Abstract
In this paper, we derive some common α-fuzzy fixed point results for fuzzy mappings under generalized almost \(\mathcal{\mathbf{F}}\)-contractions in the context of a controlled metric space, which generalize many preexisting results in the literature. As an application, we establish some multivalued fixed point results. For justification of our results, we provide a nontrivial example.
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1 Introduction
The Banach fixed point theorem (BFPT) [1] is an important tool in fixed point theory. It guarantees the existence and uniqueness of a fixed point of certain self-mappings on metric spaces. It has various applications in several branches of mathematics. There are many extensions and generalizations of the BFPT in the literature; see [2–7]. Berinde [8, 9] studied various contractive-type mappings and introduced the concept of almost contractions.
Definition 1.1
([8])
A mapping \(\mathcal{\textsl{T}} : \mathcal{W} \rightarrow \mathcal{W}\) on a metric space \((\mathcal{W}, \textsl{d})\) is called an almost contraction if there exist \(0 \leq \lambda < 1\) and \({\L } \geq 0\) such that
for all \(\omega _{1}, \omega _{2} \in \mathcal{W}\).
Further, Berinde [9] generalized Definition 1.1 in the following way.
Definition 1.2
([9])
A mapping \(\mathcal{\textsl{T}} : \mathcal{W} \rightarrow \mathcal{W}\) on a metric space \((\mathcal{W}, \textsl{d})\) is called a generalized almost contraction if there exist \(0 \leq \lambda < 1\) and \({\L } \geq 0\) such that
for all \(\omega _{1}, \omega _{2} \in \mathcal{W}\).
Wardowski [10] introduced a new type of contractions, called \(\mathcal{\mathbf{F}}\)-contractions, and established a related fixed point theorem in the context of complete metric spaces.
Definition 1.3
([10])
A mapping \(\mathcal{\textsl{T}} : \mathcal{W} \rightarrow \mathcal{W}\) on a metric space \((\mathcal{W}, \textsl{d})\) is called an \(\mathcal{\mathbf{F}}\)-contraction if there exists \(\Omega > 0\) such that
for all \(\omega _{1}, \omega _{2} \in \mathcal{W}\), where \(\mathcal{\mathbf{F}} : (0, \infty ) \rightarrow \mathbb{R}\) is a function satisfying the following axioms:
- (C1):
-
\(\mathcal{\mathbf{F}}\) is strictly nondecreasing;
- (C2):
-
for each sequence \(\{\textsl{a}_{n}\} \subset (0, \infty )\) of positive real numbers, \(\lim_{n \rightarrow \infty } \textsl{a}_{n} = 0\) if and only if \(\lim_{n \rightarrow \infty } \mathcal{\mathbf{F}}( \textsl{a}_{n}) = -\infty \);
- (C3):
-
for each sequence \(\{\textsl{a}_{n}\} \subset (0, \infty )\) such that \(\lim_{n \rightarrow \infty } \textsl{a}_{n} = 0\), there exists \(l \in (0, 1)\) such that \(\lim_{n \rightarrow \infty } (\textsl{a}_{n})^{l} \mathcal{\mathbf{F}}(\textsl{a}_{n}) = 0\).
The following works deal with F-contractions: [11–16]. Afterward, Altun et al. [17] modified Definition 1.3 by adding the following condition:
- (C4):
-
\(\mathcal{\mathbf{F}}(\inf \mathcal{\mathbf{A}}) = \inf \mathcal{\mathbf{F}}(\mathcal{\mathbf{A}})\) for all \(\mathcal{\mathbf{A}} \subset (0, \infty )\) with \(\inf \mathcal{\mathbf{A}} > 0\).
We denote by \(\mathcal{F}\) the family of all functions \(\mathcal{\mathbf{F}}\) satisfying (C1)–(C4).
Nadler [18] derived the multivalued version of Banach fixed point theorem by using the Hausdorff metric over the family of nonempty closed bounded subsets of a complete metric space. We denote by \(\textsl{CLB}(\mathcal{W})\) the family of nonempty closed bounded subsets and by \(\textsl{CLD}(\mathcal{W})\) the family of nonempty closed subsets of \(\mathcal{W}\). Recently, Kamran et al. [19] introduced the concept of an extended b-metric space, which generalized the notion of a b-metric space [20, 21] by replacing the constant with a function depending on two variables.
Definition 1.4
([19])
Let \(\mathcal{W}\) be a nonempty set, and let \(\sigma : \mathcal{W} \times \mathcal{W} \rightarrow [1, \infty )\). Then a function \(\textsl{d}_{\sigma } : \mathcal{W} \times \mathcal{W} \rightarrow [0, \infty )\) is called an extended b-metric if for all \(\omega _{1}, \omega _{2}, \omega _{3} \in \mathcal{W}\), it satisfies the following axioms:
- (i):
-
\(\textsl{d}_{\sigma }(\omega _{1}, \omega _{2}) = 0\) iff \(\omega _{1} = \omega _{2}\),
- (ii):
-
\(\textsl{d}_{\sigma }(\omega _{1}, \omega _{2}) = \textsl{d}_{\sigma }( \omega _{2}, \omega _{1})\),
- (iii):
-
\(\textsl{d}_{\sigma }(\omega _{1}, \omega _{3}) \leq \sigma (\omega _{1}, \omega _{3})[\textsl{d}_{\sigma }(\omega _{1}, \omega _{2}) + \textsl{d}_{\sigma }(\omega _{2}, \omega _{3})]\).
The pair \((\mathcal{W}, \textsl{d}_{\sigma })\) is called an extended b-metric space.
Later on, several researchers worked on fixed point results in the context of extended b-metric spaces; see [22–25]. In the same direction, Mlaiki et al. [26] gave the idea of a controlled-type metric space (for further extensions, see [27]), which generalizes the notion of a b-metric space.
Definition 1.5
([26])
Let \(\mathcal{W}\) be a nonempty set, and let \(\sigma : \mathcal{W} \times \mathcal{W} \rightarrow [1, \infty )\). Then a function \(\textsl{d}_{\sigma } : \mathcal{W} \times \mathcal{W} \rightarrow [0, \infty )\) is called a controlled metric if for all \(\omega _{1}, \omega _{2}, \omega _{3} \in \mathcal{W}\), it satisfies the following axioms:
- (i):
-
\(\textsl{d}_{\sigma }(\omega _{1}, \omega _{2}) = 0\) iff \(\omega _{1} = \omega _{2}\),
- (ii):
-
\(\textsl{d}_{\sigma }(\omega _{1}, \omega _{2}) = \textsl{d}_{\sigma }( \omega _{2}, \omega _{1})\),
- (iii):
-
\(\textsl{d}_{\sigma }(\omega _{1}, \omega _{3}) \leq \sigma (\omega _{1}, \omega _{2})\textsl{d}_{\sigma }(\omega _{1}, \omega _{2}) + \sigma ( \omega _{2}, \omega _{3})\textsl{d}_{\sigma }(\omega _{2}, \omega _{3})\).
The pair \((\mathcal{W}, \textsl{d}_{\sigma })\) is called a controlled metric space.
Remark 1.1
Every controlled metric space is a generalization of a b-metric space and is different from an extended b-metric space.
Example 1.1
Let \(\mathcal{W} = [0, \infty )\). Define \(\textsl{d}_{\sigma } : \mathcal{W} \times \mathcal{W} \rightarrow [0, \infty )\) as
Hence \((\mathcal{W}, \textsl{d}_{\sigma })\) is a controlled metric space, where \(\sigma : \mathcal{W} \times \mathcal{W} \rightarrow [1, \infty )\) is defined by
For other definitions and information on the topology induced by \(\textsl{d}_{\sigma }\), see [26]. In [28], Alamgir et al. established a Pompieu–Hausdorff metric over the family of nonempty closed subsets of a controlled metric space W as follows.
Definition 1.6
([28])
Let \(\mathcal{\mathbf{A}}\), \(\mathcal{\mathbf{B}}\) be nonempty closed subsets of a controlled metric space \((\mathcal{W}, \textsl{d}_{\sigma })\). Define \(\textsl{H}_{\sigma } : \textsl{CLD}(\mathcal{W}) \times \textsl{CLD}( \mathcal{W}) \rightarrow [0, \infty ]\) by
Theorem 1.1
([28])
Let \((\mathcal{W}, \textsl{d}_{\sigma })\) be a controlled metric space. Then the mapping \(\textsl{H}_{\sigma } : \textsl{CLD}(\mathcal{W}) \times \textsl{CLD}( \mathcal{W}) \rightarrow [0, \infty ]\) is a Pompieu–Hausdorff controlled metric on \(\textsl{CLD}(\mathcal{W})\).
On the other hand, in 1981, Heilpern [29] used fuzzy sets [30] to introduce a class of fuzzy mappings, which is a generalization of multivalued mappings and proved a fixed point theorem for fuzzy contraction mappings in metric spaces. The result introduced by Heilpern is a fuzzy generalization of the Banach fixed point theorem. Consequently, several authors studied and generalized fuzzy fixed point theorems in many directions; see [31–38]. In this paper, we prove some common α-fuzzy fixed point results for fuzzy mappings under generalized almost \(\mathcal{\mathbf{F}}\)-contractions in the context of controlled metric spaces, which generalize many preexisting results in the literature. At the end, we give an example for the justification of our main result.
2 Main results
In this section, we define fuzzy sets, fuzzy mappings, and α-fuzzy fixed points and prove some common α fuzzy fixed point results in the context of controlled metric spaces.
Definition 2.1
Let \((\mathcal{W}, \textsl{d}_{\sigma })\) be a controlled metric space with \(\sigma : \mathcal{W} \times \mathcal{W} \rightarrow [1, \infty )\). Then a fuzzy set \(\mathcal{\mathbf{A}}_{\sigma }\) in \(\mathcal{W}\) is characterized by a membership function
which assigns to every member of \(\mathcal{W}\) a membership grade in \(\mathcal{\mathbf{A}}_{\sigma }\).
We denote by \(\mathcal{\mathcal{\mathbb{F}}}_{\sigma }(\mathcal{W})\) the collection of all fuzzy sets in \(\mathcal{W}\). Let \(\mathcal{\mathbf{A}}_{\sigma } \in \mathcal{\mathcal{\mathbb{F}}}_{ \sigma }(\mathcal{W})\) and \(\alpha \in [0, 1]\). Then the α-level set of \(\mathcal{\mathbf{A}}_{\sigma }\) is denoted by \([\mathcal{\mathbf{A}}_{\sigma }]_{\alpha }\) and is defined as
where \(\overline{\mathcal{\mathbf{B}}}\) denotes the closure of \(\mathcal{\mathbf{B}}\). Clearly, \([\mathcal{\mathbf{A}}_{\sigma }]_{\alpha }\) and \([\mathcal{\mathbf{A}}_{\sigma }]_{0}\) are subsets of the controlled metric space \(\mathcal{W}\). For \(\mathcal{\mathbf{A}}_{\sigma }, \mathcal{\mathbf{B}}_{\sigma } \in \mathcal{\mathcal{\mathbf{F}}}_{\sigma }(\mathcal{W})\), a fuzzy set \(\mathcal{\mathbf{A}}_{\sigma }\) is said to be more accurate than a fuzzy set \(\mathcal{\mathbf{B}}_{\sigma }\), denoted by \(\mathcal{\mathbf{A}}_{\sigma } \subset \mathcal{\mathbf{B}}_{\sigma }\), if \(f_{\mathcal{\mathbf{A}}_{\sigma }}(\mu ) \leq f_{\mathcal{\mathbf{B}}_{ \sigma }}(\mu )\) for each \(\mu \in \mathcal{W}\). Now, for \(\mu \in \mathcal{W}\), \(\mathcal{\mathbf{A}}_{\sigma }, \mathcal{\mathbf{B}}_{\sigma } \in \mathcal{\mathcal{\mathbb{F}}}_{\sigma }(\mathcal{W})\), \(\alpha \in [0, 1]\), and \([\mathcal{\mathbf{A}}_{\sigma }]_{\alpha }, [\mathcal{\mathbf{B}}_{ \sigma }]_{\alpha } \in \textsl{CLB}(\mathcal{W})\), define
Remark 2.1
By Theorem 1.1 the function \(\textsl{H}_{\sigma } : \textsl{CLB}(\mathcal{W}) \times \textsl{CLB}( \mathcal{W}) \rightarrow [0, \infty ]\) defined by
is a generalized Hausdorff controlled fuzzy metric on \(\textsl{CLB}(\mathcal{W})\).
Definition 2.2
Let \(\mathcal{\mathbf{S}}\), \(\mathcal{\mathbf{T}}\) be fuzzy mappings from \(\mathcal{W}\) into \(\Gamma (\mathcal{W})\). Then
- (i):
-
An element \(\mu \in \mathcal{W}\) is called an α-fuzzy fixed point of \(\mathcal{\mathbf{T}}\) if there exists \(\alpha _{\mathcal{\mathbf{T}}}(\mu ) \in (0, 1]\) such that \(\mu \in [\mathcal{\mathbf{T}}\mu ]_{\alpha _{\mathcal{\mathbf{T}}}( \mu )}\).
- (ii):
-
An element \(\mu \in \mathcal{W}\) is called a common α-fuzzy fixed point of \(\mathcal{\mathbf{S}}\) and \(\mathcal{\mathbf{T}}\) if there exist \(\alpha _{\mathcal{\mathbf{S}}}(\mu ), \alpha _{\mathcal{\mathbf{T}}}( \mu ) \in (0, 1]\) such that \(\mu \in [\mathcal{\mathbf{S}}\mu ]_{\alpha _{\mathcal{\mathbf{S}}}( \mu )} \cap [\mathcal{\mathbf{T}}\mu ]_{\alpha _{\mathcal{\mathbf{T}}}( \mu )} \).
- (iii):
-
For \(\alpha = 1\), μ is called a common fixed point of fuzzy mappings.
Lemma 2.1
Let \((\mathcal{W}, \textsl{d}_{\sigma })\) be a controlled metric space, and let \(\mathcal{\mathbf{A}}, \mathcal{\mathbf{B}} \in \textsl{CLB}( \mathcal{W})\). Then for each \(\textsl{a} \in \mathcal{\mathbf{A}}\),
Proof
Let us suppose on the contrary that for each \(\textsl{a} \in \mathcal{\mathbf{A}}\),
From Definition 1.6 we have that for each \(\textsl{a} \in \mathcal{\mathbf{A}}\),
Hence from equations (4) and (5) we get
a contradiction. □
Theorem 2.1
Let \((\mathcal{W}, \textsl{d}_{\sigma })\) be a complete controlled metric space, and let \(\mathcal{\mathbf{S}}\), \(\mathcal{\mathbf{T}}\) be fuzzy mappings from \(\mathcal{W}\) into \(\Gamma (\mathcal{W})\). Suppose for each \(\omega _{1} \in \mathcal{W}\), there exist \(\alpha _{\mathcal{\mathbf{S}}}(\omega _{1}), \alpha _{ \mathcal{\mathbf{T}}}(\omega _{2}) \in (0, 1]\) such that \([\mathcal{\mathbf{S}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{S}}}( \omega _{1})}\), \([\mathcal{\mathbf{T}}\omega _{2}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{2})}\) are nonempty closed subsets of \(\mathcal{W}\). Suppose that there exist some \(\mathcal{\mathbf{F}} \in \mathcal{F}\), \(\Omega > 0\), and \({\L } \geq 0\) such that
for all \(\omega _{1}, \omega _{2} \in \mathcal{W}\) with \(\textsl{H}_{\sigma }([\mathcal{\mathbf{S}}\omega _{1}]_{\alpha _{ \mathcal{\mathbf{S}}}(\omega _{1})}, [\mathcal{\mathbf{T}}\omega _{2}]_{ \alpha _{\mathcal{\mathbf{T}}}(\omega _{2})}) > 0\), where
Then there exists a common α-fuzzy fixed point of \(\mathcal{\mathbf{S}}\) and \(\mathcal{\mathbf{T}}\).
Proof
Let us take an arbitrary \(\omega _{0} \in \mathcal{W}\). Then by the hypothesis there exists \(\alpha _{\mathcal{\mathbf{S}}}(\omega _{0}) \in (0, 1]\) such that \([\mathcal{\mathbf{S}}\omega _{0}]_{\alpha _{\mathcal{\mathbf{S}}}( \omega _{0})}\) is a nonempty closed subset of \(\mathcal{W}\). Let \(\omega _{1} \in [\mathcal{\mathbf{S}}\omega _{0}]_{\alpha _{ \mathcal{\mathbf{S}}}(\omega _{0})}\). For such \(\omega _{1}\), there exists \(\alpha _{\mathcal{\mathbf{T}}}(\omega _{1}) \in (0, 1]\) such that \([\mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{1})}\) is a nonempty closed subset of \(\mathcal{W}\). From Lemma 2.1, condition \((C1)\) of Definition 1.3, and (6) we can write
where
From condition \((C4)\) we can write
Thus we have
Then there exists \(\omega _{2} \in [\mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{1})}\) such that
For this \(\omega _{2}\), there exists \(\alpha _{\mathcal{\mathbf{S}}}(\omega _{2}) \in (0, 1]\) such that \([\mathcal{\mathbf{S}}\omega _{2}]_{\alpha _{\mathcal{\mathbf{S}}}( \omega _{2})}\) is a nonempty closed subset of \(\mathcal{W}\). From Lemma 2.1, condition \((C1)\) of Definition 1.3, and (6) we have
where
From condition \((C4)\), we can write
Then we have
Thus there exists \(\omega _{3} \in [\mathcal{\mathbf{S}}\omega _{2}]_{\alpha _{ \mathcal{\mathbf{S}}}(\omega _{2})}\) such that
This implies that
By continuing the same procedure recursively we obtain a sequence \(\{\omega _{n}\}_{n = 0}^{\infty }\) in \(\mathcal{W}\) such that \(\omega _{2n + 1} \in [\mathcal{\mathbf{S}}\omega _{2n}]_{\alpha _{ \mathcal{\mathbf{S}}}(\omega _{2n})}\), \(\omega _{2n + 2} \in [\mathcal{\mathbf{T}}\omega _{2n + 1}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{2n + 1})}\). Also,
and
for all \(n \in \mathbb{N}\). From equations (8) and (9) we have
Therefore
By taking the limit as \(n \rightarrow \infty \) in equation (10) we get \(\lim_{n \rightarrow \infty } \mathbb{F}(\textsl{d}_{\sigma }(\omega _{n}, \omega _{n + 1})) = -\infty \). Next, from condition \((C2)\) of Definition 1.3 we have
Also, by condition \((C3)\) of Definition 1.3 there exists \(l \in (0, 1)\) such that
From equation (10) we have that for all \(n \in \mathbb{N}\),
By letting \(n \rightarrow \infty \) in (11) we obtain
By equation (12) there exists \(n_{1} \in \mathbb{N}\) such that \(n (\mathcal{\mathbf{F}}(\textsl{d}_{\sigma }(\omega _{n}, \omega _{n + 1})))^{l} \leq 1\) for all \(n \geq n_{1}\). Thus, for all \(n \geq n_{1}\), we have
From the triangle inequality and equation (13) for \(m > n \geq n_{1}\), we have
Since \(\lim_{n, m \rightarrow \infty }\sigma (\omega _{n + 1}, \omega _{m})l < 1\) for all \(\omega _{n}, \omega _{m} \in \mathcal{W}\), the series \(\sum_{i = 1}^{ \infty } (\prod_{j = 1}^{i}\sigma (\omega _{j}, \omega _{m}) ) \sigma (\omega _{i}, \omega _{i + 1})\frac{1}{i^{\frac{1}{l}}}\) converges by the ratio test for each \(m \in \mathbb{N}\). Therefore, by taking the limit as \(n \rightarrow \infty \) in the above inequality we get \(\textsl{d}_{\sigma }(\omega _{n}, \omega _{m}) \rightarrow 0\). Since \(\mathcal{W}\) is complete, there exists \(\rho \in \mathcal{W}\) such that \(\lim_{n \rightarrow \infty }\omega _{n} = \rho \). Next, we prove that ρ is a fixed point of \(\mathcal{\mathbf{T}}\). Suppose on the contrary that ρ is not a fixed point of \(\mathcal{\mathbf{T}}\). Then there exist \(\mathbb{N}_{0} \in \mathbb{N}\) and a subsequence \(\{\omega _{n_{r}}\}\) of \(\{\omega _{n}\}\) such that \(\textsl{d}_{\sigma }(\omega _{2n_{r}}, [\mathcal{\mathbf{T}}\rho ]_{ \alpha _{\mathcal{\mathbf{T}}}(\rho )}) > 0\) for all \(n_{r} \geq \mathbb{N}_{0}\). As \(\textsl{d}_{\sigma }(\omega _{2n_{r}}, [\mathcal{\mathbf{T}}\rho ]_{ \alpha _{\mathcal{\mathbf{T}}}(\rho )}) > 0\) for all \(n_{r} \geq \mathbb{N}_{0}\), from Lemma 2.1, condition \((1)\) of Definition 1.3, and (6) we have
This implies that
As \(\mathcal{\mathbf{F}}\) is strictly increasing, we have
By taking the limit as \(n \rightarrow \infty \) we get
Thus \(\rho \in [\mathcal{\mathbf{T}}\rho ]_{\alpha _{\mathcal{\mathbf{T}}}( \rho )}\). By a similar procedure we can prove that \(\rho \in [\mathcal{\mathbf{S}}\rho ]_{\alpha _{\mathcal{\mathbf{S}}}( \rho )}\). Hence \(\rho \in [\mathcal{\mathbf{T}}\rho ]_{\alpha _{\mathcal{\mathbf{T}}}( \rho )} \cap [\mathcal{\mathbf{S}}\rho ]_{\alpha _{ \mathcal{\mathbf{S}}}(\rho )}\). □
Theorem 2.2
Let \((\mathcal{W}, \textsl{d}_{\sigma })\) be a complete controlled metric space, and let \(\mathcal{\mathbf{S}}\), \(\mathcal{\mathbf{T}}\) be fuzzy mappings from \(\mathcal{W}\) into \(\Gamma (\mathcal{W})\). Suppose that for each \(\omega _{1} \in \mathcal{W}\), there exist \(\alpha _{\mathcal{\mathbf{S}}}(\omega _{1}), \alpha _{ \mathcal{\mathbf{T}}}(\textsl{y}) \in (0, 1]\) such that \([\mathcal{\mathbf{S}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{S}}}( \omega _{1})}\), \([\mathcal{\mathbf{T}}\textsl{y}]_{\alpha _{ \mathcal{\mathbf{T}}}(\textsl{y})}\) are nonempty closed subsets of \(\mathcal{W}\). If there exist \(\mathcal{\mathbf{F}} \in \mathcal{F}\) and \(\Omega > 0\) such that
for all \(\omega _{1}, \omega _{2} \in \mathcal{W}\) with \(\textsl{H}_{\sigma }([\mathcal{\mathbf{S}}\omega _{1}]_{\alpha _{ \mathcal{\mathbf{S}}}(\omega _{1})}, [\mathcal{\mathbf{T}}\omega _{2}]_{ \alpha _{\mathcal{\mathbf{T}}}(\omega _{2})}) > 0\), then there exists a common α-fuzzy fixed point of \(\mathcal{\mathbf{S}}\) and \(\mathcal{\mathbf{T}}\).
Proof
By taking \({\L } = 0\) in Theorem 2.1 we get the proof. □
Corollary 2.1
Let \((\mathcal{W}, \textsl{d}_{\sigma })\) be a complete controlled metric space, and let \(\mathcal{\mathbf{T}}\) be a fuzzy mapping from \(\mathcal{W}\) into \(\Gamma (\mathcal{W})\). Suppose that for each \(\omega _{1} \in \mathcal{W}\), there exist \(\alpha _{\mathcal{\mathbf{T}}}(\omega _{1}), \alpha _{ \mathcal{\mathbf{T}}}(\omega _{2}) \in (0, 1]\) such that \([\mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{1})}\), \([\mathcal{\mathbf{T}}\omega _{2}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{2})}\) are nonempty closed subsets of \(\mathcal{W}\). If there exist \(\mathcal{\mathbf{F}} \in \mathcal{F}\), \(\Omega > 0\), and \({\L } \geq 0\) such that
for all \(\omega _{1}, \omega _{2} \in \mathcal{W}\) with \(\textsl{H}_{\sigma }([\mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{1})}, [\mathcal{\mathbf{T}}\omega _{2}]_{ \alpha _{\mathcal{\mathbf{T}}}(\omega _{2})}) > 0\), where
then there exists an α-fuzzy fixed point of \(\mathcal{\mathbf{T}}\).
Proof
By taking \(\mathcal{\mathbf{S}} = \mathcal{\mathbf{T}}\) in Theorem 2.1 we get the proof. □
Corollary 2.2
Let \((\mathcal{W}, \textsl{d}_{\sigma })\) be a complete controlled metric space, and let \(\mathcal{\mathbf{T}}\) be a fuzzy mapping from \(\mathcal{W}\) into \(\Gamma (\mathcal{W})\). Suppose that for each \(\omega _{1} \in \mathcal{W}\), there exist \(\alpha _{\mathcal{\mathbf{T}}}(\omega _{1}), \alpha _{ \mathcal{\mathbf{T}}}(\omega _{2}) \in (0, 1]\) such that \([\mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{1})}\), \([\mathcal{\mathbf{T}}\omega _{2}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{2})}\) are nonempty closed subsets of \(\mathcal{W}\). Assume there exist \(\mathcal{\mathbf{F}} \in \mathcal{F}\) and \(\Omega > 0\) such that
for all \(\omega _{1}, \omega _{2} \in \mathcal{W}\) with \(\textsl{H}_{\sigma }([\mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{1})}, [\mathcal{\mathbf{T}}\omega _{2}]_{ \alpha _{\mathcal{\mathbf{T}}}(\omega _{2})}) > 0\). Then there exists an α-fuzzy fixed point of \(\mathcal{\mathbf{T}}\).
Proof
By taking \(\mathcal{\mathbf{S}} = \mathcal{\mathbf{T}}\) and \({\L } = 0\) in Theorem 2.1 we get the proof. □
Remark 2.2
- (i):
- (ii):
- (iii):
-
Corollary 2.1 (resp., Corollary 2.2) generalizes Corollary 2.3 (resp., Corollary 2.4) of [39].
Corollary 2.3
Let \((\mathcal{W}, \textsl{d}_{\sigma })\) be a complete controlled metric space, and let \(\mathcal{\mathbf{A}}, \mathcal{\mathbf{B}} : \mathcal{W} \rightarrow \textsl{CLB}(\mathcal{W})\) be multivalued mappings. Assume that there exist \(\mathcal{\mathbf{F}} \in \mathcal{F}\), \(\Omega > 0\), and \({\L } \geq 0\) such that
for all \(\omega _{1}, \omega _{2} \in \mathcal{W}\) with \(\textsl{H}_{\sigma }(\mathcal{\mathbf{A}}\omega _{1}, \mathcal{\mathbf{B}}\omega _{2}) > 0\), where
Then there is a common fixed point of \(\mathcal{\mathbf{A}}\) and \(\mathcal{\mathbf{B}}\).
Proof
Let \(\alpha : \mathcal{W} \rightarrow (0, 1]\) be an arbitrary mapping and define the mappings \(\mathcal{\mathbf{S, \mathbf{T}}} : \mathcal{W} \rightarrow \mathcal{\mathcal{\mathbf{F}}}(\mathcal{W})\) by
and
Then we obtain
Therefore we can apply Theorem 2.1 to get a fixed point \(\rho \in \mathcal{W}\) such that
□
Corollary 2.4
Let \((\mathcal{W}, \textsl{d}_{\sigma })\) be a complete controlled metric space, and let \(\mathcal{\mathbf{A}}, \mathcal{\mathbf{B}} : \mathcal{W} \rightarrow \textsl{CLB}(\mathcal{W})\) be multivalued mappings. Assume there exist \(\mathcal{\mathbf{F}} \in \mathcal{F}\) and \(\Omega > 0\) such that
for all \(\omega _{1}, \omega _{2} \in \mathcal{W}\) with \(\textsl{H}_{\sigma }(\mathcal{\mathbf{A}}\omega _{1}, \mathcal{\mathbf{B}}\omega _{2}) > 0\). Then there exists a common fixed point of \(\mathcal{\mathbf{A}}\) and \(\mathcal{\mathbf{B}}\).
Proof
It suffices to take \({\L } = 0\) in Corollary 2.3. □
Corollary 2.5
Let \((\mathcal{W}, \textsl{d}_{\sigma })\) be a complete controlled metric space, and let \(\mathcal{\mathbf{A}} : \mathcal{W} \rightarrow \textsl{CLB}( \mathcal{W})\) be a multivalued mapping. Assume there exist \(\mathcal{\mathbf{F}} \in \mathcal{F}\), \(\Omega > 0\), and \({\L } \geq 0\) such that
for all \(\omega _{1}, \omega _{2} \in \mathcal{W}\) with \(\textsl{H}_{\sigma }(\mathcal{\mathbf{A}}\omega _{1}, \mathcal{\mathbf{A}}\omega _{2}) > 0\), where
Then there exists a fixed point of \(\mathcal{\mathbf{A}}\).
Proof
Take \(\mathcal{\mathbf{A}} = \mathcal{\mathbf{B}}\) in Corollary 2.3. □
Corollary 2.6
Let \((\mathcal{W}, \textsl{d}_{\sigma })\) be a complete controlled metric space, and let \(\mathcal{\mathbf{A}} : \mathcal{W} \rightarrow \textsl{CLB}( \mathcal{W})\) be a multivalued mapping. Assume there exist \(\mathcal{\mathbf{F}} \in \mathcal{F}\) and \(\Omega > 0\) such that
for all \(\omega _{1}, \omega _{2} \in \mathcal{W}\) with \(\textsl{H}_{\sigma }(\mathcal{\mathbf{A}}\omega _{1}, \mathcal{\mathbf{A}}\omega _{2}) > 0\). Then there exists a fixed point of \(\mathcal{\mathbf{A}}\).
Proof
Take \(\mathcal{\mathbf{A}} = \mathcal{\mathbf{B}}\) and \({\L } = 0\) in Corollary 2.3. □
Remark 2.3
- (i):
- (ii):
-
Corollary 2.4 generalizes Corollary 2.6.
- (iii):
-
Corollary 2.5 (resp., Corollary 2.6) generalizes Corollary 2.7 (resp., Corollary 2.8) of [39].
We further suppose that \(\hat{\mathcal{\mathbf{T}}}\) is a multivalued mapping induced by the fuzzy mapping \(\mathcal{\mathbf{T}} : \mathcal{W} \rightarrow \Gamma (\mathcal{W})\), that is,
Lemma 2.2
Let \((\mathcal{W}, \textsl{d}_{\sigma })\) be a complete controlled metric space, \(\mu \in \mathcal{W}\), and let \(\mathcal{\mathbf{T}}\) be a fuzzy mapping from \(\mathcal{W}\) into \(\Gamma (\mathcal{W})\) such that \(\hat{\mathcal{\mathbf{T}}}(\omega _{1})\) is a nonempty compact set for all \(\omega _{1} \in \mathcal{W}\). Then \(\mu \in \hat{\mathcal{\mathbf{T}}}(\mu )\) if and only if
for all \(\omega _{1} \in \mathcal{W}\).
Proof
Suppose that \(\mu \in \hat{\mathcal{\mathbf{T}}}(\mu )\). Then
This implies that
Conversely, suppose that
Then by the same steps we can show that \(\mu \in \hat{\mathcal{\mathbf{T}}}(\mu )\). □
Corollary 2.7
Let \((\mathcal{W}, \textsl{d}_{\sigma })\) be a complete controlled metric space, and let \(\hat{\mathcal{\mathbf{S}}}, \hat{\mathcal{\mathbf{T}}} : \mathcal{W} \rightarrow \Gamma (\mathcal{W})\) be fuzzy mappings such that for each \(\omega _{1} \in \mathcal{W}\), \(\hat{\mathcal{\mathbf{S}}}(\omega _{1})\) and \(\hat{\mathcal{\mathbf{T}}}(\omega _{1})\) are nonempty closed subsets of \(\mathcal{W}\). Assume there exist \(\mathcal{\mathbf{F}} \in \mathcal{F}\), \(\Omega > 0\), and \({\L } \geq 0\) such that
for all \(\omega _{1}, \omega _{2} \in \mathcal{W}\) with \(\textsl{H}_{\sigma }(\hat{\mathcal{\mathbf{S}}}(\omega _{1}), \hat{\mathcal{\mathbf{T}}}(\omega _{2})) > 0\), where
Then there exists \(\mu \in \mathcal{W}\) such that \(\mathcal{\mathbf{S}}(\mu )(\mu ) \geq \mathcal{\mathbf{S}}(\mu )( \omega _{1})\) and \(\mathcal{\mathbf{T}}(\mu )(\mu ) \geq \mathcal{\mathbf{T}}(\mu )( \omega _{1})\) for all \(\omega _{1} \in \mathcal{W}\).
Proof
By Corollary 2.3 there exists \(\mu \in \mathcal{W}\) such that \(\mu \in \hat{\mathcal{\mathbf{S}}}(\mu ) \cap \hat{\mathcal{\mathbf{T}}}(\mu )\). Then from Lemma 2.2 we get
for all \(\omega _{1} \in \mathcal{W}\). □
Example 2.1
Let \(\mathcal{W} = [0, 1]\). Define \(\textsl{d}_{\sigma } : \mathcal{W} \times \mathcal{W} \rightarrow [0, \infty )\) by
Then \((\mathcal{W}, \textsl{d}_{\sigma })\) is a complete controlled metric space, where \(\sigma : \mathcal{W} \times \mathcal{W} \rightarrow [1, \infty )\) is defined by
For \(\alpha \in [0, 1)\) and \(\omega _{1} \in \mathcal{W}\), define the mappings \(\mathcal{\mathbf{S}}, \mathcal{\mathbf{T}} : \mathcal{W} \rightarrow \Gamma (\mathcal{W})\) by
and
so that
Let \(\mathcal{\mathbf{F}}(\mathcal{\mathbf{T}}) = \ln ( \mathcal{\mathbf{T}})\). Then there exists \(\Omega \in (0, \ln \frac{|\omega _{2} - \omega _{1}|}{|\omega _{2} - \frac{\omega _{1}}{2}|^{\frac{1}{50}}})\) such that
for all \(\omega _{1}, \omega _{2} \in \mathcal{W}\) with \(\textsl{H}_{\sigma }([\mathcal{\mathbf{S}}\omega _{1}]_{\alpha _{ \mathcal{\mathbf{S}}}(\omega _{1})}, [\mathcal{\mathbf{T}}\omega _{2}]_{ \alpha _{\mathcal{\mathbf{T}}}(\omega _{2})}) > 0\). Hence all the axioms of Theorem 2.1 are satisfied, and therefore \(0 \in [\mathcal{\mathbf{S}}0]_{\alpha } \cap [\mathcal{\mathbf{T}}0]_{ \alpha }\).
3 Conclusion
In this work, we introduced the concept of fuzzy mappings in a more general space, called a controlled metric space. Further, we derived the existence of common α-fuzzy fixed points for two fuzzy mappings under generalized almost \(\mathcal{\mathbf{F}}\)-contractions in the setting of controlled metric spaces. Our results generalize many well-known results in the literature. For justification of the obtained results, we gave an illustrative example.
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Alamgir, N., Kiran, Q., Aydi, H. et al. Fuzzy fixed point results of generalized almost \(\mathcal{\mathbf{F}}\)-contractions in controlled metric spaces. Adv Differ Equ 2021, 476 (2021). https://doi.org/10.1186/s13662-021-03598-0
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DOI: https://doi.org/10.1186/s13662-021-03598-0