Common Fixed Point and Example for Type(β) Compatible Mappings with Implicit Relation in an Intuitionistic Fuzzy Metric Space

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  • ABSTRACT

    In this paper, we establish common fixed point theorem for type(β) compatible four mappings with implicit relations defined on an intuitionistic fuzzy metric space. Also, we present the example of common fixed point satisfying the conditions of main theorem in an intuitionistic fuzzy metric space.


  • KEYWORD

    Type(β) compatible map , Fixed point , Implicit relation

  • 1. Introduction

    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 [912].

    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.

    2. Preliminaries

    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)abcd whenever ac and bd (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, zX, 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) {xn} is said to be convergent to a point xX if, for any 0 < 𝜖 < 1 and t > 0, there exists n0N such that

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

    for all nn0.

    (b) {xn} is called a Cauchy sequence if for any 0 < 𝜖 < 1 and t > 0, there exists n0N such that

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

    for all m, nn0.

    (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, yX 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,

    image

    whenever {xn} ⊂ X such that for some xX.

    Proposition 2.5. ([15]) Let X be an IFMS with ttt and t tt for all t ∈ [0, 1]. A,B be type(𝛽) compatible maps from X into itself and let {xn} be a sequence in X such that Axn,Bxnx for some xX. 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 = {ϕM, 𝜓N}, I = [0, 1], ϕM, 𝜓N : I6R be continuous functions and ⁎, be a continuous t-norm, t-conorm. Now, we consider the following conditions ([6]):

    (I) ϕM is decreasing and 𝜓N is increasing in sixth variables.

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

    image

    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 ab = min{a, b} and ab = max{a, b},

    image

    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

    image

    then from Park [6], ϕM, 𝜓N ∈ .

    3. Main Results and Example

    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 ab = min{a, b}, ab = max{a, b} for all a, bI 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

    image

    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 {yn} ⊂ X. Since X is complete, {yn} converges to a point xX. Since {Ax2n+2}, {Bx2n+1}, {Sx2n} and {Tx2n+1} ⊂ {yn}, we have

    image

    Now, let A is continuous. Then

    image

    By Proposition 2.5,

    image

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

    image

    and by letting n → ∞, ϕM, 𝜓N are continuous, we have

    image

    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),

    image

    and, let n → ∞, we have

    image

    On the other hand, since

    image

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

    image

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

    image

    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,

    image

    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

    image

    By Proposition 2.5,

    image

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

    image

    and by n → ∞, since ϕM, 𝜓N ∈ are continuous, we have

    image

    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 zX such that Bz = x. Using (d), we have

    image

    letting n → ∞, we get

    image

    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

    image

    letting n → ∞,

    image

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

    image

    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,

    image

    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, bX. Also, let M, N be fuzzy sets on X2 × (0,∞) defined by

    image

    Let ϕM, 𝜓N : X6R be defined as in Example 2.6 and define the maps A, B, S, T : XX by Ax = x, and . Then, for some , we have

    image
    image

    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.

    4. Conclusion

    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.

    Conflict of Interest

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

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