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 [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

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

(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: [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

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

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 [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

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

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

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

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

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

(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.