Theory and Modern Applications

# Fuzzy fixed point results of generalized almost $$\mathcal{\mathbf{F}}$$-contractions in controlled metric spaces

## 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.

## 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 [27]. 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

$$\textsl{d}(\mathcal{\textsl{T}}\omega _{1}, \mathcal{ \textsl{T}} \omega _{2}) \leq \lambda \textsl{d}(\omega _{1}, \omega _{2}) + {\L } \textsl{d}(\omega _{2}, \mathcal{\textsl{T}}\omega _{1})$$
(1)

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

\begin{aligned}[b] \textsl{d}(\mathcal{\textsl{T}}\omega _{1}, \mathcal{ \textsl{T}} \omega _{2})) &\leq \lambda \textsl{d}(\omega _{1}, \omega _{2})\\ &\quad {} + {\L } \min \bigl\{ \textsl{d}\bigl( \omega _{1}, \mathcal{\textsl{T}}(\omega _{1})\bigr), \textsl{d}\bigl(\omega _{2}, \mathcal{\textsl{T}}(\omega _{2})\bigr), \textsl{d}\bigl(\omega _{1}, \mathcal{ \textsl{T}}(\omega _{2})\bigr), \textsl{d}\bigl(\omega _{2}, \mathcal{\textsl{T}}(\omega _{1})\bigr)\bigr\} \end{aligned}
(2)

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

$$\textsl{d}(\mathcal{\textsl{T}}\omega _{1}, \mathcal{ \textsl{T}} \omega _{2})>0 \quad \Longrightarrow\quad \Omega + \mathcal{ \mathbf{F}}\bigl( \textsl{d}(\mathcal{\textsl{T}}\omega _{1}, \mathcal{ \textsl{T}} \omega _{2})\bigr) \leq \mathcal{\mathbf{F}}\bigl( \textsl{d}(\omega _{1}, \omega _{2})\bigr)$$
(3)

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: [1116]. 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 [2225]. 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

$$\textsl{d}_{\sigma }(\omega _{1}, \omega _{2}) = \textstyle\begin{cases} 0 & \text{if }\omega _{1} = \omega _{2}, \\ \frac{1}{\omega _{1}} & \text{if }\omega _{1} \geq 1\text{ and }\omega _{2} \in [0, 1), \\ \frac{1}{\omega _{2}} & \text{if }\omega _{2} \geq 1\text{ and }\omega _{1} \in [0, 1), \\ 1 &\text{otherwise.} \end{cases}$$

Hence $$(\mathcal{W}, \textsl{d}_{\sigma })$$ is a controlled metric space, where $$\sigma : \mathcal{W} \times \mathcal{W} \rightarrow [1, \infty )$$ is defined by

$$\sigma (\omega _{1}, \omega _{2}) = \textstyle\begin{cases} 1 & \text{if }\omega _{1}, \omega _{2} \in [0, 1), \\ \max \{\omega _{1}, \omega _{2}\} &\text{otherwise.} \end{cases}$$

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

$$\textsl{H}_{\sigma }(\mathcal{\mathbf{A}}, \mathcal{\mathbf{B}}) = \textstyle\begin{cases} \max \{\sup_{\textsl{a} \in \mathcal{\mathbf{A}}} \textsl{d}_{\sigma }( \textsl{a}, \mathcal{\mathbf{B}}), \sup_{\textsl{b} \in \mathcal{\mathbf{B}}}\textsl{d}_{\sigma }(\textsl{b}, \mathcal{\mathbf{A}})\} & \text{if the maximum exists;} \\ \infty & \text{otherwise.} \end{cases}$$

### 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 [3138]. 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

$$\mathcal{\mathbb{F}}_{\mathcal{\mathbf{A}}_{\sigma }}: \mathcal{W} \rightarrow [0, 1],$$

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

\begin{aligned}& [\mathcal{\mathbf{A}}_{\sigma }]_{\alpha } = \bigl\{ \mu \in \mathcal{W} : \mathcal{\mathbf{A}}_{\sigma }(\mu ) \geq \alpha \bigr\} , \quad \alpha \in (0, 1], \\& [\mathcal{\mathbf{A}}_{\sigma }]_{0} = \overline{\bigl\{ \mu \in \mathcal{W} : \mathcal{\mathbf{A}}_{\sigma }(\mu ) > 0\bigr\} }, \end{aligned}

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

\begin{aligned}& \rho _{\alpha }\bigl(\mu , [\mathcal{\mathbf{A}}_{\sigma }]_{\alpha } \bigr) = \inf \bigl\{ d(\mu , \textsl{a}) : \textsl{a} \in [\mathcal{ \mathbf{A}}_{ \sigma }]_{\alpha }\bigr\} , \\& \rho _{\alpha }\bigl([\mathcal{\mathbf{A}}_{\sigma }]_{\alpha }, [ \mathcal{\mathbf{B}}_{\sigma }]_{\alpha }\bigr) = \inf \bigl\{ d( \textsl{a}, \textsl{b}) : \textsl{a} \in [\mathcal{\mathbf{A}}_{\sigma }]_{\alpha }, \textsl{b} \in [\mathcal{\mathbf{B}}_{\sigma }]_{\alpha }\bigr\} , \\& \rho \bigl([\mathcal{\mathbf{A}}_{\sigma }]_{\alpha }, [\mathcal{ \mathbf{B}}_{ \sigma }]_{\alpha }\bigr)= \sup_{\alpha } \rho _{\alpha }\bigl([ \mathcal{\mathbf{A}}_{\sigma }]_{\alpha }, [ \mathcal{\mathbf{B}}_{ \sigma }]_{\alpha }\bigr). \end{aligned}

### 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

\begin{aligned} &\textsl{H}_{\sigma }\bigl([\mathcal{\mathbf{A}}_{\sigma }]_{\alpha }, [ \mathcal{\mathbf{B}}_{\sigma }]_{\alpha }\bigr)\\ &\quad = \textstyle\begin{cases} \max \{\sup_{\textsl{a} \in [\mathcal{\mathbf{A}}_{\sigma }]_{\alpha }} d(\textsl{a}, [\mathcal{\mathbf{B}}_{\sigma }]_{\alpha }), \sup_{ \textsl{b} \in [\mathcal{\mathbf{B}}_{\sigma }]_{\alpha }}d(\textsl{b}, [ \mathcal{\mathbf{A}}_{\sigma }]_{\alpha })\} & \text{if the maximum exists,} \\ \infty & \text{otherwise,} \end{cases}\displaystyle \end{aligned}

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 $$\textsl{a} \in \mathcal{\mathbf{A}}$$,

$$\textsl{d}_{\sigma }(\textsl{a}, \mathcal{\mathbf{B}}) \leq \textsl{H}_{ \sigma }(\mathcal{\mathbf{A}}, \mathcal{\mathbf{B}}).$$

### Proof

Let us suppose on the contrary that for each $$\textsl{a} \in \mathcal{\mathbf{A}}$$,

$$\textsl{d}_{\sigma }(\textsl{a}, \mathcal{\mathbf{B}}) > \textsl{H}_{ \sigma }(\mathcal{\mathbf{A}}, \mathcal{\mathbf{B}}).$$
(4)

From Definition 1.6 we have that for each $$\textsl{a} \in \mathcal{\mathbf{A}}$$,

$$\textsl{d}_{\sigma }(\textsl{a}, \mathcal{\mathbf{B}}) \leq \textsl{H}_{ \sigma }(\mathcal{\mathbf{A}}, \mathcal{\mathbf{B}}).$$
(5)

Hence from equations (4) and (5) we get

$$\textsl{H}_{\sigma }(\mathcal{\mathbf{A}}, \mathcal{\mathbf{B}}) < \textsl{d}_{\sigma }(\textsl{a}, \mathcal{\mathbf{B}}) \leq \textsl{H}_{ \sigma }(\mathcal{\mathbf{A}}, \mathcal{\mathbf{B}}),$$

### 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

$$\Omega + \mathcal{\mathbf{F}}(\textsl{H}_{\sigma }\bigl([ \mathcal{\mathbf{S}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{S}}}( \omega _{1})}, [\mathcal{\mathbf{T}} \omega _{2}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{2})}\bigr) \leq \mathcal{\mathbf{F}}\bigl( \textsl{d}_{\sigma }(\omega _{1}, \omega _{2})\bigr) + {\L }\bigl(\textsl{M}( \omega _{1}, \omega _{2})\bigr)$$
(6)

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

\begin{aligned} \textsl{M}(\omega _{1}, \omega _{2}) &= \min \bigl\{ \textsl{d}_{\sigma }\bigl( \omega _{1}, [\mathcal{\mathbf{S}} \omega _{1}]_{\alpha _{ \mathcal{\mathbf{S}}}(\omega _{1})}\bigr), \textsl{d}_{\sigma }\bigl( \omega _{2}, [\mathcal{\mathbf{T}}\omega _{2}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{2})} \bigr), \\ &\quad \textsl{d}_{\sigma }\bigl(\omega _{1}, [ \mathcal{ \mathbf{T}}\omega _{2}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{2})}\bigr), \textsl{d}_{\sigma } \bigl(\omega _{2}, [ \mathcal{\mathbf{S}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{S}}}( \omega _{1})} \bigr)\bigr\} . \end{aligned}

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

\begin{aligned}[b] \Omega + \mathcal{\mathbf{F}}(\textsl{d}_{\sigma }\bigl(\omega _{1}, [ \mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{1})} \bigr) &\leq \Omega + \mathcal{\mathbf{F}}(\textsl{H}_{\sigma }\bigl([ \mathcal{\mathbf{S}}\omega _{0}]_{\alpha _{\mathcal{\mathbf{S}}}( \omega _{0})}, [\mathcal{\mathbf{T}} \omega _{1}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{1})}\bigr) \\ &\leq \mathcal{\mathbf{F}}\bigl( \textsl{d}_{\sigma }(\omega _{0}, \omega _{1})\bigr) + {\L }\bigl(\textsl{M}( \omega _{0}, \omega _{1}) \bigr),\end{aligned}
(7)

where

\begin{aligned} \textsl{M}(\omega _{0}, \omega _{1}) &= \min \bigl\{ \textsl{d}_{\sigma }\bigl( \omega _{0}, [\mathcal{\mathbf{S}} \omega _{0}]_{\alpha _{ \mathcal{\mathbf{S}}}(\omega _{0})}\bigr), \textsl{d}_{\sigma }\bigl( \omega _{1}, [\mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{1})} \bigr),\\ &\quad \textsl{d}_{\sigma }\bigl(\omega _{0}, [ \mathcal{ \mathbf{T}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{1})}\bigr), \textsl{d}_{\sigma } \bigl(\omega _{1}, [ \mathcal{\mathbf{S}}\omega _{0}]_{\alpha _{\mathcal{\mathbf{S}}}( \omega _{0})} \bigr)\bigr\} . \end{aligned}

From condition $$(C4)$$ we can write

$$\mathcal{\mathbf{F}}(\textsl{d}_{\sigma }\bigl(\omega _{1}, [ \mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{1})}\bigr) = \inf _{\textsl{y} \in [\mathcal{\mathbf{T}}\omega _{1}]_{ \alpha _{\mathcal{\mathbf{T}}}(\omega _{1})}} \mathcal{\mathbf{F}}\bigl( \textsl{d}_{\sigma }( \omega _{1}, \textsl{y})\bigr).$$

Thus we have

\begin{aligned} &\Omega + \inf_{\textsl{y} \in [\mathcal{\mathbf{T}}\omega _{1}]_{ \alpha _{\mathcal{\mathbf{T}}}(\omega _{1})}} \mathcal{\mathbf{F}}( \textsl{d}_{\sigma }(\omega _{1}, \textsl{y})\\ &\quad \leq \mathcal{ \mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{0}, \omega _{1})\bigr) + { \L }\min \bigl\{ \textsl{d}_{\sigma }\bigl(\omega _{0}, [\mathcal{\mathbf{S}} \omega _{0}]_{\alpha _{\mathcal{\mathbf{S}}}(\omega _{0})} \bigr), \textsl{d}_{\sigma }\bigl(\omega _{1}, [\mathcal{ \mathbf{T}}\omega _{1}]_{ \alpha _{\mathcal{\mathbf{T}}}(\omega _{1})}\bigr), \\ &\qquad \textsl{d}_{\sigma }\bigl(\omega _{0}, [\mathcal{\mathbf{T}} \omega _{1}]_{ \alpha _{\mathcal{\mathbf{T}}}(\omega _{1})}\bigr), \textsl{d}_{\sigma }\bigl( \omega _{1}, [\mathcal{\mathbf{S}}\omega _{0}]_{\alpha _{ \mathcal{\mathbf{S}}}(\omega _{0})} \bigr)\bigr\} . \end{aligned}

Then there exists $$\omega _{2} \in [\mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{1})}$$ such that

\begin{aligned} &\Omega + \mathcal{\mathbf{F}}(\textsl{d}_{\sigma }(\omega _{1}, \omega _{2}) \\ &\quad \leq \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }( \omega _{0}, \omega _{1})\bigr) + {\L }\min \bigl\{ \textsl{d}_{\sigma }(\omega _{0}, \omega _{1}), \textsl{d}_{\sigma }(\omega _{1}, \omega _{2}), \textsl{d}_{\sigma }( \omega _{0}, \omega _{2}), \textsl{d}_{\sigma }(\omega _{1}, \omega _{1}) \bigr\} \\ &\quad = \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{0}, \omega _{1})\bigr). \end{aligned}

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

\begin{aligned} \Omega + \mathcal{\mathbf{F}}(\textsl{d}_{\sigma }\bigl(\omega _{2}, [ \mathcal{\mathbf{S}}\omega _{2}]_{\alpha _{\mathcal{\mathbf{S}}}( \omega _{2})} \bigr) & \leq \Omega + \mathcal{\mathbf{F}}(\textsl{H}_{ \sigma }\bigl([ \mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{1})}, [\mathcal{\mathbf{S}} \omega _{2}]_{ \alpha _{\mathcal{\mathbf{S}}}(\omega _{2})}\bigr) \\ & \leq \Omega + \mathcal{\mathbf{F}}(\textsl{H}_{\sigma }\bigl([ \mathcal{ \mathbf{S}}\omega _{2}]_{\alpha _{\mathcal{\mathbf{S}}}( \omega _{2})}, [\mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{1})}\bigr) \\ & \leq \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{2}, \omega _{1})\bigr) + {\L }\bigl(\textsl{M}(\omega _{2}, \omega _{1})\bigr), \end{aligned}

where

\begin{aligned} \textsl{M}(\omega _{2}, \omega _{1}) &= \min \bigl\{ \textsl{d}_{\sigma }\bigl( \omega _{2}, [\mathcal{\mathbf{S}} \omega _{2}]_{\alpha _{ \mathcal{\mathbf{S}}}(\omega _{2})}\bigr), \textsl{d}_{\sigma }\bigl( \omega _{1}, [\mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{1})} \bigr),\\ &\quad \textsl{d}_{\sigma }\bigl(\omega _{2}, [ \mathcal{ \mathbf{T}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{1})}\bigr), \textsl{d}_{\sigma } \bigl(\omega _{1}, [ \mathcal{\mathbf{S}}\omega _{2}]_{\alpha _{\mathcal{\mathbf{S}}}( \omega _{2})} \bigr)\bigr\} . \end{aligned}

From condition $$(C4)$$, we can write

$$\mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }\bigl(\omega _{2}, [ \mathcal{\mathbf{S}}\omega _{2}]_{\alpha _{\mathcal{\mathbf{S}}}}( \omega _{2})\bigr)\bigr) = \inf_{\textsl{y}^{\prime } \in [\mathcal{\mathbf{S}} \omega _{2}]_{\alpha _{\mathcal{\mathbf{S}}}(\omega _{2})}} \mathcal{\mathbf{F}} \bigl(\textsl{d}_{\sigma }\bigl(\omega _{2}, \textsl{y}^{\prime } \bigr)\bigr).$$

Then we have

\begin{aligned} &\Omega + \inf_{\textsl{y}^{\prime } \in [\mathcal{\mathbf{S}}\omega _{2}]_{ \alpha _{\mathcal{\mathbf{S}}}(\omega _{2})}} \mathcal{\mathbf{F}}\bigl( \textsl{d}_{\sigma }\bigl(\omega _{2}, \textsl{y}^{\prime } \bigr)\bigr) \\ &\quad \leq \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{2}, \omega _{1})\bigr) + { \L }\min \bigl\{ \textsl{d}_{\sigma }\bigl(\omega _{2}, [\mathcal{\mathbf{S}} \omega _{2}]_{\alpha _{\mathcal{\mathbf{S}}}(\omega _{2})}\bigr), \textsl{d}_{\sigma }\bigl( \omega _{1}, [\mathcal{\mathbf{T}}\omega _{1}]_{ \alpha _{\mathcal{\mathbf{T}}}(\omega _{1})} \bigr), \\ &\qquad \textsl{d}_{\sigma }\bigl(\omega _{2}, [\mathcal{\mathbf{T}} \omega _{1}]_{ \alpha _{\mathcal{\mathbf{T}}}(\omega _{1})}\bigr), \textsl{d}_{\sigma }\bigl( \omega _{1}, [\mathcal{\mathbf{S}}\omega _{2}]_{\alpha _{ \mathcal{\mathbf{S}}}(\omega _{2})} \bigr)\bigr\} . \end{aligned}

Thus there exists $$\omega _{3} \in [\mathcal{\mathbf{S}}\omega _{2}]_{\alpha _{ \mathcal{\mathbf{S}}}(\omega _{2})}$$ such that

\begin{aligned} &\Omega + \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{2}, \omega _{3})\bigr)\\ &\quad \leq \mathcal{\mathbf{F}}\bigl( \textsl{d}_{\sigma }(\omega _{1}, \omega _{2})\bigr) + {\L }\min \bigl\{ \textsl{d}_{\sigma }(\omega _{2}, \omega _{3}), \textsl{d}_{\sigma }(\omega _{1}, \omega _{2}), \textsl{d}_{\sigma }( \omega _{2}, \omega _{2}), \textsl{d}_{\sigma }(\omega _{1}, \omega _{3}) \bigr\} . \end{aligned}

This implies that

$$\Omega + \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{2}, \omega _{3})\bigr) \leq \mathcal{\mathbf{F}}\bigl( \textsl{d}_{\sigma }(\omega _{1}, \omega _{2})\bigr).$$

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,

$$\Omega + \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }( \omega _{2n + 1}, \omega _{2n + 2})\bigr) \leq \mathcal{\mathbf{F}} \bigl(\textsl{d}_{\sigma }( \omega _{2n}, \omega _{2n + 1}) \bigr),$$
(8)

and

$$\Omega + \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }( \omega _{2n + 2}, \omega _{2n + 3})\bigr) \leq \mathcal{\mathbf{F}} \bigl(\textsl{d}_{\sigma }( \omega _{2n + 1}, \omega _{2n + 2}) \bigr)$$
(9)

for all $$n \in \mathbb{N}$$. From equations (8) and (9) we have

$$\Omega + \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{n}, \omega _{n + 1})\bigr) \leq \mathcal{\mathbf{F}}\bigl( \textsl{d}_{\sigma }( \omega _{n - 1}, \omega _{n})\bigr).$$

Therefore

\begin{aligned}[b] \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{n}, \omega _{n + 1})\bigr)& \leq \mathcal{\mathbf{F}}\bigl( \textsl{d}_{\sigma }(\omega _{n - 1}, \omega _{n})\bigr) - \Omega \leq \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }( \omega _{n - 2}, \omega _{n - 1})\bigr) - 2 \Omega \leq \cdots\\ & \leq \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{0}, \omega _{1})\bigr) - n \Omega .\end{aligned}
(10)

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

$$\lim_{n \rightarrow \infty } \textsl{d}_{\sigma }(\omega _{n}, \omega _{n + 1}) = 0.$$

Also, by condition $$(C3)$$ of Definition 1.3 there exists $$l \in (0, 1)$$ such that

$$\lim_{n \rightarrow \infty } \bigl(\textsl{d}_{\sigma }(\omega _{n}, \omega _{n + 1})\bigr)^{l} \mathcal{\mathbf{F}} \bigl(\textsl{d}_{\sigma }( \omega _{n}, \omega _{n + 1}) \bigr) = 0.$$

From equation (10) we have that for all $$n \in \mathbb{N}$$,

\begin{aligned}[b] & \bigl(\textsl{d}_{\sigma }(\omega _{n}, \omega _{n + 1})\bigr)^{l} \mathcal{\mathbf{F}}\bigl( \textsl{d}_{\sigma }(\omega _{n}, \omega _{n + 1})\bigr) - \bigl(\textsl{d}_{\sigma }(\omega _{n}, \omega _{n + 1}) \bigr)^{l} \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{0}, \omega _{1})\bigr)\\ &\quad \leq -\bigl(\textsl{d}_{\sigma }( \omega _{n}, \omega _{n + 1})\bigr)^{l} n \Omega \leq 0.\end{aligned}
(11)

By letting $$n \rightarrow \infty$$ in (11) we obtain

$$\lim_{n \rightarrow \infty } n\bigl(\textsl{d}_{\sigma }( \omega _{n}, \omega _{n + 1})\bigr)^{l} = 0.$$
(12)

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

$$\textsl{d}_{\sigma }(\omega _{n}, \omega _{n + 1}) \leq \frac{1}{n^{\frac{1}{l}}}.$$
(13)

From the triangle inequality and equation (13) for $$m > n \geq n_{1}$$, we have

\begin{aligned} \textsl{d}_{\sigma }(\omega _{n}, \omega _{m}) &\leq \sigma (\omega _{n}, \omega _{n + 1})\textsl{d}_{\sigma }( \omega _{n}, \omega _{n + 1}) + \sigma (\omega _{n + 1}, \omega _{m})\textsl{d}_{\sigma }(\omega _{n + 1}, \omega _{m}) \\ & \leq \sigma (\omega _{n}, \omega _{n + 1}) \textsl{d}_{\sigma }( \omega _{n}, \omega _{n + 1}) + \sigma (\omega _{n}, \omega _{m}) \sigma (\omega _{n + 1}, \omega _{n + 2})\textsl{d}_{\sigma }(\omega _{n + 1}, \omega _{n + 2}) \\ & \quad{} + \sigma (\omega _{n}, \omega _{m})\sigma (\omega _{n + 2}, \omega _{m})\textsl{d}_{\sigma }(\omega _{n + 2}, \omega _{m}) \\ & \quad \vdots \\ &\leq \sigma (\omega _{n}, \omega _{n + 1}) \textsl{d}_{\sigma }( \omega _{n}, \omega _{n + 1}) + \sum _{i = 1}^{m - 2} \Biggl(\prod _{j = 1}^{i}\sigma (\omega _{j}, \omega _{m}) \Biggr) \sigma (\omega _{i}, \omega _{i + 1}) \textsl{d}_{\sigma }(\omega _{i}, \omega _{i + 1}) \\ & \quad{} + \prod_{j = 1}^{m - 1} \sigma (\omega _{j}, \omega _{m}) \sigma (\omega _{m - 1}, \omega _{m}) \textsl{d}_{\sigma }(\omega _{m - 1}, \omega _{m}) \\ & \leq \sigma (\omega _{n}, \omega _{n + 1}) \textsl{d}_{\sigma }( \omega _{n}, \omega _{n + 1}) + \sum _{i = 1}^{m - 1} \Biggl(\prod _{j = 1}^{i}\sigma (\omega _{j}, \omega _{m}) \Biggr) \sigma (\omega _{i}, \omega _{i + 1}) \textsl{d}_{\sigma }(\omega _{i}, \omega _{i + 1}) \\ & \leq \sigma (\omega _{n}, \omega _{n + 1})\frac{1}{n^{\frac{1}{l}}} + \sum_{i = 1}^{\infty } \Biggl(\prod _{j = 1}^{i}\sigma (\omega _{j}, \omega _{m}) \Biggr) \sigma (\omega _{i}, \omega _{i + 1}) \frac{1}{i^{\frac{1}{l}}}. \end{aligned}

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

\begin{aligned} &\Omega + \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }\bigl(\omega _{2n_{r}}, [ \mathcal{\mathbf{T}}\rho ]_{\alpha _{\mathcal{\mathbf{T}}}(\rho )}\bigr)\bigr) \\ &\quad \leq \Omega + \mathcal{\mathbf{F}}\bigl(\textsl{H}_{\sigma }\bigl([ \mathcal{ \mathbf{S}}\omega _{2n_{r} - 1}]_{\alpha _{ \mathcal{\mathbf{S}}}(\omega _{2n_{r} - 1})}, [\mathcal{\mathbf{T}} \rho ]_{\alpha _{\mathcal{\mathbf{T}}}(\rho )}\bigr)\bigr) \\ &\quad \leq \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{2n_{r} - 1}, \rho )\bigr) + {\L }\min \bigl\{ \textsl{d}_{\sigma }\bigl( \omega _{2n_{r} - 1}, [ \mathcal{\mathbf{S}}\omega _{2n_{r} - 1}]_{\alpha _{ \mathcal{\mathbf{S}}}(\omega _{2n_{r} - 1})} \bigr), \\ & \qquad \textsl{d}_{\sigma }\bigl(\rho , [\mathcal{\mathbf{T}}\rho ]_{ \alpha _{\mathcal{\mathbf{T}}}(\rho )}\bigr), \textsl{d}_{\sigma }\bigl(\omega _{2n_{r} - 1}, [ \mathcal{\mathbf{T}}\rho ]_{\alpha _{\mathcal{\mathbf{T}}}( \rho )}\bigr), \textsl{d}_{\sigma }\bigl( \rho , [\mathcal{\mathbf{S}}\omega _{2n_{r} - 1}]_{\alpha _{\mathcal{\mathbf{S}}}(\omega _{2n_{r} - 1})}\bigr)\bigr\} \\ &\quad \leq \mathcal{\mathbf{F}}(\textsl{d}_{\sigma }(\omega _{2n_{r} - 1}, \rho ) + {\L }\min \bigl\{ \textsl{d}_{\sigma }(\omega _{2n_{r} - 1}, \omega _{2n_{r}}), \textsl{d}_{\sigma }\bigl(\rho , [\mathcal{\mathbf{T}}\rho ]_{\alpha _{ \mathcal{\mathbf{T}}}(\rho )}\bigr), \\ & \qquad \textsl{d}_{\sigma }\bigl(\omega _{2n_{r} - 1}, [ \mathcal{ \mathbf{T}}\rho ]_{\alpha _{\mathcal{\mathbf{T}}}(\rho )}\bigr), \textsl{d}_{\sigma }(\rho , \omega _{2n_{r}})\bigr\} . \end{aligned}

This implies that

\begin{aligned} \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }\bigl(\omega _{2n_{r}}, [ \mathcal{\mathbf{T}}\rho ]_{\alpha _{\mathcal{\mathbf{T}}}(\rho )}\bigr)\bigr) & \leq \mathcal{ \mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{2n_{r} - 1}, \rho )\bigr) + {\L }\min \bigl\{ \textsl{d}_{\sigma }(\omega _{2n_{r} - 1}, \omega _{2n_{r}}), \textsl{d}_{\sigma }\bigl(\rho , [\mathcal{\mathbf{T}} \rho ]_{\alpha _{\mathcal{\mathbf{T}}}(\rho )}\bigr), \\ & \quad \textsl{d}_{\sigma }\bigl(\omega _{2n_{r} - 1}, [ \mathcal{ \mathbf{T}}\rho ]_{\alpha _{\mathcal{\mathbf{T}}}(\rho )}\bigr), \textsl{d}_{\sigma }(\rho , \omega _{2n_{r}})\bigr\} - \Omega \\ & < \mathcal{\mathbf{F}}(\textsl{d}_{\sigma }(\omega _{2n_{r} - 1}, \rho ) + {\L }\min \bigl\{ \textsl{d}_{\sigma }(\omega _{2n_{r} - 1}, \omega _{2n_{r}}), \textsl{d}_{\sigma }\bigl(\rho , [\mathcal{\mathbf{T}}\rho ]_{\alpha _{ \mathcal{\mathbf{T}}}(\rho )}\bigr), \\ & \quad \textsl{d}_{\sigma }\bigl(\omega _{2n_{r} - 1}, [ \mathcal{ \mathbf{T}}\rho ]_{\alpha _{\mathcal{\mathbf{T}}}(\rho )}\bigr), \textsl{d}_{\sigma }(\rho , \omega _{2n_{r}})\bigr\} . \end{aligned}

As $$\mathcal{\mathbf{F}}$$ is strictly increasing, we have

\begin{aligned} \textsl{d}_{\sigma }\bigl(\omega _{2n_{r}}, [\mathcal{\mathbf{T}}\rho ]_{ \alpha _{\mathcal{\mathbf{T}}}(\rho )}\bigr)& < \textsl{d}_{\sigma }(\omega _{2n_{r} - 1}, \rho ) + {\L }\min \bigl\{ \textsl{d}_{\sigma }(\omega _{2n_{r} - 1}, \omega _{2n_{r}}), \textsl{d}_{\sigma }\bigl(\rho , [\mathcal{\mathbf{T}} \rho ]_{\alpha _{\mathcal{\mathbf{T}}}(\rho )}\bigr), \\ &\quad \textsl{d}_{\sigma }\bigl( \omega _{2n_{r} - 1}, [\mathcal{\mathbf{T}}\rho ]_{\alpha _{\mathcal{\mathbf{T}}}(\rho )}\bigr), \textsl{d}_{\sigma }( \rho , \omega _{2n_{r}})\bigr\} . \end{aligned}

By taking the limit as $$n \rightarrow \infty$$ we get

$$\textsl{d}_{\sigma }\bigl(\rho , [\mathcal{\mathbf{T}}\rho ]_{\alpha _{ \mathcal{\mathbf{T}}}(\rho )} \bigr) \leq 0.$$

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

$$\Omega + \mathcal{\mathbf{F}}(\textsl{H}_{\sigma }\bigl([ \mathcal{\mathbf{S}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{S}}}( \omega _{1})}, [\mathcal{\mathbf{T}} \omega _{2}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{2})}\bigr) \leq \mathcal{\mathbf{F}}\bigl( \textsl{d}_{\sigma }(\omega _{1}, \omega _{2})\bigr)$$
(14)

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

$$\Omega + \mathcal{\mathbf{F}}(\textsl{H}_{\sigma }\bigl([ \mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{1})}, [\mathcal{\mathbf{T}} \omega _{2}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{2})}\bigr) \leq \mathcal{\mathbf{F}}\bigl( \textsl{d}_{\sigma }(\omega _{1}, \omega _{2})\bigr) + {\L }\bigl(\textsl{M}( \omega _{1}, \omega _{2})\bigr)$$
(15)

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

\begin{aligned} \textsl{M}(\omega _{1}, \omega _{2}) &= \min \bigl\{ \textsl{d}_{\sigma }\bigl( \omega _{1}, [\mathcal{\mathbf{T}} \omega _{1}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{1})}\bigr), \textsl{d}_{\sigma }\bigl( \omega _{2}, [\mathcal{\mathbf{T}}\omega _{2}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{2})} \bigr),\\ &\quad \textsl{d}_{\sigma }\bigl(\omega _{1}, [ \mathcal{ \mathbf{T}}\omega _{2}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{2})}\bigr), \textsl{d}_{\sigma } \bigl(\omega _{2}, [ \mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{1})} \bigr)\bigr\} , \end{aligned}

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

$$\Omega + \mathcal{\mathbf{F}}(\textsl{H}_{\sigma }\bigl([ \mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{1})}, [\mathcal{\mathbf{T}} \omega _{2}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{2})}\bigr) \leq \mathcal{\mathbf{F}}\bigl( \textsl{d}_{\sigma }(\omega _{1}, \omega _{2})\bigr)$$
(16)

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):

Theorem 2.1 generalizes Theorem 2.1 of [39].

(ii):

Theorem 2.2 generalizes Theorem 6 of [40].

(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

$$\Omega + \mathcal{\mathbf{F}}(\textsl{H}_{\sigma }( \mathcal{\mathbf{A}}\omega _{1}, \mathcal{\mathbf{B}}\omega _{2}) \leq \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{1}, \omega _{2})\bigr) + {\L }\bigl(\textsl{M}(\omega _{1}, \omega _{2})\bigr)$$
(17)

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

$$\textsl{M}(\omega _{1}, \omega _{2}) = \min \bigl\{ \textsl{d}_{\sigma }\bigl( \omega _{1}, \mathcal{\mathbf{A}}( \omega _{1})\bigr), \textsl{d}_{\sigma }\bigl( \omega _{2}, \mathcal{\mathbf{B}}(\omega _{2})\bigr), \textsl{d}_{\sigma }\bigl( \omega _{1}, \mathcal{\mathbf{B}}( \omega _{2})\bigr), \textsl{d}_{\sigma }\bigl( \omega _{2}, \mathcal{\mathbf{A}}(\omega _{1})\bigr)\bigr\} .$$

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

$$\mathcal{\mathbf{S}}(\omega _{1}) (\mathcal{\mathbf{T}}) = \textstyle\begin{cases} \alpha & \text{if }\mathcal{\mathbf{T}} \in \mathcal{\mathbf{A}}\omega _{1}, \\ 0 & \text{if }\mathcal{\mathbf{T}} \notin \mathcal{\mathbf{A}}\omega _{1}, \end{cases}$$

and

$$\mathcal{\mathbf{T}}(\omega _{1}) (\mathcal{\mathbf{T}}) = \textstyle\begin{cases} \alpha & \text{if }\mathcal{\mathbf{T}} \in \mathcal{\mathbf{B}}\omega _{1}, \\ 0 & \text{if }\mathcal{\mathbf{T}} \notin \mathcal{\mathbf{B}}\omega _{1}. \end{cases}$$

Then we obtain

$$\begin{gathered}{} [\mathcal{\mathbf{S}}\omega _{1}]_{\alpha (\omega _{1})} = \bigl\{ \mathcal{ \mathbf{T}} : \mathcal{\mathbf{S}}(\omega _{1}) ( \mathcal{\mathbf{T}}) \geq \alpha \bigr\} = \mathcal{\mathbf{A}}\omega _{1} \quad \text{and} \\ [\mathcal{\mathbf{T}}\omega _{1}]_{\alpha ( \omega _{1})} = \bigl\{ \mathcal{\mathbf{T}} : \mathcal{\mathbf{T}}(\omega _{1}) ( \mathcal{ \mathbf{T}}) \geq \alpha \bigr\} = \mathcal{\mathbf{B}}\omega _{1}. \end{gathered}$$

Therefore we can apply Theorem 2.1 to get a fixed point $$\rho \in \mathcal{W}$$ such that

$$\rho \in [\mathcal{\mathbf{S}}\rho ]_{\alpha _{\mathcal{\mathbf{S}}}( \rho )} \cap [\mathcal{\mathbf{T}} \rho ]_{\alpha _{ \mathcal{\mathbf{T}}}(\rho )} = \mathcal{\mathbf{A}}\rho \cap \mathcal{\mathbf{B}}\rho .$$

□

### 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

$$\Omega + \mathcal{\mathbf{F}}(\textsl{H}_{\sigma }( \mathcal{\mathbf{A}}\omega _{1}, \mathcal{\mathbf{B}}\omega _{2}) \leq \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{1}, \omega _{2})\bigr)$$
(18)

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

$$\Omega + \mathcal{\mathbf{F}}(\textsl{H}_{\sigma }( \mathcal{\mathbf{A}}\omega _{1}, \mathcal{\mathbf{A}}\omega _{2}) \leq \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{1}, \omega _{2})\bigr) + {\L }\bigl(\textsl{M}(\omega _{1}, \omega _{2})\bigr)$$
(19)

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

$$\textsl{M}(\omega _{1}, \omega _{2}) = \min \bigl\{ \textsl{d}_{\sigma }\bigl( \omega _{1}, \mathcal{\mathbf{A}}( \omega _{1})\bigr), \textsl{d}_{\sigma }\bigl( \omega _{2}, \mathcal{\mathbf{A}}(\omega _{2})\bigr), \textsl{d}_{\sigma }\bigl( \omega _{1}, \mathcal{\mathbf{A}}( \omega _{2})\bigr), \textsl{d}_{\sigma }\bigl( \omega _{2}, \mathcal{\mathbf{A}}(\omega _{1})\bigr)\bigr\} .$$

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

$$\Omega + \mathcal{\mathbf{F}}(\textsl{H}_{\sigma }( \mathcal{\mathbf{A}}\omega _{1}, \mathcal{\mathbf{A}}\omega _{2}) \leq \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }(\omega _{1}, \omega _{2})\bigr)$$
(20)

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):

Corollary 2.3 generalizes Corollary 2.5 of [39].

(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,

$$\hat{\mathcal{\mathbf{T}}}(\omega _{1}) (\mathcal{\mathbf{T}}) = \Bigl\{ \mu \in \mathcal{W} : \mathcal{\mathbf{T}}(\omega _{1}) (\mu ) = \max_{ \textsl{t} \in \mathcal{W}} \mathcal{\mathbf{T}}(\omega _{1}) ( \textsl{t})\Bigr\} .$$

### 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

$$\mathcal{\mathbf{T}}(\mu ) (\mu ) \geq \mathcal{\mathbf{T}}(\mu ) ( \omega _{1})$$

for all $$\omega _{1} \in \mathcal{W}$$.

### Proof

Suppose that $$\mu \in \hat{\mathcal{\mathbf{T}}}(\mu )$$. Then

$$\hat{\mathcal{\mathbf{T}}}(\mu ) (\mu ) = \max_{\omega _{1} \in \mathcal{W}} \mathcal{ \mathbf{T}}(\mu ) (\omega _{1}).$$

This implies that

$$\hat{\mathcal{\mathbf{T}}}(\mu ) (\mu ) \geq \mathcal{\mathbf{T}}(\mu ) ( \omega _{1})\quad \text{for all } \omega _{1} \in \mathcal{W}.$$

Conversely, suppose that

$$\hat{\mathcal{\mathbf{T}}}(\mu ) (\mu ) \geq \mathcal{\mathbf{T}}(\mu ) ( \omega _{1}) \quad \text{for all } \omega _{1} \in \mathcal{W}.$$

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

$$\Omega + \mathcal{\mathbf{F}}(\textsl{H}_{\sigma }\bigl( \hat{\mathcal{\mathbf{S}}}(\omega _{1}), \hat{\mathcal{\mathbf{T}}}( \omega _{2})\bigr) \leq \mathcal{\mathbf{F}}\bigl(\textsl{d}_{\sigma }( \omega _{1}, \omega _{2})\bigr) + {\L }\bigl(\textsl{M}( \omega _{1}, \omega _{2})\bigr)$$
(21)

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

$$\textsl{M}(\omega _{1}, \omega _{2}) = \min \bigl\{ \textsl{d}_{\sigma }\bigl( \omega _{1}, \hat{\mathcal{ \mathbf{S}}}(\omega _{1})\bigr), \textsl{d}_{ \sigma }\bigl(\omega _{2}, \hat{\mathcal{\mathbf{T}}}(\omega _{2})\bigr), \textsl{d}_{\sigma }\bigl(\omega _{1}, \hat{\mathcal{\mathbf{T}}}( \omega _{2})\bigr), \textsl{d}_{\sigma }\bigl(\omega _{2}, \hat{\mathcal{\mathbf{S}}}(\omega _{1})\bigr) \bigr\} .$$

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

$$\mathcal{\mathbf{S}}(\mu ) (\mu ) \geq \mathcal{\mathbf{S}}(\mu ) ( \omega _{1})\quad \text{and}\quad \mathcal{\mathbf{T}}(\mu ) (\mu ) \geq \mathcal{\mathbf{T}}(\mu ) (\omega _{1})$$

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

$$\textsl{d}_{\sigma }(\omega _{1}, \omega _{2}) = \vert \omega _{1} - \omega _{2} \vert .$$

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

$$\sigma (\omega _{1}, \omega _{2}) = \textstyle\begin{cases} 1 & \text{if }\omega _{1}, \omega _{2} \in [0, 0.5), \\ \omega _{1} + \omega _{2} + 2 &\text{otherwise.} \end{cases}$$

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

$$\mathcal{\mathbf{S}}(\omega _{1}) (\mathcal{\mathbf{T}}) = \textstyle\begin{cases} \alpha & \text{if }0 \leq \mathcal{\mathbf{T}} \leq \frac{\omega _{1}}{50}, \\ \frac{\alpha }{2} & \text{if }\frac{\omega _{1}}{50} < \mathcal{\mathbf{T}} \leq \frac{\omega _{1}}{40}, \\ \frac{\alpha }{3} & \text{if }\frac{\omega _{1}}{40} < \mathcal{\mathbf{T}} \leq \frac{\omega _{1}}{30}, \\ \frac{\alpha }{4} & \text{if }\frac{\omega _{1}}{30} < \mathcal{\mathbf{T}} \leq 1, \end{cases}$$

and

$$\mathcal{\mathbf{T}}(\omega _{1}) (\mathcal{\mathbf{T}}) = \textstyle\begin{cases} \alpha & \text{if }0 \leq \mathcal{\mathbf{T}} \leq \frac{\omega _{1}}{20}, \\ \frac{\alpha }{4} & \text{if }\frac{\omega _{1}}{20} < \mathcal{\mathbf{T}} \leq \frac{\omega _{1}}{10}, \\ \frac{\alpha }{5} & \text{if }\frac{\omega _{1}}{10} < \mathcal{\mathbf{T}} \leq \frac{\omega _{1}}{5}, \\ \frac{\alpha }{7} & \text{if }\frac{\omega _{1}}{5} < \mathcal{\mathbf{T}} \leq 1, \end{cases}$$

so that

$$[\mathcal{\mathbf{S}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{S}}}( \omega _{1})} = \biggl[0, \frac{\omega _{1}}{50}\biggr] \quad \text{and}\quad [ \mathcal{\mathbf{T}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{T}}}( \omega _{1})} = \biggl[0, \frac{\omega _{1}}{20}\biggr].$$

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

$$\Omega + \mathcal{\mathbf{F}}(\textsl{H}_{\sigma }\bigl([ \mathcal{ \mathbf{S}}\omega _{1}]_{\alpha _{\mathcal{\mathbf{S}}}( \omega _{1})}, [\mathcal{\mathbf{T}}\omega _{2}]_{\alpha _{ \mathcal{\mathbf{T}}}(\omega _{2})}\bigr) \leq \mathcal{\mathbf{F}}\bigl( \textsl{d}_{\sigma }(\omega _{1}, \omega _{2})\bigr)$$

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.

Not applicable.

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## Acknowledgements

The authors are grateful to their Universities for their support.

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All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

### Corresponding authors

Correspondence to Hassen Aydi or Yaé Ulrich Gaba.

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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