Existence of fixed points of generalized set-valued $F$-contractions of $b$-metric spaces

1.   Introduction and preliminaries

Axioms of metric have been modified to get more general distance functions (compare ^[8]). Among the generalizations of metric, $b$-metric was initially considered by Bakhtin ^[11], Czerwik ^[14,15,16] and Berinde ^[13] to generalize the well known Banach contraction principle (shortly as BCP) ^[12]. Due to the useful applications of BCP, it has been attempted successfully by a long list of researchers to generalize in various directions. Wardowski ^[26] set up a contraction termed as $F$-contraction and obtained a generalization of BCP. After that, several authors have established different versions of $F$-contractions to generalize the results of Wardowski, for instance, see ^{[1,2,4,5,6,7,10,19,20]} and references therein. For self mappings of metric spaces, Proinov ^[21] proved that some results including Wardowski's result are equivalent to a special case of a well-known fixed point theorem of Skof ^[23]. Abbas et al. ^[3] obtained some coincidence point results for generalized set-valued $(f, L)$ -almost $F$-contractions of metric spaces along with some applications. Recently, Karapınar et al. ^[17] provided a survey on $F$- contractions in which a collection of various results of $F$-contractions are given.

Miculescu ^[18] introduced a sufficient condition for a sequence in a $b$-metric space (shortly as $b$-MS) to be Cauchy and proved some results involving set-valued contractions of a $b$-MS. After this, Suzuki ^[25] provided a sufficient condition (weaker than the one given by Miculescu) for a sequence to be Cauchy in a $b$-MS and proved some fixed point theorems for set-valued $F$-contractions.

We present new generalized set-valued $F$-contractions of a $b$-MS and extend results given in ^[3,25] and in some references therein. We provide with some examples to substantiate the main results and to proclaim that the results in this paper are proper generalizations of some existing results in the literature. We start by fixing some notations to be used in the sequel. The letters $\mathbb{N}, \; \mathbb{R}, \; \mathbb{R}^{+}$ and $\mathbb{R}_{+}$ represent the set of positive integers, real numbers, non-negative and positive real numbers, respectively and $X$ a non-empty set. Now we give some preliminary notions.

Definition 1.1. ^[14] Let $d:X\times X$ $\rightarrow \mathbb{R} ^{+}$ be a function and $s\geq 1$ a real number. Then $(X, d)$ is termed as $b$-MS if $d$ satisfies

(1) $d(r, w) = 0\Leftrightarrow r = w,$

(2) $d(r, w) = d(w, r),$

(3) $d(r, z)\leq sd(r, w)+sd(w, z),$

for all $r, w, z\in X$, where $s$ is a $b$-metric constant. For $s = 1$, $d$ is a metric.

Throughout this article, $s$ represent $b$-metric constant unless otherwise stated. Now consider the following example.

Example 1.1. ^[11,13] Consider

For all $r = \left\{ r_{j}\right\}$ and $w = \left\{ w_{j}\right\}$ in $l_{p}$, the mapping $d:l_{p}\times l_{p}\longrightarrow \mathbb{R}$ defined as

is a $b$-metric on $l_{p}$ for $s = 2^{\dfrac{1}{p}}$ as

for all $r$, $z$, $w$ in $l_{p}$. $

Let $(X, d)$ be a $b$-MS. A sequence $\{r_{j}\}$ in $(X, d)$ is Cauchy if for any given $\epsilon > 0$, there is a $J_{\epsilon }\in \mathbb{N}$ so that $d(r_{j}, r_{m}) < \epsilon$ for all $m, j\geq J_{\epsilon }$, or equivalently

for all $p\in \mathbb{N}$. A sequence $\{r_{j}\}$ in $(X, d)$ is convergent if for any given $\epsilon > 0$, there is $J_{\epsilon }\in \mathbb{N}$ and an $r$ in $X$ so that $d(r_{j}, r) < \epsilon$ for all $j\geq J_{\epsilon }$, or equivalently

and we write $r_{j}\rightarrow r$ as $j\rightarrow \infty$.

Further, a subset $E\subseteq X$ is closed if for every sequence $\{r_{j}\}$ in $E$ and $r_{j}\rightarrow r$ as $j\rightarrow \infty$, implies $r\in E$ and $E\subseteq X$ is bounded if

is finite. A $b$-MS $(X, d)$ is complete if every Cauchy sequence in $X$ converges. An et al. ^[9] explored some topological aspects of $b$-MS $(X, d)$ and asserted that $d$ is not necessarily continuous in both arguments. However, if $d$ is continuous in one variable then it is continuous in the other variable as well. Moreover, the subset

in $(X, d)$ is not an open set (in general) but if $d$ is continuous in one variable then $B_{\epsilon }(r_{0})$ is open in $X$. Throughout in this paper $b$-metric $d$ is continuous.

Let $(X, d)$ be a $b$-MS and $C_{B}(X)$ and $P(X)$ the set of non-empty, closed, bounded, and the set of non-empty subsets of $X$, respectively. For $E, G\in C_{B}(X)$, the mapping $H:C_{B}(X)\times C_{B}(X)\rightarrow \mathbb{R} ^{+}$ defined as

is Hausdorff metric on $C_{B}(X)$ generated by $d$, where

The following lemma provides important tools in connection with a $b$-MS.

Lemma 1.1. ^{[14,15,16,22]} For a $b$-MS $(X, d)$, $r, w\in X$ and $E, G\in C_{B}(X)$, the following statements hold:

(1) $(C_{B}(X), H)$ is a $b$-MS.

(2) For all $r\in E$, $d(r, G)\leq H(E, G).$

(3) For all $r, w$ in $X$, $d(r, E)\leq sd(r, w)+sd(w, E).$

(4) For $k > 1$ and $c\in E$, there is a $w\in G$ so that $d(c, w)\leq kH(E, G).$

(5) For every $k > 0$ and $c\in E$, there is a $w\in G$ so that $d(c, w)\leq H(E, G)+k.$

(6) $c\in \overline{E} = E$ if and only if $d(c, E) = 0$, where $\overline{E}$ is the closure of $E$ in $(X, d)$.

(7) For any sequence $\{r_{j}\}$ in $X$,

Now consider a mapping $F: \mathbb{R} _{+}\longrightarrow \mathbb{R}$ that satisfies:

(${\bf{A}}_{1}$) If $\lambda _{1} < \lambda _{2}$, then $F(\lambda _{1}) < F\left(\lambda _{2}\right)$, for all $\lambda _{1}, \lambda _{2}\in \mathbb{R} _{+};$

$({\bf{A}} _{2})$ For each sequence $\{\lambda _{j}\}$, $\lim\limits_{j \rightarrow \infty }\lambda _{j} = 0$ if and only if $\lim\limits_{j \rightarrow \infty }F(\lambda _{j}) = -\infty;$

$({\bf{A}} _{3})$ There is $k\in \left(0, 1\right)$ such that $\lim\limits_{\lambda \rightarrow 0^{+}}$ $\lambda ^{k}F\left(\lambda \right) = 0;$

$({\bf{A}} _{4})$ $F(\inf E) = \inf F(E)$ for all $E\subset \left(0, \infty \right)$ and $\inf E\in \left(0, \infty \right);$

$({\bf{A}} _{5})$ $F$ is upper semicontinuous;

$({\bf{A}} _{6})$ $F$ is continuous.

Note that under (A$_{1}$), (A$_{4}$) and (A$_{5}$) are equivalent (compare ^[25]). Wardowski ^[26] initiated the idea of $F$-contraction.

Definition 1.2. ^[26] Let $(X, d)$ be a metric space. A mapping $f:X\rightarrow X$ is $F$-contraction if

for all $r, w\in X$ and for some $\tau > 0$, where $F: \mathbb{R} _{+}\longrightarrow \mathbb{R}$ satisfies A$_{1}$–A$_{3}$.

Further, they obtained the existence and uniqueness of fixed point of the $F-$contraction $f$ of a complete metric space $(X, d)$. Throughout this paper, for mappings $f:X\rightarrow X$ and $S:X\rightarrow C_{B}(X)$, we use the notations $F^{ix}(f)$ and $F^{ix}(S)$ for the set of fixed points of $f$ and $S$, respectively.

Remark 1.1. For different choices of the function $F$ in Definition 1.2, one can get different contractions, for instance if $F(\kappa) = \ln (\kappa)$ for $\kappa > 0$, then the mapping $f$ becomes Banach contraction (compare ^[26] for details).

Definition 1.3. ^[25] Let a sequence $\{a_{j}\}$ be in $\mathbb{R} ^{+}$ and $\{b_{j}\}$ a sequence in $\mathbb{R} _{+}$. If there is a real number $C > 0$ such that

for all $j\in \mathbb{N}$, then we say

Lemma 1.2. ^[24,25] Let $\{r_{j}\}$ be a sequence in a $b$-MS $\left(X, d\right)$. If for $\beta > 1+\log _{2}s$,

holds. Then $\{r_{j}\}$ is a Cauchy sequence.

Lemma 1.3. ^[25] Let $\left\{ t_{j}\right\}$ be a sequence in $\mathbb{R} _{+}$. If there is a function $F: \mathbb{R} _{+}\longrightarrow \mathbb{R}$, a real number $c\in \left(0, 1\right)$ and $\tau \in \mathbb{R} _{+}$ satisfying (A$_{2}$), (A$_{3}$) and

then

holds.

Theorem 1.1. ^[25] Let $\left(X, d\right)$ be a complete $b$-MS and $S:X\longrightarrow C_{B}(X)$ a mapping. Suppose there is a function $F: \mathbb{R}_{+}\longrightarrow\mathbb{R}$,

and $\tau \in \mathbb{R} _{+}$ satisfying (A$_{2}$), (A$_{3}$), and for any $r, w\in X$ and $\eta \in Sr$, there is $\mu \in Sw$ such that either $\eta = \mu$ or

holds. Then $F^{ix}(S)$ is non-empty.

Theorem 1.2. ^[25] Let $\left(X, d\right)$ be a complete $b$-MS and amapping $S:X\longrightarrow C_{B}(X)$. Suppose there is a function $F: \mathbb{R} _{+}\longrightarrow \mathbb{R}$,

and $\tau \in \mathbb{R} _{+}$ satisfying (A$_{1}$)–(A$_{3}$), (A$_{5}$), and for any $r, w\in X$ with $Sr\neq Sw$,

holds. Then $F^{ix}(S)$ is non-empty.

Let $f:X\rightarrow X$ be single valued mapping and $S:X\rightarrow C_{B}(X)$ a set-valued mapping. For $r, w\in X$, $\eta \in Sr$ and $\mu \in Sw$, we use the following notations in the sequel.

and

for $T\in \{f, S\}$.

2.   Fixed points of generalized set-valued $F$-contractions of a $b$- metric space

We start with the following theorem.

Theorem 2.1. Let $\left(X, d\right)$ be a complete $b$-MS and $S:X\longrightarrow C_{B}(X)$. Suppose there is a function $F: \mathbb{R} _{+}\longrightarrow \mathbb{R}$,

and $\tau \in \mathbb{R} _{+}$ satisfying (A$_{2}$) and (A$_{3}$), and for any $r, w\in X$ and $\eta \in Sr$, there is $\mu \in Sw$ such that $\eta = \mu$ or

holds for some $L\geq 0$. Then $F^{ix}(S)$ is non-empty.

Proof. On contrary, consider that $F^{ix}(S)$ is empty. Fix $r_{1}\in X$ and $r_{2}\in Sr_{1}$. From our assumption $r_{1}\neq r_{2}$ and $r_{2}\notin Sr_{2}$. We can pick $r_{3}\in Sr_{2}$. As $r_{2}\neq r_{3}$, so

Similarly, we can choose a sequence $\{r_{j}\}$ in $X$ for any $j\in \mathbb{N}$ satisfying $r_{j+1}\in Sr_{j}$ and

That is

implies

Now, using Lemma 1.3,

holds. As $c\in \left(0, \frac{1}{1+\log _{2}s}\right)$, so

Hence, by Lemma 1.2, $\{r_{j}\}$ is a Cauchy sequence. Because $X$ is complete, $\{r_{j}\}$ converges to a $z\in X$, there is $z_{j}\in Sz$ such that either $z_{j} = r_{j+1}$ or

holds. Let $\{j\}$ be a sequence in $\mathbb{N}$ and $\left\{ f\left(j\right) \right\}$ be an arbitrary subsequence of $\{j\}$. Here, two cases arise:

$(1)$ $\#\left\{ j\in \mathbb{N} :z_{f\left(j\right) } = r_{f\left(j\right) +1}\right\} = \infty,$

$(2)$ $\#\left\{ j\in \mathbb{N} :z_{f\left(j\right) } = r_{f\left(j\right) +1}\right\} < \infty,$

where $\#A$ denotes the cardinality of the set $A$.

Case 1: Let $\left\{ g\left(j\right) \right\}$ be a subsequence of $\{j\}$ in $\mathbb{N}$ which satisfy $z_{f\circ g\left(j\right) } = r_{f\circ g\left(j\right) +1}.$ Since $r_{j}\rightarrow z$ as $j\rightarrow \infty$,

as $j\rightarrow \infty$. Hence,

Case 2: Let $\left\{ g\left(j\right) \right\}$ be a subsequence of $\{j\}$ in $\mathbb{N}$ such that $g\left(j\right) \notin \left\{ j\in \mathbb{N} :z_{f\left(j\right) } = r_{f\left(j\right) +1}\right\}$. This implies

That is

where

Since $r_{j}\rightarrow z$ as $j\rightarrow \infty$, therefore, $u_{j}\rightarrow 0$ as $j\rightarrow \infty$. Hence, by (A$_{2}$), we get

From (2.1) and (2.2), we obtain

Again by (A$_{2}$), we get

Hence,

Consequently, both cases imply

As $f$ was taken to be arbitrary, so

As $Sz$ is closed, we get $z\in Sz$, a contradiction. Thus $F^{ix}(S)$ is non-empty.

Consider an example to illustrate the Theorem 2.1 and to show that it is a proper generalization of some results in the literature.

Example 2.1. Let $X = \left\{ 1, 2, 3\right\}$ be a set and $d:X\times X\rightarrow \mathbb{R} ^{+}$ a mapping defined as

As

so $d$ is not a metric on $X$ but for $s = 1.12$, $d$ is a complete $b$- metric. Define $S:X\rightarrow C_{B}(X)$ and $F: \mathbb{R} _{+}\longrightarrow \mathbb{R}$ as

Note that

If $r = 3$ and $w = 1$, then $Sr = \{1, 2\}$, for $\eta = 1\in S3$, there is $\mu = 1\in S1$ such that $\eta = \mu$. For $\eta = 2$, there is $\mu = 1$ such that

If $r = 3$ and $w = 2$, then $Sr = \{1, 2\}$, for $\eta = 1$, there exists $\mu = 1\in S2$ such that $\eta = \mu$. For $\eta = 2$, there is $\mu = 1\in S2$ such that

Hence, for any $\tau \in (0, 0.51)$, $r, w\in X$ and $\eta \in Sr$, there exists $\mu \in Sw$ such that either $\eta = \mu$ or

holds for all $r, w\in X$ and for any $L\geq 0$. Hence, all the assumptions of Theorem 2.1 are met and $1\in S\left(1\right)$. $

Remark 2.1. In the above example, if $r = 3$ and $w = 1$, then $Sr = \{1, 2\}$, for $\eta = 2$, there does not exist $\mu \in S1$ such that either $\eta = \mu$ or

for any $\tau > 0$, because for $\eta = 2$, and for all $\mu \in S1 = \{1\}$, we have

that is Theorem 1.1 is not applicable in this example. Hence, Theorem 2.1 is a proper extension of Theorem 1.1.

In the following, we obtain some corollaries of Theorem 2.1.

Corollary 2.1. Let $\left(X, d\right)$ be a complete $b$-MS and $S:X\longrightarrow C_{B}(X)$ a mapping. Suppose there is a function $F: \mathbb{R}_{+}\longrightarrow\mathbb{R}$,

$\tau \in\mathbb{R}_{+}$ satisfying (A$_{2}$) and (A$_{3}$), and for any $r, w\in X$ and $\eta \in Sr$, there is $\mu \in Sw$ such that either $\eta = \mu$ or

holds. Then $F^{ix}(S)$ is non-empty.

Corollary 2.2. Let $\left(X, d\right)$ be a complete $b$-MS and $S:X\longrightarrow C_{B}(X)$ a mapping. Suppose there is a function $F: \mathbb{R}_{+}\longrightarrow\mathbb{R}$,

$\tau \in \mathbb{R} _{+}$ satisfying A$_{2}$, A$_{3}$, and for any $r, w\in X$ and $\eta \in Sr$, there is $\mu \in Sw$ such that either $\eta = \mu$ or

holds. Then $F^{ix}(S)$ is non-empty.

Following result is the corollary of Theorem 2.1 for single valued mapping.

Corollary 2.3. Let $\left(X, d\right)$ be a complete $b$-MS and a mapping $f:X\longrightarrow X$. Suppose there is a function $F: \mathbb{R} _{+}\longrightarrow \mathbb{R}$,

$\tau \in \mathbb{R} _{+}$ satisfying (A$_{2}$) and (A$_{3}$), and for any $r, w\in X$ and either $fr = fw$ or

holds for some $L\geq 0$. Then $F^{ix}(f)$ is singleton.

Proof. From Theorem 2.1, $F^{ix}(f)$ is non-empty. To check the uniqueness, assume $\kappa$ and $\varpi$ be fixed points of $f$ with $\kappa \neq \varpi$, that is $f\kappa \neq f\varpi$. Hence, from given condition, we get

implies $\tau \leq 0$, a contradiction. Hence, $F^{ix}(f)$ is singleton.

The following result is an extension of a result given in [25, Theorem 14].

Theorem 2.2. Let $\left(X, d\right)$ be a complete $b$-MS and $S:X\longrightarrow C_{B}(X)$ a mapping. Suppose there is a function $F: \mathbb{R} _{+}\longrightarrow \mathbb{R}$,

and $\tau \in \mathbb{R} _{+}$ satisfying (A$_{1}$)–(A$_{3}$) and (A$_{5}$), and for any $r, w\in X$, $r\neq w$ with $Sr\neq Sw$,

holds for some $L\geq 0$. Then $F^{ix}(S)$ is non-empty.

Proof. Let $\alpha \in Sr$ where $r, w\in X$. We have following two cases:

$(1)$ $d\left(\alpha, Sw\right) = 0,$

$(2)$ $d\left(\alpha, Sw\right) > 0.$

If $d\left(\alpha, Sw\right) = 0$ then $\alpha \in Sw$, because $Sw$ is closed. In the second case $Sr\neq Sw$. So

holds as given. As

so using (A$_{1}$) we get

From (A$_{5}$),

So, we can pick $\beta \in Sw$ fulfilling

If we replace $\tau$ with $\dfrac{\tau }{2}$ in Theorem 2.1, then we get the desired result.

Here, we obtain some corollaries of Theorem 2.2.

Corollary 2.4. Let $\left(X, d\right)$ be a complete $b$-MS and a mapping $S:X\longrightarrow C_{B}(X)$. Suppose there is a function $F: \mathbb{R} _{+}\longrightarrow \mathbb{R}$,

and $\tau \in \mathbb{R} _{+}$ satisfying (A$_{1}$)–(A$_{3}$) and (A$_{5}$), and for any $r, w\in X$, $r\neq w$ with $Sr\neq Sw$,

holds. Then $F^{ix}(S)$ is non-empty.

Corollary 2.5. Let $\left(X, d\right)$ be a complete $b$-MS and a mapping $S:X\longrightarrow C_{B}(X)$. Suppose there is a function $F: \mathbb{R} _{+}\longrightarrow \mathbb{R}$,

and $\tau \in \mathbb{R} _{+}$ satisfying (A$_{1}$)–(A$_{3}$) and (A$_{5}$), and for any $r, w\in X$, $r\neq w$ with $Sr\neq Sw$,

holds. Then $F^{ix}(S)$ is non-empty.

In the following, we obtain another result for a new set-valued $F$- contractions of a $b$-MS.

Theorem 2.3. Let $\left(X, d\right)$ be a complete $b$-MS and $S:X\rightarrow C_{B}(X)$ a set-valued mapping. Suppose there is a function $F: \mathbb{R} _{+}\longrightarrow \mathbb{R}$,

$\tau \in \mathbb{R} _{+}$ satisfying (A$_{1}$)–(A$_{3}$), (A$_{6}$) and

for all $r, w\in X$ with $Sr\neq Sw$ and for some $L\geq 0$. Then $F^{ix}(S)$ is non-empty.

Proof. Let $r_{0}\in X$ and $r_{j+1}\in Sr_{j}$ for all $j\in \mathbb{N}$. If $r_{j} = r_{j+1}$ for some $j\in \mathbb{N}$, then $r_{j}\in Sr_{j}$ and there is nothing to prove further. Now suppose $r_{j}\neq r_{j+1}$ for all $j\in \mathbb{N}$. As $F$ is right continuous at $H\left(Sr_{j}, Sr_{j+1}\right)$ for each $j\in \mathbb{N}$, so there is a $k > 1$ such that

Moreover, there exists $r_{j+2}\in Sr_{j+1}$ such that

Now, from (2.3)–(2.5) and (A$_{1}$) we get

If

then

implies $\tau \leq 0$, a contradiction. So

which further implies

we get

for any $j\in \mathbb{N}$. By Lemma 1.3,

holds. Since

by Lemma 1.2, $\{r_{j}\}$ is a Cauchy sequence. Since $X$ is complete, $\{r_{j}\}$ converges to some $w\in X$.

Now we will show that $w\in Sw$. On contrary assume that $w\notin Sw$, that is $d\left(w, Sw\right) > 0$. As

By (A$_{1}$), we get

On taking limit as $j$ tends to $\infty$ and by the continuity of $F$, we get

implies $2\tau \leq 0$, a contradiction. Hence, $S$ has a fixed point.

Here is an example to explain the Theorem 2.3.

Example 2.2. Let $X = \left\{ a, b, c, \rho, e\right\}$ and set a mapping $d:X\times X\rightarrow \mathbb{R} ^{+}$ by

As

so $d$ is not a metric on $X.$ For any $s\geq \dfrac{12}{11}$, $d$ is a complete $b$-metric. Set $L = 2$ and define a mapping $S:X\rightarrow C_{B}(X)$ as

For $r\in \left\{ a, b\right\}$ and $w\in \left\{ c, \rho, e\right\},$ there are following cases:

If $r = a$, $w = c$, then

If $r = a$, $w = \rho$, then

If $r = a$, $w = e$, then

If $r = b$, $w = c$, then

If $r = b$, $w = \rho$, then

If $r = b$, $w = e$, then

Hence, for any $\tau \in \left(0, 0.5492\right)$,

holds for all $r, w\in X$ and for any $L = 2.$ That is, all the assumptions of Theorem 2.3 are met, so there exists a fixed point $a\in S\left(a\right).$

Following result is the metric version of Theorem 2.3.

Corollary 2.6. Let $\left(X, d\right)$ be a complete metric space and $S:X\rightarrow C_{B}(X)$ a set-valued mapping. Suppose there is a function $F: \mathbb{R} _{+}\longrightarrow \mathbb{R}$, $c\in \left(0, 1\right)$, $\tau \in \mathbb{R} _{+}$ satisfying (A$_{1}$)–(A$_{3}$), (A$_{6}$) and

for all $r, w\in X$ with $Sr\neq Sw$ and for some $L\geq 0$. Then $F^{ix}(S)$ is non-empty.

Proof. Take $s = 1$ in Theorem 2.3.

Abbas et al. ^[3] proved a coincidence point theorem for generalized set-valued $\left(f, L\right)$-almost $F$-contraction and the following is one of the corollary of their main result.

Corollary 2.7. ^[3] Let $\left(X, d\right)$ be a complete metric space and $F: \mathbb{R} _{+}\longrightarrow \mathbb{R}$ be a function with $\tau \in \left(0, \infty \right)$ satisfying (A$_{1}$)–(A$_{3}$) and (A$_{6}$). There is a mapping $S:X\rightarrow C_{B}(X)$ with $L\geq0$, satisfying

for every $r, w\in X$ with $Sr\neq Sw$. Then $F^{ix}(S)$ is non-empty.

Remark 2.2. Note that the Corollary 2.7 is the corollary of Corollary 2.6.

In the next example, we show that the Corollary 2.6 properly generalize the Corollary 2.7.

Example 2.3. Let $X = \left\{ 1, 2, 3, 4, 5\right\}$ and set a mapping $d:X\times X\rightarrow \mathbb{R} ^{+}$ as

so $d$ is a metric on $X$ and it is a complete metric. Set $L = 2$ and define a mapping $S:X\longrightarrow CL\left(X\right)$ as

For $r\in \left\{ 1, 2\right\}$ and $w\in \left\{ 3, 4, 5\right\}$, there is following cases: $

If $r = 1$, $w = 3$, then

If $r = 1$, $w = 4$, then

If $r = 1$, $w = 5$, then

If $r = 2$, $w = 3$, then

If $r = 2$, $w = 4$, then

If $r = 2$, $w = 5$, then

Hence, for all $\tau \in (0, 0.3467]$,

holds for all $r, w\in X$ and for any $L = 2$. Hence, all the assumptions of Corollary 2.6 are satisfied, so there exist a fixed point $1\in S\left(1\right)$. On the other hand, if $r = 1$ and $w = 3$, then

Hence, there is no $\tau > 0$ and $L\geq 0$ such that

That is

for all $\tau > 0$ and $L\geq 0$. Hence, Corollary 2.7 is not applicable in this example.

Remark 2.3. Above example provides a situation where Corollary 2.7 is not applicable while Corollary 2.6 is applicable. Note that Corollary 2.6 is a consequence of our main result (Theorem 2.3).

3.   Conclusions

In this paper, we have introduced generalized set-valued $F$-contractions of $b$-metric spaces and presented the results about the existence of non-empty fixed point sets of newly introduced mappings. Our results improve some already existing very important results in the literature. Examples show that the new results offer proper generalizations. It is worth noting that by setting $b$-metric constant equal to one, we obtain some specific cases showing notable enhancement of existing results yet in metric spaces (see Corollary 2.6 and Example 2.3 above). It would be interesting to investigate these results in the framework of asymmetric distance spaces.

Acknowledgments

The authors are grateful for the funding provided by Basque Government for the grant number IT1207-19.

Conflict of interest

The authors declare that they have no conflicts of interest.