A novel framework of complex valued fuzzy metric spaces and fixed point theorems 1

1 Introduction and Preliminaries

The concept of fuzzy set was introduced by Zadeh [3] in a seminal paper in 1965. After then the study of fuzzy sets has been very widespread. Many authors [4–6] have contributed towards some introductory and essential text in fuzzy sets.

Kramosil and Michalek [7] pioneered the notion of a fuzzy metric space by generalizing the concept of the probabilistic metric space to the fuzzy situation. Grabiec [8] tagged on Kramosil and Michalek and obtained the fuzzy version of Banach contraction principle. George and Veeramani [9] modified the concept of fuzzy metric spaces due to Kramosil and Michalek [7] and defined the Hausdorff topology of fuzzy metric spaces. This proved a milestone in fixed point theory of fuzzy metric space and afterward a flood of papers appeared for fixed point theorems in fuzzy metric spaces.

The extension of crisp sets to fuzzy sets, in terms of membership functions, is mathematically comparable to the extension of the set of integers I to the set of real numbers R. That is, expanding the range of the membership function μ _s  (x) from {0, 1} to [0, 1] is mathematically analogous to the extension of I to R. Of course, the expansion of the number set did not end with real numbers. Historically, the introduction of real numbers was followed by their extension to the set of complex numbers C. The result of such an extension, in the context of set theory, is the complex fuzzy set, i.e., a fuzzy set pigeonholed by a complex-valued membership function.

Fuzzy complex numbers and fuzzy complex analysis were first introduced by Buckley [10–13]. Acknowledging the Buckley’s work some authors continued research in fuzzy complex numbers. One can see in [14–20]. In this series Ramot et al. [1] extended fuzzy sets to complex fuzzy sets. According to Ramot et al. [1], the complex fuzzy set is characterized by a membership function, whose range is not limited to [0, 1] but extended to the unit circle in the complex plane. Membership in a complex fuzzy set remains “as fuzzy” as membership in a traditional fuzzy set.

Definition 1.1. [1] A Complex Fuzzy Set S, defined on a universe of discourse U, is characterized by a membership function μ _s  (x) that assigns every element x ∈ U, a complex valued grade of membership in S. The values μ _s  (x) lie within the unit circle in the complex plane, and are thus of the form μ s ( x ) = r s ( x ) . e iw s ( x ) , ( i = - 1 ) , where r _s  (x) and w _s  (x) both real-valued, with r _s  (x) ∈ [0, 1].

The complex fuzzy set S, may be represented as the set of ordered pairs, given by S = { ( x , μ s ( x ) ) | x ∈ U } .

Clearly, each complex grade of membership is defined by an amplitude term r _s  (x) and a phase term w _s  (x). Notice that the amplitude term r _s  (x) is equal to |μ _s  (x) |, the amplitude of μ _s  (x). Ramot et al. [1] claimed that complex fuzzy sets are generalizations of ordinary fuzzy sets, it can be possible to represent any ordinary fuzzy set in terms of a complex fuzzy set. If any ordinary fuzzy set S is characterized by the real-valued membership function λ _s  (x), then S can be transformed into a complex fuzzy set is by setting the amplitude term r _s  (x) equal to λ _s  (x), and the phase term equal to zero for all x. From this study, it is possible to deduce that the amplitude term is basically equivalent to the traditional real-valued grade of membership, while the phase term is the distinguishing factor between ordinary and complex fuzzy sets. Thus we can say without a phase term, the complex fuzzy set effectively reduces to a conventional fuzzy set.

After Ramot et al. [1], contributions from Dong Qiu et al. [17] and Nilamber Sethi et al. [15] in the field of complex fuzzy set are also important and notable.

Recently Azam et al. [21] introduced the concept of complex valued metric spaces. They defined a partial order ‘≾’ on the set of complex numbers C for comparing two complex numbers as follows.

Let C be the set of complex numbers and z₁, z₂ ∈ C then z₁ ≾ z₂ if and only if Re ( z 1 ) ≤ Re ( z 2 ) and Im ( z 1 ) ≤ Im ( z 2 ) . It follows that z₁ ≾ z₂, if one the following conditions are satisfied:

Re (z₁) = Re (z₂) and Im (z₁) = Im (z₂);

In particular, we write z₁ ⋨ z₂ if z₁ ≠ z₂ and one of (C2), (C3) and (C4) is satisfied while z₁ ≺ z₂ if only (C4) is satisfied. Note that 0 ≾ z 1 ⋨ z 2 ⇒ | z 1 | < | z 2 | , z 1 ≾ z 2 , z 2 ≺ z 3 ⇒ z 1 ≺ z 3 . Moreover in a paper, Rouzkard and Imdad [22] recognized the utility of complex valued metric spaces and given some beautiful remarks. They established that the idea of complex valued metric spaces is intended to define rational expressions which are not meaningful in cone metric spaces and thus many results of analysis cannot be generalized to cone metric spaces. The definition of cone metric space banks on the underlying Banach space which is not a division Ring. However, in complex valued metric spaces, we can study improvements of analysis involving divisions. Utilizing the concept of Azam et al. [21], Verma et al. [23] defined ’max’ functions for complex numbers with partial order relation ≾.

Definition 1.2. [23] The ’max’ functions for complex numbers with partial order relation ‘≾’ is defined as

max {z₁, z₂} = z₂ ⇔ z₁ ≾ z₂ .

On the similar lines ’min’ function can also be defined as:

min {z₁, z₂} = z₁ ⇔ z₁ ≾ z₂.

The purpose of this paper is manifold. Firstly, utilizing the idea of complex fuzzy sets due to Ramot et al. [1], we define the notion of complex valued fuzzy metric spaces with the help of continuous t-norms. Secondly, In Section 3, we define a Hausdorff topology on complex valued fuzzy metric space and introduce the concept of Cauchy sequences in a complex valued fuzzy metric space. In Section 4, the complex valued fuzzy version of vital Banach contraction principle is proved. In next section some fixed point theorems through rational expression are established. Finally, the Jungck type fixed point results for complex valued fuzzy metric spaces is demonstrated.

2 Rudiments of complex valued fuzzy metric spaces

First of all we define complex valued continuous t-norm.

Definition 2.1. A binary operation * : r _s e ^iθ  × r _s e ^iθ →r _s e ^iθ , wherein r _s  ∈ [0, 1] and a fix θ ∈ [ 0 , π 2 ], is called complex valued continuous t-norm if it satisfies the following conditions:

* is associative and commutative,

Example 2.1.a * b =min (a, b).

Example 2.2.a * b = max (a + b - e ^iθ , 0), for a fix θ ∈ [ 0 , π 2 ].

Example 2.3. a * b = { min { a , b } , if max { a , b } = e i θ ; 0 , otherwise , for a fix θ ∈ [ 0 , π 2 ] .

Where ‘min’ and ‘max’ are defined in definition (1).

Note: Throughout this article, θ is taken to be fixed in [ 0 , π 2 ] with the assumption that the complex fuzzy set S = {(x, μ _s  (x)) |x ∈ U} interacts with other complex fuzzy sets in view of the partial ordering due to Azam et al. [21].

Finally, we define complex valued fuzzy metric spaces as follows.

Definition 2.2. The triplet (X, M, *) is said to be complex valued fuzzy metric space if X is an arbitrary non empty set, * is a complex valued continuous t-norm and M : X × X × (0, ∞) → r _s e ^iθ is a complex valued fuzzy set, where r _s  ∈ [0, 1] and θ ∈ [ 0 , π 2 ], satisfying the following conditions:

M (x, y, t) ≻0,

Remark 2.1. If we take θ = 0 then complex valued fuzzy metric simply goes to real valued fuzzy metric.

Example 2.4. (Induced complex valued fuzzy metric) Let (X, d) be a metric space. Let a * b =min {a, b}, for all a, b ∈ r _s e ^iθ , where r _s  ∈ [0, 1] and θ ∈ [ 0 , π 2 ]. For each t > 0, x, y ∈ X, If we define M ( x , y , t ) = e i θ kt n kt n + md ( x , y ) , where k, m, n ∈ R⁺. Then (X, M, *) is a complex valued fuzzy metric space.

In above example by choosing k = m = n = 1, one gets M ( x , y , t ) = e i θ t t + d ( x , y ) . This complex valued fuzzy metric induced by a metric d is referred to as a standard complex valued fuzzy metric.

Example 2.5. Let X = R. We define a * b = min {a, b} , ∀ a, b ∈ r _s e ^iθ , where r _s  ∈ [0, 1] and θ ∈ [ 0 , π 2 ]. Furthermore for all x, y ∈ X and t ∈ (0, ∞), we define M ( x , y , t ) = e i θ e - | x - y | t . Then (X, M, *) is a complex valued fuzzy metric space.

Example 2.6. Let X = N. We define a * b =max(a + b - e ^iθ , 0) , ∀ a, b ∈ r _s e ^iθ , where r _s  ∈ [0, 1]and θ ∈ [0, π/2] and M ( x , y , t ) = { e i θ x y , if x ≤ y ; e i θ y x , if y ≤ x , for all x, y ∈ X and t ∈ (0, ∞).

Then (X,M,*) is a complex valued fuzzy metric space.

Remark 2.2. Note that the above function M (x, y, t) in Example 2.6 is not a complex valued fuzzy metric with the t- norm defined as a * b = min⥹{a, b}, for all a, b ∈ r _s e ^iθ , where r _s  ∈ [0, 1] and θ ∈ [ 0 , π 2 ].

Example 2.7. Let X = {x _n } ∪ {1}, where {x _n } _n∈N⊂(0, ∞) is increasing sequence converging to 1. For all t > 0 and n, m ∈ N, we define M (x _n , x _n , t) =e ^iθ ,   M (1, 1, t) = e ^iθ , M (x _n , x _m , t) = e ^iθ    min{x _n , x _m } and M ( x n , 1 , t ) = { e i θ min { x n , t } , if 0 < t < 1 ; e i θ x n , if t > 1 . Then (X, M, *) is a complex valued fuzzy metric space, where a * b = min (a, b) , ∀ a, b ∈ r _s e ^iθ , where r _s  ∈ [0, 1] and θ ∈ [ 0 , π 2 ].

3 Topology Induced by a complex valued fuzzy metric

Definition 3.1.Open Ball - Let (X, M, *) be a complex valued fuzzy metric space. We define an open ball B (x, r, t) with centre x ∈ X and radius r ∈ C,   0 ≺ r ≺ e ^iθ , t > 0 as B ( x , r , t ) = { y ∈ X : M ( x , y , t ) ≻ e i θ - r } , where θ ∈ [ 0 , π 2 ] .

A point x ∈ X is called an interior point of set A ⊂ X, whenever there exists r ∈ C, 0 ≺ r ≺ e ^iθ such that B ( x , r , t ) = { y ∈ X : M ( x , y , t ) ≻ e i θ - r } ⊂ A , where θ ∈ [ 0 , π 2 ] .

The subset A of X is called open whenever each element of A is an interior point of A.

Remark 3.1. In a complex valued fuzzy metric space(X, M, *), whenever M (x, y, t) ≻ e ^iθ  - r for x, y ∈ X, t > 0, r ∈ C and 0 ≺ r ≺ e ^iθ , we can find a t₀, 0 < t₀ < t such that M (x, y, t₀) ≻ e ^iθ  - r, for a fix θ ∈ [ 0 , π 2 ].

Remark 3.2. For any r₁ ≻ r₂, we can find a r₃, such that r₁ * r₃ ≿ r₂, where r₁, r₂, r₃ ∈ r _s e ^iθ .

Proposition 3.1. In complex valued fuzzy metric spaces, every open ball is an open set.

Proof. Consider an open ball B (x, r, t). To show B (x, r, t) to be open we show that at every point of B (x, r, t), there exists an open ball contained in B (x, r, t).

Let y ∈ B (x, r, t) ⇒ M (x, y, t) ≻ e ^iθ  - r, where r ∈ C and 0 ≺ r ≺ e ^iθ , thus we can find a t₀, 0 < t₀ < t such that M (x, y, t₀) ≻ e ^iθ  - r.

Let r₀ = M (x, y, t₀) ≻ e ^iθ  - r.Then we canfind ^′s′, where 0 ≺ s ≺ e ^iθ , such that r₀ ≻ e ^iθ  - s ≻ e ^iθ  - r .

For given ′ r 0 ′ and ^′s′, where r₀ ≻ e ^iθ  - s, we can find r₁, 0 ≺ r₁ ≺ e ^iθ such that r₀ * r₁ ≿ e ^iθ  - s.

Consider the ball B (y, e ^iθ  - r₁, t - t₀). We assert that B (y, e ^iθ  - r₁, t - t₀) ⊂ B (x, r, t).

Let z ∈ B (y, e ^iθ  - r₁, t - t₀) ⇒ M (y, z, t - t₀) ≻ e ^iθ  - (e ^iθ  - r₁) = r₁. i . e .   M (y, z, t - t₀) ≻ r₁ .

Consider M ( x , z , t ) ≿ M ( x , y , t 0 ) * M ( y , z , t - t 0 ) ≿ r 0 * r 1 ≿ e i θ - s ≻ e i θ - r . i.e. M (x, z, t) ≻ e ^iθ  - r.

Which amounts to say that z ∈ B (x, r, t). Hence B (y, e ^iθ  - r₁, t - t₀) ⊂ B (x, r, t).

This shows that B (x, r, t) is an open set.

In view of partial ordering due to Azam et al. [21], increasing and decreasing functions for the set of complex numbers, are defined.

Definition 3.2. Let X be any non empty set. A function f : X → C is called increasing function if f (x₁) ≻ f (x₂) whenever x₁ > x₂,   x₁, x₂ ∈ X.

Example 3.1. Let X = N. Consider the function f : X → C defined as f (x) = xe^iπ/3, ∀ x ∈ X, then f (x) is an increasing function.

Example 3.2. Consider the function f : C → C defined as f (z) =2z - 5,    ∀ z ∈ C, then f (z) is an increasing function.

Definition 3.3. A function f : X → C is called decreasing function if f (x₁) ≺ f (x₂), wherein x₁ > x₂ .   x₁, x₂ ∈ X.

Example 3.3. Let X = N. Consider the function f : X → C defined as f (x) = xe^4πi/3, ∀ x ∈ X, then f (x) is a decreasing function.

Example 3.4. Consider the function f : C → C defined as f (z) =3i - z,    ∀ z ∈ C, then f (z) is a decreasing function.

Lemma 3.1.M (x, y, .) is non-decreasing function for allx, y ∈ X.

Proof. Suppose M (x, y, t) ≻ M (x, y, s) for some 0 < t < s.

It can be seen that M ( x , y , t ) * M ( y , y , s - t ) ≾ M ( x , y , t + s - t ) i . e . M ( x , y , t ) * M ( y , y , s - t ) ≾ M ( x , y , s ) but M (y, y, s - t) = e ^iθ  .

Implies M (x, y, t) * e ^iθ  ≾ M (x, y, s) ≺ M (x, y, t) (by assumption) Thus we have M (x, y, t) ≺ M (x, y, t).

Leads to contradiction, therefore for all x, y ∈ X, M (x, y, .) is non-decreasing.

Topology on X- Let (X, M, *) be a complex valued fuzzy metric space. We define τ = { A ⊂ X : x ∈ A if and only if there exists t > 0 and r ∈ C , 0 ≺ r ≺ e i θ , θ ∈ [ 0 , π 2 ] such that B ( x , r , t ) ⊂ A } . Then τ is a topology on X.

Theorem 3.1. Every complex valued fuzzy metric space is Hausdorff.

Proof. Let (X, M, *) be a complex valued fuzzy metric space. Let x, y be two distinct points of X. Then 0 ≺ M ( x , y , t ) ≺ e i θ . Let M (x, y, t) = r, for some r ∈ C then 0 ≺ r ≺ e ^iθ . For each r₀  (r ≺ r₀ ≺ e ^iθ ), we can find a r₁  (r₁ ≺ e ^iθ ) such that r₁ * r₁ ≿ r₀ .

Now consider two open balls B ( x , e i θ - r 1 , t 2 ) and B ( y , e i θ - r 1 , t 2 ). Certainly B ( x , e i θ - r 1 , t 2 ) ∩ B ( y , e i θ - r 1 , t 2 ) = φ .

If not then there exists p ∈ B ( x , e i θ - r 1 , t 2 ) ∩ B ( y , e i θ - r 1 , t 2 ).

Now consider r = M ( x , y , t ) ≿ M ( x , p , t 2 ) * M ( p , y , t 2 ) ≻ r 1 * r 1 ≿ r 0 . Which is a contradiction. Therefore (X, M, *) is Hausdorff.

Definition 3.4. Boundedness - Let (X, M, *) be a complex valued fuzzy metric space.A subset ^′A′ of X is said to be F _c bounded if and only if there exist t > 0 and r ∈ C,   0 ≺ r ≺ e ^iθ such that M ( x , y , t ) ≻ e i θ - r , for all x , y ∈ A .

Next theorem is proved for the convergence of the sequences in complex valued fuzzy metric spaces.

Theorem 3.2. Let (X, M, *) be a complex valued fuzzy metric space and τ be the topology induced by complex valued fuzzy metric. Then for a sequence {x _n } ∈ X we have x _n  → x if and only if M (x _n , x, t) → e ^iθ as n→ ∞ or |M (x _n , x, t) |→1 as n→ ∞.

Proof. For each t > 0, suppose x _n  → x. Then for r ∈ C,   0 ≺ r ≺ e ^iθ , there exist n₀ ∈ N such that x n ∈ B ( x , r , t ) for all n ≥ n 0 . So that M (x _n , x, t) ≻ e ^iθ  - r, this implies e ^iθ  - M (x _n , x, t) ≺ r.

On making n→ ∞, one gets M ( x n , x , t ) → e i θ . Or | M ( x n , x , t ) | → 1 as n → ∞ . Conversely, if for each t > 0 , M (x _n , x, t) → e ^iθ as n→ ∞. Then for 0 ≺ r ≺ e ^iθ , there exists n₀ ∈ N such that e ^iθ  - M (x _n , x, t) ≺ r,    ∀ n ≥ n₀.

So that M (x _n , x, t) ≺ e ^iθ  - r,    ∀ n ≥ n₀.

This implies x _n  ∈ B (x, r, t) and hence x _n  → x.

Definition 3.5. Cauchy sequence - A sequence x _n in a complex valued fuzzy metric space (X, M, *) is a Cauchy sequence if and only if lim n → ∞ M ( x n + p , x n , t ) = e i θ , p > 0 , t > 0 or lim n → ∞ | M ( x n + p , x n , t ) | = 1 , p > 0 , t > 0 .

Definition 3.6. A complex valued fuzzy metric space in which every Cauchy sequence is convergent, is called complex valued complete fuzzy metric spaces.

A point x ∈ X is called limit point of a subset A of X whenever there exists r ∈ C, 0 ≺ r ≺ e ^iθ , such that B ( x , r , t ) ∩ ( A / X ) ≠ φ . A subset B of X is closed whenever each limit point of B belongs to B.

Definition 3.7. Closed Ball - Let (X, M, *) be a complex valued fuzzy metric space. We define a closed ball B [x, r, t] with centre x ∈ X and radius r ∈ C   (0 ≺ r ≺ e ^iθ ) and for all t > 0 by B [ x , r , t ] = { y ∈ X : M ( x , y , t ) ∈ e i θ - r } .

Lemma 3.2. In complex valued fuzzy metric spaces, every closed ball is closed set.

Proof. Let (X, M, *) be a complex valued fuzzy metric space and let B [x, r, t] be closed ball in X.

Let z ∈ B [ x , r , t ] ¯. Since X is first countable then there exists a sequence {z _n } in B [x, r, t] such that z _n  → z. Therefore M (z _n , z, t) → e ^iθ as n→ ∞ for all t > 0.

For a given є > 0, M ( x , z , t + є ) ≿ M ( x , z n , t ) * M ( z n , z , #x0454 ) . Hence M ( x , z , t + #x0454; ) ≿ lim n → ∞ M ( x , z n , t ) * lim n → ∞ M ( z n , z , #x0454; ) . ≿ ( e i θ - r ) * e i θ = e i θ - r . (If M (x, z _n , t) is bounded then sequence {z _n } has a sub-sequence, which can be again denoted by {z _n } for which lim n → ∞ M ( x , z n , t ) exists).

In a particular case for n ∈ N, taking #x0454; = 1 n.

Then M ( x , z , t + 1 n ) ≿ e i θ - r .

Hence M ( x , z , t ) = lim n → ∞ M ( x , z , t + 1 n ) ≿ e i θ - r . Which leads to z ∈ B [ x , r , t ] . This implies B [ x , r , t ] ¯ ⊆ B [ x , r , t ] . But B [ x , r , t ] ⊆ B [ x , r , t ] ¯ always.

Thus we have B [ x , r , t ] ¯ = B [ x , r , t ] . Therefore B [x, r, t] is a closed set.

4 Banach contraction principle in complex valued fuzzy metric spaces

We begin with the following observations that are helpful in demonstration of results in this section and further sections as well.

Lemma 4.1.Let (X, M, *) be a complex valued complete fuzzy metric space such that lim t → ∞ M ( x , y , t ) = e i θ , for all x , y ∈ X , if M ( x , y , kt ) ≿ M ( x , y , t ) , for all x, y ∈ X, 0 < k < 1, t ∈ (0, ∞) then x = y.

Proof. Suppose there exists k ∈ (0, 1), such that M ( x , y , kt ) ≿ M ( x , y , t ) , ∀ x , y ∈ X , t ∈ ( 0 , ∞ ) so that   M ( x , y , t ) ≿ M ( x , y , t k ).

Repeated application gives M ( x , y , t ) ≿ M ( x , y , t k n ), for some positive integer n.

On making n→ ∞, reduces to M (x, y, t) ≿ e ^iθ this implies M (x, y, t) = e ^iθ . Thus we have x = y .

Lemma 4.2.Let {y _n } be a sequence in a complex valued fuzzy metric space (X, M, *) with lim t → ∞ M ( x , y , t ) = e i θ,for all x, y ∈ X. If there exists a number k ∈ (0, 1) such that M ( y n + 1 , y n + 2 , kt ) ≿ M ( y n , y n + 1 , t ) , for all t > 0 and n = 0, 1, 2, 3, . . . .

Then {y _n } is a Cauchy sequence in X.

Proof. For n = 0, we have M (y₁, y₂, t) ≿ M (y₀,y 1 , t k ), for all t > 0 and k ∈ (0, 1).

By induction, one gets M ( y n + 1 , y n + 2 , t ) ≿ M ( y 0 , y 1 , t k n + 1 ) , for all n.

Thus for any positive integer p and using (CF4), we have M ( y n , y n + p , t ) ≿ M ( y n , y n + 1 , t p ) * . . . * ( p times ) * . . . * M ( y n + p - 1 , y n + p , t p ) ≿ M ( y 0 , y 1 , t pk n ) * . . . * ( p times ) * . . . * M ( y 0 , y 1 , t pk n + p - 1 ) . Which on letting n→ ∞, reduces to

lim n → ∞ M ( y n , y n + p , t ) ≿ e i θ * e i θ * . . . * e i θ   (since k < 1 and lim n → ∞ M ( x , y , t ) = e i θ).

Which implies that lim n → ∞ M ( y n , y n + p , t ) ≿ e i θ.

This necessitates that {y _n } is Cauchy sequence in X.

The complex valued fuzzy version of Banach contraction principle runs as follows.

Theorem 4.1. Let (X, M, *) be a complex valued complete fuzzy metric space such that lim t → ∞ M ( x , y , t ) = e i θ , for all x , y ∈ X and t > 0 .(1) Let T : X → X be a mapping satisfying

M ( Tx , Ty , kt ) ≿ M ( x , y , t ) , for all x , y ∈ X and 0 < k < 1 .(2) Then T has a unique fixed point.

Proof. Suppose T satisfies condition (4.2).x₀ ∈ X be an arbitrary point and define the sequence {x _n } by x_n+1 = Tx _n , n = 0, 1, 2, . . .

Employing condition (4.2), with x = x _n and y = x_n+1, we have M ( x n , x n + 1 , t ) = M ( Tx n - 1 , Tx n , t ) ≿ M ( x n - 1 , x n , t k ) that is M ( x n , x n + 1 , t ) ≿ M ( x n - 1 , x n , t / k ) .

In view of Lemma 4.2, we have {x _n } is Cauchy sequence in X.

Since X is complete, then essentially x _n  → u as n→ ∞, where u ∈ X.

Consider M ( Tu , u , t ) ≿ M ( Tu , x n + 1 , t 2 ) * M ( x n + 1 , u , t 2 ) = M ( Tu , Tx n , t 2 ) * M ( x n + 1 , u , t 2 ) ≿ M ( u , x n , t 2 k ) * M ( x n + 1 , u , t 2 ) . Letting n→ ∞, we have M ( Tu , u , t ) ≿ M ( u , u , t 2 k ) * M ( u , u , t 2 ) .

Now by (CF2), we have M (Tu, u, t) ≿ e ^iθ  * e ^iθ  .

Or M (Tu, u, t) = e ^iθ  .

This definitely implies Tu = u

Thus u is a fixed point of T.

Now for uniqueness of fixed point, assume w ∈ X be another fixed point of T such that w ≠ u .

One knows that e i θ ≿ M ( u , w , t ) = M ( Tu , Tw , t ) ≿ M ( u , w , t k ) = M ( Tu , Tw , t k ) ≿ M ( u , w , t k 2 ) ≿ M ( u , w , t k 3 ) ≿ . . . ≿ M ( u , w , t k n ) thus we obtain e i θ ≿ M ( u , w , t k n ).

Since k < 1 . then on making n→ ∞, one gets u = w . Thus T has a unique fixed point.

Following example demonstrates the validity of hypothesis of Theorem 4.1.

Example 4.1. Let X = R. Define the metric d by d (x, y) = |x - y| on X. Let a * b = min {a, b} , ∀ a, b ∈ r _s e ^iθ , where r s ∈ [ 0 , 1 ] and θ ∈ [ 0 , π 2 ] .

For each t > 0 and x, y ∈ X, if we define M ( x , y , t ) = e i θ t t + d ( x , y ) . then certainly (X, M, *) is a complex valued complete fuzzy metric space. Obviously here lim t → ∞ M ( x , y , t ) = e i θ, for all x, y ∈ X.

Now we define a self mapping T on X by Tx = x 3 .

By a routine calculation, one can verify that T satisfies the condition M ( Tx , Ty , kt ) ≿ M ( x , y , t ) , ∀ x , y ∈ X , for k = 1 2.

Thus all the conditions of Theorem 4.1 are satisfied and x = 0 is the unique fixed point of T.

In this section, some theorems are established by using rational inequality in complex valued fuzzy metric spaces.

Theorem 5.1. Let (X, M, *) be a complex valued complete fuzzy metric space such that lim t → ∞ M ( x , y , t ) = e i θ , for all x , y ∈ X and t > 0 . Let f and g be two self mappings of complex valued fuzzy metric space (X, M, *). Further f and g have a unique common fixed point provided the involved maps satisfy the following condition

M ( fx , gy , kt ) ≿ min { ( M ( y , gy , t ) M ( x , fx , t ) ( M ( x , y , t ) , M ( x , y , t ) } , for all x, y ∈ X and k ∈ ( 0 , 1 2 ) .

Proof. Let x₀ be any arbitrary point in X, define x_2n+1 = fx_2n and x_2n+2 = gx_2n+1n = 0, 1, 2 . . . In what follows, we discuss two cases to get the desired fixed point.

Case I: When x _n  ≠ x_n+1, for n = 0, 1, 2, . . . By (5.2) with x = x_2n and y = x_2n+1, we have

M ( x 2 n + 1 , x 2 n + 2 , kt ) ≿ M ( fx 2 n , gx 2 n + 1 , kt ) ≿ min { M ( x 2 n + 1 , gx 2 n + 1 , t ) M ( x 2 n , fx 2 n , t ) ( M ( x 2 n , x 2 n + 1 , t ) , M ( x 2 n , x 2 n + 1 , t ) } = min { M ( x 2 n + 1 , x 2 n + 2 , t ) M ( x 2 n , x 2 n + 1 , t ) ( M ( x 2 n , x 2 n + 1 , t ) , M ( x 2 n , x 2 n + 1 , t ) } = min { M ( x 2 n + 1 , x 2 n + 2 , t ) , M ( x 2 n , x 2 n + 1 , t ) } . Now suppose min {M (x_2n+1, x_2n+2, t), M (x_2n,x_2n+1, t)} = M (x_2n+1, x_2n+2, t) , utilizing (5.3), one obtains M (x_2n+1, x_2n+2, kt) ≿ M (x_2n+1, x_2n+2, t) .

By Lemma 4.1, we have x_2n+1 = x_2n+2.

This leads to a contradiction, therefore by (5.3), we must have M ( x 2 n + 1 , x 2 n + 2 , kt ) ≿ M ( x 2 n , x 2 n + 1 , t ) , ∀ t > 0 . Similarly we can obtain M ( x 2 n , x 2 n + 1 , kt ) ≿ M ( x 2 n - 1 , x 2 n , t ) , ∀ t > 0 . In general, one gets M ( x n + 1 , x n + 2 , kt ) ≿ M ( x n , x n + 1 , t ) , ∀ t > 0 . Thus, in view of Lemma 4.2, {x _n } is a Cauchy sequence in X.

Since X is complete, there exists some u ∈ X such that x _n  → u as n→ ∞.

Implying thereby the convergence of {x_2n} and {x_2n+1} being sub-sequences of the convergent sequence {x _n }. Then x_2n → u and x_2n+1 → u as n→ ∞. Now we claim that u is a fixed point of f. Setting x = u and y = x_2n+1 in Inequality (5.2), one yields M ( fu , u , t ) ≿ M ( fu , x 2 n + 2 , t 2 ) * M ( x 2 n + 2 , u , t 2 ) = M ( fu , gx 2 n + 1 , t 2 ) * M ( x 2 n + 2 , u , t 2 ) ≿ min { M ( x 2 n + 1 , gx 2 n + 1 , t 2 k ) M ( u , fu , t 2 k ) M ( u , x 2 n + 1 , t 2 k ) ,

M ( u , x 2 n + 1 , t 2 k ) } * M ( x 2 n + 2 , u , t 2 ) = min { M ( x 2 n + 1 , x 2 n + 2 , t 2 k ) M ( u , fu , t 2 k ) M ( u , x 2 n + 1 , t 2 k ) , M ( u , x 2 n + 1 , t 2 k ) } * M ( x 2 n + 2 , u , t 2 ) . Which on making n→ ∞, reduces to M ( fu , u , t ) ≿ min { ( M ( u , u , t 2 k ) M ( u , fu , t 2 k ) M ( u , u , t 2 k ) , M ( u , u , t 2 k ) } * M ( u , u , t 2 ) = min { M ( u , fu , t 2 k ) , e i θ } = M ( u , fu , t 2 k ) . Since k ∈ ( 0 , 1 2 ) then in view of Lemma 4.1, we have fu = u .

Thus u is a fixed point of mapping f .

Arguing the same, one can show that u is a fixed point of mapping g.

Hence u is a common fixed point of mappings f and g .

To prove uniqueness of common fixed point, let w ∈ X be another common fixed point of mappings f and g i.e. w = fw = gw, then using condition (5.2) with x = u and y = w, one gets M ( u , w , kt ) = M ( fu , gw , kt ) ≿ min { M ( w , gw , t ) M ( u , fu , t ) ( M ( u , w , t ) , M ( u , w , t ) } = min { ( M ( w , w , t ) M ( u , u , t ) ( M ( u , w , t ) , M ( u , w , t ) } = min { ( e i θ e i θ ) ( M ( u , w , t ) , M ( u , w , t ) } . Since M (u, w, t) ∈ r _s e ^iθ , r _s  ∈ [0, 1] and θ ∈ [ 0 , π 2 ], also M (u, w, t) ≾ e ^iθ , then we certainly have min { ( e i θ e i θ ) ( M ( u , w , t ) , M ( u , w , t ) } = M ( u , w , t ) . Therefore we obtain M ( u , w , kt ) ≿ M ( u , w , t ) , which yields u = w, in view of Lemma 4.1. Thus u is the unique common fixed point of fand g.

Case II: When x _n  = x_n+1,   n = 0, 1, 2, . . . Implies that {x _n } is a constant sequence and so convergent. The rest of the proof can be completed on the lines of Case I. This concludes the proof.

The validity of Theorem 5.1 is substantiated by the following example.

Example 5.1 Let X = { 0 } ∪ { 1 n : n ∈ N } with the metric d defined by d ( x , y ) = | x - y | , ∀ x , y ∈ X . For all x, y ∈ X and t ∈ (0, ∞), we define M ( x , y , t ) = e i θ t t + d ( x , y ) . Clearly (X, M, *) is complex valued complete fuzzy metric space with t-norm ‘*’ defined as a * b = min {a, b} where a, b ∈ r _s e ^iθ , for a fixed θ ∈ [ 0 , π 2 ] and r _s  ∈ [0, 1].

Certainly here lim t → ∞ M ( x , y , t ) = e i θ, for all x, y ∈ X, t ∈ (0, ∞). Define f ( x ) = g ( x ) = x 3. By routine calculation, one can easily verify that f and g satisfy the condition M (fx, gy, kt) ≿ min{ ( M ( y , gy , t ) M ( x , fx , t ) M ( x , y , t ) , M ( x , y , t ) } , for all x, y ∈ X and for k = 2 5 ∈ ( 0 , 1 2 ).

Thus all the conditions of Theorem 5.1 are satisfied. Further x = 0 remains fixed under mappings f  &  g and is indeed unique.

Restricting Theorem 5.1 to single mapping (by setting f = g), we deduce the following corollary:

Corollary 5.1. Let (X, M, *) be a complex valued complete fuzzy metric space such that lim t → ∞ M ( x , y , t ) = e i θ , for all x , y ∈ X and t > 0 . Let f be a self mappings satisfying

M ( fx , fy , kt ) ≿ min { ( M ( y , fy , t ) M ( x , fx , t ) ( M ( x , y , t ) , M ( x , y , t ) } , for all x, y ∈ X and k ∈ ( 0 , 1 2 ) . Then f has a unique common fixed point.

Next theorem is obtained for the inequality involving control function.

We define a class of control functions Φ = {φ/φ : r _s e ^iθ  → r _s e ^iθ } such that [(i)] φ is a continuous function, [(ii)] φ (e ^iθ ) = e ^iθ and [(iii)] φ (a) ≻ a,   ∀ a ∈ r _s e ^iθ , where θ ∈ [ 0 , π 2 ] and r _s  ∈ [0, 1] .

Theorem 5.2.Let (X, M, *) be a complex valued complete fuzzy metric space such that lim t → ∞ M ( x , y , t ) = e i θ , for all x , y ∈ X and t > 0 . Let f and g be two self mappings satisfying M ( fx , gy , kt ) ≿ φ μ ( x , y , t ) , where

μ ( x , y , t ) = min { ( M ( y , gy , t ) M ( x , fx , t ) ( M ( x , y , t ) , M ( x , y , t ) } , for all x, y ∈ X and k ∈ ( 0 , 1 2 ) .

Then f and ghave a unique common fixed point.

Proof. Proof can be obtained immediately as a consequence of aforesaid control function and Theorem 5.1.

6 Jungck type fixed point theorem in complex valued fuzzy metric spaces

The following theorem was the generalization of Banach’s contraction principle in metric spaces given by Jungck [2].

Theorem 6.1.Let f be a continuous mapping of a complete metric space (X, d) into itself and g : X → X be a mapping satisfying the following conditions:

g (X) ⊆ f (X),

Our next theorem is established as complex valued fuzzy version of theorem of Jungck [2].

Theorem 6.2.Let (X, M, *) be a complex valued complete fuzzy metric space and f, g : X → X be a mappings satisfying the following conditions:

g (X) ⊆ f (X),

Proof. Let x₀ ∈ X. Since g (X) ⊆ f (X) then we can find x₁ such that fx₁ = gx₀ .

By induction, one can define a sequence {x _n } in X such that fx _n  = gx_n-1 .

Now consider M ( fx n + 1 , fx n , t ) = M ( gx n , gx n - 1 , t ) ≿ M ( fx n , fx n - 1 , t k ) .

Thus in view of Lemma 4.2, {fx _n } is a Cauchy sequence and due to completeness of X,   {fx _n } converges to a point y ∈ X so that {gx_n-1} = {fx _n } also converges to point y.

It can be shown by Condition (6.1), that continuity of f implies continuity of g.

Therefore {gfx _n } converges to gy.

Since f and g commute on X, then gfx _n  = fgx _n , and so fgx _n converges to fy.

By the uniqueness of limit, we have fy = gy ⇒ ffy = fgy.

Consider M ( gy , ggy , t ) ≿ M ( fy , fgy , t k ) ≿ M ( gy , ggy , t k ) ≿ … ≿ M ( gy , ggy , t k n ) which on letting n→ ∞, give rise M ( gy , ggy , t ) = e i θ . which yields gy = ggy and therefore gy = ggy = fgy.

This implies that gy is a common fixed point of f and g.

For the uniqueness of common fixed point, let u and v be two distinct common fixed points of fand g.

Using Condition (5.2) with x = u and y = v, one gets e i θ ≿ M ( u , v , t ) = M ( gu , gv , t ) ≿ M ( fu , fv , t k ) = M ( u , v , t k ) ≿ . . . ≿ M ( u , v , t k n ) . Which on making n→ ∞, yields that u = v .

This completes the proof.

With a view to demonstrate the validity of hypothesis of Theorem 6.2, we adopt the followingexample.

Example 6.1. Let X = [2, 20] with the metric d defined by d ( x , y ) = | x - y | , ∀ x , y ∈ X . For all x, y ∈ X and t ∈ (0, ∞), we define M ( x , y , t ) = e i θ t t + d ( x , y ) . Clearly (X, M, *) is complex valued complete fuzzy metric space with t-norm ‘*’ defined as a * b = min{a, b} where a, b ∈ r _s e ^iθ , for a fixed θ ∈ [ 0 , π 2 ] and r _s  ∈ [0, 1]. Here lim t → ∞ M ( x , y , t ) = e i θ , for all x , y ∈ X . Define the mappings f, g : X → X as f ( x ) = { 2 , x = 2 ; 5 , 2 < x ≤ 5 ; 3 , x ∈ ( 5 , 20 ] and g ( x ) = { 2 , x = 2 or x > 5 ; 3 , 2 < x ≤ 5 .

Clearly, one concludes that g (x) ⊆ f (x).

In view of the verification of Condition (6.1), subsequent cases are discussed, for k = 1 2 ∈ ( 0 , 1 ).

Case I: when x = 2 and y = 2, M ( gx , gy , t 2 ) = e i θ t 2 t 2 + | 2 - 2 | = e i θ ≿ e i θ t t + | 2 - 2 | = M ( fx , fy , t ) .

Case II: when x = 2 and y ∈ (2, 5], M ( gx , gy , t 2 ) = e i θ t 2 t 2 + | 2 - 3 | ≿ e i θ t t + | 2 - 5 | ≿ e i θ t t + 3 = M ( fx , fy , t ) .

Case III: when x = 2 and y ∈ (5, 20], M ( gx , gy , t 2 ) = e i θ t 2 t 2 + 0 ≿ e i θ t t + | 2 - 3 | = M ( fx , fy , t ) .

Case IV: when x ∈ (2, 5] and y = 2, M ( gx , gy , t 2 ) = e i θ t 2 t 2 + | 3 - 2 | ≿ e i θ t t + | 5 - 2 | ≿ e i θ t t + 3 = M ( fx , fy , t ) .

Case V: when x ∈ (2, 5] and y ∈ (2, 5], M ( gx , gy , t 2 ) = e i θ t 2 t 2 + | 3 - 3 | ≿ e i θ t t + | 5 - 5 | = M ( fx , fy , t ) .

Case VI: when x ∈ (2, 5] and y ∈ (5, 20], M ( gx , gy , t 2 ) = e i θ t 2 t 2 + | 3 - 2 | ≿ e i θ t t + | 5 - 3 | = M ( fx , fy , t ) .

Case VII: when x > 5 and y = 2, M ( gx , gy , t 2 ) = e i θ t 2 t 2 + | 2 - 2 | ≿ e i θ t t + | 3 - 2 | = M ( fx , fy , t ) .

Case VIII: when x > 5 and y ∈ (2, 5], M ( gx , gy , t 2 ) = e i θ t 2 t 2 + | 3 - 2 | ≿ e i θ t t + | 3 - 5 | = M ( fx , fy , t ) .

Case IX: when x > 5 and y > 5, M ( gx , gy , t 2 ) = e i θ t 2 t 2 + | 2 - 2 | ≿ e i θ t t + | 3 - 3 | = M ( fx , fy , t ) .

Thus all the conditions of Theorem 6.2 are satisfied and x = 2 is the unique fixed point of f and g.