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Fixedpoint results for fuzzy generalized βFcontraction mappings in fuzzy metric spaces and their applications
Fixed Point Theory and Algorithms for Sciences and Engineering volume 2023, Article number: 8 (2023)
Abstract
In this paper, we introduce fuzzy generalized βFcontractions as a generalization of fuzzy Fcontractions with admissible mappings. We deduce sufficient conditions for the existence and uniqueness of fixed points for fuzzy generalized βFcontractions in complete strong fuzzy metric spaces. Our results generalize several fixedpoint results from the literature. We present an application of our main result.
1 Introduction and preliminary results
Fixed point theory was initiated by Banach [1] with his celebrated Banach contraction principle in 1922. It has been generalized into many directions by mathematicians (see [2]) and served as an important tool to solve problems in various fields (see [3–5], and [6]). Inspired by the concept of fuzzy set by Zadeh [7], Kramosil and Michálek [8] defined fuzzy metric spaces as a generalization of probabilistic metric spaces. Later, George and Veeramani [9] modified the notion of a fuzzy metric space to obtain Hausdorff topology. The study of fixedpoint theorems in fuzzy metric space was initiated by Grabiec [10]. He introduced a fuzzy version of the Banach contraction principle in a fuzzy metric space. Subsequently, mathematicians proposed various contractive conditions and studied the existence of fixed points of these contractions in fuzzy metric spaces. For example, see [11–14], and [15].
In 2012, Wardowski [16] introduced the concept of an Fcontraction as a new type of contraction in complete metric spaces. After that, several mathematicians extended Fcontractions to various spaces (see [17–23], and [24]). Recently, Huang et al. [25] introduced fuzzy Fcontractions, a generalization of Fcontractions with simpler conditions in fuzzy metric spaces and presented some fixedpoint theorems for fuzzy Fcontractions. Recent works related to fuzzy Fcontractions can be found in [26–28], and references therein. Samet et al. [29] introduced the notion of an αadmissible contractive mapping and proved some fixedpoint theorems in the setting of metric spaces. Gopal and Vetro [30] adopted the idea and introduced αadmissible and βadmissible fuzzy contractive mappings, which generalized some fixedpoint results in fuzzy metric spaces.
In this paper, motivated by the works of Gopal and Vetro [30] and Huang et al. [25], we introduce generalized fuzzy Fcontractions with admissible property in a fuzzy metric space. We prove some fixed point theorems for such types of contractive mappings and provide an application to illustrate our results.
Before we proceed to our main results, we recall some definitions and notions that will be used throughout the rest of the paper.
Definition 1
([31])
A binary operation \(* : [ 0, 1 ] \times [ 0, 1 ] \to [ 0, 1] \) is a continuous triangular norm (tnorm for short) if the following conditions hold:

T1.
\(a * 1 = a \) for all \(a \in [0,1 ] \);

T2.
∗ is associative and commutative;

T3.
\(a * b \leq c * d \) whenever \(a \leq c \) and \(b \leq d \), where \(a, b , c ,d \in [ 0,1 ] \);

T4.
∗ is continuous.
Some commonly seen continuous tnorms are \(\min \{ a, b \}\), \(a \cdot b \), and \(\max \{ a+ b  1, 0 \} \).
Definition 2
([9])
The 3tuple \(( X, M , * ) \) is said to be a fuzzy metric space if X is an arbitrary nonempty set, ∗ is a continuous tnorm, and M is a fuzzy set on \(X \times X \times ( 0 , \infty ) \) satisfying the following conditions:

FMS1.
\(M ( x, y, t ) > 0 \);

FMS2.
\(M ( x, y, t ) = 1 \) if and only if \(x = y \);

FMS3.
\(M ( x, y, t ) = M ( y, x, t ) \);

FMS4.
\(M ( x, z, t+ s ) \geq M ( x, y, t ) * M ( y, z, s ) \);

FMS5.
\(M ( x, y, \cdot ) : (0, \infty ) \rightarrow ( 0, 1 ] \) is continuous
for all \(x, y \in X \).
In the definition above, if we replace the triangular inequality with the condition

FMS4*.
\(M ( x, z, t ) \geq M ( x, y, t ) * M ( y, z, t ) \) for all \(x, y, z \in X \) and \(t > 0 \),
then \(( X, M , * ) \) is called a strong fuzzy metric space.
Definition 3
([9])
Let \(( X , M , * ) \) be a fuzzy metric space, and let \(\{ x_{n}\} \) be a sequence in X. Then

1.
\(\{ x_{n} \} \) is convergent if there exists \(x \in X \) such that \(\lim_{n \to \infty} M ( x_{n}, x , t ) = 1 \) for all \(t > 0 \);

2.
\(\{ x_{n} \} \) is a Cauchy sequence if for all \(\varepsilon \in (0, 1 ) \) and \(t >0 \), there exists \(n_{0} \in \mathbb{N} \) such that \(M ( x_{n} , x_{m}, t ) > 1  \varepsilon \) for all \(n, m \geq n_{0} \);

3.
\(( X, M , * ) \) is complete if every Cauchy sequence is convergent.
Remark 1
George and Veeramani [9] mentioned that for a fuzzy metric space \(( X, M , * ) \), \(\{ x_{n} \} \) converges to \(x \in X \) if and only if \(\lim_{n \to \infty} x_{n} = x\).
For the rest of this paper, we denote by \(\mathcal{F} \) the class of all mappings \(F : [ 0, 1 ] \to \mathbb{R} \) such that for all \(x, y \in [ 0 , 1 ] \), \(x < y \) implies \(Fx < Fy \). In other words, F is strictly increasing on \([ 0 , 1 ] \).
Definition 4
([25])
Let \(( X , M , * ) \) be a fuzzy metric space, and let \(F \in \mathcal{F}\). The mapping \(g: X \to X \) is said to be a fuzzy Fcontraction if there exists \(\tau \in ( 0 , 1 ) \) such that
for all \(x, y \in X \) such that \(x \neq y \) and \(t > 0 \).
Definition 5
([30])
A mapping \(g : X \to X \) is said to be a βadmissible mapping if there exists a function \(\beta : X \times X \times ( 0, \infty ) \to (0, \infty ) \) such that for all \(x, y \in X \) and \(t > 0 \),
The rest of the paper is organized as follows. In Sect. 2, we present our main results of fuzzy generalized βFcontraction mappings in a fuzzy metric space together with an example. Next, we provide an application of our result in finding the solution of the integral equation in Sect. 3. Finally, Sect. 4 concludes the paper.
2 Main results
We start this section by defining fuzzy generalized βFcontraction.
Definition 6
Let \(( X, M , * ) \) be a fuzzy metric space. Let \(\beta : X \times X \times ( 0, \infty ) \to (0, \infty ) \) be a function, and let \(F \in \mathcal{F}\). A function \(g : X \to X \) is said to be a fuzzy generalized βFcontraction if there exists \(\tau \in ( 0, 1 ) \) such that for all \(x , y \in X\), \(x \neq y\), and \(t > 0 \),
where \(N ( x , y, t ) = \min \{ M ( x, y , t), M ( x , gx , t ) , M ( y , gy , t ) \} \).
Definition 7
Let \(( X, M , * ) \) be a fuzzy metric space. Let \(\beta : X \times X \times ( 0, \infty ) \to (0, \infty ) \) be a function, and let \(F \in \mathcal{F}\). A function \(g : X \to X \) is said to be a fuzzy βFcontraction if there exists \(\tau \in ( 0, 1 ) \) such that for all \(x , y \in X\), \(x \neq y\), and \(t > 0 \),
If in Definition 6 we take \(N ( x , y ,t ) = M ( x, y ,t )\) for all \(x ,y \in X \) and \(t > 0 \), then we obtain Definition 7. Therefore we can say that a fuzzy βFcontraction is a fuzzy generalized βFcontraction. However, the converse is false. Furthermore, if we let \(\beta (x, y ,t ) = 1 \) for all \(x ,y \in X \), \(x \neq y \), and \(t > 0 \), then we obtain Definition 4.
Theorem 1
Let \(( X, M , * ) \) be a complete strong fuzzy metric space, and let \(g : X \to X \) be a fuzzy generalized βFcontraction, where \(F\in \mathcal{F}\). Assume that the following conditions hold:

1.
g is βadmissible;

2.
there exists \(x_{0} \in X \) such that \(\beta (x_{0}, gx_{0}, t ) \leq 1 \) for all \(t > 0 \);

3.
for each sequence \(\{ x_{n} \} \) in X such that \(\beta ( x_{n} , x _{n+1}, t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{ 0 \}\) and \(t > 0 \), there exists \(k_{0} \in \mathbb{N} \cup \{ 0 \}\) such that \(\beta ( x_{m}, x_{n}, t ) \leq 1 \) for all \(n > m \geq k_{0}\) and \(t > 0 \);

4.
if \(\{ x_{n} \} \) is a sequence in X such that \(\beta ( x_{n} , x _{n+1}, t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{ 0 \}\) and \(t > 0 \) and if \(x_{n} \to x \) as \(n \to \infty \), then \(\beta ( x_{n} , x , t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{0 \} \).
Then g has a fixed point.
Proof
Let \(x_{0} \in X \) be such that \(\beta (x_{0}, gx_{0}, t ) \leq 1\) for all \(t > 0 \). We define a sequence \(\{ x_{n} \} \) by \(x_{n+1} = gx_{n} \) for all \(n \in \mathbb{N} \cup \{0 \} \). If there exists \(n \in \mathbb{N} \cup \{0 \} \) such that \(x_{n+1} = x_{n} \), then \(x_{n} \) is a fixed point of g, and the proof is complete. Now assume that \(x_{n} \neq x_{n+1} \) for all \(n \in \mathbb{N} \cup \{0 \} \). Since \(\beta (x_{0}, gx_{0}, t ) = \beta ( x_{0}, x_{1}, t ) \leq 1 \) for all \(t > 0 \) and g is βadmissible mapping, this implies that
Continuing this process, we conclude that \(\beta (x_{n}, x_{n+1}, t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{0 \} \) and \(t > 0 \). Since g is a fuzzy generalized βFcontraction, there exists \(\tau \in ( 0 , 1 ) \) satisfying (1). It is clear that \(\tau \beta (x_{n} , x_{n+1}, t ) < 1 \) for all \(n \in \mathbb{N} \cup \{0 \} \) and \(t > 0 \). Now substituting \(x = x_{n1} \) and \(y =x_{n} \), we get
for all \(t > 0 \), where
If \(\min \{ M (x_{n1}, x_{n} , t), M ( x_{n}, x_{n+1} , t ) \} = M ( x_{n}, x_{n+1} , t ) \), then
which leads to a contradiction. So we have \(\min \{ M (x_{n1}, x_{n} , t), M ( x_{n}, x_{n+1} , t ) \} = M (x_{n1}, x_{n} , t) \) and
which implies
Since F is a strictly increasing function, we get \(M ( x_{n}, x_{n+1} , t ) > M (x_{n1}, x_{n} , t) \) for all \(n \in \mathbb{N} \cup \{0 \} \) and \(t > 0 \). Hence for all \(t > 0\), the sequence \(\{ M ( x_{n}, x_{n+1} , t ) \} \) is strictly increasing in the interval \([ 0, 1 ] \) and bounded above. This implies that \(\{ M ( x_{n}, x_{n+1} , t ) \} \) is convergent, that is, for any \(t > 0 \), there exists \(A ( t ) \in [0, 1 ] \) such that \(\lim_{n \to \infty} M ( x_{n}, x_{n+1} , t ) = A (t) \). We claim that \(A ( t ) = 1 \) for all \(t > 0 \). Suppose that \(A ( t_{0} ) < 1 \) for some \(t_{0} > 0 \). Taking the limit in both sides of (3), we obtain
which leads to a contraction. Therefore we have
for all \(t > 0\). Now we will show that \(\{x_{n}\} \) is a Cauchy sequence. Suppose that \(\{x_{n}\} \) is not a Cauchy sequence. Then by condition 3 there exist \(k_{0} \in \mathbb{N} \cup \{0 \} \), \(\varepsilon > 0 \), two subsequences \(\{ x_{n_{k}}\} \), \(\{ x_{m_{k}}\} \) of \(\{ x_{n} \} \), and \(s > 0 \) such that \(n_{k} > m_{k} \geq k \) for all \(k \in \mathbb{N} \cup \{0\} \) with \(k \geq k_{0} \) and
Let \(n_{k} \) be the smallest integer that satisfies the above inequality. Then
Since \(( X, M , * ) \) is a strong fuzzy metric space, by (FMS4*) we have
Taking the limit as \(k \to \infty \), by (4) and (T1) we obtain
This implies that \(\lim_{k \to \infty} M ( x_{n_{k}} , x_{m_{k}} , s ) = 1  \varepsilon \). Again, by (FMS4*) we get
Letting \(k \to \infty \), by (4) and (T1) we have
Since \(\tau < 1 \) and \(\beta (x_{n_{k}}, x_{m_{k}}, t ) \leq 1 \) for all \(k \in \mathbb{N} \cup \{0\} \) with \(k \geq k_{0} \), this implies that \(\tau \beta (x_{n_{k}}, x_{m_{k}}, t ) < 1 \) for all \(k \in \mathbb{N} \cup \{0\} \) such that \(k \geq k_{0} \). Using (1), we get
where
Letting \(k \to \infty \), we get
Therefore letting \(k \to \infty \) in both sides of (5), we obtain
This implies that \(F ( 1  \varepsilon ) = 0 \), which contradicts the fact that \(F ( 1  \varepsilon ) > 0 \). Thus \(\{x_{n}\} \) is a Cauchy sequence. Since \(( X, M, * ) \) is complete, there exists \(x \in X \) such that \(\lim_{n \to \infty} M ( x_{n}, x , t ) = 1 \) for all \(t > 0 \) or, in other words, \(\lim_{n \to \infty}x_{n} = x\). Since \(\beta (x_{n}, x_{n+1}, t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{0 \} \) and \(t > 0 \), by condition 4 we have \(\beta ( x_{n} , x , t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{0 \} \). Now we will show that x is a fixed point of g. Suppose that \(gx \neq x \), i.e., \(M ( x, gx , t ) < 1 \). Then by (1) we obtain
for all \(t > 0 \), where
Letting \(n \to \infty \) in this inequality, we have
which leads to a contradiction. Thus \(gx = x \), which means that x is a fixed point of g. □
Corollary 1
Let \(( X, M , * ) \) be a complete strong fuzzy metric space, and let \(g : X \to X \) be a fuzzy βFcontraction, where \(F\in \mathcal{F}\). Assume that the following conditions hold:

1.
g is βadmissible;

2.
there exists \(x_{0} \in X \) such that \(\beta (x_{0}, gx_{0}, t ) \leq 1 \) for all \(t > 0 \);

3.
for each sequence \(\{ x_{n} \} \) in X such that \(\beta ( x_{n} , x _{n+1}, t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{ 0 \}\) and \(t > 0 \), there exists \(k_{0} \in \mathbb{N} \cup \{ 0 \}\) such that \(\beta ( x_{m}, x_{n}, t ) \leq 1 \) for all \(n > m \geq k_{0}\) and \(t > 0 \);

4.
if \(\{ x_{n} \} \) is a sequence in X such that \(\beta ( x_{n} , x _{n+1}, t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{ 0 \}\), \(t > 0 \) and if \(x_{n} \to x \) as \(n \to \infty \), then \(\beta ( x_{n} , x , t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{0 \} \).
Then g has a fixed point.
Proof
Letting \(N ( x , y, t ) = M ( x, y , t )\) in (1), the remaining proof follows the same lines as Theorem 1. □
Remark 2
Huang et al. [25] mentioned that their results are applicable to strong fuzzy metric spaces. Therefore, if we let \(\beta ( x, y , t ) = 1 \) for all \(x, y \in X \) and \(t > 0\) in the corollary above, then we obtain the main results of Huang et al. [25] in the context of strong fuzzy metric spaces. Hence we successfully generalize some fixed point results in the existing literature.
Now we provide examples to illustrate Theorem 1.
Example 1
Let X be a set of positive real numbers. Let ∗ be a product continuous tnorm, that is, \(a * b = a\cdot b\) for all \(a, b \in X \). We define a fuzzy set as follows:
for \(x, y \in X \) and \(t > 0 \). We can easily verify that \(( X , M, * ) \) is a complete strong fuzzy metric space. Let \(g (x ) = 1/x \) for all \(x \in X\), and let \(F ( y ) =\ln (y) \) for \(0 < y \leq 1 \) and \(\tau = 1/2 \). Also, define
First of all, we show that g is a βadmissible mapping. Let \(x, y\in X \) be such that \(\beta ( x ,y ,t ) = 1 / 2 \leq 1 \). Then \(x \leq y \), which implies that \(g(x) = 1/x \geq 1/y = g(y)\). It follows that \(\beta ( gx, gy, t ) =1 \leq 1 \). Now suppose \(\beta ( x ,y , t ) = 1 \leq 1 \), which means that \(y \leq x \), and thus \(g( y ) = 1 / y \geq 1/x = g(x) \). It follows that \(\beta ( gx , gy, t ) = 1/2 \leq 1 \). Thus g is a βadmissible mapping. Next, we proceed to check whether the contractive condition of Theorem 1 is satisfied or not. Let \(x , y \in X\). We will consider the following cases.
Case 1
\(x \leq y \) where \(x \geq 1 \). We have \(1 \leq x \leq y\), \(gx =\frac{1}{x} \leq x \), \(gy = \frac{1}{y} \leq y \), \(x^{2} \leq y ^{2} \), and
This implies that
Figure 1 shows the illustration of Case 1 on 3D view.
Case 2
\(y \leq x \) where \(y \geq 1 \). We have \(1 \leq y \leq x\), \(gx =\frac{1}{x} \leq x\), \(gy = \frac{1}{y} \leq y\), \(y^{2} \leq x^{2} \), and
This implies that
Figure 2 shows the illustration of Case 2 on 3D view.
Case 3
\(x \leq y \) where \(y < 1 \). We have \(x \leq y < 1\), \(gx =\frac{1}{x} \geq x\), \(gy = \frac{1}{y} \geq y \), \(y^{2} \geq x^{2} \), and
This implies that
Figure 3 shows the illustration of Case 3 on 3D view.
Case 4
\(y \leq x \) where \(x < 1 \). We have \(y \leq x < 1\), \(gx =\frac{1}{x} \geq x\), \(gy = \frac{1}{y} \geq y \), \(x^{2} \geq y^{2} \) and
This implies that
Figure 4 shows the illustration of Case 4 on 3D view.
Case 5
\(x \leq y \) where \(x < 1 \) and \(y \geq 1 \). We have \(gx =\frac{1}{x} \geq 1 > x\), \(gy = \frac{1}{y} \leq 1 \leq y \), and
This implies that
Figure 5 shows the illustration of Case 5 on 3D view.
Case 6
\(y \leq x \) where \(y < 1 \) and \(x \geq 1 \). We have \(gx =\frac{1}{x} \leq 1 \leq x\), \(gy = \frac{1}{y} \geq 1 > y\), and
This implies that
Figure 6 shows the illustration of Case 6 on 3D view.
Hence all the conditions in Theorem 1 are satisfied. Therefore g has a fixed point. In fact, we can see that \(1 \in X \) is a fixed point of g.
The following example shows that the assumption “strong” in Theorem 1 is not superfluous.
Example 2
Let \(X = [ 0, \infty ) \), and let ∗ be a minimum continuous tnorm. We define the fuzzy set M by
for \(x,y \in X \) and \(t > 0 \). Then \(( X, M , * ) \) is a complete but not strong fuzzy metric space. For instance, take \(x = 3\), \(y = 0.5\), \(z = 8\), and \(t = 3 \). Then we have \(M ( x, z, t ) = M ( 3, 8, 3 ) = 0.2857 \), \(M ( x, y, t ) = M ( 3, 0.5, 3 ) = 0.5455 \), and \(M ( y, z , t ) = M ( 0.5, 8 , 3 ) = 0.375 \). Checking with condition FMS4*, it is clear that on the righthand side, we get
However, \(0.2857 \ngeq 0.375 \). Therefore \(( X, M , * ) \) is not strong. Now define the mapping \(g : X \to X \) by
for \(x \in X \). Let \(F ( y ) = y \) for \(y \in [ 0,1] \), and let \(\tau = 9/10 \). Also, define \(\beta : X \times X \times ( 0, \infty ) \to ( 0, \infty ) \) as
for \(x, y \in X \) and \(t > 0 \). It is easy to show that g is a fuzzy generalized Fcontraction and conditions 1, 2, 3, and 4 of Theorem 1 hold. Observe that the mapping g does not have any fixed point.
In [25] the following theorem is obtained using Definition 4.
Theorem 2
([25])
Let \(( X, M, * ) \) be a complete fuzzy metric space such that
for all \(x, y \in X \). If \(g : X \to X \) is a continuous fuzzy Fcontraction, then g has a unique fixed point.
The following example shows that Theorem 1 we obtained above is applicable but not Theorem 2 in [25].
Example 3
Let \(X = [ 0, \infty ) \), and let ∗ be a product continuous tnorm. We define the fuzzy set M by
for \(x, y \in X \), and \(t > 0 \). Then \(( X , M, * ) \) is a complete strong fuzzy metric space.
Define the selfmapping \(g : X \to X \) by
for \(x \in X \). Let \(F ( y ) = \frac{1}{ \ln (y)} \) for \(0 < y \leq 1 \), and let \(\tau = 1/2 \). Also, let \(\beta : X \times X \times ( 0, \infty ) \to (0, \infty ) \) be defined as
for \(t > 0 \). It is clear that g is a fuzzy generalized βFcontraction. Let \(x, y \in X \) be such that \(\beta (x, y, t ) \leq 1 \). Based on the definition of β, it follows that \(x, y \in [ 0, 1 ]\). In the case where \(x \in [ 0, 1 ] \), we have \(x^{2} \leq x \), which implies
Thus by the definition of g we obtain \(gx, gy \in [ 0, 1 ] \), which means that \(\beta ( gx, gy, t ) = \frac{1}{2} \leq 1 \). This shows that g is βadmissible. In addition, there exists \(x_{0} \in X \) such that \(\beta ( x_{0}, gx_{0}, t ) \leq 1 \) for all \(t > 0 \). Indeed, let \(x = 1/2\). Then for all \(t > 0 \),
Consider a sequence \(\{ x_{n} \} \) in X such that \(\beta ( x_{n} , x_{n+1} , t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{ 0 \}\) and \(t > 0 \) and \(\lim_{n \to \infty} x_{n} = x \). Also, let \(k_{0} = 1 \) where \(n > m \geq k_{0} \). By the definition of β it follows that \(x_{n} \in [ 0, 1 ] \) for all \(n \in \mathbb{N} \cup \{ 0 \} \). Assume that \(x > 1 \). Then \(x_{n} < x \), and we get
for all \(n \in \mathbb{N} \cup \{ 0 \} \) and \(t > 0 \), which contradicts the assumption that \(\lim_{n \to \infty} x_{n} = x \). Thus we have \(x \in [ 0, 1 ] \). Hence \(\beta ( x_{n}, x, t ) = 1/2 \leq 1 \) and \(\beta ( x_{n}, x_{m}, t ) \leq 1 \) for all \(m, n \in \mathbb{N} \cup \{ 0 \} \). Consequently, all the hypotheses in Theorem 1 are satisfied. This implies that g has a fixed point, which is \(x = 0 \).
However, g is not a fuzzy Fcontraction. To see this, consider \(x = 1\), \(y = 400 \), and \(t = 30 \). It follows that \(gx = 1/4 \) and \(gy = 1600 \). Checking the inequality in the Definition 4, we have
on the lefthand side and
on the righthand side. Since \(0.0094 \ngeq 0.0752 \), we conclude that Theorem 2 of [25] does not hold for this example.
In the next theorem, we present a result by using a weaker condition compared with (1).
Theorem 3
Let \(( X, M , * ) \) be a complete strong fuzzy metric space with \(a*b=\min \{a,b\}\) for all \(a,b \in [0, 1]\). Moreover, let \(\beta : X \times X \times ( 0, \infty ) \to (0, \infty ) \), \(F \in \mathcal{F}\), and \(g: X\to X\) be such that there exists \(\tau \in ( 0, 1 ) \) satisfying
for all \(x , y \in X \) and \(t > 0 \), where
In addition, assume that the following conditions hold:

1.
g is βadmissible;

2.
there exists \(x_{0} \in X \) such that \(\beta (x_{0}, gx_{0}, t ) \leq 1 \) for all \(t > 0 \);

3.
for each sequence \(\{ x_{n} \} \) in X such that \(\beta ( x_{n} , x _{n+1}, t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{ 0 \}\) and \(t > 0 \), there exists \(k_{0} \in \mathbb{N} \cup \{ 0 \}\) such that \(\beta ( x_{m}, x_{n}, t ) \leq 1 \) for all \(n > m \geq k_{0}\) and \(t > 0 \);

4.
if \(\{ x_{n} \} \) is a sequence in X such that \(\beta ( x_{n} , x _{n+1}, t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{ 0 \}\) and \(t > 0 \) and if \(x_{n} \to x \) as \(n \to \infty \), then \(\beta ( x_{n} , x , t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{0 \} \).
Then g has a fixed point.
Proof
Let \(x_{0} \in X \) be such that \(\beta (x_{0}, gx_{0}, t ) \leq 1\) for all \(t > 0 \). We define the sequence \(\{ x_{n} \} \) by \(x_{n+1} = gx_{n} \) for \(n \in \mathbb{N} \cup \{0 \} \). If there exists \(n \in \mathbb{N} \cup \{0 \} \) such that \(x_{n+1} = x_{n} \), then \(x_{n} \) is a fixed point of g, and the proof is complete. Now assume that \(x_{n} \neq x_{n+1} \) for all \(n \in \mathbb{N} \cup \{0 \} \). Since \(\beta (x_{0}, gx_{0}, t ) = \beta ( x_{0}, x_{1}, t ) \leq 1 \) for all \(t > 0 \) and g is βadmissible mapping, this implies that
Continuing this process, we conclude that \(\beta (x_{n}, x_{n+1}, t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{0 \} \) and \(t > 0 \). By the hypothesis there exists \(\tau \in ( 0 , 1 ) \) that satisfies (6). It is clear that \(\tau \beta (x_{n} , x_{n+1}, t ) < 1 \) for all \(n \in \mathbb{N} \cup \{0 \} \) and \(t > 0 \). Using (6), we have
for all \(t > 0 \), where
Since the tnorm is minimum, by (FMS4*) we obtain
So we have
for all \(n \in \mathbb{N} \cup \{0 \} \) and \(t > 0 \). If \(\min \{ M (x_{n1}, x_{n} , t), M ( x_{n}, x_{n+1} , t ) \} = M ( x_{n}, x_{n+1} , t ) \), then
which leads to a contradiction. So we have \(\min \{ M (x_{n1}, x_{n} , t), M ( x_{n}, x_{n+1} , t ) \} = M (x_{n1}, x_{n} , t) \) and
which implies
Since F is a strictly increasing function, we get \(M ( x_{n}, x_{n+1} , t ) > M (x_{n1}, x_{n} , t) ) \) for all \(n \in \mathbb{N} \cup \{0 \} \) and \(t > 0 \). Hence for all \(t > 0\), the sequence \(\{ M ( x_{n}, x_{n+1} , t ) \} \) is strictly increasing in the interval \([ 0, 1 ] \) and bounded above. This implies that \(\{ M ( x_{n}, x_{n+1} , t ) \} \) is convergent, that is, for any \(t > 0 \), there exists \(A ( t ) \in [0, 1 ] \) such that \(\lim_{n \to \infty} M ( x_{n}, x_{n+1} , t ) = A (t) \). We claim that \(A ( t ) = 1 \) for all \(t > 0 \). Suppose that \(A ( t_{0} ) < 1 \) for some \(t_{0} > 0 \). Taking the limit in both sides of (7), we obtain
which leads to a contradiction. Therefore we have
for all \(t > 0\). Now we will show that \(\{x_{n}\} \) is a Cauchy sequence. Suppose that \(\{x_{n}\} \) is not a Cauchy sequence. Then by condition 3 there exist \(k_{0} \in \mathbb{N} \cup \{0 \} \), \(\varepsilon > 0 \), two subsequences \(\{ x_{n_{k}}\} \), \(\{ x_{m_{k}}\} \) of \(\{ x_{n} \} \), and \(s > 0 \) such that \(n_{k} > m_{k} \geq k \) for all \(k \in \mathbb{N} \cup \{0\} \) with \(k \geq k_{0} \) and
Let \(n_{k} \) be the smallest integer satisfying the above inequality. This means that
Using (FMS4*) and the minimum tnorm, we have
Taking the limit as \(k \to \infty \), by (8) we obtain
This implies that \(\lim_{k \to \infty} M ( x_{n_{k}} , x_{m_{k}} , s ) = 1  \varepsilon \). Again, by (FMS4*) and minimum tnorm we get
Letting \(k \to \infty \), by (8) we have
Since \(\tau < 1 \) and \(\beta (x_{n_{k}}, x_{m_{k}}, t ) \leq 1 \) for all \(k \in \mathbb{N} \cup \{0\} \) with \(k \geq k_{0} \), this implies that \(\tau \beta (x_{n_{k}}, x_{m_{k}}, t ) < 1 \) for all \(k \in \mathbb{N} \cup \{0\} \) with \(k \geq k_{0} \). Using (6), we get
where
Since
and
we have
Letting \(k \to \infty \), we get
Therefore taking the limit as \(k \to \infty \) in both sides of (9), we obtain
This implies that \(F ( 1  \varepsilon ) = 0 \), which contradicts the fact that \(F ( 1  \varepsilon ) > 0 \). Thus \(\{x_{n}\} \) is a Cauchy sequence. Since \(( X, M, * ) \) is complete, there exists \(x \in X \) such that \(\lim_{n \to \infty} M ( x_{n}, x , t ) = 1 \) for all \(t > 0 \) or, in other words, \(\lim_{n \to \infty}x_{n} = x\). Together with the fact that \(\beta (x_{n}, x_{n+1}, t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{0 \} \) and \(t > 0 \), which we obtained above, by condition 4 we have \(\beta ( x_{n} , x , t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{0 \} \). Now we will show that x is a fixed point of g. Suppose that \(gx \neq x \), that is, \(M ( x, gx , t ) < 1 \). Then by (6) we get
for all \(t > 0 \), where
Taking the limit as \(n \to \infty \) in the inequality above, we have
which leads to a contradiction. Thus \(gx = x \), which means that x is a fixed point of g. □
Remark 3
We can obtain Corollary 1 from Theorem 3 by letting \(N^{*} ( x , y, t ) = M ( x, y , t )\) for \(x, y \in X\) and \(t > 0 \) in (6).
Remark 4
Note that Example 2 can be used to show that the assumption “strong” in Theorem 3 is not superfluous.
Note that the results above do not guarantee that the fixed points are unique. To obtain the uniqueness of the fixed point, we will consider the following condition:

(U)
For any pair of fixed points \(x , y \in X \), \(\beta (x, y , t ) \leq 1 \) for all \(t > 0 \).
Theorem 4
In addition to the hypotheses of Theorem 1, suppose that condition (U) holds. Then g has a unique fixed point.
Proof
The existence of a fixed point follows from Theorem 1. Assume that \(x , y \in X\) are fixed points of g such that \(x \neq y \), that is, \(M ( x , y , t ) < 1 \). Considering condition (U), we have \(\beta ( x , y , t ) \leq 1 \) for all \(t > 0 \). Since g is a fuzzy generalized βFcontraction mapping, by (1) we obtain
for all \(t > 0 \), where
Since \(gx = x \) and \(gy = y \), we have
It follows that
which leads to a contradiction. Hence we conclude that \(x = y \). □
Corollary 2
In addition to the hypotheses of Corollary 1, suppose that condition (U) holds. Then g has a unique fixed point.
Theorem 5
In addition to the hypotheses of Theorem 3, suppose that condition (U) holds. Then g has a unique fixed point.
Proof
The existence of fixed point follows from Theorem 3. Assume that \(x , y \in X\) are a fixed point of g such that \(x \neq y \), that is, \(M ( x , y , t ) < 1 \). By condition (U) we have \(\beta ( x , y , t ) \leq 1 \) for all \(t > 0 \). By (6) we obtain
for all \(t > 0 \), where
Since \(gx = x \) and \(gy = y \), we have
It follows that
which leads to a contradiction. Hence we conclude that \(x = y \). □
3 Applications
In this section, we present an application of our result in finding the solution for an integral equation. Let us define the class of functions \(\Phi := \{ \phi ( t ) : ( 0, \infty ) \to ( 0, \infty ) \}\) such that \(\int ^{t} _{0} \phi (u) \,du \) is increasing and continuous for all \(t > 0 \). The following theorem is our fixedpoint result for the Wardowskitype contraction via βadmissible mappings of the integral type in a fuzzy metric space.
Theorem 6
Let \(( X , M , * ) \) be a complete fuzzy metric space, and let \(g : X \to X \) be a fuzzy generalized βFcontraction. Let ϕ be a mapping in Φ such that
where \(N ( x , y, t ) = \min \{ M ( x, y , t), M ( x , gx , t ) , M ( y , gy , t ) \} \) for \(x , y\in X \) and \(t > 0 \). Suppose that the following conditions hold:

1.
g is βadmissible;

2.
there exists \(x_{0} \in X \) such that \(\beta (x_{0}, gx_{0}, t ) \leq 1 \) for all \(t > 0 \);

3.
for each sequence \(\{ x_{n} \} \) in X such that \(\beta ( x_{n} , x _{n+1}, t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{ 0 \}\) and \(t > 0 \), there exists \(k_{0} \in \mathbb{N} \cup \{ 0 \}\) such that \(\beta ( x_{m}, x_{n}, t ) \leq 1 \) for all \(n > m \geq k_{0}\) and \(t > 0 \);

4.
if \(\{ x_{n} \} \) is a sequence in X such that \(\beta ( x_{n} , x _{n+1}, t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{ 0 \}\) \(t > 0 \) and if \(x_{n} \to x \) as \(n \to \infty \), then \(\beta ( x_{n} , x , t ) \leq 1 \) for all \(n \in \mathbb{N} \cup \{0 \} \).

5.
For any pair of fixed points \(x , y \in X \), \(\beta (x, y , t ) \leq 1 \) for all \(t > 0 \).
Then g has a unique fixed point.
Proof
Let \(\phi ( t ) = 1 \). Now apply Theorem 1 followed by Theorem 4. Then we obtain the desired result. □
4 Conclusions
In this paper, we have introduced fuzzy generalized βFcontractions as an extension of fuzzy Fcontractions. We proved several fixedpoint results for this contraction and its variation under the setting of complete strong fuzzy metric spaces. In addition, we provided a sufficient condition to obtain the uniqueness of a fixed point for this contraction. At the end of this paper, we showed an application of our main result in finding the solution for an integral equation. As mentioned in Remark 2, our work extends and generalizes the results of [25]. This is supported by Example 3, which shows that our result is applicable, but not the result in [25].
Since the debut of Fcontractions in metric spaces, they received considerable attention from other researchers. Lately, Gautam et al. [26] introduced Kannan Fcontractions as an extension of Fcontractions. They obtained common fixedpoint results for Kannan Fcontractions in quasipartial bmetric spaces and gave an application of their result in functional equations. Now, if we consider fuzzy a metric space, it is interesting to see if we can incorporate Kannan’s contractive condition for fuzzy generalized βFcontractions. We will end this paper with an open problem: “Can we obtain the existence and uniqueness of a fixed point for a Kannantype fuzzy generalized Fcontractive mapping under a fuzzy metric space setting?”.
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Acknowledgements
The authors appreciate the support of their institutions. They gratefully acknowledge the Ministry of Higher Education Malaysia, Universiti Malaysia Terengganu, and Fundamental Research Grant Scheme (FRGS) for their financial support.
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This work was supported by the Ministry of Higher Education Malaysia and Universiti Malaysia Terengganu under the Fundamental Research Grant Scheme (FRGS) Project Code FRGS/1/2021/STG06/UMT/02/1 and Vote no. 59659.
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Conceptualization: Zabidin Salleh, Koon Sang Wong; Methodology: Koon Sang Wong; Formal analysis and investigation: Koon Sang Wong, Zabidin Salleh, Che Mohd Imran Che Taib; Writing  original draft preparation: Koon Sang Wong; Writing  review and editing: Zabidin Salleh, Che Mohd Imran Che Taib; Funding acquisition: Zabidin Salleh; Resources: Zabidin Salleh, Che Mohd Imran Che Taib; Validation and Visualization: Che Mohd Imran Che Taib; Supervision: Zabidin Salleh. All authors reviewed the manuscript.
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Wong, K.S., Salleh, Z. & Che Taib, C.M.I. Fixedpoint results for fuzzy generalized βFcontraction mappings in fuzzy metric spaces and their applications. Fixed Point Theory Algorithms Sci Eng 2023, 8 (2023). https://doi.org/10.1186/s1366302300746x
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DOI: https://doi.org/10.1186/s1366302300746x