In this article, recognizing the innovative concept of complex valued fuzzy set due to Ramot et al. [1], we introduce the notion of complex valued fuzzy metric spaces. Several allied topological aspects for complex valued fuzzy metric spaces are also defined which fortify the concept. Moreover the complex valued fuzzy version of some celebrated and essential results from metric spaces including Banach contraction principle and the main theorem of Jungck [2] are established. Additionally some fixed point theorems satisfying certain rational expressions are proved.
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 μs (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 μs (x) that assigns every element x ∈ U, a complex valued grade of membership in S. The values μs (x) lie within the unit circle in the complex plane, and are thus of the form
where rs (x) and ws (x) both real-valued, with rs (x) ∈ [0, 1].
The complex fuzzy set S, may be represented as the set of ordered pairs, given by
Clearly, each complex grade of membership is defined by an amplitude term rs (x) and a phase term ws (x). Notice that the amplitude term rs (x) is equal to |μs (x) |, the amplitude of μs (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 λs (x), then S can be transformed into a complex fuzzy set is by setting the amplitude term rs (x) equal to λs (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
It follows that z1 ≾ z2, if one the following conditions are satisfied:
Re (z1) = Re (z2) and Im (z1) = Im (z2);
Re (z1) < Re (z2) and Im (z1) = Im (z2);
Re (z1) = Re (z2) and Im (z1) < Im (z2);
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
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 .
z1≾ max {z2, z3} ⇒ z1 ≾ z2 or z1 ≾ z3.
On the similar lines ’min’ function can also be defined as:
min {z1, z2} = z1 ⇔ z1 ≾ z2.
min {z1, z2} ≾ z3 ⇒ z1 ≾ z3 or z2 ≾ z3.
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.
Rudiments of complex valued fuzzy metric spaces
First of all we define complex valued continuous t-norm.
Definition 2.1. A binary operation * : rseiθ × rseiθ→rseiθ, wherein rs ∈ [0, 1] and a fix , is called complex valued continuous t-norm if it satisfies the following conditions:
* is associative and commutative,
* is continuous,
a * eiθ = a, ∀ a ∈ rseiθ, where rs ∈ [0, 1] ,
a * b ≾ c * d whenever a ≾ c and b ≾ d, for all a, b, c, d ∈ rseiθ, where rs ∈ [0, 1] .
Example 2.1.a * b =min (a, b).
Example 2.2.a * b = max (a + b - eiθ, 0), for a fix .
Example 2.3.
for a fix
Where ‘min’ and ‘max’ are defined in definition (1).
Note: Throughout this article, θ is taken to be fixed in with the assumption that the complex fuzzy set S = {(x, μs (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, ∞) → rseiθ is a complex valued fuzzy set, where rs ∈ [0, 1] and , satisfying the following conditions:
M (x, y, t) ≻0,
M (x, y, t) = eiθ for all t > 0 ⇔ x = y,
M (x, y, t) = M (y, x, t),
M (x, y, t) * M (y, z, s) ≿ M (x, z, t + s),
M (x, y, .) : (0, ∞) → rseiθ is continuous,
for all x, y, z ∈ X, s, t > 0, rs ∈ [0, 1] and Also (M, *) is called a complex valued fuzzy metric.
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 ∈ rseiθ, where rs ∈ [0, 1] and . For each t > 0, x, y ∈ X, If we define
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
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 ∈ rseiθ, where rs ∈ [0, 1] and . Furthermore for all x, y ∈ X and t ∈ (0, ∞), we define
Then (X, M, *) is a complex valued fuzzy metric space.
Example 2.6. Let X = N. We define a * b =max(a + b - eiθ, 0) , ∀ a, b ∈ rseiθ, where rs ∈ [0, 1]and θ ∈ [0, π/2] and
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 ∈ rseiθ, where rs ∈ [0, 1] and .
Example 2.7. Let X = {xn} ∪ {1}, where {xn} n∈N⊂(0, ∞) is increasing sequence converging to 1. For all t > 0 and n, m ∈ N, we define M (xn, xn, t) =eiθ, M (1, 1, t) = eiθ, M (xn, xm, t) = eiθ min{xn, xm} and
Then (X, M, *) is a complex valued fuzzy metric space, where a * b = min (a, b) , ∀ a, b ∈ rseiθ, where rs ∈ [0, 1] and .
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 ≺ eiθ, t > 0 as
where
A point x ∈ X is called an interior point of set A ⊂ X, whenever there exists r ∈ C, 0 ≺ r ≺ eiθ such that
where
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) ≻ eiθ - r for x, y ∈ X, t > 0, r ∈ C and 0 ≺ r ≺ eiθ, we can find a t0, 0 < t0 < t such that M (x, y, t0) ≻ eiθ - r, for a fix .
Remark 3.2. For any r1 ≻ r2, we can find a r3, such that r1 * r3 ≿ r2, where r1, r2, r3 ∈ rseiθ.
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) ≻ eiθ - r, where r ∈ C and 0 ≺ r ≺ eiθ, thus we can find a t0, 0 < t0 < t such that M (x, y, t0) ≻ eiθ - r.
Let r0 = M (x, y, t0) ≻ eiθ - r.Then we canfind ′s′, where 0 ≺ s ≺ eiθ, such that r0 ≻ eiθ - s ≻ eiθ - r .
For given and ′s′, where r0 ≻ eiθ - s, we can find r1, 0 ≺ r1 ≺ eiθ such that r0 * r1 ≿ eiθ - s.
Consider the ball B (y, eiθ - r1, t - t0). We assert that B (y, eiθ - r1, t - t0) ⊂ B (x, r, t).
Let z ∈ B (y, eiθ - r1, t - t0) ⇒ M (y, z, t - t0) ≻ eiθ - (eiθ - r1) = r1. i . e . M (y, z, t - t0) ≻ r1 .
Consider
i.e. M (x, z, t) ≻ eiθ - r.
Which amounts to say that z ∈ B (x, r, t). Hence B (y, eiθ - 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) = xeiπ/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) = xe4πi/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
but M (y, y, s - t) = eiθ .
Implies M (x, y, t) * eiθ ≾ 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
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
Let M (x, y, t) = r, for some r ∈ C then 0 ≺ r ≺ eiθ. For each r0 (r ≺ r0 ≺ eiθ), we can find a r1 (r1 ≺ eiθ) such that r1 * r1 ≿ r0 .
Now consider two open balls and . Certainly
If not then there exists .
Now consider
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 Fc bounded if and only if there exist t > 0 and r ∈ C, 0 ≺ r ≺ eiθ such that
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 {xn} ∈ X we have xn → x if and only if M (xn, x, t) → eiθ as n→ ∞ or |M (xn, x, t) |→1 as n→ ∞.
Proof. For each t > 0, suppose xn → x. Then for r ∈ C, 0 ≺ r ≺ eiθ, there exist n0 ∈ N such that
So that M (xn, x, t) ≻ eiθ - r, this implies eiθ - M (xn, x, t) ≺ r.
On making n→ ∞, one gets
Or
Conversely, if for each t > 0 , M (xn, x, t) → eiθ as n→ ∞. Then for 0 ≺ r ≺ eiθ, there exists n0 ∈ N such that eiθ - M (xn, x, t) ≺ r, ∀ n ≥ n0.
So that M (xn, x, t) ≺ eiθ - r, ∀ n ≥ n0.
This implies xn ∈ B (x, r, t) and hence xn → x.
Definition 3.5. Cauchy sequence - A sequence xn in a complex valued fuzzy metric space (X, M, *) is a Cauchy sequence if and only if
or
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 ≺ eiθ, such that
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 ≺ eiθ) and for all t > 0 by
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 . Since X is first countable then there exists a sequence {zn} in B [x, r, t] such that zn → z. Therefore M (zn, z, t) → eiθ as n→ ∞ for all t > 0.
For a given є > 0,
Hence
(If M (x, zn, t) is bounded then sequence {zn} has a sub-sequence, which can be again denoted by {zn} for which exists).
In a particular case for n ∈ N, taking .
Then
Hence
Which leads to
This implies But always.
Thus we have Therefore B [x, r, t] is a closed set.
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 if
for all x, y ∈ X, 0 < k < 1, t ∈ (0, ∞) then x = y.
Proof. Suppose there exists k ∈ (0, 1), such that
so that .
Repeated application gives , for some positive integer n.
On making n→ ∞, reduces to M (x, y, t) ≿ eiθ this implies M (x, y, t) = eiθ. Thus we have
Lemma 4.2.Let {yn} be a sequence in a complex valued fuzzy metric space (X, M, *) with ,for all x, y ∈ X. If there exists a number k ∈ (0, 1) such that
for all t > 0 and n = 0, 1, 2, 3, . . . .
Then {yn} is a Cauchy sequence in X.
Proof. For n = 0, we have M (y1, y2, t) ≿ M (y0,, for all t > 0 and k ∈ (0, 1).
By induction, one gets for all n.
Thus for any positive integer p and using (CF4), we have
Which on letting n→ ∞, reduces to
(since k < 1 and ).
Which implies that .
This necessitates that {yn} 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 thatLet T : X → X be a mapping satisfying
Then T has a unique fixed point.
Proof. Suppose T satisfies condition (4.2).x0 ∈ X be an arbitrary point and define the sequence {xn} by xn+1 = Txn, n = 0, 1, 2, . . .
Employing condition (4.2), with x = xn and y = xn+1, we have
In view of Lemma 4.2, we have {xn} is Cauchy sequence in X.
Since X is complete, then essentially xn → u as n→ ∞, where u ∈ X.
Consider
Letting n→ ∞, we have
Now by (CF2), we have M (Tu, u, t) ≿ eiθ * eiθ .
Or M (Tu, u, t) = eiθ .
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
thus we obtain .
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 ∈ rseiθ, where
For each t > 0 and x, y ∈ X, if we define
then certainly (X, M, *) is a complex valued complete fuzzy metric space. Obviously here , for all x, y ∈ X.
Now we define a self mapping T on X by
By a routine calculation, one can verify that T satisfies the condition
for .
Thus all the conditions of Theorem 4.1 are satisfied and x = 0 is the unique fixed point of T.
Fixed point theorems through rational expression
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 thatLet 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
for all x, y ∈ X and
Proof. Let x0 be any arbitrary point in X, define x2n+1 = fx2n and x2n+2 = gx2n+1n = 0, 1, 2 . . . In what follows, we discuss two cases to get the desired fixed point.
Case I: When xn ≠ xn+1, for n = 0, 1, 2, . . . By (5.2) with x = x2n and y = x2n+1, we have
Now suppose min {M (x2n+1, x2n+2, t), M (x2n,x2n+1, t)} = M (x2n+1, x2n+2, t) , utilizing (5.3), one obtains M (x2n+1, x2n+2, kt) ≿ M (x2n+1, x2n+2, t) .
By Lemma 4.1, we have x2n+1 = x2n+2.
This leads to a contradiction, therefore by (5.3), we must have
Similarly we can obtain
In general, one gets
Thus, in view of Lemma 4.2, {xn} is a Cauchy sequence in X.
Since X is complete, there exists some u ∈ X such that xn → u as n→ ∞.
Implying thereby the convergence of {x2n} and {x2n+1} being sub-sequences of the convergent sequence {xn}. Then x2n → u and x2n+1 → u as n→ ∞. Now we claim that u is a fixed point of f. Setting x = u and y = x2n+1 in Inequality (5.2), one yields
Which on making n→ ∞, reduces to
Since 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
Since M (u, w, t) ∈ rseiθ, rs ∈ [0, 1] and , also M (u, w, t) ≾ eiθ, then we certainly have
Therefore we obtain
which yields u = w, in view of Lemma 4.1. Thus u is the unique common fixed point of fand g.
Case II: When xn = xn+1, n = 0, 1, 2, . . . Implies that {xn} 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 with the metric d defined by
For all x, y ∈ X and t ∈ (0, ∞), we define
Clearly (X, M, *) is complex valued complete fuzzy metric space with t-norm ‘*’ defined as a * b = min {a, b} where a, b ∈ rseiθ, for a fixed and rs ∈ [0, 1].
Certainly here , for all x, y ∈ X, t ∈ (0, ∞). Define . By routine calculation, one can easily verify that f and g satisfy the condition M (fx, gy, kt) ≿ min for all x, y ∈ X and for .
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 thatLet f be a self mappings satisfying
for all x, y ∈ X and 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 : rseiθ → rseiθ} such that [(i)] φ is a continuous function, [(ii)] φ (eiθ) = eiθ and [(iii)] φ (a) ≻ a, ∀ a ∈ rseiθ, where and rs ∈ [0, 1] .
Theorem 5.2.Let (X, M, *) be a complex valued complete fuzzy metric space such that
Let f and g be two self mappings satisfying
where
for all x, y ∈ X and
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.
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),
g commutes with f,
there exists 0 < k < 1 such that, for all x, y ∈ X,
Then f and g have a unique common fixed pointin 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),
f is continuous,
M (gx, gy, kt) ≿ M (fx, fy, t) , forallx, y ∈ X, 0 < k < 1, t ∈ (0, ∞) . (6.1)
Then f and g have a unique common fixed point in X provided f and g commute on 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 {xn} in X such that fxn = gxn-1 .
Now consider
Thus in view of Lemma 4.2, {fxn} is a Cauchy sequence and due to completeness of X, {fxn} converges to a point y ∈ X so that {gxn-1} = {fxn} also converges to point y.
It can be shown by Condition (6.1), that continuity of f implies continuity of g.
Therefore {gfxn} converges to gy.
Since f and g commute on X, then gfxn = fgxn, and so fgxn converges to fy.
By the uniqueness of limit, we have fy = gy ⇒ ffy = fgy.
Consider
which on letting n→ ∞, give rise
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
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
For all x, y ∈ X and t ∈ (0, ∞), we define
Clearly (X, M, *) is complex valued complete fuzzy metric space with t-norm ‘*’ defined as a * b = min{a, b} where a, b ∈ rseiθ, for a fixed and rs ∈ [0, 1]. Here
Define the mappings f, g : X → X as
Clearly, one concludes that g (x) ⊆ f (x).
In view of the verification of Condition (6.1), subsequent cases are discussed, for .
Case I: when x = 2 and y = 2,
Case II: when x = 2 and y ∈ (2, 5],
Case III: when x = 2 and y ∈ (5, 20],
Case IV: when x ∈ (2, 5] and y = 2,
Case V: when x ∈ (2, 5] and y ∈ (2, 5],
Case VI: when x ∈ (2, 5] and y ∈ (5, 20],
Case VII: when x > 5 and y = 2,
Case VIII: when x > 5 and y ∈ (2, 5],
Case IX: when x > 5 and y > 5,
Thus all the conditions of Theorem 6.2 are satisfied and x = 2 is the unique fixed point of f and g.
Footnotes
Acknowledgments
The authors are very grateful to Professor Dorel Mihet for his fruitful observations on the first draft of this paper and for the valuable suggestions in particular which brought several improvements. Poom Kumam was supported by the Theoretical and Computational Science Center (TaCS-Center) (Project Grant No. TaCS2558-1).
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