On Common Fixed Point for Single and Set-Valued 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
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ABSTRACT
In this paper, we introduce the notion of single and set-valued maps satisfying OWC property in IFMS using implicit relation. Also, we obtain common fixed point theorems for single and set-valued maps satisfying OWC properties in IFMS using implicit relation.
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KEYWORD
Common fixed point , Occasionally weakly compatible map , Implicit relation.
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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 semi-compatible map and obtained some fixed point using this maps in IFMS.
In this paper, we introduce the notion of single and set-valued maps satisfying occasionally weakly compatible (OWC) property in IFMS using implicit relation. Also, we obtain common fixed point theorems for single and set-valued maps satisfying OWC property in IFMS using implicit relation.
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 non-nearness betweenx andy with respect tot , respectivelyThrough out this paper,
X will represent the IFMS andCB (X ), the set of all non-empty 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.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 set-valued 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 non-increasing int 2 andt 5 for allt > 0.ψ (t 1,t 2,t 3,t 4,t 5) is non-decreasing 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 2u =Au . Suppose thatA 2u ≠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 2u =Au =Bv . Similarly, we can show thatB 2v =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 set-valued 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 set-valued 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.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 semi-compatible map and obtained some fixed point using this maps in IFMS.
In this paper, we introduce the notion of single and set-valued maps satisfying OWC property in IFMS using implicit relation. Also, we obtain common fixed point theorems for single and set-valued 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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