Zadeh [1] introduced the concept of fuzzy sets in 1965 and in the next decade, Grabiec [2] obtained the Banach contraction principle in setting of fuzzy metric spaces, Also, Altun and Turkoglu [3] proved some fixed theorems using implicit relations in fuzzy metric spaces. Furthermore, Park et al. [4] defined the intuitionistic fuzzy metric space, and Park et al. [5] proved a fixed point theorem of Banach for the contractive mapping of a complete intuitionistic fuzzy metric space. Recently, Park [6, 7], Park et al. [8] obtained a unique common fixed point theorem for type(𝛼) and type(𝛽) compatible mappings defined on intuitionistic fuzzy metric space. Also, authors proved the fixed point theorem using compatible properties in many articles [9–12].

In this paper, we will obtain a unique common fixed point theorem and example for this theorem under the type(𝛽) compatible four mappings with implicit relations defined on intuitionistic fuzzy metric space.

We will give some definitions, properties of the intuitionistic fuzzy metric space X as following:

Let us recall (see [13]) that a continuous t–norm is a binary operation ⁎ : [0, 1] × [0, 1] → [0, 1] which satisfies the following conditions:(a)⁎ is commutative and associative; (b)⁎ is continuous; (c)a ⁎ 1 = a for all a ∈ [0, 1]; (d)a ⁎ b ≤ c ⁎ d whenever a ≤ c and b ≤ d (a, b, c, d ∈ [0, 1]).

Similarly, a continuous t–conorm is a binary operation : [0, 1] × [0, 1] → [0, 1] which satisfies the following conditions:

(a) is commutative and associative; (b) is continuous; (c) a 0 = a for all a ∈ [0, 1]; (d) a b ≥ c d whenever a ≤ c and b ≤ d (a, b, c, d ∈ [0, 1]).

Definition 2.1. ([14]) The 5..tuple (X,M,N, ⁎, ) is said to be an intuitionistic fuzzy metric space (IFMS) if X is an arbitrary set, ⁎ is a continuous t–norm, is a continuous t–conorm and M,N are fuzzy sets on X² × (0,∞) satisfying the following conditions; for all x, y, z ∈ X, such that

(a) M(x,y,t) > 0, (b) M(x,y,t) = 1 x = y, (c) M(x,y,t) = M(y, x, t), (d) M(x,y,t) ⁎ M(y, z, s) ≤ M(x, z, t + s), (e) M(x,y,⋅) : (0,∞) → (0, 1] is continuous, (f) N(x,y,t) > 0, (g) N(x,y,t) = 0 x = y, (h) N(x,y,t) = N(y, x, t), (i) N(x,y,t) N(y, z, s) ≥ N(x, z, t + s), (j) N(x,y,⋅) : (0,∞) → (0, 1] is continuous.

Note that (M, N) is called an IFM on X. The functions M(x, y, t) and N(x, y, t) denote the degree of nearness and the degree of non-nearness between x and y with respect to t, respectively.

Definition 2.2. ([6]) Let X be an IFMS.

(a) {x_n} is said to be convergent to a point x ∈ X if, for any 0 < 𝜖 < 1 and t > 0, there exists n₀ ∈ N such that

M(xn, x, t) > 1 – 𝜖, N(xn, x, t) < 𝜖

for all n ≥ n₀.

(b) {x_n} is called a Cauchy sequence if for any 0 < 𝜖 < 1 and t > 0, there exists n₀ ∈ N such that

M(xn, xm, t) > 1 – 𝜖, N(xn, xm, t) < 𝜖

for all m, n ≥ n₀.

(c) X is complete if every Cauchy sequence converges in X.

Lemma 2.3. ([8]) Let X be an IFMS. If there exists a number k ∈ (0, 1) such that for all x, y ∈ X and t > 0,

M(x, y, kt) ≥ M(x, y, t), N(x, y, kt) ≤ N(x, y, t),

then x = y.

Definition 2.4. ([7]) Let A,B be mappings from IFMS X into itself. The mappings are said to be type(𝛽) compatible if for all t > 0,

whenever {x_n} ⊂ X such that for some x ∈ X.

Proposition 2.5. ([15]) Let X be an IFMS with t ⁎ t ≥ t and t t ≤ t for all t ∈ [0, 1]. A,B be type(𝛽) compatible maps from X into itself and let {x_n} be a sequence in X such that Ax_n,Bx_n → x for some x ∈ X. Then we have the following

(a) BBxn → Ax if A is continuous at x, (b) AAxn → Bx if B is continuous at x, (c) ABx = BAx and Ax = Bx if A and B are continuous at x.

Implicit relations on fuzzy metric spaces have been used in many articles ([3, 16]). Let = {ϕ_M, 𝜓_N}, I = [0, 1], ϕ_M, 𝜓_N : I⁶ → R be continuous functions and ⁎, be a continuous t-norm, t-conorm. Now, we consider the following conditions ([6]):

(I) ϕ_M is decreasing and 𝜓_N is increasing in sixth variables.

(II) If, for some k ∈ (0, 1), we have

for any fixed t > 0, any nondecreasing functions u, v : (0,∞) → I with 0 < u(t), v(t) ≤ 1, and any nonincreasing functions x, y : (0,∞) → I with 0 < x(t), y(t) ≤ 1, then there exists h ∈ (0, 1) with

u(ht) ≥ v(t) ⁎ u(t), x(ht) ≤ y(t) x(t).

(III) If, for some k ∈ (0, 1), we have

ϕM(u(kt), u(t), 1, 1, u(t), u(t)) ≥ 1

for any fixed t > 0 and any nondecreasing function u : (0,∞) → I, then u(kt) ≥ u(t). Also, if, for some k ∈ (0, 1), we have

𝜓N(x(kt), x(t), 0, 0, x(t), x(t)) ≤ 1

for any fixed t > 0 and any nonincreasing function x : (0,∞) → I, then x(kt) ≤ x(t).

Example 2.6. ([6]) Let a⁎b = min{a, b} and ab = max{a, b},

Also, let t > 0, 0 < u(t), v(t), x(t), y(t) ≤ 1, k ∈ (0, ½ ) where u, v : [0,∞) → I are nondecreasing functions and x, y : [0,∞) → I are nonincreasing functions. Now, suppose

then from Park [6], ϕ_M, 𝜓_N ∈ .

Now, we will prove some common fixed point theorem for four mappings on complete IFMS as follows:

Theorem 3.1. Let X be a complete intuitionistic fuzzy metric space with a⁎b = min{a, b}, ab = max{a, b} for all a, b ∈ I and A, B, S and T be mappings from X into itself satisfying the conditions:

(a) S(X) ⊆ B(X) and T(X) ⊆ A(X), (b) One of the mappings A, B, S, T is continuous, (c) A and S as well as B and T are type(𝛽) compatible (d) There exist k ∈ (0, 1) and ϕM, 𝜓N ∈ such that

for all x, y ∈ X and t > 0.

Then A, B, S and T have a unique common fixed point in X.

Proof. Let x₀ be an arbitrary point of X. Then from Theorem 3.1 of ([6]), we can construct a Cauchy sequence {y_n} ⊂ X. Since X is complete, {y_n} converges to a point x ∈ X. Since {Ax_2n+2}, {Bx_2n+1}, {Sx_2n} and {Tx_2n+1} ⊂ {y_n}, we have

Now, let A is continuous. Then

By Proposition 2.5,

Using condition (d), we have, for any t > 0,

and by letting n → ∞, ϕ_M, 𝜓_N are continuous, we have

Therefore, by (III), we have

M(Ax, x, kt) ≥ M(Ax, x, t),

N(Ax, x, kt) ≤ N(Ax, x, t).

Hence Ax = x from Lemma 2.3. Also, we have, by condition (d),

and, let n → ∞, we have

On the other hand, since

ϕ_M is nonincreasing and 𝜓_N is nondecreasing in the fifth variable, we have, for any t > 0,

which implies that Sx = x. Since S(X) ⊆ B(X), there exists a point y ∈ X such that By = x. Using condition (d), we have

which implies that x = Ty. Since By = Ty = x and B, T are type(𝛽) compatible, we have TTy = BBy. Hence Tx = TTy = BBy = Bx. Therefore, from (d), we have, for any t > 0,

From (III), we have

M(x, Tx, kt) ≥ M(x, Tx, t),

M(x, Tx, kt) ≤ M(x, Tx, t).

Therefore, we have x = Tx = Bx from Lemma 2.3. Hence x is a common fixed point of A,B, S and T. The same result holds if we assume that B is continuous instead of A.

Now, suppose that S is continuous. Then

By Proposition 2.5,

Using (d), we have for any t > 0,

and by n → ∞, since ϕ_M, 𝜓_N ∈ are continuous, we have

Thus, we have, from (III),

M(Sx, x, kt) ≥ M(Sx, x, t),

N(Sx, x, kt) ≤ N(Sx, x, t).

Hence Sx = x by Lemma 2.3. Since S(X) ⊆ B(X), there exists a point z ∈ X such that Bz = x. Using (d), we have

letting n → ∞, we get

which implies that x = Tz. Since Bz = Tz = x and B, T are type(𝛽) compatible, we have TBz = BBz and so Tx = TBz = BBz = Bx. Thus, we have

letting n → ∞,

Thus, x = Tx = Bx. Since T(X) ⊆ A(X), there exists w ∈ X such that Aw = x. Thus, from (d),

Hence we have x = Sw = Aw. Also, since A, S are type(𝛽) compatible,

x = Sx = SSw = AAw = Ax.

Hence x is a common fixed point of A,B, S and T. The same result holds if we assume that T is continuous instead of S.

Finally, suppose that A, B, S and T have another common fixed point u. Then we have, for any t > 0,

Therefore, from (III), x = u. This completes the proof.

Example 3.2. Let X be a intuitionistic fuzzy metric space with X = [0, 1], ⁎, be t-norm and t-conorm defined by

a ⁎ b = min{a, b}, a b = max{a, b}

for all a, b ∈ X. Also, let M, N be fuzzy sets on X² × (0,∞) defined by

Let ϕ_M, 𝜓_N : X⁶ → R be defined as in Example 2.6 and define the maps A, B, S, T : X → X by Ax = x, and . Then, for some , we have

Thus the condition (d) of Theorem 3.1 is satisfied. Also, it is obvious that the other conditions of the theorem are satisfied. Therefore 0 is the unique common fixed point of A, B, S and T.

Park et al. [4, 5] defined an IFMS and proved uniquely existence fixed point for the mappings satisfying some properties in an IFMS. Also, Park et al. [8] studied the type(𝛼) compatible mapping, and Park [7] proved some properties of type(𝛽) compatibility in an IFMS.

In this paper, we obtain a unique common fixed point and example for type(𝛽) compatible mappings under implicit relations in an IFMS. This paper attempted to develop a proof method according to some conditions based on the fundamental properties and results in this space. I think that this results will be extended and applied to the other spaces, and further research should be conducted to determine how to combine the collaborative learning algorithm with our proof method in the future.

No potential conflict of interest relevant to this article was reported.