Common Fixed Point and Example for Type(β ) Compatible Mappings with Implicit Relation in an Intuitionistic Fuzzy Metric Space
 Author: Park Jong Seo
 Organization: Park Jong Seo
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 14, Issue1, p66~72, 25 March 2014

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 [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.
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 alla ∈ [0, 1]; (d)a ⁎b ≤c ⁎d whenevera ≤c andb ≤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) ifX is an arbitrary set, ⁎ is a continuoust –norm, is a continuoust –conorm andM,N are fuzzy sets onX ^{2} × (0,∞) satisfying the following conditions; for allx, 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 onX . The functions M(x, y, t ) and N(x, y, t ) denote the degree of nearness and the degree of nonnearness betweenx andy with respect tot , respectively.Definition 2.2. ([6]) LetX be an IFMS.(a) {
x_{n} } is said to be convergent to a pointx ∈X if, for any 0 < 𝜖 < 1 andt > 0, there existsn _{0} ∈N such thatM(xn, x, t) > 1 – 𝜖, N(xn, x, t) < 𝜖
for all
n ≥n _{0}.(b) {
x_{n} } is called a Cauchy sequence if for any 0 < 𝜖 < 1 andt > 0, there existsn _{0} ∈N such thatM(xn, xm, t) > 1 – 𝜖, N(xn, xm, t) < 𝜖
for all
m, n ≥n _{0}.(c)
X is complete if every Cauchy sequence converges inX .Lemma 2.3. ([8]) LetX be an IFMS. If there exists a numberk ∈ (0, 1) such that for allx, y ∈X andt > 0,M(x, y, kt) ≥ M(x, y, t), N(x, y, kt) ≤ N(x, y, t),
then
x =y .Definition 2.4. ([7]) LetA,B be mappings from IFMSX into itself. The mappings are said to be type(𝛽) compatible if for allt > 0,whenever {
x_{n} } ⊂X such that for somex ∈X .Proposition 2.5. ([15]) LetX be an IFMS witht ⁎t ≥t andt t ≤t for allt ∈ [0, 1].A,B be type(𝛽) compatible maps fromX into itself and let {x_{n} } be a sequence inX such thatAx_{n},Bx_{n} →x for somex ∈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} :I ^{6} →R be continuous functions and ⁎, be a continuous tnorm, tconorm. 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 havefor any fixed
t > 0, any nondecreasing functionsu, v : (0,∞) →I with 0 <u (t ),v (t ) ≤ 1, and any nonincreasing functionsx, y : (0,∞) →I with 0 <x (t ),y (t ) ≤ 1, then there existsh ∈ (0, 1) withu(ht) ≥ v(t) ⁎ u(t), x(ht) ≤ y(t) x(t).
(III) If, for some
k ∈ (0, 1), we haveϕM(u(kt), u(t), 1, 1, u(t), u(t)) ≥ 1
for any fixed
t > 0 and any nondecreasing functionu : (0,∞) →I , thenu (kt ) ≥u (t ). Also, if, for somek ∈ (0, 1), we have𝜓N(x(kt), x(t), 0, 0, x(t), x(t)) ≤ 1
for any fixed
t > 0 and any nonincreasing functionx : (0,∞) →I , thenx (kt ) ≤x (t ).Example 2.6. ([6]) Leta ⁎b = min{a, b } anda b = max{a, b },Also, let
t > 0, 0 <u (t ),v (t ),x (t ),y (t ) ≤ 1,k ∈ (0, ½ ) whereu, v : [0,∞) →I are nondecreasing functions andx, y : [0,∞) →I are nonincreasing functions. Now, supposethen from Park [6],
ϕ _{M}, 𝜓_{N} ∈ .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. LetX be a complete intuitionistic fuzzy metric space witha ⁎b = min{a, b },a b = max{a, b } for alla, b ∈I andA, B, S andT be mappings fromX 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 andT have a unique common fixed point inX .Proof. Letx _{0} be an arbitrary point ofX . Then from Theorem 3.1 of ([6]), we can construct a Cauchy sequence {y_{n} } ⊂X . SinceX is complete, {y_{n} } converges to a pointx ∈X . Since {Ax _{2n+2}}, {Bx _{2n+1}}, {Sx _{2n}} and {Tx _{2n+1}} ⊂ {y_{n} }, we haveNow, let
A is continuous. ThenBy Proposition 2.5,
Using condition (d), we have, for any
t > 0,and by letting
n → ∞,ϕ _{M}, 𝜓_{N} are continuous, we haveTherefore, 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 haveOn the other hand, since
ϕ _{M} is nonincreasing and 𝜓_{N} is nondecreasing in the fifth variable, we have, for anyt > 0,which implies that
Sx =x . SinceS (X ) ⊆B (X ), there exists a pointy ∈X such thatBy =x . Using condition (d), we havewhich implies that
x =Ty . SinceBy =Ty =x andB, T are type(𝛽) compatible, we haveTTy =BBy . HenceTx =TTy =BBy =Bx . Therefore, from (d), we have, for anyt > 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. Hencex is a common fixed point ofA,B, S andT . The same result holds if we assume thatB is continuous instead ofA .Now, suppose that
S is continuous. ThenBy Proposition 2.5,
Using (d), we have for any
t > 0,and by
n → ∞, sinceϕ _{M}, 𝜓_{N} ∈ are continuous, we haveThus, 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. SinceS (X ) ⊆B (X ), there exists a pointz ∈X such thatBz =x . Using (d), we haveletting
n → ∞, we getwhich implies that
x =Tz . SinceBz =Tz =x andB, T are type(𝛽) compatible, we haveTBz =BBz and soTx =TBz =BBz =Bx . Thus, we haveletting
n → ∞,Thus,
x =Tx =Bx . SinceT (X ) ⊆A (X ), there existsw ∈X such thatAw =x . Thus, from (d),Hence we have
x =Sw =Aw . Also, sinceA, S are type(𝛽) compatible,x = Sx = SSw = AAw = Ax.
Hence
x is a common fixed point ofA,B, S andT . The same result holds if we assume thatT is continuous instead ofS .Finally, suppose that
A, B, S andT have another common fixed pointu . Then we have, for anyt > 0,Therefore, from (III),
x =u . This completes the proof.Example 3.2. LetX be a intuitionistic fuzzy metric space withX = [0, 1], ⁎, be tnorm and tconorm defined bya ⁎ b = min{a, b}, a b = max{a, b}
for all
a, b ∈X . Also, letM, N be fuzzy sets onX ^{2} × (0,∞) defined byLet
ϕ _{M}, 𝜓_{N} :X ^{6} →R be defined as in Example 2.6 and define the mapsA, B, S, T :X →X byAx =x , and . Then, for some , we haveThus 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 andT .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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