Some Common Fixed Points for Type(β) Compatible Maps in an Intuitionistic Fuzzy Metric Space

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  • ABSTRACT

    Previously, Park et al. (2005) defined an intuitionistic fuzzy metric space and studied several fixed-point theories in this space. This paper provides definitions and describe the properties of type(β) compatible mappings, and prove some common fixed points for four self-mappings that are compatible with type(β) in an intuitionistic fuzzy metric space. This paper also presents an example of a common fixed point that satisfies the conditions of Theorem 4.1 in an intuitionistic fuzzy metric space.


  • KEYWORD

    Compatible map , Type(β) compatible map , Fixed point

  • 1. Introduction

    Grabiec [1] demonstrated the Banach contraction theorem in the fuzzy metric spaces introduced by Kramosil and Michalek [2]. Park [3-5], Park and Kim [6] also proved a fixed-point theorem in a fuzzy metric space.

    Recently, Park et al. [7] defined an intuitionistic fuzzy metric space while Park et al. [8] proved a fixed-point Banach theorem for the contractive mapping of a complete intuitionistic fuzzy metric space. Park et al. [9] defined a type(α) compatible map and obtained results for five mappings using a type(α) compatibility map in intuitionistic fuzzy metric spaces. Furthermore, Park [10] introduced a type(β) compatible mapping and proved some of the properties of the type(β) compatibility mapping in an intuitionistic fuzzy metric space.

    This paper proves some common fixed points for four self-mappings that satisfy type(β) compatibility mapping in intuitionistic fuzzy metric space, while it also provides an example in the given conditions for an intuitionistic fuzzy metric space.

    2. Preliminaries

    First, some definitions and properties of the intuitionistic fuzzy metric space X are provided, as follows.

    Let us recall ([11]) 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 * bc * d 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) abcd whenever ac and bd (a, b, c, d ∈ [0, 1]).

      >  Definition 2.1.

    [12] The 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 in X2 × (0,∞), which satisfy 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 referred to as an intuitionistic fuzzy metric on X. The functions M(x, y, t) and N(x, y, t) denote the degree of proximity and the degree of non-proximity between x and y with respect to t, respectively.

      >  Example 2.2.

    [13] Let (X, d) be a metric space. Denote a * b = ab and ab = min{1, a+b} for all a, b ∈ [0, 1] and let Md, Nd be the fuzzy sets on X2 × (0,∞), which are defined as follows :

    image

    for k, m, nR+(m ≥ 1). Thus, (X, Md, Nd,*,?) is an intuitionistic fuzzy metric space, i.e., the intuitionistic fuzzy metric space induced by the metric d.

      >  Definition 2.3.

    [13] Let X be an intuitionistic fuzzy metric space.

    (a) {xn} is said to be convergent to a point xX by limn→∞ xn = x if

    image

    for all t > 0.

    (b) {xn} is a Cauchy sequence if

    image

    for all t > 0 and p > 0.

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

    In this paper, X is considered to be the intuitionistic fuzzy metric space with the following condition:

    image

    for all x, yX and t > 0.

      >  Lemma 2.4.

    [6] Let {xn} be a sequence in an intuitionistic fuzzy metric space X with the condition (1). If there exists a number k ∈ (0,1) such that for all x, yX and t > 0,

    image

    for all t > 0 and n = 1, 2, 3 …, then {xn} is a Cauchy sequence in X.

      >  Lemma 2.5.

    [14] Let X be an intuitionistic fuzzy metric space. 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.

    3. Properties of type(β) compatible mappings and an example

    This section introduces type(α) and type(β) compatible maps in an intuitionistic fuzzy metric space, and it also presents an example of the relations of type(β) compatible maps.

      >  Definition 3.1.

    [14] Let A,B be mappings from the intuitionistic fuzzy metric space X into itself. These mappings are said to be compatible if

    image

    for all t > 0, whenever {xn} ⊂ X such that limn→∞ Axn = limn→∞ Bxn = x for some xX.

      >  Definition 3.2.

    ([10]) Let A,B be mappings from the intuitionistic fuzzy metric space X into itself. The mappings are said to be type(β) compatible if

    image

    for all t > 0, whenever {xn} ⊂ X such that limn→∞ Axn = limn→∞ Bxn = x for some xX.

      >  Proposition 3.3.

    [10] Let X be an intuitionistic fuzzy metric space and A,B be the continuous mappings from X into itself. Thus, A and B are compatible if they are type(β) compatible.

      >  Proposition 3.4.

    [10] Let X be an intuitionistic fuzzy metric space and A,B be mappings from X into itself. If A,B are type(β) compatible and Ax = Bx for some xX, then ABx = BBx = BAx = AAx.

      >  Proposition 3.5.

    [10] Let X be an intuitionistic fuzzy metric space and A,B be type(β) compatible mappings from X into itself. Let {xn} ⊂ X so limn→∞ Axn = limn→∞ Bxn = x for some xX, then

    (a)limn→∞ BBxn = Ax if A is continuous at x ∈ X,

    (b)limn→∞ AAxn = Bx if B is continuous at x ∈ X,

    (c)ABx = BAx and Ax = Bx if A and B are continuous at x ∈ X.

      >  Example 3.6.

    Let X = [0,∞) with the metric d defined by d(x, y) = |xy| and for each t > 0, let Md,Nd be fuzzy sets on X2 × [0,∞), which are defined as follows

    image

    for all x, yX. Clearly, (X,Md,Nd, *, ?) is an intuitionistic fuzzy metric space where *, ? are defined by a * b = min{a, b} and ab = max{a, b} for all a, b ∈ [0, 1]. Let us define A,B : XX as

    image

    Thus, A,B are discontinuous at x = 1. Let {xn} ⊂ X be defined by

    image

    Next, we have limn→∞ Axn = limn→∞ Bxn = 1

    Furthermore,

    image

    and

    image

    Therefore, A,B are type(β) compatible but they are not compatible.

    4. Main Results and Example

    This section proves the main theorem and presents an example using the given conditions in an intuitionistic fuzzy metric space.

      >  Theorem 4.1.

    Let X be a complete intuitionistic fuzzy metric space where t * tt, ttt for all t ∈ [0, 1]. Let A,B,S and T be mappings from X into itself so:

    (a) AT(X) ∪ BS(X) ⊂ ST(X);

    (b) there exists k ∈ (0, 1) so for all x, y ∈ X and t > 0,

    M2(Ax,By,kt) * [M(Sx, Ax, kt)M(Ty,By, kt)]

    *M2(Ty,By, kt) + aM(Ty,By, kt)M(Sx,By, 2kt)

    ≥ [pM(Sx, Ax, t) + qM(Sx, Ty, t)]M(Sx,By, 2kt),

    N2(Ax,By,kt) ? [N(Sx, Ax, kt)N(Ty,By, kt)] ?N2(Ty,By, kt) + aM(Ty,By, kt)N(Sx,By, 2kt) ≤ [pN(Sx, Ax, t) + qN(Sx, Ty, t)]N(Sx,By, 2kt),

    where 0 < p, q < 1, 0 ≤ a < 1 such that p + qa = 1;

    (c) S and T are continuous and ST = TS;

    (d) the pairs (A, S) and (B, T) are type(β) compatible.

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

    Proof. Let x0 be an arbitrary point of X. Using (a), we can construct an {xn} ⊂ X as follows:

    ATx2n = STx2n+1, BSx2n+1 = STx2n+2, n = 0, 1, 2,…. Next, let zn = STxn. Using (b), we obtain

    M2(ATx2n,BSx2n+1, kt) * [M(STx2n, ATx2n, kt) ×M(TSx2n+1,BSx2n+1, kt)] *M2(TSx2n+1, BSx2n+1, kt) + aM(TSx2n+1,BSx2n+1, kt) ×M(STx2n,BSx2n+1, 2kt) ≥ [pM(STx2n+1, ATx2n, t) +qM(STx2n, TSx2n+1, t)] ×M(STx2n,BSx2n+1, 2kt),

    N2(ATx2n,BSx2n+1, kt) ? [N(STx2n, ATx2n, kt) ×N(TSx2n+1,BSx2n+1, kt)] ? N2(TSx2n+1, BSx2n+1, kt) + aN(TSx2n+1,BSx2n+1, kt) ×N(STx2n,BSx2n+1, 2kt) ≤ [pN(STx2n+1, ATx2n, t) +qN(STx2n, TSx2n+1, t)] ×N(STx2n,BSx2n+1, 2kt)

    and

    M2(STx2n+1, STx2n+2, kt) * [M(z2n, STx2n+1, kt) ×M(z2n+1, STx2n+2, kt)] *M2(z2n+1, STx2n+2, kt) +aM(z2n+1, STx2n+2, kt)M(z2n, STx2n+2, 2kt) ≥ [pM(z2n, STx2n+1, t) + qM(z2n, z2n+1, t)] ×M(z2n, STx2n+2, 2kt),

    N2(STx2n+1, STx2n+2, kt) ? [N(z2n, STx2n+1, kt) ×N(z2n+1, STx2n+2, kt)] ? N2(z2n+1, STx2n+2, kt) +aN(z2n+1, STx2n+2, kt)N(z2n, STx2n+2, 2kt) ≤ [pN(z2n, STx2n+1, t) + qN(z2n, z2n+1, t)] ×N(z2n, STx2n+2, 2kt).

    Then,

    M2(z2n+1, z2n+2, kt) *[M(z2n, z2n+1, kt)M(z2n+1, z2n+2, kt)] +aM(z2n+1, z2n+2, kt)M(z2n, z2n+2, 2kt) ≥ [p + q]M(z2n, z2n+1, t)M(z2n, z2n+2, 2kt),

    N2(z2n+1, z2n+2, kt) ?[N(z2n, z2n+1, kt)N(z2n+1, z2n+2, kt)] +aN(z2n+1, z2n+2, kt)N(z2n, z2n+2, 2kt) ≤ [p + q]N(z2n, z2n+1, t)N(z2n, z2n+2, 2kt),

    and

    M2(z2n+1, z2n+2, kt)M(z2n+1, z2n+2, kt)] +aM(z2n+1, z2n+2, kt)M(z2n, z2n+2, 2kt) ≥ [p + q]M(z2n, z2n+1, t)M(z2n, z2n+2, 2kt),

    N2(z2n+1, z2n+2, kt)N(z2n+1, z2n+2, kt)] +aN(z2n+1, z2n+2, kt)N(z2n, z2n+2, 2kt) ≤ [p + q]N(z2n, z2n+1, t)N(z2n, z2n+2, 2kt).

    Therefore, it follows that

    z2n+1M(z2n+1, z2n+2, kt) ≥ M(z2n, z2n+1, t),

    N(z2n+1, z2n+2, kt) ≤ N(z2n, z2n+1, t)

    for all t > 0 and k ∈ (0, 1). In general, for m = 1, 2, …, we have

    M(zm+1, zm+2, kt) ≥ M(zm, zm+1, t),

    N(zm+1, zm+2, kt) ≤ N(zm, zm+1, t)

    Thus, {zn} is a Cauchy sequence in X and, because X is complete, {zn} converges to a point zX. Since {ATx2n}, {BSx2n+1} are subsequences of {zn}, limn→∞ ATx2n = z = limn→∞ BSx2n+1.

    Let yn = TXn, un = Sxn for n = 1, 2,…. Thus, we have Ay2nz, Sy2nz, Tu2n+1z and Bu2n+1z. Furthermore,

    M(AAy2n, SSy2n, t) → 1,

    M(BBu2n+1, TT2n+1, t) → 1,

    N(AAy2n, SSy2n, t) → 0,

    N(BBu2n+1, TT2n+1, t) → 0

    as n →∞. Based on the continuity of T and Proposition 3.4, we obtain TBu2n+1Tz, BBu2n+1Tz.

    Next, by taking x = y2n, y = Bu2n+1 in (b), for n →∞ we obtain,

    M2(z, Tz, kt) * [M(z, z, kt)M(Tz, Tz, kt)] *M2(Tz, Tz, kt) + aM(Tz, Tz, kt)M(z, Tz, 2kt) ≥ [pM(z, z, t) + qM(z, Tz, t)]M(z, Tz, 2kt),

    N2(z, Tz, kt) ? [N(z, z, kt)N(Tz, Tz, kt)] ?N2(Tz, Tz, kt) + aN(Tz, Tz, kt)N(z, Tz, 2kt) ≤ [pN(z, z, t) + qN(z, Tz, t)]N(z, Tz, 2kt),

    then

    M2(z, Tz, kt) + aM(z, Tz, 2kt) ≥ [p + qM(z, Tz, t)]M(z, Tz, 2kt), N2(z, Tz, kt) ≤ qN(z, Tz, t)N(z, Tz, 2kt).

    Since M(x, y, ?) is nondecreasing and N(x, y, ?) is nonincreasing for all x, yX, we obtain

    M(z, Tz, kt) + a ≥ p + qM(z, Tz, t),

    N(z, Tz, kt) ≤ qN(z, Tz, t)

    and

    image

    Thus, z = Tz. Similarly, we have z = Sz.

    Next, by taking x = y2n and y = z in condition (b), for n→∞ we obtain

    M(z,Bz, kt) *M(z,Bz, kt) +aM(z,Bz, kt)M(z,Bz, 2kt) ≥ (p + q)M(z,Bz, 2kt),

    N(z,Bz, kt) ? N(z,Bz, kt) +aN(z,Bz, kt)N(z,Bz, 2kt) ≤ 0.

    Thus,

    M(z,Bz, kt) + aM(z,Bz, kt) ≥ p + q,

    N(z,Bz, kt) + aN(z,Bz, kt) ≤ 0.

    Therefore,

    M(z,Bz, kt) ≥ 1,

    N(z,Bz, kt) ≤ 0

    for all t > 0 and k ∈ (0, 1). Thus, z = Bz. Similarly, we obtain z = Az. Therefore, z is a common fixed point of A,B,S and T.

    Let w be another common fixed point of A,B,S and T.

    Using condition (b), we have

    M2(z,w, kt) * [M(z, z, kt)M(w,w, kt)] *M2(w,w, kt) + aM(w,w, kt)M(z,w, 2kt) ≥ [pM(z, z, t) + qM(z,w, t)]M(z,w, 2kt),

    N2(z,w, kt) ? [N(z, z, kt)N(w,w, kt)] ?N2(w,w, kt) + aN(w,w, kt)N(z,w, 2kt) ≤ [pN(z, z, t) + qN(z,w, t)]M(z,w, 2kt).

    Thus,

    M2(z,w, kt) +M(z,w, 2kt) ≥ (p + qM(z,w, t))M(z,w, 2kt), N2(z,w, kt) ≤ qM(z,w, t)M(z,w, 2kt),

    Therefore,

    M(z,w, kt) ≤ M(z,w, 2kt),

    N(z,w, kt) ≥ N(z,w, 2kt),

    so

    image

    Thus, z = w. This means that A,B,S and T have a unique common fixed point.

      >  Corollary 4.2.

    Let X be a complete intuitionistic fuzzy metric space where t * tt, ttt for all t ∈ [0, 1] and let A,B be mappings from X into itself such that:

    (e) A(X) ⊂ S(X),

    (f) there exists k ∈ (0,1) so for all x, y ∈ X and t > 0,

    M2(Ax, Ay, kt) * [M(Sx, Ax, kt)M(Sy, Ay, kt)] M2(Sy, Ay, kt) + aM(Sy, Ay, kt)M(Sx, Ay, 2kt) ≥ [pM(Sx, Ax, t) + qM(Sx, Sy, t)]M(Sx, Ay, 2kt),

    N2(Ax, Ay, kt) ? [N(Sx, Ax, kt)N(Sy, Ay, kt)kt)] ?N2(Sy, Ay, kt) + aM(Sy, Ay, kt)N(Sx, Ay, 2kt) ≤ [pN(Sx, Ax, t) + qN(Sx, Sy, t)]N(Sx, Ay, 2kt),

    where 0 < p, q < 1, 0 ≤ a < 1 such that p + qa = 1,

    (g) S is continuous,

    (h) A and S are type(β) compatible.

    Thus, A and S have a unique common fixed point in X.

    Proof. Therefore, if we enter A = B and S = T into Theorem 4.1, all of the conditions of Theorem 4.1 are satisfied. Thus, the proof of this corollary follows from Theorem 4.1.

      >  Example 4.3.

    Let

    image

    with the metric d defined by d(x, y) = |xy| and for each t > 0, let Md,Nd be fuzzy sets on X2 × [0,∞), which are defined as follows

    image

    for all x, yX. Clearly, (X,Md,Nd, *, ?) is a complete intuitionistic fuzzy metric space where *, ? are defined by a * b = min{a, b} and ab = max{a, b} for all a, b ∈ [0, 1]. Let A,B,S and T be maps from X into itself, which are defined by

    image

    for all xX. Then,

    image

    Furthermore, ST = TS and S, T are continuous. If we take

    image

    and t = 1, the condition (b) of Theorem 4.1 is satisfied. Moreover, A, S are type(β) compatible if limn→∞ xn = 0 where {xn} ⊂ X such that limn→∞ Axn = limn→∞ Sxn = 0 for some 0 ∈ X.

    Similarly, B, T are type(β) compatible. Thus,

    M(0,B0, kt) + aM(0,B0, kt) ≥ p + q,

    N(0,B0, kt) + aN(0,B0, kt) ≤ 0.

    Therefore, M(0,B0, kt) ≥ 1 and N(0,B0, kt) ≤ 0 for all t > 0 and k ∈ (0, 1). Thus, 0 = B0. Similarly, we obtain 0 = A0. Therefore, 0 is a common fixed point of A,B,S and T.

    Let w be another common fixed point of A,B,S and T. Then,

    M2(0,w,kt) +M(0,w, 2kt) ≥ (p + qM(0,w,t))M(0,w,2kt),

    M2(0,w, kt) ≤ qM(0,w, t)M(0,w, 2kt).

    Therefore, because

    M(0,w, kt) ≤ M(0,w, 2kt),

    N(0,w, kt) ≥ N(0,w, 2kt),

    Thus,

    image

    Therefore, 0 = w. Thus, A,B,S and T have a unique common fixed point 0.

    5. Conclusion

    Park et al. [7] defined an intuitionistic fuzzy metric space and Park et al. [8] proved a fixed-point Banach theorem for the contractive mapping of a complete intuitionistic fuzzy metric space. Park et al. [9] defined a type(α) compatible mapping and obtained results for five mappings using type(α) compatibility in intuitionistic fuzzy metric spaces. Furthermore, Park [10] introduced type(β) compatible mapping and proved some properties of type(β) compatibility in an intuitionistic fuzzy metric space. In this paper, we proved some common fixed points for four self-mappings that satisfy type(β) compatibility and we provided an example in the given conditions for an intuitionistic fuzzy metric space.

    This paper attempted to develop a method to provide a proof based on the fundamental concepts and properties defined in the new space. I think that the results of this paper will be extended to the intuitionistic M-fuzzy metric space and other spaces. Further research should be conducted to determine how to combine the collaborative learning algorithm with our proof method in the future.

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