On Common Fixed Point for Single and Set-Valued Maps Satisfying OWC Property in IFMS using Implicit Relation

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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.


  • KEYWORD

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

  • 1. Introduction

    Several authors [15] 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)abcd whenever ac and bd (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)abcd whenever ac and bd (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 X2 × (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 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 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, BCB(X) and for every t > 0, denote

    image

    If A consists of a single point a, we write

    δM(A, B, t) = δM(a, B, t), δN (A, B, t) = δN (a, B, t).

    Furthermore, if B consists of a single point b, we write

    δM(A, B, t) = M(a, b, t), δN (A, B, t) = N(a, b, t).

    It follows immediately from definition that

    image

    Also, δM(A, B, t) = 1 and δN (A, B, t) = 0 if and only if A = B = {a} for al A, BCB(X).

    Definition 2.2. Let X be an IFMS, A : XX and B : XCB(X).

    (a) A point xX is called a coincidence point of hybrid maps A and B if x = AxBx.

    (b) Hybrid maps A and B are said to be compatible if ABxCB(X) for all xX and

    image

    whenever {xn} is a sequence in X such that BxnDCB(X) and AxnxD.

    (c) Hybrid maps A and B are said to be weakly compatible if ABx = BAx whenever AxBx.

    (d) Hybrid maps A and B are said to be occasionally weakly compatible (OWC) if there exists some points xX such that AxBx and ABxBAx.

    Example 2.3. Let X = [0, ∞) with ab = min{a, b}, ab = max{a, b} for all a, b ∈ [0, 1] and for all t > 0,

    image

    Define the maps A : XX and B : XCB(X) by

    image

    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 tt = t and tt = t for all t ∈ [0, 1]. Also, let A, B : XX and S, T : XCB(X) be single and set-valued mappings such that the hybrid pairs (A, S) and (B, T) are OWC satisfying

    image

    for every x, yX, 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(t1, t2, t3, t4, t5) is non-increasing in t2 and t5 for all t > 0. ψ(t1, t2, t3, t4, t5) is non-decreasing in t2 and t5 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, vX such that AuSu, ASuSAu and BvTv, BTvTBv.

    First, we prove that Au = Bv. As AuSu and BvTv, so,

    image

    If AuBv, then δM(Su, Tv, t) < 1 and δN(Su, Tv, t) > 0. Using (1) for x = u and y = v, we have

    image

    That is,

    image

    Also, ϕ, ψ satisfies (b), so

    δM(Su, Tv, t) = 1 and δN(Su, Tv, t) = 0.

    This is a contradiction which gives Au = Bv

    Now, we prove that A2u = Au. Suppose that A2uAu, then δM(SAu, Tv, t) < 1 and δN(SAu, Tvt) > 0. Also, using (1) for x = Au and y = v, we get

    image

    Also, AuSu and ASuSAu, so AAuASuSAu, BvTv and BTvTBv, hence

    image

    Therefore

    image

    But ϕ, ψ satisfies (b), so,

    δM(SAu, Tv, t) = 1 and δN (SAu, Tv, t) = 0,

    a contradiction and hence A2u = Au = Bv. Similarly, we can show that B2v = Bv.

    Let Au = Bv = z, then Az = z = Bz, zSz and zTz. Therefore z is a fixed point of A, B, S and T.

    Finally, we prove the uniqueness of the fixed point. Let zz0 be another fixed point of A, B, S and T, then by (1), we have,

    image

    From (b), we get

    image

    This is a contradiction. Hence z = z0. 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+, ab = min{a, b} and ab = max{a, b} for all a, b ∈ [0, 1] such that for all t > 0,

    image

    Define the maps A, B, S and T on X by

    image

    Define ϕ : [0, 1] [0, 1], ψ : [0, 1] → [0, 1] as

    image

    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, tt = t and tt = t for all t ∈ [0, 1] and let A : XX and S, T : XCB(X) be single and set-valued mappings such that the hybrid pair (A, S) and (A, T) are OWC satisfying

    image

    for every x, yX, 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, tt = t and tt = t for all t ∈ [0, 1] and let A : XX and S : XCB(X) be single and set-valued mappings such that the hybrid pair (A, S) is OWC satisfying

    image

    for every x, yX, 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.

  • 1. Deng Z. 1982 “Fuzzy pseudo-metric spaces,” [Journal of Mathe maticalAnalysis and Applications] Vol.86 P.74-95 google doi
  • 2. Grabiec M. 1988 “Fixed points in fuzzy metric spaces,” [Fuzzy Sets and Systems] Vol.27 P.385-389 google doi
  • 3. Kaleva O., Seikkala S. 1984 “On fuzzy metric spaces,” [Fuzzy Sets and Systems] Vol.12 P.215-229 google doi
  • 4. Kubiaczyk I., Sharma S. 2003 “Common coincidence point in fuzzy metric spaces,” [Journal of Fuzzy Mathematics] Vol.11 P.1-5 google
  • 5. Singh B., Chauhan M. S. 2000 “Common fixed points of compatible maps in fuzzy metric spaces,” [Fuzzy Sets and Systems] Vol.115 P.471-475 google doi
  • 6. Jungck G. 1986 “Compatible mappings and common fixed points,” [International Journal of Mathematics and Mathematical Sciences] Vol.9 P.771-779 google doi
  • 7. Vijayaraju P., Sajath Z. M. I. 2011 “Common fixed points of single and multivalued maps in fuzzy metric spaces,” [Applied Mathematics] Vol.2 P.595-599 google doi
  • 8. Park J. H., Park J. S., Kwun Y. C. 2006 “A common fixed point theorem in the intuitionistic fuzzy metric space,” [Proceedings of the 2nd International Conference on Advances in Natural Computation (ICNC) and 3rd International Conference on Fuzzy Systems and Knowledge Discovery (FSKD)] P.293-300 google
  • 9. Park J. S. 2011 “Some properties for the compatible mappings in intuitionistic fuzzy metric space,” [Far East Journal of Mathematical Sciences] Vol.50 P.79-86 google
  • 10. Park J. S. 2009 “On a common fixed point for occasionally weakly semi-compatible hybrid mappings in an intuitionistic fuzzy metric space,” [JP Journal of Fixed Point Theory and Applications] Vol.4 P.1-10 google
  • 11. Schweizer B., Sklar A. 1960 “Statistical metric spaces,” [Pacific Journal of Mathematics] Vol.10 P.313-334 google doi
  • 12. Park J. S. 2012 “Fixed point theorems for weakly compatible functions using (JCLR) property in intuitionistic fuzzy metric space,” [International Journal of Fuzzy Logic and Intelligent Systems] Vol.12 P.296-299 google doi
  • 13. Park J. S. 2011 “Some common fixed point theorems using compatible maps in intuitionistic fuzzy metric space,” [International Journal of Fuzzy Logic and Intelligent Systems] Vol.11 P.108-112 google doi
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