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.

• KEYWORD

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

• 1. Introduction

Several authors  studied and developed the various concepts in different direction and proved some fixed point in fuzzy metric space. Also, Jungck  introduced the concept of compatible maps, and Vijayaraju and Sajath  obtained some common fixed point theorems in fuzzy metric space. Recently, Park et.a..  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  proved some properties for several types compatible maps, and Park  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 () 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. () 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

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, 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

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,

Define the maps A : XX and B : XCB(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 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

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,

If AuBv, 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 A2uAu, then δM(SAu, Tv, t) ＜ 1 and δN(SAu, Tvt) ＞ 0. Also, using (1) for x = Au and y = v, we get

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

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+, ab = min{a, b} and ab = 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, 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

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

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.  introduced the IFMS, and proved common fixed point theorem in IFMS. Also, Park  proved some properties for several types compatible maps, and Park  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

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