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

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.

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 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, B ∈ CB(X) and for every t > 0, denote

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

Also, δM(A, B, t) = 1 and δN (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 {xn} is a sequence in X such that Bxn → D ∈ CB(X) and Axn → 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.

 

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

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(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, 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 δM(Su, Tv, t) < 1 and δN(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 A2u = Au. Suppose that A2u ≠ Au, then δM(SAu, Tv, t) ＜ 1 and δN(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 A2u = Au = Bv. Similarly, we can show that B2v = 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 ≠ z0 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 = 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+, 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.

 

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.