Proof
Case 1 σ < σ 0 .For i ∈ N, σ i < σ i 0 , and since F is nondecreasing, For an arbitrary σ > σ 0 , there exists m ∈ N such that σ < σ 1/2 m 0 , and we can repeat the above process m-times to get lim m → ∞ ∞ i m F σ i 0.
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 x n T n x 0 n ∈ N .Then {x n } is a Cauchy sequence.
Proof.We assume F x N x 0 , T x 0 , 1/x and G x M x 0 , T x 0 , 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 Since σ < 1, ∞ n 1 σ n < ∞, for any ε 0 > 0 there exists n 0 such that ∞ n n 0 σ n < ε 0 .For the above ε 0 > 0, if m > n > n 0 and t > ε 0 , Since X is complete, there exists some y ∈ X such that lim n → ∞ x n 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.
g a♦0 a for all a ∈ 0, 1 ; h a♦b ≤ c♦d whenever a ≤ c and b ≤ d, and a, b, c, d ∈ 0, 1 .
x 0 , Tx 0 , 1 k/σ i , N x n , x m , t ≤ N x n , x m , ε 0 ≤ according to 4.4 , we have lim n → ∞ M x n , x m , t1 and lim n → ∞ N x n , x m , t 0 for m > n.So {x n } is Cauchy sequence.
y n , t n , and we also denote this product by i n i 1 N x i , y i , t i .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: 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 a * b ab and a * b min{a, b}, and basic examples of t-conorms are a♦b max{a, b} and a♦b min{1, a b}.