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 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 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, B ∈ CB(X) and for every t > 0, denote
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
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
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,
Define the maps A : X → X and B : X → CB(X) by
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.
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
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,
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
That is,
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 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
Also, Au ∈ Su and ASu ∈ SAu, so AAu ∈ ASu ⊆ SAu, Bv ∈ Tv and BTv ⊆ TBv, hence
Therefore
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 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,
From (b), we get
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,
Define the maps A, B, S and T on X by
Define ϕ : [0, 1] → [0, 1], ψ : [0, 1] → [0, 1] as
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
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
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.
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.