On Common Fixed Point for Single and Set-Valued Maps Satisfying OWC Property in IFMS using Implicit Relation
On Common Fixed Point for Single and Set-Valued Maps Satisfying OWC Property in IFMS using Implicit Relation
International Journal of Fuzzy Logic and Intelligent Systems. 2015. Jun, 15(2): 132-136
Copyright © 2015, Korean Institute of Intelligent Systems
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
• Received : January 27, 2015
• Accepted : May 27, 2015
• Published : June 30, 2015
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Jong Seo, Park

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.
Keywords
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 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.
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 all a ∈ [0, 1], (d) a b c d whenever a c and b d ( a , b , c , d ∈ [0, 1]). Also, a continuous t −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 all a ∈ [0, 1], (d) a b c d whenever a c and b 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) if X is an arbitrary set, ∗ is a continuous t −norm, ⋄ is a continuous t −conorm and M , N are fuzzy sets on X 2 × (0, ∞) satisfying the following conditions; for all x , y , z in X and all s , t ∈ (0, ∞),
• (a)M(x,y,t) ＞ 0,
• (b)M(x,y,t) = 1 if and only ifx=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 ifx=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
Through out this paper, X will represent the IFMS and CB ( X ), the set of all non-empty closed and bounded subsets of X . For A , B CB ( X ) and for every t > 0, denote
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If A consists of a single point a , we write
δ 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 point b , we write
δ M ( A , B , t ) = M ( a , b , t ), δ N ( A , B , t ) = N ( a , b , t ).
It follows immediately from definition that
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Also, δ M ( A , B , t ) = 1 and δ N ( A , B , t ) = 0 if and only if A = B = { a } for al A , B CB ( X ).
Definition 2.2 . Let X be an IFMS, A : X X and B : X CB ( X ).
(a) A point x X is called a coincidence point of hybrid maps A and B if x = Ax Bx .
(b) Hybrid maps A and B are said to be compatible if ABx CB ( X ) for all x X and
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whenever { x n } is a sequence in X such that Bx n D CB ( X ) and Ax n x D .
(c) Hybrid maps A and B are said to be weakly compatible if ABx = BAx whenever Ax Bx .
(d) Hybrid maps A and B are said to be occasionally weakly compatible (OWC) if there exists some points x X such that Ax Bx and ABx BAx .
Example 2.3 . Let X = [0, ∞) with a b = min{ a , b }, a b = max{ a , b } for all a , b ∈ [0, 1] and for all t > 0,
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Define the maps A : X X and B : X CB ( X ) by
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Here 1 is a coincidence point of A and B , but A and B are not weakly compatible as BA (1) = [1, 5] ≠ AB (1) = [2, 5]. Also, A and B are OWC hybrid maps as A and B are weakly compatible at x = 0 as A (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 . Let X be an IFMS with t t = t and t t = t for all t ∈ [0, 1]. Also, let A , B : X X and S , T : X CB ( X ) be single and set-valued mappings such that the hybrid pairs ( A , S ) and ( B , T ) are OWC satisfying
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for every x , y X , t > 0.
Also, let implicit relation Φ = { ϕ , ψ } such that ϕ : [0, 1] 5 → [0, 1] and ψ : [0, 1] 5 → [0, 1] continuous functions satisfying
(a) ϕ ( t 1 , t 2 , t 3 , t 4 , t 5 ) is non-increasing in t 2 and t 5 for all t > 0. ψ ( t 1 , t 2 , t 3 , t 4 , t 5 ) is non-decreasing in t 2 and t 5 for all t > 0.
(b) ϕ ( t , t , 1, 1, t ) ≥ 0 implies that t = 1, and ψ ( t , t , 0, 0, t ) ≤ 1 implies that t = 0 for all t > 0.
Then A , B , S and T have a unique common fixed point in X .
Proof Since the hybrid pairs ( A , S ) and ( B , T ) are OWC maps, there exist two elements u , v X such that Au Su , ASu SAu and Bv Tv , BTv TBv .
First, we prove that Au = Bv . As Au Su and Bv Tv , so,
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If Au Bv , then δ M ( Su , Tv , t ) < 1 and δ N ( Su , Tv , t ) > 0. Using (1) for x = u and y = v , we have
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That is,
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Also, ϕ , ψ satisfies (b), so
• δM(Su,Tv,t) = 1 andδN(Su,Tv,t) = 0.
This is a contradiction which gives Au = Bv
Now, we prove that A 2 u = Au . Suppose that A 2 u Au , then δ M ( SAu , Tv , t ) ＜ 1 and δ N ( SAu , Tvt ) ＞ 0. Also, using (1) for x = Au and y = v , we get
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Also, Au Su and ASu SAu , so AAu ASu SAu , Bv Tv and BTv TBv , hence
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Therefore
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But ϕ , ψ satisfies (b), so,
• δM(SAu,Tv,t) = 1 andδN(SAu,Tv,t) = 0,
a contradiction and hence A 2 u = Au = Bv . Similarly, we can show that B 2 v = Bv .
Let Au = Bv = z , then Az = z = Bz , z Sz and z Tz . Therefore z is a fixed point of A , B , S and T .
Finally, we prove the uniqueness of the fixed point. Let z z 0 be another fixed point of A , B , S and T , then by (1), we have,
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From (b), we get
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This is a contradiction. Hence z = z 0 . Therefore z is unique common fixed point of A , B , S and T .
Example 3.2 . Let X be an IFMS in which X = R + , a b = min{ a , b } and a b = max{ a , b } for all a , b ∈ [0, 1] such that for all t > 0,
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Define the maps A , B , S and T on X by
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Define ϕ : [0, 1] [0, 1], ψ : [0, 1] → [0, 1] as
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Here the pairs ( A , S ) and ( B , T ) are OWC and the contractive condition is satisfied. Hence 1 is a unique common fixed point of A , B , S and T .
Corollary 3.3 . Let X be an IFMS, t t = t and t t = t for all t ∈ [0, 1] and let A : X X and S , T : X CB ( X ) be single and set-valued mappings such that the hybrid pair ( A , S ) and ( A , T ) are OWC satisfying
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for every x , y X , t ＞ 0 and ϕ , ψ are satisfies (a) and (b), respectively in Theorem 3.1. Then A , S and T have a unique common fixed point in X .
Proof Suppose that A = B in Eq. (1) of Theorem 3.1, then we get this corollary.
Corollary 3.4 . Let X be an IFMS, t t = t and t t = t for all t ∈ [0, 1] and let A : X X and S : X CB ( X ) be single and set-valued mappings such that the hybrid pair ( A , S ) is OWC satisfying
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for every x , y X , t ＞ 0 and ϕ , ψ are functions satisfying (a) and (b), respectively in Theorem 3.1. Then A and S have a unique common fixed point in X .
Proof Suppose that A = B and S = 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 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.
Acknowledgements
This author is supported by Chinju National University of Education Research Fund in 2014.
BIO
Jong Seo Park received the B.S., M.S. and Ph.D. degrees in mathematics from Dong-A University, Busan, Korea, in 1983, 1985 and 1995, respectively. He is currently Professor in Chinju National University of Education, Jinju, Korea. His research interests include fuzzy mathematics, fuzzy fixed point theory and fuzzy differential equation, etc.
E-mail: parkjs@cue.ac.kr
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