On Common Fixed Point for Single and SetValued Maps Satisfying OWC Property in IFMS using Implicit Relation
 Author: Park Jong Seo
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 15, Issue2, p132~136, 25 June 2015

ABSTRACT
In this paper, we introduce the notion of single and setvalued maps satisfying OWC property in IFMS using implicit relation. Also, we obtain common fixed point theorems for single and setvalued maps satisfying OWC properties in IFMS using implicit relation.

KEYWORD
Common fixed point , Occasionally weakly compatible map , Implicit relation.

1. Introduction
Several authors [1–5] studied and developed the various concepts in different direction and proved some fixed point in fuzzy metric space. Also, Jungck [6] introduced the concept of compatible maps, and Vijayaraju and Sajath [7] obtained some common fixed point theorems in fuzzy metric space. Recently, Park et.a.. [8] introduced the intuitionistic fuzzy metric space (IFMS), Park [12, 13] studied the compatible and weakly compatible maps in IFMS, and proved common fixed point theorem in IFMS. Also, Park [9] proved some properties for several types compatible maps, and Park [10] defined occasionally weakly semicompatible map and obtained some fixed point using this maps in IFMS.
In this paper, we introduce the notion of single and setvalued maps satisfying occasionally weakly compatible (OWC) property in IFMS using implicit relation. Also, we obtain common fixed point theorems for single and setvalued maps satisfying OWC property in IFMS using implicit relation.
2. Preliminaries
In this part, we recall some definitions, properties and known results in the IFMS as follows : Let us recall ([11]) that a continuous
t −norm is an 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]). Also, a continuoust −conorm is an 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 alla ∈ [0, 1], (d)a ⋄b ≥c ⋄d whenevera ≤c andb ≤d (a ,b ,c ,d ∈ [0, 1]).Definition 2.1 . ([8]) 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 inX and alls ,t ∈ (0, ∞),(a)M(x, y, t) ＞ 0, (b)M(x, y, t) = 1 if and only if 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 if and only if 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 functionsM (x ,y ,t ) andN (x ,y ,t ) denote the degree of nearness and the degree of nonnearness betweenx andy with respect tot , respectivelyThrough out this paper,
X will represent the IFMS andCB (X ), the set of all nonempty closed and bounded subsets ofX . ForA ,B ∈CB (X ) and for everyt > 0, denoteIf
A consists of a single pointa , we writeδ _{M}(A ,B ,t ) =δ _{M}(a ,B ,t ),δ ^{N} (A ,B ,t ) =δ ^{N} (a ,B ,t ).Furthermore, if
B consists of a single pointb , we writeδ _{M}(A ,B ,t ) =M (a ,b ,t ),δ ^{N} (A ,B ,t ) =N (a ,b ,t ).It follows immediately from definition that
Also,
δ _{M}(A ,B ,t ) = 1 andδ ^{N} (A ,B ,t ) = 0 if and only ifA =B = {a } for alA ,B ∈CB (X ).Definition 2.2 . LetX be an IFMS,A :X →X andB :X →CB (X ).(a) A point
x ∈X is called a coincidence point of hybrid mapsA andB ifx =Ax ∈Bx .(b) Hybrid maps
A andB are said to be compatible ifABx ∈CB (X ) for allx ∈X andwhenever {
x _{n}} is a sequence inX such thatBx _{n} →D ∈CB (X ) andAx _{n} →x ∈D .(c) Hybrid maps
A andB are said to be weakly compatible ifABx =BAx wheneverAx ∈Bx .(d) Hybrid maps
A andB are said to be occasionally weakly compatible (OWC) if there exists some pointsx ∈X such thatAx ∈Bx andABx ⊆BAx .Example 2.3 . LetX = [0, ∞) witha ∗b = min{a ,b },a ⋄b = max{a ,b } for alla ,b ∈ [0, 1] and for allt > 0,Define the maps
A :X →X andB :X →CB (X ) byHere 1 is a coincidence point of
A andB , butA andB are not weakly compatible asBA (1) = [1, 5] ≠AB (1) = [2, 5]. Also,A andB are OWC hybrid maps asA andB are weakly compatible atx = 0 asA (0) ∈B (0) and 0 =AB (0) ⊆BA (0) = {0}. Hence weakly compatible hybrid maps are OWC, but the converse is not true in general.3. Main Results
Theorem 3.1 . LetX be an IFMS witht ∗t =t andt ⋄t =t for allt ∈ [0, 1]. Also, letA ,B :X →X andS ,T :X →CB (X ) be single and setvalued mappings such that the hybrid pairs (A ,S ) and (B ,T ) are OWC satisfyingfor every
x ,y ∈X ,t > 0.Also, let implicit relation Φ = {
ϕ ,ψ } such thatϕ : [0, 1]^{5} → [0, 1] andψ : [0, 1]^{5} → [0, 1] continuous functions satisfying(a)
ϕ (t _{1},t _{2},t _{3},t _{4},t _{5}) is nonincreasing int _{2} andt _{5} for allt > 0.ψ (t _{1},t _{2},t _{3},t _{4},t _{5}) is nondecreasing int _{2} andt _{5} for allt > 0.(b)
ϕ (t ,t , 1, 1,t ) ≥ 0 implies thatt = 1, andψ (t ,t , 0, 0,t ) ≤ 1 implies thatt = 0 for allt > 0.Then
A ,B ,S andT have a unique common fixed point inX .Proof Since the hybrid pairs (A ,S ) and (B ,T ) are OWC maps, there exist two elementsu ,v ∈X such thatAu ∈Su ,ASu ⊆SAu andBv ∈Tv ,BTv ⊆TBv .First, we prove that
Au =Bv .As Au ∈Su andBv ∈Tv , so,If
Au ≠Bv , thenδ _{M}(Su ,Tv ,t ) < 1 andδ ^{N}(Su ,Tv ,t ) > 0. Using (1) forx =u andy =v , we haveThat is,
Also,
ϕ ,ψ satisfies (b), soδM(Su, Tv, t) = 1 and δN(Su, Tv, t) = 0.
This is a contradiction which gives
Au =Bv Now, we prove that
A ^{2}u =Au . Suppose thatA ^{2}u ≠Au , thenδ _{M}(SAu ,Tv ,t ) ＜ 1 andδ ^{N}(SAu ,Tvt ) ＞ 0. Also, using (1) forx =Au andy =v , we getAlso,
Au ∈Su andASu ∈SAu , soAAu ∈ASu ⊆SAu ,Bv ∈Tv andBTv ⊆TBv , henceTherefore
But
ϕ ,ψ satisfies (b), so,δM(SAu, Tv, t) = 1 and δN (SAu, Tv, t) = 0,
a contradiction and hence
A ^{2}u =Au =Bv . Similarly, we can show thatB ^{2}v =Bv .Let
Au =Bv =z , thenAz =z =Bz ,z ∈Sz andz ∈Tz . Thereforez is a fixed point ofA ,B ,S andT .Finally, we prove the uniqueness of the fixed point. Let
z ≠z _{0} be another fixed point ofA ,B ,S andT , then by (1), we have,From (b), we get
This is a contradiction. Hence
z =z _{0}. Thereforez is unique common fixed point ofA ,B ,S andT .Example 3.2 . LetX be an IFMS in whichX =R ^{+},a ∗b = min{a ,b } anda ⋄b = max{a ,b } for alla ,b ∈ [0, 1] such that for allt > 0,Define the maps
A ,B ,S andT onX byDefine
ϕ : [0, 1]→ [0, 1], ψ : [0, 1] → [0, 1] asHere the pairs (
A ,S ) and (B ,T ) are OWC and the contractive condition is satisfied. Hence 1 is a unique common fixed point ofA ,B ,S andT .Corollary 3.3 . LetX be an IFMS,t ∗t =t andt ⋄t =t for allt ∈ [0, 1] and letA :X →X andS ,T :X →CB (X ) be single and setvalued mappings such that the hybrid pair (A ,S ) and (A ,T ) are OWC satisfyingfor every
x ,y ∈X ,t ＞ 0 andϕ ,ψ are satisfies (a) and (b), respectively in Theorem 3.1. ThenA ,S andT have a unique common fixed point inX .Proof Suppose thatA =B in Eq. (1) of Theorem 3.1, then we get this corollary.Corollary 3.4 . LetX be an IFMS,t ∗t =t andt ⋄t =t for allt ∈ [0, 1] and letA :X →X andS :X →CB (X ) be single and setvalued mappings such that the hybrid pair (A ,S ) is OWC satisfyingfor every
x ,y ∈X ,t ＞ 0 andϕ ,ψ are functions satisfying (a) and (b), respectively in Theorem 3.1. ThenA andS have a unique common fixed point inX .Proof Suppose thatA =B andS =T in Eq. (1) of Theorem 3.1, then we get this corollary.4. Conclusion
Park et.al. [8] introduced the IFMS, and proved common fixed point theorem in IFMS. Also, Park [9] proved some properties for several types compatible maps, and Park [10] defined occasionally weakly semicompatible map and obtained some fixed point using this maps in IFMS.
In this paper, we introduce the notion of single and setvalued maps satisfying OWC property in IFMS using implicit relation. Also, we obtain common fixed point theorems for single and setvalued maps satisfying OWC property in IFMS using implicit relation.
Conflict of Interest No potential conflict of interest relevant to this article was reported.

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