Fixed-Point and Random Fixed-Point Theorems in Preordered Sets Equipped with a Distance Metric

Abstract

This paper explores fixed points for both contractive and non-contractive mappings in traditional b-metric spaces, preordered b-metric spaces, and random b-metric spaces. Our findings provide insights into the behavior of mappings under various constraints and extend our approach to include coincidence and common fixed-point theorems in these spaces. We present new examples and graphical representations for the first time, offering novel results and enhancing several related findings in the literature, while broadening the scope of earlier works of Ran and Reurings, Nieto and Rodríguez-López, Górnicki, and others.

1. Introduction

Fixed-point theorems offer a strong foundation for grasping and addressing solutions to linear and nonlinear problems found in the realms of biology, engineering, and physical sciences.

Bakhtin introduced the concept of b-metric spaces, originally called quasi-metric spaces, in [1]. Later, Czerwik [2,3] expanded on this concept to develop generalizations of Banach’s fixed-point theorem. Among the numerous nontrivial generalizations of standard metric spaces, b-metric spaces hold considerable importance. Berinde and Păcurar authored a survey paper that explores the origins and related aspects of b-metric spaces [4]. Fagin et al. [5] conducted an in-depth study on the relaxation of the triangle inequality in b-metric spaces, referring to this modification as nonlinear elastic matching (NEM). They highlighted its broad applicability in areas such as trademark shape analysis [6] and ice floe measurement [7].

The synergy between the order and metric structure of the space provides significant advantages in fixed-point theory. Ran and Reurings [8] presented a fixed-point theorem that extends the Banach contraction principle to partially ordered metric spaces. Initially, their approach relied on the use of a continuous function. Later, Nieto and Rodríguez-López [9] achieved a similar result by replacing the requirement of continuity for the nonlinear operator with monotonicity. The key aspect of this theorem is that the contractivity condition of the nonlinear map holds only for elements that are comparable within the partial order. A comprehensive resource on applications of fixed-point theory concerning monotone mappings is available in [10]. A crucial result in fixed-point theory for monotone mappings is the classical Knaster–Tarski theorem (also known as the Abian–Brown theorem) [11].

In the work of Górnicki [12], a theorem was developed for preordered metric spaces, where the preordered binary relation is defined as less strict than a partial order (also see [13]). The theorem is notable for requiring the Kannan-type condition [14] on the map to hold only for elements that are comparable, rather than universally across the entire set. In 2019, Górnicki [12] demonstrated several random fixed-point theorems for single-valued operators that are asymptotically regular and satisfy certain Kannan’s type conditions. Górnicki’s mappings include a variety of contractive mapping classes [14,15,16,17,18,19], featuring one of the broadest forms of contractions, the Ćirić contraction [20].

In numerous scenarios, mathematical models or equations used to describe phenomena across various fields involve parameters with unknown values. In such instances, it is often more practical to treat these equations as random operator equations. An intriguing aspect of nonlinear analysis is the combination of deterministic fixed-point theorems with randomness in nonlinear mappings. The study of fixed-point theorems for random operators began with the Prague school of probability research. Hanš [21] developed random fixed-point theorems for contraction mappings within a separable metric space.

It has been demonstrated that a deterministic fixed-point theorem can generally correspond to a random fixed-point theorem when the underlying measurable space $(\nabla ,\wp )$ is a Suslin family (see [22] for definitions). However, it remains an open question whether this correspondence holds when the measurable space is not a Suslin family. In [23], Nieto et al. established the random version of the classical Banach contraction principle in partially ordered metric spaces.

This paper is structured as follows: Section 2 presents a concise overview of the necessary definitions and lemmas for the later sections. Section 3 demonstrates fixed-point theorems, coincidence fixed-point theorems, and common fixed-point theorems in b-metric spaces under both contractive and non-contractive conditions, making use of asymptotic regularity and sequences approximating fixed points. In Section 4, we delve into fixed-point theorems and common fixed-point theorems within the context of preordered b-metric spaces. Section 5 explores random fixed-point theorems and common fixed-point theorems. The final section concludes this paper. Various graphical plots are included to illustrate the examples discussed.

2. Preliminaries

This section covers fundamental definitions and important results that are essential for understanding the subsequent sections.

Definition 1.

Let $\mathcal{Z}$ be a nonempty set and $q\in [1,\infty ).$ We say that a function $\mathcal{D}:\mathcal{Z}\times \mathcal{Z}\to {\mathbb{R}}_{\ge 0}$ is a b-metric if it satisfies the following properties:

$$\begin{array}{cc}\hfill \mathcal{D}(x,y)& =0ifandonlyifx=yforanyx,y\in \mathcal{Z};\hfill \\ \hfill \mathcal{D}(x,y)& =\mathcal{D}(y,x)foranyx,y\in \mathcal{Z};\hfill \\ \hfill \mathcal{D}(x,y)& \le q\left[\mathcal{D}\right(x,z)+\mathcal{D}(y,z\left)\right]foranyx,y,z\in \mathcal{Z}.\hfill \end{array}$$

The triplet $(\mathcal{Z},\mathcal{D},q)$ is said to constitute a b-metric space (b-MS).

Remark 1.

For $q=1,$ a b-MS reduces to a metric space.

Definition 2.

A preordered b-MS is a quadruplet $(\mathcal{Z},\mathcal{D},q,⪯)$, where $\mathcal{Z}$ is a b-MS with respect to the b-metric $\mathcal{D}$ and ⪯ is a binary relation that is reflexive and transitive.

Definition 3.

Let $(\mathcal{Z},\mathcal{D},q)$ be a b-MS, $\mathcal{B}$ a nonempty subset of $\mathcal{Z}$, and let ⪯ be a binary relation on $\mathcal{Z}.$ Then, the quadruplet $(\mathcal{Z},\mathcal{D},q,⪯)$ is said to be nondecreasing regular if for all sequences ${\left(z_n\right)}_{n\in \mathbb{N}}\subseteq \mathcal{B}$, such that $z_n\to z\in \mathcal{B}$ and $z_n⪯z_{n+1}$ for all $n\in \mathbb{N}$; thus, we have that $z_n⪯z$ for all $n\in \mathbb{N}.$

Definition 4

([24]). Let $(\mathcal{Z},\mathcal{D},q)$ be a b-metric space. A mapping $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ satisfying the condition

$$\begin{array}{c}\hfill {\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}({\mathcal{S}}^{n+1}x,{\mathcal{S}}^{n}x)=0forallx\in \mathcal{Z},\end{array}$$

is called asymptotically regular.

Definition 5

([25]). Let $(\mathcal{Z},\mathcal{D},q)$ be a b-metric space and let $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ be a mapping. A sequence ${\left(x_n\right)}_{n\in \mathbb{N}}\subseteq \mathcal{Z}$ satisfying the condition

$$\begin{array}{c}\hfill {\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}(\mathcal{S}{x}_n,x_n)=0\end{array}$$

is said to be an approximating fixed-point sequence of $\mathcal{S}.$

Lemma 1

([26]). Let ${\left(r_n\right)}_{n\in \mathbb{N}}$ be a sequence of non-negative real numbers satisfying

$$\begin{array}{c}\hfill r_{n+1}\le (1-{\delta }_n)r_n+{\delta }_n{\mathsf{\Psi }}_n,n\in {\mathbb{N}}_0,\end{array}$$

where ${\delta }_n\in (0,1)$ and ${\left({\mathsf{\Psi }}_n\right)}_{n\in \mathbb{N}}\subseteq \mathbb{R},$ such that the following conditions hold:

1. 

${\displaystyle \sum _{n\in \mathbb{N}}}{\delta }_n=\infty$;

2. 

${\displaystyle \underset{n\to \infty }{lim}}sup{\mathsf{\Psi }}_n\le 0$ or ${\displaystyle \sum _{n\in \mathbb{N}}}\left|{\delta }_n{\mathsf{\Psi }}_n\right|=\infty .$

Then, ${\displaystyle \underset{n\to \infty }{lim}}{r}_n=0.$

Definition 6

([27]). Let $(\mathcal{Z},\mathcal{D},q)$ be a b-MS with $q\ge 1$ and $\mathcal{S}$ be a self-mapping on $\mathcal{Z}$. Then, $\mathcal{S}$ is said to be k-continuous $(k\in \mathbb{N})$, if ${\displaystyle \underset{n\to \infty }{lim}}{\mathcal{S}}^k{x}_n=\mathcal{S}z$, whenever ${\left(x_n\right)}_{n\in \mathbb{N}}$ is a sequence in $\mathcal{Z}$ such that ${\displaystyle \underset{n\to \infty }{lim}}{\mathcal{S}}^{k-1}{x}_n=z.$

Definition 7

([28]). Let $(\mathcal{Z},\mathcal{D},q)$ be a b-MS with $q\ge 1$ and $\mathcal{S}$ be a self-mapping on $\mathcal{Z}$. Then, the set $\mathcal{O}(S,x)=\{{\mathcal{S}}^{n}x:n\in {\mathbb{N}}_0\}$ is said to be the orbit of $\mathcal{S}$ at x.

Definition 8

([28]). Let $(\mathcal{Z},\mathcal{D},q)$ be a b-MS with $q\ge 1$ and $\mathcal{S}$ be a self-mapping on $\mathcal{Z}$. Then, $\mathcal{S}$ is said to be orbitally continuous at a point $\vartheta \in \mathcal{Z}$ if for any sequence ${\left(x_n\right)}_{n\in \mathbb{N}}\subset \mathcal{O}(S,x)$ for some $x\in \mathcal{Z},{\displaystyle \underset{n\to \infty }{lim}}{x}_n=\vartheta$ implies ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{S}{x}_n=\mathcal{S}\vartheta .$

Definition 9

([29]). The Dottie number is a constant that is the unique real root of the equation $cosv=v,$ where the argument of the cosine function is in radians.

Lemma 2

([30]). Every sequence ${\left(z_n\right)}_{n\in \mathbb{N}}$ of elements from a b-MS $(\mathcal{Z},\mathcal{D})$ of constant $q\ge 1$, having the property that there exists a $\lambda \in (0,1)$, such that

$$\begin{array}{c}\hfill \mathcal{D}(z_{n+1},z_n)\le \lambda \mathcal{D}(z_n,z_{n-1}),\end{array}$$

for every natural number n, is Cauchy in $\mathcal{Z}$.

Definition 10.

Let ∇ be a nonempty set with ℘ a sigma-algebra on $\nabla (sothat$ the pair $(\nabla ,\wp )$ forms a measurable space), and let ($\mathcal{Z},\mathcal{D},q)$ be a b-metric space. We call a mapping $\mathcal{S}:\nabla \to 2^{\mathcal{Z}}$ a ℘-measurable if for any open subset $\mathcal{B}$ of $\mathcal{Z}$, the set ${\mathcal{S}}^{-1}\left(\mathcal{B}\right):=\{w\in \nabla :\mathcal{S}\left(w\right)\cap \mathcal{B}\ne \varnothing \}$ belongs to the sigma-algebra ℘.

Definition 11.

Let ∇ be a nonempty set with ℘ a sigma-algebra on ∇, and let ($\mathcal{Z},\mathcal{D},q)$ be a b-metric space. We call a mapping $\mathcal{S}:\nabla \times \mathcal{Z}\to \mathcal{Z}$ a random operator if for each $z\in \mathcal{Z}$, $\mathcal{S}(·,z):\nabla \to \mathcal{Z}$ is measurable.

Definition 12.

Let the pair $(\nabla ,\wp )$ form a measurable space, ($\mathcal{Z},\mathcal{D},q)$ be a b-metric space and $\mathcal{S}:\nabla \times \mathcal{Z}\to \mathcal{Z}$ a random operator. Then, a mapping $u:\nabla \to \mathcal{Z}$ is said to be a random fixed point of random operator $\mathcal{S}$ if $\mathcal{S}(w,u(w\left)\right)=u\left(w\right)$ for all $w\in \nabla .$ In this scenario, the set of fixed points of $\mathcal{S}\left(w\right)$ is a set-valued map defined by $\mathcal{F}\left(\mathcal{S}\right(w\left)\right)=\{u\in \mathcal{Z}:\mathcal{S}(w,u)=u\}.$

Definition 13.

Let $(\nabla ,\wp )$ be a measurable space and $(\mathcal{Z},\mathcal{D},q)$ a b-metric space. A random operator $\mathcal{S}:\nabla \times \mathcal{Z}\to \mathcal{Z}$ is referred to as asymptotically regular if for each fixed $\omega \in \nabla ,$

$$\begin{array}{c}\hfill \mathcal{D}({\mathcal{S}}^n(w,z),{\mathcal{S}}^{n+1}(w,z))\to 0\mathrm{as}n\to \infty ,\end{array}$$

where ${\mathcal{S}}^n(w,z)$ is the value at z of the $n^{th}$ iterate of $\mathcal{S}(w,·),thatis,{\mathcal{S}}^n(w,z)=\mathcal{S}(w,{\mathcal{S}}^{n-1}(w,z))$, for all $z\in \mathcal{Z}$.

Definition 14.

A measurable function $u:\nabla \to \mathcal{Z}$ is referred to as a selector for a measurable set-valued operator $\mathcal{S}:\nabla \to 2^{\mathcal{Z}}$ if $u\left(w\right)\in \mathcal{S}\left(w\right)$ for all $w\in \nabla .$

Lemma 3

([31]). Let $(\nabla ,\wp )$ be a measurable space, $(\mathcal{Z},\mathcal{D})$ a separable b-metric space, and ${\mathcal{Z}}_1$ be a b-metric space. If $\mathcal{S}:\nabla \times \mathcal{Z}\to {\mathcal{Z}}_1$ is a function, such that it is measurable in the first component and continuous in the second component, and $u:\nabla \to \mathcal{Z}$ is measurable, then $\mathcal{S}(·,u(·\left)\right):\nabla \to \mathcal{Z}$ is a measurable function.

Lemma 4

([22]). Let $(\nabla ,\wp )$ be a measurable space and $(\mathcal{Z},\mathcal{D})$ be a complete and separable b-MS, and $\mathcal{S}:\nabla \to \mathfrak{C}\left(\mathcal{Z}\right)$ a measurable map, where $\mathfrak{C}\left(\mathcal{Z}\right)$ denote the set of all nonempty closed subsets of $\mathcal{Z}.$ Then, $\mathcal{S}$ has a measurable selection.

3. Fixed-Point Theorems in b-Metric Space

We begin with the following result:

Theorem 1.

If $(\mathcal{Z},\mathcal{D},q)$ is a complete b-metric space and a mapping $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ is an asymptotically regular mapping such that there exists a positive $M<{\displaystyle \frac{1}{q}}$ satisfying

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,\mathcal{S}y)\le M\left[\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,\mathcal{S}y)+\mathcal{D}(x,y\left)\right]forallx,y\in \mathcal{Z},\end{array}$$

(1)

then $\mathcal{F}\left(\mathcal{S}\right)$ has a single element.

Proof. 

Let $x\in \mathcal{Z}$ and define a sequence ${\left(x_n\right)}_{n\in \mathbb{N}},$ where $x_n={\mathcal{S}}^{n}x$.

For $m>n(\in \mathbb{N}),$

$$\begin{array}{cc}\hfill \mathcal{D}({\mathcal{S}}^{n+1}x,{\mathcal{S}}^{m+1}x)& \le M\{\mathcal{D}({\mathcal{S}}^{n}x,{\mathcal{S}}^{n+1}x)+\mathcal{D}({\mathcal{S}}^{m}x,{\mathcal{S}}^{m+1}x)+\mathcal{D}({\mathcal{S}}^{n}x,{\mathcal{S}}^{m}x)\}\hfill \\ \hfill & \le M\{\mathcal{D}({\mathcal{S}}^{n}x,{\mathcal{S}}^{n+1}x)+\mathcal{D}({\mathcal{S}}^{m}x,{\mathcal{S}}^{m+1}x)\}+Mq\mathcal{D}({\mathcal{S}}^{n}x,{\mathcal{S}}^{n+1}x)\hfill \\ \hfill & +M{q}^2\mathcal{D}({\mathcal{S}}^{n+1}x,{\mathcal{S}}^{m+1}x)+M{q}^2\mathcal{D}({\mathcal{S}}^{m+1}x,{\mathcal{S}}^{m}x)\hfill \end{array}$$

so

$$\begin{array}{c}\hfill (1-M{q}^2)\mathcal{D}({\mathcal{S}}^{n+1}x,{\mathcal{S}}^{m+1}x)\le (M+Mq)\mathcal{D}({\mathcal{S}}^{n}x,{\mathcal{S}}^{n+1}x)+(M+M{q}^2)\mathcal{D}({\mathcal{S}}^{m}x,{\mathcal{S}}^{m+1}x).\end{array}$$

Letting n tend to infinity shows that ${\left({\mathcal{S}}^{n}x\right)}_{n\in \mathbb{N}}={\left(x_n\right)}_{n\in \mathbb{N}}$ is Cauchy in $\mathcal{Z}$, owing to the asymptotic regularity of the mapping $\mathcal{S}.$ The completeness of $\mathcal{Z}$ implies

$$\begin{array}{c}\hfill {\displaystyle \underset{n\to \infty }{lim}}{\mathcal{S}}^{n}x=\mathcal{V},\mathrm{for}\mathrm{some}\mathcal{V}\in \mathcal{Z}.\end{array}$$

We claim that $\mathcal{V}$ is a fixed point of $\mathcal{S}.$ Now,

$$\begin{array}{cc}\hfill \mathcal{D}(\mathcal{V},\mathcal{S}\mathcal{V})& \le q[\mathcal{D}(\mathcal{V},{\mathcal{S}}^{n+1}x)+\mathcal{D}({\mathcal{S}}^{n+1}x,\mathcal{S}\mathcal{V})]\hfill \\ \hfill & \le q\mathcal{D}(\mathcal{V},{\mathcal{S}}^{n+1}x)+qM[\mathcal{D}({\mathcal{S}}^{n}x,{\mathcal{S}}^{n+1}x)+\mathcal{D}(\mathcal{V},\mathcal{S}\mathcal{V})+\mathcal{D}({\mathcal{S}}^{n}x,\mathcal{V})].\hfill \end{array}$$

With n tending to infinity,

$$\begin{array}{c}\hfill (1-qM)\mathcal{D}(\mathcal{V},\mathcal{S}\mathcal{V})\le 0.\end{array}$$

Since $M<{\displaystyle \frac{1}{q}},\mathcal{V}\in \mathcal{F}\left(\mathcal{S}\right).$

Assume that $\mathcal{S}$ has another fixed point $\mathcal{U}\ne \mathcal{V}.$ Then,

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{U},\mathcal{V})\le M\left\{\mathcal{D}\right(\mathcal{U},\mathcal{S}\left(\mathcal{U}\right))+\mathcal{D}((\mathcal{V},\mathcal{S}(\mathcal{V}\left)\right)+\mathcal{D}(\mathcal{U},\mathcal{V})\}<\mathcal{D}(\mathcal{U},\mathcal{V}),\end{array}$$

which is a contradiction. Therefore, $\mathcal{F}\left(\mathcal{S}\right)=\left\{\mathcal{V}\right\}.$ □

We now prove a fixed-point theorem for generalized mappings:

Theorem 2.

If $(\mathcal{Z},\mathcal{D},q)$ is a complete b-MS and $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ is an asymptotically regular mapping and if there exist $0\le M<{\displaystyle \frac{1}{q^2}}$ and $0\le K<{\displaystyle \frac{1}{q}}$ satisfying

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,\mathcal{S}y)\le M·\mathcal{D}(x,y)+K·\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,\mathcal{S}y\left)\right\}forallx,y\in \mathcal{Z},\end{array}$$

(2)

then $\mathcal{F}\left(\mathcal{S}\right)=\left\{\widehat{x}\right\},$ where $\widehat{x}\in \mathcal{Z}$ and the sequence of forward iterates ${\left({\mathcal{S}}^{n}x\right)}_{n\in \mathbb{N}}$ converges to $\widehat{x}$ for each $x\in \mathcal{Z}.$

Proof. 

Pick some arbitrary $x_0\in \mathcal{Z}$ and define a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ defined as $x_{n+1}=\mathcal{S}{x}_n,$ where $n\in {\mathbb{N}}_0.$ Estimate the b-metric distance between $x_n$ and $x_{n+p}$ (for some positive $p)$ as follows:

$$\begin{array}{cc}\hfill \mathcal{D}(x_{n+p},x_n)& \le q[\mathcal{D}(x_{n+p},x_{n+p+1})+\mathcal{D}(x_{n+p+1},x_n)]\hfill \\ \hfill & \le q\mathcal{D}(x_{n+p},x_{n+p+1})+q^2[\mathcal{D}(x_{n+p+1},x_{n+1})+\mathcal{D}(x_{n+1},x_n)]\hfill \\ \hfill & =q\mathcal{D}(x_{n+p},x_{n+p+1})+q^2\mathcal{D}(x_{n+1},x_n)+q^2\mathcal{D}(\mathcal{S}{x}_{n+p},\mathcal{S}{x}_n)\hfill \\ \hfill & \le q\mathcal{D}(x_{n+p},x_{n+p+1})+q^2\mathcal{D}(x_{n+1},x_n)+q^{2}M\mathcal{D}(x_{n+p},x_n)\hfill \\ \hfill & +K{q}^2\{\mathcal{D}(x_{n+p},x_{n+p+1})+\mathcal{D}(x_n,x_{n+1})\}.\hfill \end{array}$$

On shifting the like terms aside, we have

$$\begin{array}{cc}\hfill (1-q^{2}M)\mathcal{D}(x_{n+p},x_n)& \le q\mathcal{D}(x_{n+p},x_{n+p+1})+q^2\mathcal{D}(x_{n+1},x_n)+\hfill \\ \hfill & K{q}^2\{\mathcal{D}(x_{n+p},x_{n+p+1})+\mathcal{D}(x_n,x_{n+1})\}.\hfill \end{array}$$

Applying the limit as n approaches infinity, the summands on the RHS of the aforementioned inequality vanish since $\mathcal{S}$ is asymptotically regular. Therefore, the sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ is a Cauchy sequence in $\mathcal{Z}$. The completeness of the space $\mathcal{Z}$ ensures the existence of some element $\widehat{x}\in \mathcal{Z}$ which is the limit of the sequence, i.e., ${\displaystyle \underset{n\to \infty }{lim}}{x}_n=\widehat{x}.$ Now, let us prove that $\widehat{x}\in \mathcal{F}\left(\mathcal{S}\right).$

$$\begin{array}{cc}\hfill \mathcal{D}(\mathcal{S}\widehat{x},\widehat{x})& \le q[\mathcal{D}(\mathcal{S}\widehat{x},x_{n+1})+\mathcal{D}(x_{n+1},\widehat{x})]\hfill \\ \hfill & =q[\mathcal{D}(\mathcal{S}\widehat{x},\mathcal{S}{x}_n)+\mathcal{D}(x_{n+1},\widehat{x})]\hfill \\ \hfill & \le q[M\mathcal{D}(\widehat{x},x_n)+K\{\mathcal{D}(\widehat{x},\mathcal{S}\widehat{x})+\mathcal{D}(x_n,\mathcal{S}{x}_n)\}]+q\mathcal{D}(x_{n+1},\widehat{x}),\hfill \end{array}$$

so

$$\begin{array}{cc}\hfill (1-Kq)\mathcal{D}(\mathcal{S}\widehat{x},\widehat{x})& \le q[M\mathcal{D}(\widehat{x},x_n)+K\mathcal{D}(x_n,\mathcal{S}{x}_n)+\mathcal{D}(x_{n+1},\widehat{x})].\hfill \end{array}$$

The three terms on the RHS vanish as n tends to infinity since $x_n\to x.$ Therefore, $\widehat{x}\in \mathcal{F}\left(\mathcal{S}\right).$ For uniqueness, let $x^{+}$ be another element in $\mathcal{F}\left(\mathcal{S}\right)$, distinct from $\widehat{x}.$

$$\begin{array}{cc}\hfill \mathcal{D}(\widehat{x},x^{+})& =\mathcal{D}(\mathcal{S}\widehat{x},\mathcal{S}{x}^{+})\hfill \\ \hfill & \le M\mathcal{D}(\widehat{x},x^{+})+K[\mathcal{D}(\widehat{x},\mathcal{S}\widehat{x})+\mathcal{D}(x^{+},\mathcal{S}{x}^{+})]=M\mathcal{D}(\widehat{x},x^{+}).\hfill \end{array}$$

Since $M<1,$ $\widehat{x}$ and $x^{+}$ coincide in $\mathcal{F}\left(\mathcal{S}\right)$. Thus, $\left|\mathcal{F}\right(\mathcal{S}\left)\right|=1.$

The proof for the convergence of the sequence of successive approximations to $\widehat{x}$ is immediate. □

Example 1.

Let $(\mathcal{Z},\mathcal{D},q)$ be a b-metric space where $\mathcal{Z}=[0,1]\cup [3,{\displaystyle \frac{10}{3}}]$, $\mathcal{D}(u,v)={|u-v|}^2$ for all $u,v\in \mathcal{Z}.$ Here, the quasi-index is $q=2.$ Consider the mapping $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ given by $\mathcal{S}\left(u\right)=0foru\in [0,1]$ and $\mathcal{S}\left(u\right)=1foru\in [3,{\displaystyle \frac{10}{3}}].$ It is immediate that $\mathcal{S}$ is an asymptotically regular mapping. If both $u,v\in E=[0,1]$ or $F=[3,{\displaystyle \frac{10}{3}}],$ then $\mathcal{D}(\mathcal{S}u,\mathcal{S}v)=0.$ In this case, (2) holds for the aforementioned values of the parameters M and K. Now, consider $u\in E$ and $v\in F$; thus, (2) holds for any non-negative $M<{\displaystyle \frac{1}{4}}$ and $K={\displaystyle \frac{1}{4}}.$ We verify this example with Figure 1, which shows the behavior of the self-mapping $\mathcal{S}$ on $E\cup F$.

We now provide an example to demonstrate that the converse of Theorem 2 does not hold.

Example 2.

In the context of the complete b-metric space $(\mathcal{Z},\mathcal{D},q)$, where $\mathcal{Z}=[0,1]$, $\mathcal{D}(u,v)={(u-v)}^2$, and the mapping $S:\mathcal{Z}\to \mathcal{Z}$ is defined as $S\left(x\right)={\displaystyle \frac{x}{3}}+{\displaystyle \frac{2}{3}}sin\left(\pi x\right)$, several observations can be made. Firstly, it is evident that S is an asymptotically regular continuous mapping on $[0,1]$. Moreover, it possesses a unique fixed point at zero. However, despite these properties, it does not satisfy the condition expressed by (2) for all $x,y\in \mathcal{Z}$ (Figure 2). It is notable that even when considering the supremum of permissible values for parameters, with $M={\displaystyle \frac{1}{2}}$ and $K={\displaystyle \frac{1}{4}}$, the mapping fails to fulfill (2).

Corollary 1.

Let $(\mathcal{Z},\mathcal{D},q)$ be a complete b-MS and $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ be an asymptotically regular mapping satisfying (2) with $0\le M<{\displaystyle \frac{1}{q^2}}$ and $K=0.$ Then, $\mathcal{F}\left(\mathcal{S}\right)=\left\{\widehat{x}\right\}\in \mathcal{Z}$ and ${\mathcal{S}}^{n}x\to \widehat{x}$ for each $x\in \mathcal{Z}.$

The following corollary marks a notable advancement in the theory of Kannan-type contractions within b-metric spaces. Unlike previous results, such as those found in [32], which impose various constraints on the contraction ratio, this corollary extends the framework by relaxing these limitations and offering broader applicability for fixed-point theorems in b-metric spaces.

Corollary 2.

If $(\mathcal{Z},\mathcal{D},q)$ is a complete b-MS and $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ is an asymptotically regular mapping and if there exists $0\le K<{\displaystyle \frac{1}{q}}$ satisfying

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,\mathcal{S}y)\le K·\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,\mathcal{S}y\left)\right\}forallx,y\in \mathcal{Z},\end{array}$$

(3)

then $\mathcal{F}\left(\mathcal{S}\right)=\left\{\widehat{x}\right\},$ where $\widehat{x}\in \mathcal{Z}$, and the sequence of forward iterates ${\left({\mathcal{S}}^{n}x\right)}_{n\in \mathbb{N}}$ converges to $\widehat{x}$ for each $x\in \mathcal{Z}.$

Now, we present a different version of Theorem 2, where we extend the range of K to infinity.

Theorem 3.

Let $\mathcal{Z}$ be a complete b-MS with a continuous b-metric $\mathcal{D}$ and coefficient q and $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ be a continuous asymptotically regular mapping. If there exist $0\le M<{\displaystyle \frac{1}{q^2}}$ and $0\le K<\infty$ satisfying

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,\mathcal{S}y)\le M·\mathcal{D}(x,y)+K·\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,\mathcal{S}y\left)\right\}forallx,y\in \mathcal{Z},\end{array}$$

(4)

then $\mathcal{F}\left(\mathcal{S}\right)=\left\{\widehat{x}\right\},$ where $\widehat{x}\in \mathcal{Z}$, and the sequence of forward iterates ${\left({\mathcal{S}}^{n}x\right)}_{n\in \mathbb{N}}$ converges to $\widehat{x}$ for each $x\in \mathcal{Z}.$

Proof. 

We can follow the similar steps of Theorem 2 until the convergence of ${\left(x_n\right)}_{n\in \mathbb{N}}$ with the following: $x_n\stackrel{n\to \infty }{\to }\widehat{x}\Rightarrow \mathcal{S}{x}_n\stackrel{n\to \infty }{\to }\mathcal{S}\widehat{x}\Rightarrow x_{n+1}\stackrel{n\to \infty }{\to }\mathcal{S}\widehat{x}.$ From the uniqueness of the limit, $\widehat{x}\in \mathcal{F}\left(\mathcal{S}\right).$ □

Corollary 3.

Let $\mathcal{Z}$ be a complete b-MS with a continuous b-metric $\mathcal{D}$ and coefficient q and let $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ be a continuous asymptotically regular mapping satisfying (4) with $M=0$ and $0\le K<\infty .$ Then, $\mathcal{F}\left(\mathcal{S}\right)=\left\{\widehat{x}\right\}\in \mathcal{Z}$ and ${\mathcal{S}}^{n}x\stackrel{n\to \infty }{\to }\widehat{x}$ for each $x\in \mathcal{Z}.$

Remark 2.

Corollary 3 represents a significant advancement of the existing Kannan-type contraction results. In this extension, the contraction ratio can attain any non-negative real number.

Corollary 4.

Let $(\mathcal{Z},\mathcal{D})$ be a complete MS (taking $q=1)$ and $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ be a continuous and asymptotically regular mapping satisfying (4) with $M\in [0,1)$ and $K\in [0,\infty ).$ Then, $\mathcal{F}\left(\mathcal{S}\right)=\left\{\widehat{x}\right\}\subseteq \mathcal{Z}$ and ${\mathcal{S}}^{n}x\to \widehat{x}$ for each $x\in \mathcal{Z}.$

Remark 3.

Theorem 2.6 in Górnicki’s article [12] is derived as a Corollary 4 of our result.

Corollaries 3 and 4 represent a significant extension of Kannan’s theorem, applying it to a b-MS and a MS, respectively. These corollaries open up new avenues of exploration by broadening the scope of Kannan’s theorem into these distinct contexts.

Example 3.

Let $(\mathbb{R},\mathcal{D},q)$ be a b-metric space where $\mathcal{D}(u,v)={|u-v|}^2$ for all $u,v\in \mathbb{R}.$ Here, the quasi-index is two $(q=2).$ Let $\mathcal{G}:[0,{\displaystyle \frac{\pi }{2}}]\to [0,{\displaystyle \frac{\pi }{2}}]$ be defined as $\mathcal{G}\left(v\right)=sinv.$ Then, the mapping $\mathcal{G}$ is a smooth function. For some $v_0\in [0,{\displaystyle \frac{\pi }{2}}]$, construct a sequence ${\left(v_n\right)}_{n\in \mathbb{N}}$ defined as $v_n={\mathcal{G}}^n{v}_0=sin{v}_{n-1}.$ Estimate the distance between the ${(n+1)}^{th}$ iterate and the preceding one as follows:

$$\begin{array}{c}\hfill \mathcal{D}({\mathcal{G}}^{n+1}{v}_0,{\mathcal{G}}^n{v}_0)=\mathcal{D}(sin{v}_n,sin{v}_{n-1})=|sin{v}_n-sin{v}_{n-1}{|}^2=|cos{\kappa }_n{|}^2·{|v_n-v_{n-1}|}^2,\end{array}$$

where ${\kappa }_n\in (v_{n-1},v_n).$ Letting n grow without bound, the above distance converges to zero since $|cos{\kappa }_n|<1$ for all $n\in \mathbb{N}$ implies that it is asymptotically regular. $\mathcal{G}$ defined on $[0,{\displaystyle \frac{\pi }{2}}]$ is not a contraction. With the mean-value theorem,

$$\begin{array}{c}\hfill |\mathcal{D}(sinv,sinu)|={|sinv-sinu|}^2={|cos\kappa |}^2·{|\mathcal{D}(v,u)|}^2={|cos\kappa |}^2·{|v-u|}^4.\end{array}$$

For both u and v tending to 0, it drives $cos\kappa$ towards 1, yet it is not feasible to achieve a universal κ satisfying the contraction inequality. $\mathcal{D}\left(\mathcal{G}\right(v),\mathcal{G}(u\left)\right)\le \beta \mathcal{D}(v,u)forsome\beta \in [0,1)$ holds for all $u,v\in [0,{\displaystyle \frac{\pi }{2}}].$ Therefore, $\mathcal{G}$ is not a contraction on the given interval $[0,{\displaystyle \frac{\pi }{2}}]$ in this b-MS. The mapping $\mathcal{G}$ satisfies the inequality $\mathcal{D}(\mathcal{G}v,\mathcal{G}u)\le M\mathcal{D}(u,v)+K\left(\mathcal{D}\right(u,\mathcal{G}u)+\mathcal{D}(v,\mathcal{G}v\left)\right)$ for $M={\displaystyle \frac{1}{8}}$ and $K=8$ for all $v,u\in [0,{\displaystyle \frac{\pi }{2}}].$ We emphasize that this example is significant because it introduces constraints on the parameter that are not typically required in Kannan-type inequalities within a b-metric space, where the parameter does not need to satisfy conditions like $K<1.$ The conditions outlined in Theorem 3 ensure the existence of a unique fixed point. This is further illustrated through graphical representations, as shown in Figure 3 and Figure 4.

Example 4.

Let $(\mathbb{R},\mathcal{D},q)$ be a b-metric space where $\mathcal{D}(u,v)={|u-v|}^2$ for all $u,v\in \mathbb{R}.$ Let $\mathcal{G}:[0,{\displaystyle \frac{\pi }{2}}]\to [0,{\displaystyle \frac{\pi }{2}}]$ be defined as $\mathcal{G}\left(v\right)=cosv.$ Then, the mapping $\mathcal{G}$ is a smooth function (therefore continuous), an asymptotically regular mapping. The cosine mapping is (such as the sin function) not a contraction on the same interval but for $M={\displaystyle \frac{1}{8}}$ and $K={\displaystyle \frac{1}{4}}$, it satisfies $\mathcal{D}(\mathcal{G}u,\mathcal{G}v)\le M\mathcal{D}(u,v)+K\left[\mathcal{D}\right(u,\mathcal{G}u)+\mathcal{D}(v,\mathcal{G}v\left)\right]$ for all $u,v\in [0,{\displaystyle \frac{\pi }{2}}]$ (see Figure 5). Therefore, $\mid \mathcal{F}\left(\mathcal{G}\right)\mid =1.$

Let us advance by addressing a key aspect of this function and its stability. We demonstrate that the Dottie number serves as a universally attracting fixed point within this b-metric space. Despite $\mathcal{G}$ not being a contraction, we observe that orbits surrounding the Dottie number are drawn towards it, indicating its stability in the space. Let $x_0$ be the Dottie number and x be its nearby point. Then, $\mathcal{G}x$ will be closer to $x_0$ than x if $\mathcal{D}(\mathcal{G}x,x_0)<\mathcal{D}(x,x_0),$ that is, the ratio ${\displaystyle \frac{\mathcal{D}(\mathcal{G}x,\mathcal{G}{x}_0)}{\mathcal{D}(x,x_0)}}<1.$ Since $\mathcal{G}$ is a smooth function, as the nearby point x moves closer to the Dottie number, that is, as $x\to x_0,$ we get $\mathcal{D}({\mathcal{G}}^{\prime }{x}_0,0)<1.$ Since ${(\mid {\mathcal{G}}^{\prime }{x}_0\mid )}^2=\mid {(sin0.73908513321516)}^2\mid =0.45375316586<1.$ Therefore, the Dottie number is an attracting fixed point in this b-metric space. This is represented using Figure 6.

Theorem 4.

If $(\mathcal{Z},\mathcal{D},q)$ is a complete b-metric space and $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ is an asymptotically regular mapping. Suppose that there exist $\mathfrak{c}\in (0,{\displaystyle \frac{1}{q^2}})$ and $\mathfrak{l}\ge 0$, such that the mapping $\mathcal{S}$ satisfies

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,\mathcal{S}y)\le \mathfrak{c}·\mathcal{D}(x,y)+\mathfrak{l}·\mathcal{D}(x,\mathcal{S}x),\end{array}$$

(5)

then $\mathcal{F}\left(\mathcal{S}\right)=\left\{\vartheta \right\}$ for some $\vartheta \in \mathcal{Z}.$ Further, ${\mathcal{S}}^{n}z\to \vartheta$ as $n\to \infty$ for any $z\in \mathcal{Z}.$

Proof. 

Pick an arbitrary element, say $z_0\in \mathcal{Z}$, using it we define a sequence ${\left(z_n\right)}_{n\in \mathbb{N}}$ given by $z_{n+1}=\mathcal{S}{z}_n,n\in {\mathbb{N}}_0.$ For some natural numbers $m\mathrm{and}n$ with $m>n,$ we obtain

$$\begin{array}{cc}\hfill \mathcal{D}(z_m,z_n)& \le q·\mathcal{D}(z_m,z_{m+1})+q^2·\mathcal{D}(z_{n+1},z_n)+q^2·\mathcal{D}(\mathcal{S}{z}_m,\mathcal{S}{z}_n)\hfill \\ \hfill & \le q·\mathcal{D}(z_m,z_{m+1})+q^2·\mathcal{D}(z_{n+1},z_n)+q^2·\mathfrak{c}·\mathcal{D}(z_m,z_n)+q^2·\mathfrak{l}·\mathfrak{D}(z_m,z_{m+1}).\hfill \end{array}$$

Upon simplification, this yields

$$\begin{array}{cc}\hfill (1-q^2·\mathfrak{c})\mathcal{D}(z_m,z_n)& \le q·\mathcal{D}(z_m,z_{m+1})+q^2·\mathcal{D}(z_{n+1},z_n)+q^2·\mathfrak{l}·\mathfrak{D}(z_m,z_{m+1}).\hfill \end{array}$$

Letting n tend to infinity yields a Cauchy sequence ${\left(z_n\right)}_{n\in \mathbb{N}}$ in $\mathcal{Z}.$ Since the space $\mathcal{Z}$ is complete, there exists $\vartheta \in \mathcal{Z}$ such that the limit of ${\left(z_n\right)}_{n\in \mathbb{N}}$ is $\vartheta .$ In order to establish that the limit $\vartheta$ is an element of $\mathcal{F}\left(\mathcal{S}\right)$, one has

$$\begin{array}{cc}\hfill \mathcal{D}(\vartheta ,\mathcal{S}\vartheta )& \le q·[\mathcal{D}(\vartheta ,z_{n+1})+\mathcal{D}(\mathcal{S}{z}_n,\mathcal{S}\vartheta )]\hfill \\ \hfill & \le q·\mathcal{D}(\vartheta ,z_{n+1})+q·[\mathfrak{c}·\mathcal{D}(z_n,\vartheta )+\mathfrak{l}·\mathcal{D}(z_{n+1},z_n)]\stackrel{n\to \infty }{\to }0.\hfill \end{array}$$

Therefore, $\vartheta \in \mathcal{F}\left(\mathcal{S}\right).$ Assume that $\mathcal{F}\left(\mathcal{S}\right)$ contains two distinct elements, $\vartheta$ and $\widehat{\vartheta }$. Estimate the distance

$$\begin{array}{cc}\hfill \mathcal{D}(\vartheta ,\widehat{\vartheta })& \le \mathfrak{c}·\mathcal{D}(\vartheta ,\widehat{\vartheta })+\mathfrak{l}·\mathcal{D}(\vartheta ,\mathcal{S}\vartheta ).\hfill \end{array}$$

That is,

$$\begin{array}{c}\hfill (1-\mathfrak{c})·\mathcal{D}(\vartheta ,\widehat{\vartheta })\le 0.\end{array}$$

Since the multiplicand is always positive, $\mathcal{F}\left(\mathcal{S}\right)=\left\{\vartheta \right\}.$ □

Setting $q=1$ yields the following result in the setting of a metric space.

Corollary 5.

If $(\mathcal{Z},\mathcal{D})$ is a complete metric space and $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ is an asymptotically regular mapping. Suppose that there exist $\mathfrak{c}\in (0,1)$ and $\mathfrak{l}\ge 0$, such that the mapping $\mathcal{S}$ satisfies (5), then $\mathcal{F}\left(\mathcal{S}\right)=\left\{\vartheta \right\}$ for some $\vartheta \in \mathcal{Z}.$ Further, ${\mathcal{S}}^{n}z\stackrel{n\to \infty }{\to }\vartheta$ for any $z\in \mathcal{Z}.$

Since the b-metric $\mathcal{D}$ has the virtue that $\mathcal{D}(x,y)=\mathcal{D}(y,x)$, we have the following dual theorem:

Theorem 5.

Let $(\mathcal{Z},\mathcal{D},q)$ be a complete b-MS and $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ an asymptotically regular mapping. Suppose that there exist $\mathfrak{c}\in (0,{\displaystyle \frac{1}{q^2}})$ and $\mathfrak{l}\ge 0$, such that the mapping $\mathcal{S}$ satisfies

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,\mathcal{S}y)\le \mathfrak{c}·\mathcal{D}(x,y)+\mathfrak{l}·\mathcal{D}(y,\mathcal{S}y),\end{array}$$

then $\mathcal{F}\left(\mathcal{S}\right)=\left\{\vartheta \right\}$ for some $\vartheta \in \mathcal{Z}.$ Further, ${\displaystyle \underset{n\to \infty }{lim}}{\mathcal{S}}^{n}z=\vartheta$ for any $z\in \mathcal{Z}.$

Proof. 

The same steps can be followed as in Theorem 4. □

Remark 4.

It is important to note that in Theorem 4, continuity of the mapping was not assumed, yet $\mathfrak{l}$ can still take values in the range $[0,\infty )$.

We now give an example which illustrates Theorem 4.

Example 5.

Let $\mathbb{Z}={\mathbb{R}}^2$ with the usual b-metric $\mathcal{D}(u,v)={|a-c|}^2+{|b-d|}^2$, where $u=(a,b)\in {\mathbb{R}}^2$ and $v=(c,d)\in {\mathbb{R}}^2$. Suppose that $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ is a mapping given by $\mathcal{S}(s,t)=({\displaystyle \frac{s}{4}},{\displaystyle \frac{t}{5}}).$ It is immediate that $\mathcal{S}$ satisfies the inequality (5) for all $\mathfrak{c}\in [{\displaystyle \frac{1}{16}},1)$ and $\mathfrak{l}\in [0,\infty )$ (see Figure 7). Moreover, $\mathcal{S}$ has a unique fixed point.

Figure 8 illustrates the behavior of $\mathcal{S}$ on points in the two-dimensional plane ${\mathbb{R}}^2$. The mapping $\mathcal{S}$ is defined as $\mathcal{S}(s,t)=\left({\displaystyle \frac{s}{4}},{\displaystyle \frac{t}{5}}\right)$, where $(s,t)$ is a point in ${\mathbb{R}}^2$. The quiver plot depicts the displacement vectors induced by $\mathcal{S}$ at various points in the plane. Additionally, the red point marks the unique fixed point of $\mathcal{S}$, found using fixed-point iteration starting from the initial guess $(1,1)$. The blue circles represent the trajectory followed by the iteration process, demonstrating the convergence to the fixed point. The iteration process begins at the initial guess $(1,1)$ and iterates through the following points: $(0.25,0.2)$, $(0.0625,0.04)$, $(0.015625,0.008)$, $(0.00390625,0.0016)$, and $(0.00097656,0.00032)$.

In the next theorem, we employ an approximating fixed-point sequence approach instead of asymptotic regularity of the self-mapping S.

Theorem 6.

Let $(\mathcal{Z},\mathcal{D},q)$ be a complete b-metric space and $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ be a mapping that satisfies (1) with $M<{\displaystyle \frac{1}{q^2}}$. Suppose that ${\left(x_n\right)}_{n\in \mathbb{N}}$ is an approximating fixed-point sequence of $\mathcal{S},$ then $\mathcal{S}$ has a unique fixed point ϑ and it is given by the following:

$${\displaystyle \underset{n\to \infty }{lim}}{x}_n=\vartheta .$$

Proof. 

By substituting $x=x_n$ and $y=x_{n+p}$ (for some $p\in \mathbb{N}$) in (1), the inequality

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}{x}_n,\mathcal{S}{x}_{n+p})\le M[\mathcal{D}(x_n,\mathcal{S}{x}_n)+\mathcal{D}(x_{n+p},\mathcal{S}{x}_{n+p})+\mathcal{D}(x_n,x_{n+p})],\end{array}$$

holds for all $n\in {\mathbb{N}}_0,p\in \mathbb{N}.$ Now, we claim that ${\left(\mathcal{S}{x}_n\right)}_{n\in \mathbb{N}}$ is a Cauchy sequence in $\mathcal{Z}$. In virtue of the relaxed triangle inequality, we have

$$\begin{array}{cc}\hfill & (1-M·q^2)\mathcal{D}(\mathcal{S}{x}_n,\mathcal{S}{x}_{n+p})\le M(1+q)\mathcal{D}(x_n,\mathcal{S}{x}_n)+M(1+q^2)\mathcal{D}(x_{n+p},\mathcal{S}{x}_{n+p}).\hfill \end{array}$$

Since ${\left(x_n\right)}_{n\in \mathbb{N}}$ is an approximating fixed-point sequence of $\mathcal{S}$, both terms on the right-hand side vanish as n approaches infinity. Thus, ${\left(\mathcal{S}{x}_n\right)}_{n\in \mathbb{N}}$ is a Cauchy sequence. Again, several implementations of the relaxed triangle inequality yield

$$\begin{array}{cc}\hfill & \mathcal{D}(x_n,x_{n+p})\le q\mathcal{D}(x_n,\mathcal{S}{x}_n)+q^2\mathcal{D}(x_{n+p},\mathcal{S}{x}_{n+p})+q^2\mathcal{D}(\mathcal{S}{x}_n,\mathcal{S}{x}_{n+p})\hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & {\displaystyle \frac{1}{q^2}}\mathcal{D}(\mathcal{S}{x}_n,\mathcal{S}{x}_{n+p})-{\displaystyle \frac{1}{q}}\mathcal{D}(x_n,\mathcal{S}{x}_n)-\mathcal{D}(x_{n+p},\mathcal{S}{x}_{n+p})\le \mathcal{D}(x_n,x_{n+p}).\hfill \end{array}$$

With the Squeeze theorem on limits, it follows that ${\left(x_n\right)}_{n\in \mathbb{N}}$ is Cauchy in $\mathcal{Z}$. The completeness of $\mathcal{Z}$ ensures that sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ is convergent, denoted by $\vartheta .$ The limit is an element of $\mathcal{F}\left(\mathcal{S}\right)$ and follows from

$$\begin{array}{cc}\hfill \mathcal{D}(\vartheta ,\mathcal{S}\vartheta )& \le q[\mathcal{D}(\vartheta ,x_{n+1})+\mathcal{D}(x_{n+1},\mathcal{S}\vartheta )]\hfill \\ \hfill & \le q\mathcal{D}(\vartheta ,x_{n+1})+q^2[\mathcal{D}(x_{n+1},\mathcal{S}{x}_{n+1})+\mathcal{D}(\mathcal{S}{x}_{n+1},\mathcal{S}\vartheta )]\hfill \\ \hfill & \le q\mathcal{D}(\vartheta ,x_{n+1})+q^2\mathcal{D}(x_{n+1},\mathcal{S}{x}_{n+1})+q^{2}M[\mathcal{D}(x_{n+1},\vartheta )\hfill \\ \hfill & +\mathcal{D}(\mathcal{S}{x}_{n+1},x_{n+1})+\mathcal{D}(\mathcal{S}\vartheta ,\vartheta )].\hfill \end{array}$$

Shifting the fifth summand in the left side and tending n to infinity yields $(1-M{q}^2)\mathcal{D}(\vartheta ,\mathcal{S}\vartheta )$ $\le 0.$ Since $q^{2}M<1,$ $\vartheta$ lies in $\mathcal{F}\left(\mathcal{S}\right).$

Assume that $\vartheta$ and $\widehat{\vartheta }$ are two distinct fixed points of $\mathcal{S}.$ Then,

$$\begin{array}{cc}\hfill \mathcal{D}(\vartheta ,\widehat{\vartheta })& =\mathcal{D}(\mathcal{S}(\vartheta ),\mathcal{S}(\widehat{\vartheta }))\le M[\mathcal{D}(\vartheta ,\widehat{\vartheta })+\mathcal{D}(\vartheta ,\mathcal{S}(\vartheta ))+\mathcal{D}(\widehat{\vartheta },\mathcal{S}(\widehat{\vartheta }))]<\mathcal{D}(\vartheta ,\widehat{\vartheta }).\hfill \end{array}$$

Therefore, $\mathcal{F}\left(\mathcal{S}\right)=\left\{\vartheta \right\}.$ □

The following theorem is an extension of Theorem 6.

Theorem 7.

Let $(\mathcal{Z},\mathcal{D},q)$ be a complete b-metric space and $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ be a self-mapping on $\mathcal{Z}$ satisfying

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,\mathcal{S}y)\le M\mathcal{D}(x,y)+K\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,\mathcal{S}y\left)\right\}forallx,y\in \mathcal{Z},\end{array}$$

(6)

where $M\in [0,{\displaystyle \frac{1}{q^2}})$ and $K\in [0,{\displaystyle \frac{1}{q^2}}).$ Suppose that ${\left(x_n\right)}_{n\in \mathbb{N}}$ is an approximating fixed-point sequence of $\mathcal{S},$ then $\mathcal{F}\left(\mathcal{S}\right)=\left\{\vartheta \right\}\subseteq \mathcal{Z}$ and for each $x\in \mathcal{Z},{\left({\mathcal{S}}^{n}x\right)}_{n\in \mathbb{N}}$ converges to ϑ.

Proof. 

We take an approximating sequence, say ${\left(x_n\right)}_{n\in \mathbb{N}}$, of $\mathcal{S}$, that is ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}(x_n,\mathcal{S}{x}_n)=0$. We assert that ${\left(x_n\right)}_{n\in \mathbb{N}}$ is Cauchy in $\mathcal{Z}.$ Estimate the distance between $x_m$ and $x_n$ (for some $m>n)$ as follows:

$$\begin{array}{cc}\hfill \mathcal{D}(x_n,x_m)& \le q[\mathcal{D}(x_n,\mathcal{S}{x}_n)+\mathcal{D}(\mathcal{S}{x}_n,x_m)]\hfill \\ \hfill & \le q\mathcal{D}(x_n,\mathcal{S}{x}_n)+q^2[\mathcal{D}(\mathcal{S}{x}_n,\mathcal{S}{x}_m)+\mathcal{D}(\mathcal{S}{x}_m,x_m)]\hfill \\ \hfill & \le q\mathcal{D}(x_n,\mathcal{S}{x}_n)+q^2\mathcal{D}(\mathcal{S}{x}_m,x_m)+q^2[M·\mathcal{D}(x_n,x_m)\hfill \\ \hfill & +K·\{\mathcal{D}(x_n,\mathcal{S}{x}_n)+\mathcal{D}(x_m,\mathcal{S}{x}_m)\}].\hfill \end{array}$$

Simplifying the inequality yields

$$\begin{array}{c}\hfill (1-M{q}^2)\mathcal{D}(x_n,x_m)\le (q+K{q}^2)\mathcal{D}(x_n,\mathcal{S}{x}_n)+(q^2+K{q}^2)\mathcal{D}(x_m,\mathcal{S}{x}_m).\end{array}$$

As n approaches infinity, all the summands on the RHS vanish. Therefore, ${\left(x_n\right)}_{n\in \mathbb{N}}$ is a Cauchy sequence in $\mathcal{Z}$. There must exist $\vartheta \in \mathcal{Z}$, which is the limit of this Cauchy sequence, owing to the completeness of the space. Now, let us show that this limit is in $\mathcal{F}\left(\mathcal{S}\right).$

$$\begin{array}{cc}\hfill \mathcal{D}(\mathcal{S}\vartheta ,\vartheta )& \le q[\mathcal{D}(\mathcal{S}\vartheta ,x_n)+\mathcal{D}(x_n,\vartheta )]\hfill \\ \hfill & \le q^2[\mathcal{D}(\mathcal{S}\vartheta ,\mathcal{S}{x}_n)+\mathcal{D}(\mathcal{S}{x}_n,x_n)]+q·\mathcal{D}(x_n,\vartheta )\hfill \\ \hfill & \le q^{2}M\mathcal{D}(\vartheta ,x_n)+q^{2}K[\mathcal{D}(\vartheta ,\mathcal{S}\vartheta )+\mathcal{D}(x_n,\mathcal{S}{x}_n)]+q^2\mathcal{D}(\mathcal{S}{x}_n,x_n)+q·\mathcal{D}(x_n,\vartheta ).\hfill \end{array}$$

Upon simplification, we have

$$\begin{array}{cc}\hfill (1-K·q^2)\mathcal{D}(\mathcal{S}\vartheta ,\vartheta )& \le (q^{2}M+q)·\mathcal{D}(\vartheta ,x_n)+q^2(K+1)·\mathcal{D}(x_n,\mathcal{S}{x}_n)\to 0,\hfill \end{array}$$

as n gets infinitely large, since ${\displaystyle \underset{n\to \infty }{lim}}{x}_n=\vartheta$ and $\left(x_n\right)$ is an approximating fixed point-sequence of $\mathcal{S}.$ Therefore, $\vartheta \in \mathcal{F}\left(\mathcal{S}\right).$ In order to show that $\mathcal{F}\left(\mathcal{S}\right)=\left\{\vartheta \right\},$

$$\begin{array}{c}\hfill \mathcal{D}(\vartheta ,\widehat{\vartheta })\le M\mathcal{D}(\vartheta ,\widehat{\vartheta })+K\{\mathcal{D}(\vartheta ,\mathcal{S}\vartheta )+\mathcal{D}(\widehat{\vartheta },\mathcal{S}\widehat{\vartheta })\}<\mathcal{D}(\vartheta ,\widehat{\vartheta }),\end{array}$$

which is not possible. Thus, $\vartheta$ is the only element in $\mathcal{F}\left(\mathcal{S}\right)$. □

In the following theorem, we extend the range for the parameter K by making $\mathcal{S}$ a continuous mapping.

Theorem 8.

Let $(\mathcal{Z},\mathcal{D},q)$ be a complete b-MS with a continuous b-metric $\mathcal{D}$ and $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ be a continuous mapping satisfying

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,\mathcal{S}y)\le M\mathcal{D}(x,y)+K\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,\mathcal{S}y\left)\right\}forallx,y\in \mathcal{Z},\end{array}$$

(7)

where $M\in [0,{\displaystyle \frac{1}{q^2}})$ and $K\in [0,\infty ).$ Suppose that ${\left(x_n\right)}_{n\in \mathbb{N}}$ is an approximating fixed-point sequence of $\mathcal{S},$ then $\mathcal{F}\left(\mathcal{S}\right)=\left\{\vartheta \right\}\in \mathcal{Z}$ and for each $x\in \mathcal{Z},{\left({\mathcal{S}}^{n}x\right)}_{n\in \mathbb{N}}$ converges to ϑ.

Proof. 

Like in the preceding theorem, we obtain the following:

$$\begin{array}{cc}\hfill (1-M{q}^2)\mathcal{D}(x_n,x_m)& \le (q+q^{2}K)·\mathcal{D}(x_n,\mathcal{S}{x}_n)+(q^2+q^{2}K)·\mathcal{D}(x_m,\mathcal{S}{x}_m).\hfill \end{array}$$

${\left(x_n\right)}_{n\in \mathbb{N}}$ is Cauchy and, in turn, converges to some $\vartheta \in \mathcal{Z}.$ Since the b-metric and $\mathcal{S}$ are continuous, $\mathcal{D}(\vartheta ,\mathcal{S}\vartheta )={\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}(x_n,\mathcal{S}{x}_n)=0.$ This implies that $\vartheta \in \mathcal{F}\left(\mathcal{S}\right).$ The cardinality one of $\mathcal{F}\left(\mathcal{S}\right)$ is immediate following inequality (7). □

It is crucial to highlight that Theorem 8 represents a substantial generalization of the Kannan mapping within a b-MS framework. Specifically, when the parameter M is set to 0, the theorem reduces to the classical Kannan mapping, where the parameter K ranges over the interval $[0,\infty )$.

Remark 5.

It is crucial to note that the parameter K in Theorem 8 is independent of the quasi-index of the space.

We now proceed to prove some coincidence point and common fixed-point theorems in b-metric spaces.

Theorem 9.

Suppose that $(\mathcal{Z},\mathcal{D},q)$ is a b-MS and $T,\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ are mappings satisfying $cl\left(T\mathcal{Z}\right)\subseteq \mathcal{S}\mathcal{Z}$ and

$$\begin{array}{c}\hfill \mathcal{D}(Tx,Ty)\le \varphi \left(\mathcal{D}\right(\mathcal{S}x,\mathcal{S}y\left)\right)+B\mathcal{D}(Tx,\mathcal{S}x)+C\mathcal{D}(Ty,\mathcal{S}y),\end{array}$$

(8)

where $\varphi :{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ is continuous and $\varphi \left(0\right)=0$ and $B,C$ are non-negative with $C\in [0,1).$ Moreover, assume that the b-metric is continuous. If $\mathcal{S}$ and T satisfy the E.A. property [33], i.e., there exists a sequence ${\left(z_n\right)}_{n\in \mathbb{N}}$ in $\mathcal{Z}$, such that ${\displaystyle \underset{n\to \infty }{lim}}T{z}_n={\displaystyle \underset{n\to \infty }{lim}}\mathcal{S}{z}_n$, then $\mathcal{S}$ and T have a coincidence point.

Proof. 

From the given E.A. property of mappings $\mathcal{S}$ and T, there exists a sequence $\left(z_n\right)$ and $u\in \mathcal{Z}$, such that ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{S}{z}_n={\displaystyle \underset{n\to \infty }{lim}}T{z}_n=u.$ Since u is the limit of sequence $T{z}_n,u\in cl\left(T\mathcal{Z}\right).$ Therefore, there exists some $\widehat{t}\in \mathcal{Z}$, such that $u=\mathcal{S}\widehat{t}.$ Taking $x=z_n$ and $y=\widehat{t}$ into (8), one obtains the following inequality:

$$\begin{array}{c}\hfill \mathcal{D}(T{z}_n,T\widehat{t})\le \varphi \left(\mathcal{D}(\mathcal{S}{z}_n,\mathcal{S}\widehat{t})\right)+B\mathcal{D}(T{z}_n,\mathcal{S}{z}_n)+C\mathcal{D}(T\widehat{t},\mathcal{S}\widehat{t}).\end{array}$$

Tending n to infinity, one has

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}\widehat{t},T\widehat{t})\le \varphi \left(\mathcal{D}(\mathcal{S}\widehat{t},\mathcal{S}\widehat{t})\right)+B\mathcal{D}(u,u)+C\mathcal{D}(T\widehat{t},\mathcal{S}\widehat{t}).\end{array}$$

Therefore, $(1-C)\mathcal{D}(T\widehat{t},S\widehat{t})\le 0.$ This implies that $T\widehat{t}$ and $S\widehat{t}$ coincide, i.e., $T\widehat{t}=S\widehat{t}$. □

In Theorem 9, if $\varphi$ is the constant function, say $\varphi \left(y\right)=A$, for all $y\in {\mathbb{R}}_{\ge 0},$ then we obtain the following result.

Corollary 6.

Suppose that $(\mathcal{Z},\mathcal{D},q)$ is a b-MS and $T,\mathcal{S}:X\to X$ are mappings satisfying $cl\left(T\mathcal{Z}\right)\subseteq \mathcal{S}\mathcal{Z}$ and

$$\begin{array}{c}\hfill \mathcal{D}(Tx,Ty)\le A\mathcal{D}(\mathcal{S}x,\mathcal{S}y)+B\mathcal{D}(Tx,\mathcal{S}x)+C\mathcal{D}(Ty,\mathcal{S}y),\end{array}$$

(9)

where $A,B,C$ are non-negative with $C\in [0,1).$ Moreover, assume that $\mathcal{D}$ is continuous. If $\mathcal{S}$ and T satisfy the E.A. property, then $\mathcal{S}$ and T have a coincidence point.

If all the constants coincide, that is, $A=B=C$, then we obtain the following:

Corollary 7.

Suppose that $(\mathcal{Z},\mathcal{D},q)$ is a b-MS and $T,S:X\to X$ are mappings satisfying $cl\left(TX\right)\subseteq SX$ and

$$\begin{array}{c}\hfill \mathcal{D}(Tx,Ty)\le A\left[\mathcal{D}\right(\mathcal{S}x,\mathcal{S}y)+\mathcal{D}(Tx,\mathcal{S}x)+\mathcal{D}(Ty,\mathcal{S}y\left)\right],\end{array}$$

(10)

where $A\in [0,1)$. Moreover, assume that $\mathcal{D}$ is continuous. If $\mathcal{S}$ and T satisfy the E.A. property, then $\mathcal{S}$ and T have a coincidence point.

The following theorems on common fixed points of two self-mappings can be established easily on the lines of proof of Theorems 7 and 8.

Theorem 10.

Let $(\mathcal{Z},\mathcal{D},q)$ be a complete b-MS and $U,\mathcal{S}$ be self-mappings on $\mathcal{Z}$ satisfying

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,Uy)\le M\mathcal{D}(x,y)+K\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,Uy\left)\right\}forallx,y\in \mathcal{Z},\end{array}$$

(11)

where $M\in [0,{\displaystyle \frac{1}{q^2}})$ and $K\in [0,{\displaystyle \frac{1}{q^2}}).$ Suppose that ${\left(x_n\right)}_{n\in \mathbb{N}}$ is an approximating fixed-point sequence of both U and $\mathcal{S}$, then $\mathcal{F}\left(\mathcal{S}\right)\cap \mathcal{F}\left(\mathcal{U}\right)=\left\{\vartheta \right\}\subseteq \mathcal{Z}$ and the approximating fixed-point sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ converges to $\vartheta .$

Theorem 11.

Let $(\mathcal{Z},\mathcal{D},q)$ be a complete b-MS with continuous b-metric $\mathcal{D}$ and coefficient q, and $U,\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ be continuous mappings satisfying

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,Uy)\le M\mathcal{D}(x,y)+K\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,Uy\left)\right\}forallx,y\in \mathcal{Z},\end{array}$$

(12)

where $M\in [0,{\displaystyle \frac{1}{q^2}})$ and $K\in [0,\infty ).$ Suppose that there exists a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ such that $\mathcal{D}(x_n,U{x}_n)$ and $\mathcal{D}(x_n,\mathcal{S}{x}_n)$ tend to 0 as $n\to \infty .$ Then, $\mathcal{F}\left(\mathcal{S}\right)\cap \mathcal{F}\left(\mathcal{U}\right)=\left\{\vartheta \right\}\subseteq \mathcal{Z}$ and for each $x\in \mathcal{Z},{\left({\mathcal{S}}^{n}x\right)}_{n\in \mathbb{N}}$ converges to ϑ.

On setting $M=0$, we get the following theorems (of Kannan-type) involving two mappings in a b-MS.

Corollary 8.

Let $(\mathcal{Z},\mathcal{D},q)$ be a complete b-metric space and let $U,\mathcal{S}$ be self-mappings on $\mathcal{Z}$ satisfying

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,Uy)\le K\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,Uy\left)\right\}forallx,y\in \mathcal{Z},\end{array}$$

(13)

where $K\in [0,{\displaystyle \frac{1}{q^2}}).$ Suppose that ${\left(x_n\right)}_{n\in \mathbb{N}}$ is an approximating fixed-point sequence of both $\mathcal{S}$ and U, then $\mathcal{F}\left(\mathcal{S}\right)\cap \mathcal{F}\left(\mathcal{U}\right)=\left\{\vartheta \right\}\in \mathcal{Z}$ and the approximating fixed-point sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ converges to $\vartheta .$

Corollary 9.

Let $(\mathcal{Z},\mathcal{D},q)$ be a complete b-MS, where the b-metric $\mathcal{D}$ is continuous and $\mathcal{S},U:\mathcal{Z}\to \mathcal{Z}$ be continuous mappings satisfying

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,Uy)\le K\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,Uy\left)\right\}forallx,y\in \mathcal{Z},\end{array}$$

(14)

where $K\in [0,\infty ).$ Suppose that there exists a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$, such that $\mathcal{D}(x_n,U{x}_n)$ and $\mathcal{D}(x_n,\mathcal{S}{x}_n)$ tend to 0 as $n\to \infty .$ Then, $\mathcal{F}\left(\mathcal{S}\right)\cap \mathcal{F}\left(\mathcal{U}\right)=\left\{\vartheta \right\}\in \mathcal{Z}$ and for each $x\in \mathcal{Z},{\left({\mathcal{S}}^{n}x\right)}_{n\in \mathbb{N}}$ converges to ϑ.

Theorem 12.

Suppose that $(\mathcal{Z},\mathcal{D},q)$ is a complete b-MS, $\mathcal{S}$ and U are asymptotically regular mappings on $\mathcal{Z}$ satisfying the following inequality:

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,Uy)\le M\mathcal{D}(x,y)+K\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,Uy\left)\right\}forallx,y\in \mathcal{Z},\end{array}$$

(15)

where $M\in [0,1)$ and $K\in [0,\infty ).$ Suppose further that $\mathcal{S}$ and U are either k-continuous for some $k\ge 1$ or orbitally continuous. Then, $\mathcal{F}\left(U\right)=\mathcal{F}\left(\mathcal{S}\right)=\left\{\vartheta \right\}.$ Further, ${\displaystyle \underset{n\to \infty }{lim}}{U}^{n}x={\displaystyle \underset{n\to \infty }{lim}}{\mathcal{S}}^{n}x=\vartheta$ for any $x\in \mathcal{Z}$.

Proof. 

Let us define a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ as $x_n={\mathcal{S}}^{n}x$ for any $x\in \mathcal{Z}$. We aim to establish that it is Cauchy in $\mathcal{Z}.$ We estimate the b-metric distance between $x_m$ and $x_n$, where $m>n,$ as follows:

$$\begin{array}{cc}\hfill \mathcal{D}(x_m,x_n)& =\mathcal{D}({\mathcal{S}}^{m}x,{\mathcal{S}}^{n}x)\le q[\mathcal{D}({\mathcal{S}}^{m}x,U^{n}x)+\mathcal{D}(U^{n}x,{\mathcal{S}}^{n}x)].\hfill \end{array}$$

Utilizing (15), one obtains

$$\begin{array}{c}\mathcal{D}(x_n,x_m)\le Mq·\mathcal{D}({\mathcal{S}}^{m-1}x,U^{n-1}x)+Kq·[\mathcal{D}({\mathcal{S}}^{m}x,{\mathcal{S}}^{m-1}x)+\mathcal{D}(U^{n}x,U^{n-1}x)]\hfill \\ \hfill +q·\mathcal{D}(U^{n}x,{\mathcal{S}}^{n}x).\end{array}$$

(16)

Let us make some statements for our proof.

Firstly, we prove that ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}({\mathcal{S}}^{n}x,U^{n}x)=0.$ We consider the nontrivial scenario where the self-mappings U and $\mathcal{S}$ are distinct.

• Case 1 $(M=0)$: In this case, (15) reduces to

  $$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,Uy)\le K\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,Uy\left)\right\},\end{array}$$

  for all $x,y\in \mathcal{Z}.$ Replacing x and y by ${\mathcal{S}}^{n}x$ and $U^{n}x$, respectively, in the last inequality, we obtain

  $$\begin{array}{c}\hfill \mathcal{D}({\mathcal{S}}^{n+1}x,U^{n+1}x)\le K\{\mathcal{D}({\mathcal{S}}^{n}x,{\mathcal{S}}^{n+1}x)+\mathcal{D}(U^{n}x,U^{n+1}x)\},\end{array}$$

  for all $x\in \mathcal{Z}.$ Since the mappings are asymptotically regular, ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}({\mathcal{S}}^{n+1}x,U^{n+1}x)=0.$
• Case 2 $(M\ne 0)$: Let $r_n=\mathcal{D}({\mathcal{S}}^{n}x,U^{n}x)$, ${\delta }_n=1-M$ and ${\mathsf{\Psi }}_n={\displaystyle \frac{K}{1-M}}·[\mathcal{D}({\mathcal{S}}^{n}x,{\mathcal{S}}^{n+1}x)+\mathcal{D}(U^{n}x,U^{n+1}x)].$ Observe that ${\mathsf{\Psi }}_n\stackrel{n\to \infty }{\to }0$, and ${\displaystyle \sum _n}{\delta }_n=\infty .$ Following Lemma 1, we conclude that ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}({\mathcal{S}}^{n}x,U^{n}x)=0$.

From the above cases, ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}({\mathcal{S}}^{n}x,U^{n}x)=0$ holds for all $x\in \mathcal{Z}.$

Estimate the distance, as follows:

$$\begin{array}{cc}\hfill \mathcal{D}({\mathcal{S}}^{m}x,U^{n}x)& \le q·[\mathcal{D}({\mathcal{S}}^{m}x,U^{m}x)+\mathcal{D}(U^{m}x,U^{n}x)]\hfill \\ \hfill & \le q·\mathcal{D}({\mathcal{S}}^{m}x,U^{m}x)+q^2[\mathcal{D}(U^{m}x,U^{m-1}x)+\mathcal{D}(U^{m-1}x,U^{n}x)]\hfill \\ \hfill & \le q·\mathcal{D}({\mathcal{S}}^{m}x,U^{m}x)+q^2·\mathcal{D}(U^{m}x,U^{m-1}x)+q^3·\mathcal{D}(U^{m-1}x,U^{m-2}x)\hfill \\ \hfill & +\cdots +q^n·\mathcal{D}(U^{n+1}x,U^{n}x).\hfill \end{array}$$

Since U is asymptotically regular and ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}({\mathcal{S}}^{n}x,U^{n}x)=0$, we obtain $\mathcal{D}({\mathcal{S}}^{m}x,U^{n}x)\stackrel{n\to \infty }{\to }0.$

Keeping the limit obtained and tending n to infinity in (16) yields ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}({\mathcal{S}}^{n}x,{\mathcal{S}}^{m}x)=0.$ That is, ${\left(x_n\right)}_{n\in \mathbb{N}}$ is a Cauchy sequence in $\mathcal{Z}.$ The completeness of $\mathcal{Z}$ ensures the existence of some $\vartheta \in \mathcal{Z}$ such that ${\displaystyle \underset{n\to \infty }{lim}}{x}_n=\vartheta .$ In addition, ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}(U^{n}x,\vartheta )=0$ follows from the following inequality.

$$\begin{array}{c}\hfill \mathcal{D}(U^{n}x,\vartheta )\le q·[\mathcal{D}(U^{n}x,{\mathcal{S}}^{n}x)+\mathcal{D}({\mathcal{S}}^{n}x,\vartheta )].\end{array}$$

It remains to show that $\vartheta \in \mathcal{F}\left(U\right),\mathcal{F}\left(\mathcal{S}\right).$

Since $\mathcal{S}$ is k-continuous,

$$\begin{array}{c}\hfill {\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}({\mathcal{S}}^{k-1}{x}_n,\vartheta )=0\Rightarrow {\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}({\mathcal{S}}^k{x}_n,\mathcal{S}\vartheta )=0.\end{array}$$

Since the limit is unique in a b-MS, $\vartheta \in \mathcal{F}\left(\mathcal{S}\right).$ Now, if $\mathcal{S}$ is orbitally continuous, as follows:

$$\begin{array}{c}\hfill {\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}(\mathcal{S}{x}_n,\mathcal{S}\vartheta )=0,\mathrm{whenever}{\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}(x_n,\vartheta )=0.\end{array}$$

Again, $\vartheta \in \mathcal{F}\left(\mathcal{S}\right).$ Based on the same line of arguments, $\vartheta \in \mathcal{F}\left(U\right).$ Consequently, $\vartheta \in \mathcal{F}\left(\mathcal{S}\right)\cap \mathcal{F}\left(U\right).$ The proof for $\left|\mathcal{F}\right(\mathcal{S})\cap \mathcal{F}(U\left)\right|=1$ is easy and straightforward. □

Corollary 10.

Suppose that $(\mathcal{Z},\mathcal{D},q)$ is a complete b-MS, U and $\mathcal{S}$ are self-mappings on $\mathcal{Z}$ for which $U^{\widehat{t}}$ and ${\mathcal{S}}^{\tilde{t}}$ are asymptotically regular for some positive integers $\widehat{t}$ and $\tilde{t}$, respectively. Suppose that the mappings satisfy the following condition:

$$\begin{array}{c}\hfill \mathcal{D}({\mathcal{S}}^{\tilde{t}}x,U^{\widehat{t}}y)\le M\mathcal{D}(x,y)+K\{\mathcal{D}(x,{\mathcal{S}}^{\tilde{t}}x)+\mathcal{D}(y,U^{\widehat{t}}y)\}forallx,y\in \mathcal{Z},\end{array}$$

(17)

where $M\in [0,1)$ and $K\in [0,\infty )$. Then, $\mathcal{F}\left(U\right)=\mathcal{F}\left(\mathcal{S}\right)=\left\{\vartheta \right\}$ provided that $U^{\widehat{t}}$ and ${\mathcal{S}}^{\tilde{t}}$ are either k-continuous for some $k\ge 1$ or orbitally continuous.

Theorem 13.

Suppose that $(\mathcal{Z},\mathcal{D},q)$ be a complete b-MS, and $\mathcal{S}$ and U are asymptotically regular mappings on $\mathcal{Z}$ satisfying the following inequality:

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,Uy)\le M\mathcal{D}(x,y)+K\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,Uy\left)\right\}forallx,y\in \mathcal{Z},\end{array}$$

(18)

where $M\in [0,1)$ and $K\in [0,{\displaystyle \frac{1}{q^2}}).$ Then, $\mathcal{F}\left(U\right)=\mathcal{F}\left(\mathcal{S}\right)=\left\{\vartheta \right\}.$ Further, ${\displaystyle \underset{n\to \infty }{lim}}{U}^{n}x={\displaystyle \underset{n\to \infty }{lim}}{\mathcal{S}}^{n}x=\vartheta$ for any $x\in \mathcal{Z}$.

Proof. 

We will follow the similar steps of Theorem 12 until the convergence of sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ to $\vartheta$ and ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}(U^{n}x,\vartheta )=0$. We are left to establish that $\vartheta \in \mathcal{F}\left(\mathcal{S}\right),\mathcal{F}\left(U\right).$ Estimate the distance between $\vartheta$ and $\mathcal{S}\vartheta$, as follows:

$$\begin{array}{cc}\hfill \mathcal{D}(\vartheta ,\mathcal{S}\vartheta )& \le q[\mathcal{D}(\vartheta ,{\mathcal{S}}^{n}x)+\mathcal{D}({\mathcal{S}}^{n}x,\mathcal{S}\vartheta )]\hfill \\ \hfill & \le q\mathcal{D}(\vartheta ,{\mathcal{S}}^{n}x)+q^2[\mathcal{D}({\mathcal{S}}^{n}x,U^{n}x)+\mathcal{D}(U^{n}x,\mathcal{S}\vartheta )]\hfill \\ \hfill & \le q\mathcal{D}(\vartheta ,{\mathcal{S}}^{n}x)+q^2\mathcal{D}({\mathcal{S}}^{n}x,U^{n}x)+M{q}^2\mathcal{D}(\vartheta ,U^{n-1}x)\hfill \\ \hfill & +K{q}^2[\mathcal{D}(U^{n-1}x,U^{n}x)+\mathcal{D}(\vartheta ,\mathcal{S}\vartheta )].\hfill \end{array}$$

Since ${\displaystyle \underset{n\to \infty }{lim}}{x}_n=\vartheta$, ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}(U^{n}x,\vartheta )=0$, ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}({\mathcal{S}}^{n}x,U^{n}x)=0$ and ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}(U^{n-1}x,U^{n}x)=0,$ the four summands on the $RHS$ except $K{q}^2\mathcal{D}(\vartheta ,\mathcal{S}\vartheta )$ reduce to zero as n approaches infinity. Therefore, we have

$$\begin{array}{c}\hfill (1-K{q}^2)\mathcal{D}(\vartheta ,\mathcal{S}\vartheta )\le 0.\end{array}$$

Since $K\in [0,{\displaystyle \frac{1}{q^2}})$, $\vartheta \in \mathcal{F}\left(\mathcal{S}\right).$ In an analogous manner, we can establish that $\vartheta \in \mathcal{F}\left(U\right).$ That is, $\vartheta \in \mathcal{F}\left(U\right)\cap \mathcal{F}\left(\mathcal{S}\right).$ We claim that $\left|\mathcal{F}\right(\mathcal{U})\cap \mathcal{F}(\mathcal{S}\left)\right|=\left\{\vartheta \right\}.$ On the contrary, assume that there exists $\widehat{\vartheta }(\ne \vartheta )$ in $\mathcal{Z}$ such that $\mathcal{F}\left(\mathcal{U}\right)\cap \mathcal{F}\left(\mathcal{S}\right)$ contains $\widehat{\vartheta }$. Selecting $x=\vartheta$ and $y=\widehat{\vartheta }$ and placing them into (18), we obtain

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}\vartheta ,U\widehat{\vartheta })\le M\mathcal{D}(\vartheta ,\widehat{\vartheta })+K\{\mathcal{D}(\vartheta ,\mathcal{S}\vartheta )+d(\widehat{\vartheta },U\widehat{\vartheta })\}.\end{array}$$

The second and third terms vanish and we get

$$\begin{array}{c}\hfill (1-M)\mathcal{D}(\vartheta ,\widehat{\vartheta })\le 0.\end{array}$$

Since $M\in (0,1),$ $\vartheta =\widehat{\vartheta }.$ Therefore, we get $\mathcal{F}\left(\mathcal{S}\right)\cap \mathcal{F}\left(U\right)=\left\{\vartheta \right\}.$ □

Corollary 11.

Suppose that $(\mathcal{Z},\mathcal{D},q)$ is a complete b-MS, U and $\mathcal{S}$ are self-mappings on $\mathcal{Z}$ for which $U^{\widehat{t}}$ and ${\mathcal{S}}^{\tilde{t}}$ are asymptotically regular for some positive integers $\widehat{t}$ and $\tilde{t}$, respectively. Suppose that the mappings satisfy following condition:

$$\begin{array}{c}\hfill \mathcal{D}({\mathcal{S}}^{\tilde{t}}x,U^{\widehat{t}}y)\le M\mathcal{D}(x,y)+K\{\mathcal{D}(x,{\mathcal{S}}^{\tilde{t}}x)+\mathcal{D}(y,U^{\widehat{t}}y)\}forallx,y\in \mathcal{Z},\end{array}$$

(19)

where $M\in [0,1)$ and $K\in [0,{\displaystyle \frac{1}{q^2}})$. Then, $\mathcal{F}\left(U\right)=\mathcal{F}\left(\mathcal{S}\right)=\left\{\vartheta \right\}$.

Example 6.

Consider $\mathcal{Z}=l^{{\displaystyle \frac{1}{2}}}$ a classical complete b-metric space with the b-metric given by $\mathcal{D}(u,v)={\left({\sum }_{j\in \mathbb{N}}{|u_j-v_j|}^{{\displaystyle \frac{1}{2}}}\right)}^2$ for all $u,v\in l^{{\displaystyle \frac{1}{2}}}.$ Let $v=(v_1,v_2,\dots )\in l^{{\displaystyle \frac{1}{2}}}$ be an arbitrary element in $l^{{\displaystyle \frac{1}{2}}}$. The mappings $\mathcal{S},\mathcal{U}:\mathcal{Z}\to \mathcal{Z}$ are defined as follows:

$$\mathcal{S}v=\left\{\begin{array}{cc}\left({\displaystyle \frac{2v_1}{3}},0,0,\dots \right)\hfill & for{v}_1\ge 0,\hfill \\ \left({\displaystyle \frac{v_1}{2}},0,0,\dots \right)\hfill & for{v}_1<0.\hfill \end{array}\right.$$

and

$$\mathcal{U}v=\left\{\begin{array}{cc}(0,0,0,\dots )\hfill & for{v}_1\le 0,\hfill \\ (-1,0,0,\dots )\hfill & for{v}_1\ge 0.\hfill \end{array}\right.$$

We can choose $0\le M<1$ and $K\ge 2$, such that both the mappings $\mathcal{S}$ and U satisfy the following inequality:

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}u,\mathcal{U}v)\le M\mathcal{D}(u,v)+K\left[\mathcal{D}\right(u,\mathcal{S}u)+\mathcal{D}(v,Uv\left)\right]\end{array}$$

holds for all $u,v\in \mathcal{Z}$. Therefore, the mappings satisfy all the hypotheses of Theorem 12. We conclude that $\mid \mathcal{F}\left(\mathcal{S}\right)\mid =\mid \mathcal{F}\left(U\right)\mid =1.$

Remark 6.

The significance of the above example lies in demonstrating that all the conditions required by the theorem are met for mappings defined within infinite-dimensional spaces.

In the following section, we build upon previous findings by examining Kannan-type mappings within preordered spaces. Unlike partial orders, the preordered binary relation is less stringent, providing a broader framework for analysis. This extension allows us to explore the behavior and properties of these mappings under a weaker relational structure.

4. Fixed-Point Theorems in Preordered b-Metric Space

We begin with the following outcome:

Theorem 14.

Let $(\mathcal{Z},\mathcal{D},q,⪯)$ be a preordered b-MS and let $\mathcal{S}$ be a self-mapping on $\mathcal{Z}$. Suppose that the following conditions hold:

• $\mathcal{Z}$ is complete;
• $\mathcal{S}$ is monotone;
• $\mathcal{D}and\mathcal{S}$ are continuous on their respective domains;
• There exists $x_0\in \mathcal{Z}$ such that $x_0⪯\mathcal{S}{x}_0$;
• ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}({\mathcal{S}}^{n}x,{\mathcal{S}}^{n+1}x)=0$;
• The following inequality holds for $x,y\in \mathcal{Z}$ with $x⪯y$;

  $$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,\mathcal{S}y)\le M\mathcal{D}(x,y)+K\{\mathcal{D}(x,\mathcal{S}x)+\mathcal{D}(y,\mathcal{S}y)\}forsomeM\in [0,{\displaystyle \frac{1}{q^2}}),K\in [0,\infty ).\end{array}$$

Then, $\mathcal{F}\left(\mathcal{S}\right)\ne \varnothing$, and if for all $(x,y)\in \mathcal{Z}\times \mathcal{Z}$, there exists $w\in \mathcal{Z}$ such that

$$\begin{array}{c}\hfill x⪯wandy⪯w,\end{array}$$

(20)

then $\mathcal{F}\left(\mathcal{S}\right)=\left\{u\right\}$ for some $u\in \mathcal{Z}.$

Moreover, for each $x_0\in \mathcal{Z}$ such that $x_0⪯\mathcal{S}{x}_0,$ the sequence ${\left({\mathcal{S}}^n{x}_0\right)}_{n\in \mathbb{N}}$ converges to the element of $\mathcal{F}\left(\mathcal{S}\right).$

Proof. 

Let $x_0\in \mathcal{Z}$ satisfy $x_0⪯\mathcal{S}{x}_0$. Construct a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}\in \mathcal{Z}$ as

$$\begin{array}{c}\hfill x_n=\mathcal{S}{x}_{n-1},n\in \mathbb{N}.\end{array}$$

(21)

Since $\mathcal{S}$ is monotone, one has

$$\begin{array}{c}\hfill x_0⪯\mathcal{S}{x}_0=x_1\Rightarrow x_1=\mathcal{S}{x}_0⪯\mathcal{S}{x}_1=x_2.\end{array}$$

By symmetry, we obtain $x_n⪯x_{n+1}$ for all $n\in {\mathbb{N}}_0.$ Choose some $m\in \mathbb{N}$ such that $m>n$; thus, we have

$$\begin{array}{cc}\hfill \mathcal{D}(x_n,x_m)& \le q[\mathcal{D}(x_n,x_{n+1})+\mathcal{D}(x_{n+1},x_m)]\hfill \\ \hfill & \le q\mathcal{D}(x_n,x_{n+1})+q^2[\mathcal{D}(x_{n+1},x_{m+1})+\mathcal{D}(x_{m+1},x_m)]\hfill \\ \hfill & \le q\mathcal{D}(x_n,x_{n+1})+q^2\mathcal{D}(x_{m+1},x_m)+M·q^2\mathcal{D}(x_n,x_m)\hfill \\ \hfill & +K·q^2[\mathcal{D}(x_n,x_{n+1})+\mathcal{D}(x_m,x_{m+1})].\hfill \end{array}$$

Simplifying the above inequality yields

$$\begin{array}{cc}\hfill (1-M·q^2)\mathcal{D}(x_n,x_m)& \le q·(1+K·q)·\mathcal{D}(x_n,x_{n+1})+q^2·(1+K)·\mathcal{D}(x_m,x_{m+1})].\hfill \end{array}$$

With n growing without bounds, all the terms on the RHS vanish owing to the hypothesis that $\mathcal{S}$ is asymptotically regular; therefore, ${\left(x_n\right)}_{n\in \mathbb{N}}$ is Cauchy in $\mathcal{Z}$. We know that our space $\mathcal{Z}$ is complete. As a result, this Cauchy sequence must converge, with the limit being $\widehat{u}.$ Since $\mathcal{S}$ is continuous, $x_n\stackrel{n\to \infty }{\to }u\Rightarrow \mathcal{S}{x}_n\stackrel{n\to \infty }{\to }\mathcal{S}u\Rightarrow x_{n+1}\stackrel{n\to \infty }{\to }\mathcal{S}u.$ We assume that there exists some $\widehat{v}\in \mathcal{F}\left(\mathcal{S}\right)$ with $\widehat{v}\ne \widehat{u}$. By hypothesis, there exists some $w_0\in \mathcal{Z}$ such that $\widehat{u}⪯w_0$ and $\widehat{v}⪯w_0$. Construct a sequence ${\left(w_n\right)}_{n\in \mathbb{N}}$, where $w_n=\mathcal{S}{w}_{n-1}$, letting it be the sequence consisting of successive approximations of $\mathcal{S}$ with the initial point as $w_0.$ Following the monotone property of $\mathcal{S},$ we get $\widehat{v}⪯w_1$ and $\widehat{u}⪯w_1$; moreover, with induction, we obtain $\widehat{v}⪯w_n$ and $\widehat{u}⪯w_n$ for all $n\ge 0.$

Case 1

If $\widehat{v}=w_{n_0}$ for some $n_0\in {\mathbb{N}}_0,$ then $\widehat{v}=\mathcal{S}\widehat{v}=\mathcal{S}{w}_{n_0}=w_{n_0+1}$ and inductively $w_n=\widehat{v}$ for all $n\ge n_0$, that is, ${\left(w_n\right)}_{n\in \mathbb{N}}$ is an eventually constant sequence and its limit is $\widehat{v}.$

Case 2

If $\widehat{v}⪯w_n$ for all $n\in {\mathbb{N}}_0$, then

$$\begin{array}{cc}\hfill \mathcal{D}(\widehat{v},w_{n+1})& =\mathcal{D}(\mathcal{S}\widehat{v},\mathcal{S}{w}_n)\hfill \\ \hfill & \le M·\mathcal{D}(\widehat{v},w_n)+K·\{\mathcal{D}(\widehat{v},\mathcal{S}\widehat{v})+\mathcal{D}(w_n,\mathcal{S}{w}_n)\}\hfill \\ \hfill & \le M·q·[\mathcal{D}(\widehat{v},w_{n+1})+\mathcal{D}(w_{n+1},w_n)]+K·\{\mathcal{D}(\widehat{v},\mathcal{S}\widehat{v})\hfill \\ \hfill & +\mathcal{D}({\mathcal{S}}^n{w}_0,{\mathcal{S}}^{n+1}{w}_0)\}.\hfill \end{array}$$

Upon simplification, we have

$$\begin{array}{c}\hfill (1-M·q)\mathcal{D}(\widehat{v},w_{n+1})\le M·q·\mathcal{D}(w_{n+1},w_n)+K·\{\mathcal{D}(\widehat{v},\mathcal{S}\widehat{v})+\mathcal{D}({\mathcal{S}}^n{w}_0,{\mathcal{S}}^{n+1}{w}_0)\}.\end{array}$$

Since $\mathcal{S}$ is asymptotically regular, all the terms on the $RHS$ become zero on letting n tend to infinity. Therefore, ${\left(w_n\right)}_{n\in \mathbb{N}}$ converges to $\widehat{v}.$

Thus, from the uniqueness of the limit in $\mathcal{Z}$, it follows that $\widehat{u}=\widehat{v},$ so $\mathcal{F}\left(\mathcal{S}\right)=\left\{\widehat{u}\right\}.$ □

Remark 7.

We can weaken the condition that $\mathcal{S}$ is continuous by requiring that $\mathcal{S}$ is either k-continuous or orbitally continuous in Theorem 14.

Theorem 15.

Let $(\mathcal{Z},\mathcal{D},q,⪯)$ be a preordered b-MS and let $\mathcal{S}$ be a self-map on $\mathcal{Z}$. Suppose the map and the space satisfy the following conditions:

1. 

$(\mathcal{Z},\mathcal{D},q)$ is complete;

2. 

$\mathcal{S}$ is monotone;

3. 

$(\mathcal{Z},\mathcal{D},q⪯)$ is nondecreasing regular;

4. 

There exists some $x_0\in \mathcal{Z}$ such that $x_0⪯\mathcal{S}{x}_0$;

5. 

$\mathcal{S}$ is asymptotically regular;

6. 

For some $0\le M<{\displaystyle \frac{1}{q^2}}$ and $0\le K<{\displaystyle \frac{1}{q}}$, the following inequality holds:

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,\mathcal{S}y)\le M·\mathcal{D}(x,y)+K·\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,\mathcal{S}y\left)\right\}\end{array}$$

(22)

for all $x,y\in \mathcal{Z},$ with $x⪯y$.

Then, $\mathcal{F}\left(\mathcal{S}\right)\ne \varnothing ,$ and $\left|\mathcal{F}\right(\mathcal{S}\left)\right|=1$, if (20) is satisfied. Furthermore, for each $x_0\in \mathcal{Z}$ such that $x_0⪯\mathcal{S}{x}_0,$ the sequence ${\left({\mathcal{S}}^n{x}_0\right)}_{n\in \mathbb{N}}$ converges to the element of $\mathcal{F}\left(\mathcal{S}\right).$

Proof. 

Following the proof of Theorem 14, we have a monotone sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ as $x_n=\mathcal{S}{x}_{n-1}$ which is Cauchy and in turn converges to some $\widehat{u}\in \mathcal{Z}.$ With the hypothesis that $(\mathcal{Z},\mathcal{D},q,⪯)$ is a nondecreasing regular space, we have $x_n⪯\widehat{u}$ for all $n\in \mathbb{N}$. Now, we claim that $\widehat{u}\in \mathcal{F}\left(\mathcal{S}\right)$. Estimate the distance between $\widehat{u}$ and its image, as follows:

$$\begin{array}{cc}\hfill \mathcal{D}(\widehat{u},\mathcal{S}\widehat{u})& \le q[\mathcal{D}(\widehat{u},x_{n+1})+\mathcal{D}(x_{n+1},\mathcal{S}\widehat{u})]\hfill \\ \hfill & \le q\mathcal{D}(\widehat{u},x_{n+1})+M·q\mathcal{D}({\mathcal{S}}^n{x}_0,\widehat{u})+K·q[\mathcal{D}({\mathcal{S}}^n{x}_0,{\mathcal{S}}^{n+1}{x}_0)+\mathcal{D}(\widehat{u},\mathcal{S}\widehat{u})]\hfill \end{array}$$

Upon simplification, we have the following:

$$\begin{array}{cc}\hfill (1-K·q)\mathcal{D}(\widehat{u},\mathcal{S}\widehat{u})& \le q\mathcal{D}(\widehat{u},x_{n+1})+M·q\mathcal{D}({\mathcal{S}}^n{x}_0,\widehat{u})+K·q\mathcal{D}({\mathcal{S}}^n{x}_0,{\mathcal{S}}^{n+1}{x}_0).\hfill \end{array}$$

Since ${\left(x_n\right)}_{n\in \mathbb{N}}$ converges to $\widehat{u}$, $\mathcal{S}$ is asymptotically regular and $K<{\displaystyle \frac{1}{q}}$, as n approaches ∞ results in $\widehat{u}\in \mathcal{F}\left(\mathcal{S}\right).$ The proof for $\mid \mathcal{F}\left(\mathcal{S}\right)\mid =1$ is immediate. □

Example 7.

Let $\mathcal{Z}=[-1,1]$ be endowed with the b-metric $\mathcal{D}(u,v)={|u-v|}^2$ for all $u,v\in \mathcal{Z}.$ Endow the space $\mathcal{Z}$ with the partial order

$$\begin{array}{c}\hfill u⪯v\iff (u=voru<v\le 0).\end{array}$$

Consider a mapping $\mathcal{S}:\mathcal{Z}\to \mathcal{Z}$ defined as $\mathcal{S}u={\displaystyle \frac{u}{4}}$ for $u\in [-1,0]$ and $\mathcal{S}u=u^e$ for $u\in (0,1].$ We can check that ${\displaystyle \underset{m\to \infty }{lim}}\mathcal{D}({\mathcal{S}}^m\left(u\right),{\mathcal{S}}^{m+1}\left(u\right))=0$ (see Figure 9). Choose arbitrary $u,v\in \mathcal{Z}$ such that $u⪯v$. If $u=v,$ then the inequality is obviously true. If the elements $u,v$ do not coincide, then with the partial ordering $u<v\le 0$ the inequality

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}u,\mathcal{S}v)\le K\left(\mathcal{D}\right(u,\mathcal{S}\left(u\right))+\mathcal{D}(v,\mathcal{S}\left(v\right)\left)\right)\end{array}$$

holds for all $u,v\in [-1,0]$ with $M=0$ and $K={\displaystyle \frac{1}{9}}$ (see Figure 10). Thus, with this example, we have illustrated Theorem 15, showing that the preordering and the inequality are satisfied for elements that are comparable under the given binary relation defined on the space. It is important to note that the inequality is not satisfied for all elements in the space $\mathcal{Z}.$ If $u=0$ and $v=1$, the inequality is given by

$$\mathcal{D}(\mathcal{S}u,\mathcal{S}v)=1>\mathcal{D}(\mathcal{S}u,u)+\mathcal{D}(\mathcal{S}v,v)=0.$$

That is, the Kannan-type inequality does not hold for this pair of elements.

Remark 8.

It is crucial to note that if we do not impose the condition given by (20) on the elements of $\mathcal{Z}\times \mathcal{Z}$ in Theorems 14 and 15, there exists some map $\mathcal{S}$ whose fixed-point set $\mathcal{F}\left(\mathcal{S}\right)$ is not a singleton set. Consider $\mathcal{Z}=\{e_1,e_2,\cdots ,e_k\}\subseteq {\mathbb{R}}^k$, where $e_k=(0,0,\dots ,1,\dots ,0)$ where 1 is at the $k^{th}$ slot with the order

$$\begin{array}{c}\hfill u⪯v\iff u_j\le t_{j}forallj,\end{array}$$

where $u_j$ and $t_j$ are the jth respective components in the two elements $u,v$ of $\mathcal{Z}$. Then, $(\mathcal{Z},⪯)$ is a set with the preorder defined as above. Moreover, two elements are not comparable in the set unless they are the same. With the identity mapping $\mathcal{I}$ on a complete b-metric space $(\mathcal{Z},\mathcal{D},q)$ (where $\mathcal{D}(u,v)={\left({\displaystyle \sum _{j\in {\mathbb{N}}_k}}|u_j-v_j|\right)}^2,u=(u_1,\dots ,u_k),v=(v_1,v_2,v_3,\dots ,v_k)\in {\mathbb{R}}^k)$, all the hypotheses of Theorem 15 are satisfied, that is, $\mathcal{I}$ is asymptotically regular, non-decreasing. Also, the inequality

$$\begin{array}{cc}\hfill \mathcal{D}(\mathcal{I}(u_1,u_2,\dots ,u_k),\mathcal{I}(v_1,v_2,\dots ,v_k))\le K& [\mathcal{D}((u_1,u_2,\dots ,u_k),\mathcal{I}(u_1,u_2,\dots ,u_k))\hfill \\ \hfill & +\mathcal{D}((v_1,v_2,\dots ,v_k),\mathcal{I}(v_1,v_2,\dots ,v_k))]\hfill \end{array}$$

is true for all $K\in [0,\infty ),$ where $u_1,\dots ,u_k,v_1,\dots ,v_k\in \mathcal{Z}.$ The set $\mathcal{F}\left(\mathcal{I}\right)$ has k distinct elements since $e_j⪯e_j(=\mathcal{I}{e}_j)$ holds for all $j\in \{1,2,\dots ,k\}.$ The non-decreasing condition is also satisfied since any monotone non-decreasing sequence which converges must be a constant sequence and since the upper bound is the limit of the sequence.

Example 8.

Let $\mathcal{Z}=\{e_1,e_2,\dots ,e_k\}\subseteq l^{{\displaystyle \frac{1}{2}}}$ (a sequence space which forms a complete b-metric space with respect to $\mathcal{D}(u,v)={\left({\displaystyle \sum _j}\mid u_j-v_j{\mid }^{{\displaystyle \frac{1}{2}}}\right)}^{2}forallu=(u_1,u_2,\dots ),v=(v_1,v_2,\dots )),$ where $e_k$ represents an element of $l^{{\displaystyle \frac{1}{2}}}$ whose kth component is 1 and remaining zero. Now, we consider the order

$$\begin{array}{c}\hfill v⪯t\iff v_j⪯t_j,\end{array}$$

where $v=(v_1,v_2,\dots )$, $t=(t_1,t_2,\dots )\in \mathcal{Z}.$

Thus, $\mathcal{Z}$ is a set with partial order ⪯, such that two elements are comparable if and only if the elements coincide. Consider the dilation map $\mathcal{G}:l^{{\displaystyle \frac{1}{2}}}\to l^{{\displaystyle \frac{1}{2}}}$ defined as $\mathcal{G}(u_1,u_2,u_3,\dots )=({\displaystyle \frac{u_1}{2}},{\displaystyle \frac{u_2}{2}},{\displaystyle \frac{u_3}{2}},\dots )$ for all $u=(u_1,u_2,\dots )\in l^{{\displaystyle \frac{1}{2}}}.$ It is immediate that the map $\mathcal{G}$ is continuous, monotone and asymptotically regular. Moreover, $(\mathcal{Z},\mathcal{D},⪯)$ is nondecreasing regular (since the limit of the sequence is the upper bound of the monotone sequence). This example is vital in the sense that this space is an infinite dimensional space where all the hypotheses of Theorem 15 are true but the uniqueness of the fixed point is violated.

Corollary 12.

Let $(\mathcal{Z},\mathcal{D},q,⪯)$ be a preordered complete b-MS and let $\mathcal{S}$ be a self-map on $\mathcal{Z}$. Suppose the map and the space satisfy all the hypotheses of Theorem 14 and instead of inequality (22), the following inequality holds for some $0\le K<\infty$

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,\mathcal{S}y)\le K·\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,\mathcal{S}y\left)\right\}\end{array}$$

(23)

for all $x,y\in \mathcal{Z},$ with $x⪯y$.

Then, $\mathcal{F}\left(\mathcal{S}\right)\ne \varnothing ,$ and $\left|\mathcal{F}\right(\mathcal{S}\left)\right|=1$, if (20) is satisfied. Furthermore, for each $x_0\in \mathcal{Z}$ such that $x_0⪯\mathcal{S}{x}_0,$ the sequence ${\left({\mathcal{S}}^n{x}_0\right)}_{n\in \mathbb{N}}$ converges to the element of $\mathcal{F}\left(\mathcal{S}\right).$

From Theorem 15, the subsequent corollary can be derived.

Corollary 13.

Let $(\mathcal{Z},\mathcal{D},⪯)$ be a preordered metric space and let $\mathcal{S}$ be a self-map on $\mathcal{Z}$. Suppose the map and the space satisfy the following conditions:

1. 

$(\mathcal{Z},\mathcal{D})$ is complete;

2. 

$\mathcal{S}$ is monotone;

3. 

$(\mathcal{Z},\mathcal{D},⪯)$ be nondecreasing regular;

4. 

There exists some $x_0\in \mathcal{Z}$ such that $x_0⪯\mathcal{S}{x}_0$;

5. 

$\mathcal{S}$ is asymptotically regular;

6. 

For some $0\le M<1$ and $0\le K<1$, the following inequality holds:

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,\mathcal{S}y)\le M·\mathcal{D}(x,y)+K·\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,\mathcal{S}y\left)\right\}\end{array}$$

(24)

for all $x,y\in \mathcal{Z},$ with $x⪯y$.

Then, $\mathcal{F}\left(\mathcal{S}\right)\ne \varnothing ,$ and $\left|\mathcal{F}\right(\mathcal{S}\left)\right|=1$, if (20) is satisfied. Furthermore, for each $x_0\in \mathcal{Z}$ such that $x_0⪯\mathcal{S}{x}_0,$ the sequence ${\left({\mathcal{S}}^n{x}_0\right)}_{n\in \mathbb{N}}$ converges to the element of $\mathcal{F}\left(\mathcal{S}\right).$

Upon setting $M=0$ in (24), we get a Kannan-type contraction in a preordered metric space.

Corollary 14.

Let $(\mathcal{Z},\mathcal{D},⪯)$ be a preordered MS and let $\mathcal{S}$ be a self-map on $\mathcal{Z}$. Suppose the map and the space satisfy the following conditions:

1. 

$(\mathcal{Z},\mathcal{D})$ is complete;

2. 

$\mathcal{S}$ is monotone;

3. 

$(\mathcal{Z},\mathcal{D},⪯)$ be nondecreasing regular;

4. 

There exists some $x_0\in \mathcal{Z}$ such that $x_0⪯\mathcal{S}{x}_0$;

5. 

$\mathcal{S}$ is asymptotically regular;

6. 

For some $0\le K<1$, the following inequality holds:

$$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,\mathcal{S}y)\le K·\left\{\mathcal{D}\right(x,\mathcal{S}x)+\mathcal{D}(y,\mathcal{S}y\left)\right\}\end{array}$$

(25)

for all $x,y\in \mathcal{Z},$ with $x⪯y$.

Then, $\mathcal{F}\left(\mathcal{S}\right)\ne \varnothing ,$ and $\left|\mathcal{F}\right(\mathcal{S}\left)\right|=1$, if (20) is satisfied. Furthermore, for each $x_0\in \mathcal{Z}$ such that $x_0⪯\mathcal{S}{x}_0,$ the sequence ${\left({\mathcal{S}}^n{x}_0\right)}_{n\in \mathbb{N}}$ converges to the element of $\mathcal{F}\left(\mathcal{S}\right).$

Remark 9.

As a special case of our result, we have the above generalization of Theorem 3.8 of [12].

We derive the aforementioned Kannan-type contractions involving a pair of mappings in b-metric spaces. The proof of the following theorem is omitted.

Theorem 16.

Let $(\mathcal{Z},\mathcal{D},q,⪯)$ be a preordered b-MS and let $\mathcal{S}$ and U be two self-mappings on $\mathcal{Z}$. Suppose that the following conditions hold:

• The space is complete;
• $\mathcal{D}$ is continuous;
• $\mathcal{S}$ and U are monotone and continuous;
• There exists $x_0\in \mathcal{Z}$ such that $x_0⪯\mathcal{S}{x}_0,$
• ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}({\mathcal{S}}^{n}x,{\mathcal{S}}^{n+1}x)=0forallx\in \mathcal{Z}$;
• ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}(U^{n}x,U^{n+1}x)=0forallx\in \mathcal{Z}$;
• The following inequality holds for $x,y\in \mathcal{Z}$ with $x⪯y$:

  $$\begin{array}{c}\hfill \mathcal{D}(\mathcal{S}x,Uy)\le M\mathcal{D}(x,y)+K\{\mathcal{D}(x,\mathcal{S}x)+\mathcal{D}(y,Uy)\}forsomeM\in [0,{\displaystyle \frac{1}{q}}),K\in [0,\infty ).\end{array}$$
  Then, $\mathcal{F}\left(\mathcal{S}\right),\mathcal{F}\left(U\right)\ne \varnothing$, and if for all $(x,y)\in \mathcal{Z}\times \mathcal{Z}$, there exists $w\in \mathcal{Z}$ such that

  $$\begin{array}{c}\hfill x⪯wandy⪯w,\end{array}$$

  (26)
  then $\mathcal{F}\left(\mathcal{S}\right)=\mathcal{F}\left(U\right)=\left\{u\right\}$ for some $u\in \mathcal{Z}.$

Moreover, for each $x_0\in \mathcal{Z}$ such that $x_0⪯\mathcal{S}{x}_0,$ the sequence ${\left({\mathcal{S}}^n{x}_0\right)}_{n\in \mathbb{N}}$ converges to the element of $\mathcal{F}\left(\mathcal{S}\right).$

Remark 10.

For $q=1$, the results translate to the context of a preordered metric space for a pair of mappings.

5. Random Fixed-Point Theorems in b-Metric Space

In this section, we use similar concepts and definitions related to random mapping and measurability as those used in [12].

The ensuing fixed-point theorem represents a randomized version of Theorem 3.

Theorem 17.

Let $(\nabla ,\wp )$ be a measurable space, $(\mathcal{Z},\mathcal{D},q)$ be a complete and separable b-metric space with a continuous b-metric, and let $\mathcal{S}:\nabla \times \mathcal{Z}\to \mathcal{Z}$ be a random continuous operator. Assume that $\mathcal{S}$ is asymptotically regular and there exist functions $M:\nabla \to \mathbb{R}$ with the range of the function M, $R\left(M\right)=[0,{\displaystyle \frac{1}{q^2}})$ and $K:\nabla \to [0,\infty )$ such that for each $w\in \nabla$, $\mathcal{S}$ satisfies the following inequality:

$$\begin{array}{c}\hfill \mathcal{D}\left(\mathcal{S}\right(w,x),\mathcal{S}(w,y\left)\right)\le M\left(w\right)\mathcal{D}(x,y)+K\left(w\right)\left\{\mathcal{D}\right(x,\mathcal{S}(w,x))+\mathcal{D}(y,\mathcal{S}(w,y)\left)\right\},\end{array}$$

(27)

for all $x,y\in \mathcal{Z}.$ Then, $\mathcal{S}$ has a unique random fixed point.

Proof. 

Fix a measurable function $z_0:\nabla \to \mathcal{Z}$. If $\mathcal{S}(w,z_0\left(w\right))=z_0\left(w\right)$ for each $w\in \nabla$, then $z_0$ is a random fixed point of $\mathcal{S}.$ Suppose that, for some $w\in \nabla ,$ $\mathcal{S}(w,z_0\left(w\right))\ne z_0\left(w\right).$ We define the sequence

$$\begin{array}{c}\hfill u_n\left(w\right)=\mathcal{S}(w,u_{n-1}\left(w\right)),\end{array}$$

with the starting point $u_0\left(w\right)=z_0\left(w\right)$, for all $w\in \nabla$ and $n\in \mathbb{N}.$ Choose some natural number m such that $m>n$ and estimate the distance between $u_n\left(w\right)$ and $u_m\left(w\right)$ as follows:

$$\begin{array}{cc}\hfill \mathcal{D}(u_n\left(w\right),u_m\left(w\right))& \le q[\mathcal{D}(u_n\left(w\right),u_{n+1}\left(w\right))+\mathcal{D}(u_{n+1}\left(w\right),u_m\left(w\right))]\hfill \\ \hfill & \le q\mathcal{D}(u_n\left(w\right),u_{n+1}\left(w\right))+q^2\mathcal{D}(u_{n+1}\left(w\right),u_{m+1}\left(w\right))\hfill \\ \hfill & +q^2\mathcal{D}(u_{m+1}\left(w\right),u_m\left(w\right))\hfill \\ \hfill & \le q\mathcal{D}(u_n\left(w\right),u_{n+1}\left(w\right))+q^2\mathcal{D}(u_{m+1}\left(w\right),u_m\left(w\right))\hfill \\ \hfill & +q^2[M\left(w\right)\mathcal{D}(u_n\left(w\right),u_m\left(w\right))+K\left(w\right)\{\mathcal{D}(u_n\left(w\right),u_{n+1}\left(w\right))\hfill \\ \hfill & +\mathcal{D}(u_m\left(w\right),u_{m+1}\left(w\right))\left\}\right].\hfill \end{array}$$

Simplifying the above inequality yields the following:

$$\begin{array}{cc}\hfill (1-q^{2}M\left(w\right))\mathcal{D}(u_n\left(w\right),u_m\left(w\right))& \le q\mathcal{D}(u_n\left(w\right),u_{n+1}\left(w\right))+q^2\mathcal{D}(u_{m+1}\left(w\right),u_m\left(w\right))\hfill \\ \hfill & +q^{2}K\left(w\right)\{\mathcal{D}(u_n\left(w\right),u_{n+1}\left(w\right))+\mathcal{D}(u_m\left(w\right),u_{m+1}\left(w\right))\}\hfill \\ \hfill & =q\mathcal{D}({\mathcal{S}}^n(w,u_0\left(w\right)),{\mathcal{S}}^{n+1}(w,u_0\left(w\right)))\hfill \\ \hfill & +q^2\mathcal{D}({\mathcal{S}}^m(w,u_0\left(w\right)),{\mathcal{S}}^{m+1}(w,u_0\left(w\right)))\hfill \\ \hfill & +q^{2}K\left(w\right)\{\mathcal{D}({\mathcal{S}}^n(w,u_0\left(w\right)),{\mathcal{S}}^{n+1}(w,u_0\left(w\right)))\hfill \\ \hfill & +\mathcal{D}({\mathcal{S}}^m(w,u_0\left(w\right)),{\mathcal{S}}^{m+1}(w,u_0\left(w\right)))\}.\hfill \end{array}$$

Letting $n\to \infty$ forces ${\left(u_n\left(w\right)\right)}_{n\in \mathbb{N}}$ to be a Cauchy sequence in $\mathcal{Z}$ for each fixed $w\in \nabla .$ The completeness of the space ensures the existence of $\widehat{u}\left(w\right)\in \mathcal{Z},w\in \nabla ,$ such that $\widehat{u}\left(w\right)={\displaystyle \underset{n\to \infty }{lim}}{u}_n\left(w\right).$ It is immediate to observe that for each natural number $n,$ the function $w\to u_n\left(w\right)$ is measurable; therefore, the pointwise limit $\widehat{u}:\nabla \to \mathcal{Z}$ is also measurable. We aim to establish that the pointwise limit is a random fixed point of $\mathcal{S}$. For each fixed $w\in \nabla ,$

$$\begin{array}{c}\hfill \mathcal{D}(u_n\left(w\right),\mathcal{S}(w,u_n\left(w\right))\stackrel{n\to \infty }{⟶}\mathcal{D}(\widehat{u}\left(w\right),\mathcal{S}(w,\widehat{u}\left(w\right)),\end{array}$$

this step follows on account of the continuity of the b-metric and the self-mapping $\mathcal{S}$. However, the uniqueness of the random fixed point still remains to be shown. Suppose that $\widehat{\widehat{z}}\left(w\right)(\ne \widehat{u}\left(w\right))$ is another random fixed point of $\mathcal{S}.$ In light of (27),

$$\begin{array}{cc}\hfill \mathcal{D}(\widehat{u}\left(w\right),\widehat{\widehat{z}}\left(w\right))& =\mathcal{D}(\mathcal{S}(w,\widehat{u}\left(w\right)),\mathcal{S}(w,\widehat{\widehat{z}}\left(w\right)))\hfill \\ \hfill & \le M\left(w\right)\mathcal{D}(\widehat{u}\left(w\right),\widehat{\widehat{z}}\left(w\right))+K\left(w\right)\left\{\mathcal{D}\right(\widehat{u}\left(w\right),\mathcal{S}(w,\widehat{u}\left(w\right))+\mathcal{D}(\widehat{\widehat{z}}\left(w\right),\mathcal{S}(w,\widehat{\widehat{z}}\left(w\right))\}\hfill \\ \hfill & <\mathcal{D}(\widehat{u}\left(w\right),\widehat{\widehat{z}}\left(w\right)),\hfill \end{array}$$

which is not possible. □

Theorem 4.8 of [12] is obtained as a corollary of the above theorem.

Corollary 15.

Let $(\nabla ,\wp )$ be a measurable space, $(\mathcal{Z},\mathcal{D})$ be a complete and separable metric space, and $\mathcal{S}:\nabla \times \mathcal{Z}\to \mathcal{Z}$ be a random continuous operator that is asymptotically regular. Assume that there exist functions $M:\nabla \to \mathbb{R}$ with $R\left(M\right)=[0,1)$ and $K:\nabla \to [0,\infty )$ such that for each $w\in \nabla$, $\mathcal{S}$ satisfies the following inequality:

$$\begin{array}{c}\hfill \mathcal{D}\left(\mathcal{S}\right(w,x),\mathcal{S}(w,y)\le M(w\left)\mathcal{D}\right(x,y)+K(w\left)\right\{\mathcal{D}(x,\mathcal{S}(w,x\left)\right)+\mathcal{D}(y,\mathcal{S}(w,y\left)\right)\},\end{array}$$

for all $x,y\in \mathcal{Z}.$ Then, $\mathcal{S}$ has a unique random fixed point.

On setting M as the zero function, we obtain the following result:

Corollary 16.

Let $(\nabla ,\wp )$ be a measurable space, $(\mathcal{Z},\mathcal{D})$ be a complete and separable metric space, and $\mathcal{S}:\nabla \times \mathcal{Z}\to \mathcal{Z}$ be a random continuous operator which is asymptotically regular. Assume that there exists a function $K:\nabla \to [0,\infty )$ such that for each $w\in \nabla$, $\mathcal{S}$ satisfies the following inequality:

$$\begin{array}{c}\hfill \mathcal{D}\left(\mathcal{S}\right(w,x),\mathcal{S}(w,y\left)\right)\le K\left(w\right)\left\{\mathcal{D}\right(x,\mathcal{S}(w,x))+\mathcal{D}(y,\mathcal{S}(w,y)\left)\right\},\end{array}$$

for all $x,y\in \mathcal{Z}.$ Then, $\mathcal{S}$ has a unique random fixed point.

In the following theorem, we drop the condition that $\mathcal{S}$ is a continuous random operator.

Theorem 18.

Let $(\nabla ,\wp )$ be a measurable space, $(\mathcal{Z},\mathcal{D},q)$ be a complete and separable b-metric space, and let $\mathcal{S}:\nabla \times \mathcal{Z}\to \mathcal{Z}$ be a random operator satisfying asymptotic regularity. Assume that there exist functions $M:\nabla \to \mathbb{R}$ with $R\left(M\right)=[0,{\displaystyle \frac{1}{q^2}})$ and $K:\nabla \to \mathbb{R}$ with $R\left(K\right)=[0,{\displaystyle \frac{1}{q}})$ such that for each $w\in \nabla$, $\mathcal{S}$ satisfies the following inequality:

$$\begin{array}{c}\hfill \mathcal{D}\left(\mathcal{S}\right(w,x),\mathcal{S}(w,y\left)\right)\le M\left(w\right)\mathcal{D}(x,y)+K\left(w\right)\left\{\mathcal{D}\right(x,\mathcal{S}(w,x))+\mathcal{D}(y,\mathcal{S}(w,y)\left)\right\},\end{array}$$

(28)

for all $x,y\in \mathcal{Z}.$ Then, $\mathcal{S}$ has a unique random fixed point.

Proof. 

We prove until the convergence of the sequence ${\left(u_n\left(w\right)\right)}_{n\in \mathbb{N}}$ to the mapping $\widehat{u}\left(w\right)$ following the preceding theorem. Let us show that the limit is the random fixed point of the random operator, that is, $\widehat{u}\left(w\right)=\mathcal{S}(w,\widehat{u}\left(w\right)).$ Estimate the b-metric distance between $\widehat{u}\left(w\right)$ and $\mathcal{S}(w,\widehat{u}\left(w\right))$, as follows:

$$\begin{array}{cc}\hfill \mathcal{D}(\widehat{u}\left(w\right),\mathcal{S}(w,\widehat{u}\left(w\right)))& \le q[\mathcal{D}(\widehat{u}\left(w\right),\mathcal{S}(w,u_n\left(w\right)))+\mathcal{D}(\mathcal{S}(w,u_n\left(w\right)),\mathcal{S}(w,\widehat{u}\left(w\right)))]\hfill \\ \hfill & \le q\mathcal{D}(\widehat{u}\left(w\right),u_{n+1}\left(w\right))+q[M\left(w\right)d(u_n\left(w\right),\widehat{u}\left(w\right))\hfill \\ \hfill & +K\left(w\right)\{d(u_n\left(w\right),\mathcal{S}(w,u_n\left(w\right)))+\mathcal{D}(\widehat{u}\left(w\right),\mathcal{S}(w,\widehat{u}\left(w\right))\}]\hfill \end{array}$$

On simplification, we have the following:

$$\begin{array}{cc}\hfill (1-qK\left(w\right))\mathcal{D}(\widehat{u}\left(w\right),\mathcal{S}(w,\widehat{u}\left(w\right)))& \le q\mathcal{D}(\widehat{u}\left(w\right),u_{n+1}\left(w\right))+q[M\left(w\right)d(u_n\left(w\right),\widehat{u}\left(w\right))\hfill \\ \hfill & +K\left(w\right)d(u_n\left(w\right),\mathcal{S}(w,u_n\left(w\right)))]\hfill \end{array}$$

Tending n to infinity yields $\widehat{u}\left(w\right)$ as a random fixed point of $\mathcal{S}.$ The remaining is immediate. □

On setting $q=1$ in the above theorem, we obtain the result in a metric space.

Corollary 17.

Let $(\nabla ,\wp )$ be a measurable space, $(\mathcal{Z},\mathcal{D})$ be a complete and separable metric space, and $\mathcal{S}:\nabla \times \mathcal{Z}\to \mathcal{Z}$ a random operator which is asymptotically regular. Assume that there exist functions $M:\nabla \to \mathbb{R}$ with $R\left(M\right)=[0,1)$ and $K:\nabla \to \mathbb{R}$ with $R\left(K\right)=[0,1)$ such that for each $w\in \nabla$, $\mathcal{S}$ satisfies the following inequality:

$$\begin{array}{c}\hfill \mathcal{D}\left(\mathcal{S}\right(w,x),\mathcal{S}(w,y\left)\right)\le M\left(w\right)\mathcal{D}(x,y)+K\left(w\right)\left\{\mathcal{D}\right(x,\mathcal{S}(w,x))+\mathcal{D}(y,\mathcal{S}(w,y)\left)\right\},\end{array}$$

(29)

for all $x,y\in \mathcal{Z}.$ Then, there exists a unique random fixed point of $\mathcal{S}$.

In the following theorems, we prove the random versions of fixed-point theorems where we have a pair of asymptotically regular mappings in a b-metric space.

Theorem 19.

Let $(\nabla ,\wp )$ be a measurable space, $(\mathcal{Z},\mathcal{D},q)$ be a complete and separable b-metric space with a continuous b-metric, and let $\mathcal{S},U:\nabla \times \mathcal{Z}\to \mathcal{Z}$ be asymptotically regular random continuous operators. Assume that there exist functions $M:\nabla \to \mathbb{R}$ with $R\left(M\right)=[0,{\displaystyle \frac{1}{q^2}})$ and $K:\nabla \to [0,\infty )$ such that for each $w\in \nabla$, $\mathcal{S}$ satisfies the following inequality:

$$\begin{array}{c}\hfill \mathcal{D}\left(\mathcal{S}\right(w,x),U(w,y)\le M(w\left)\mathcal{D}\right(x,y)+K(w\left)\right\{\mathcal{D}(x,\mathcal{S}(w,x\left)\right)+\mathcal{D}(y,U(w,y\left)\right)\},\end{array}$$

for all $x,y\in \mathcal{Z}.$ Then, $\mathcal{S}$ and U have a unique random fixed point.

Proof. 

Combining the proofs of Theorems 12 and 17, we can write the proof of this result, omitting the proof. □

Theorem 20.

Let $(\nabla ,\wp )$ be a measurable space, $(\mathcal{Z},\mathcal{D},q)$ be a complete and separable b-metric space and $U,S:\nabla \times \mathcal{Z}\to \mathcal{Z}$ be random operators satisfying asymptotic regularity. Assume that there exist functions $M:\nabla \to \mathbb{R}$ with $R\left(M\right)=[0,{\displaystyle \frac{1}{q^2}})$ and $K:\nabla \to \mathbb{R}$ with $R\left(K\right)=[0,{\displaystyle \frac{1}{q}})$ such that for each $w\in \nabla$, S and U satisfying the following inequality:

$$\begin{array}{c}\hfill \mathcal{D}\left(U\right(w,x),S(w,y\left)\right)\le M\left(w\right)\mathcal{D}(x,y)+K\left(w\right)\left\{\mathcal{D}\right(x,U(w,x))+\mathcal{D}(y,S(w,y)\left)\right\},\end{array}$$

for all $x,y\in \mathcal{Z}.$ Then, U and S have a unique random fixed point.

Proof. 

By amalgamating the proofs of Theorems 13 and 18, we can establish that S and U have a unique random fixed point. □

Remark 11.

It is worth noting that the aforementioned results remain valid even if continuity is replaced by weaker notions of continuity (see Bisht [15]).

Theorem 21.

Let $(\nabla ,\wp )$ be a measurable space, $(\mathcal{Z},\mathcal{D},q⪯)$ be a complete and separable preordered b-MS, and $\mathcal{S}:\nabla \times \mathcal{Z}\to \mathcal{Z}$ be an asymptotically regular continuous random operator. Suppose that the following assertions hold:

1. 

$\mathcal{D}$ is continuous;

2. 

For each $w\in \nabla$, the function $\mathcal{S}(w,·)$ satisfies the following:

$$\begin{array}{c}\hfill (u,v\in \mathcal{Z}\mathrm{and}u⪯v)\Rightarrow \mathcal{S}(w,u)⪯\mathcal{S}(w,v),\end{array}$$

that is, $\mathcal{S}$ is a monotone operator with respect to $\mathcal{Z}$ for each $w\in \nabla$;

3. 

There exists a random variable $u_0:\nabla \to \mathcal{Z}$ with the following property

$$\begin{array}{c}\hfill u_0\left(w\right)⪯\mathcal{S}(w,u_0\left(w\right))or{u}_0\left(w\right)⪰\mathcal{S}(w,u_0\left(w\right))\end{array}$$

for each $w\in \nabla$;

4. 

There exist functions $M:\nabla \to [0,{\displaystyle \frac{1}{q^2}})$ and $K:\nabla \to [0,\infty )$ such that for each $w\in \nabla$,

$$\begin{array}{c}\mathcal{D}\left(\mathcal{S}\right(w,u),\mathcal{S}(w,v\left)\right)\le M\left(w\right)·\mathcal{D}(u,v)+K\left(w\right)·\left\{\mathcal{D}\right(u,\mathcal{S}(w,u))\hfill \\ +\mathcal{D}(v,\mathcal{S}(w,v\left)\right)\}\hfill \end{array}$$

(30)

for every comparable $u,v\in \mathcal{Z}$ (that is $u⪯v$ or $v⪯u$).

Then, there exists a random variable $u:\nabla \to \mathcal{Z}$ such that $u\in \mathcal{F}\left(\mathcal{S}\right)$, that is, u a random fixed point of $\mathcal{S}.$ Moreover, $\mid \mathcal{F}\left(\mathcal{S}\right)\mid =1$ if for every $u,v\in \mathcal{Z},$ there exists $z\in \mathcal{Z}$ that is comparable to u and v.

Proof. 

Fix a measurable function $u_0:\nabla \to \mathcal{Z}.$ If for each $w\in \nabla ,$ $\mathcal{S}(w,u_0\left(w\right))=u_0\left(w\right),$ then $u_0\in \mathcal{F}\left(\mathcal{S}\right)$ (random fixed point of $\mathcal{S}$). Suppose that for some $w\in \nabla$, $\mathcal{S}(w,u_0\left(w\right))\ne u_0\left(w\right).$ Design a sequence

$$\begin{array}{c}\hfill z_0\left(w\right)=u_0\left(w\right)\mathrm{and}{z}_n\left(w\right)=\mathcal{S}(w,z_{n-1}\left(w\right)),\end{array}$$

for all $w\in \nabla$ and $n\in \mathbb{N}.$ Since $\mathcal{S}$ is a monotone operator with assertion (3), we have

$$\begin{array}{c}\hfill z_0\left(w\right)=u_0\left(w\right)⪯\mathcal{S}(w,u_0\left(w\right))=z_1\left(w\right)\end{array}$$

implies

$$\begin{array}{c}\hfill z_1\left(w\right)=\mathcal{S}(w,z_0\left(w\right))⪯\mathcal{S}(w,z_1\left(w\right))=z_2\left(w\right).\end{array}$$

With the Principle of Mathematical Induction, we obtain

$$\begin{array}{c}\hfill z_0\left(w\right)⪯z_1\left(w\right)⪯z_2\left(w\right)⪯\cdots ⪯z_n\left(w\right)⪯z_{n+1}\left(w\right)⪯\cdots ,\end{array}$$

or

$$\begin{array}{c}\hfill z_0\left(w\right)⪰z_1\left(w\right)⪰z_2\left(w\right))⪰\cdots ⪰z_n\left(w\right))⪰z_{n+1}\left(w\right)⪰\cdots .\end{array}$$

For some natural number $m>n,$ estimate the b-metric distance between $z_n\left(w\right)$ and $z_m\left(w\right)$, as follows:

$$\begin{array}{cc}\hfill \mathcal{D}(z_n\left(w\right),z_m\left(w\right))& \le q[\mathcal{D}(z_n\left(w\right),z_{n+1}\left(w\right))+\mathcal{D}(z_{n+1}\left(w\right),z_m\left(w\right))]\hfill \\ \hfill & \le q\mathcal{D}(z_n\left(w\right),z_{n+1}\left(w\right))+q^2[\mathcal{D}(z_{n+1}\left(w\right),z_{m+1}\left(w\right))+\mathcal{D}(z_{m+1}\left(w\right),z_m\left(w\right))]\hfill \\ \hfill & \le (q+q^{2}K\left(w\right))\left(\mathcal{D}(z_n\left(w\right),z_{n+1}\left(w\right))\right)\hfill \\ \hfill & +q^2(1+K\left(w\right))\left(\mathcal{D}(z_m\left(w\right),z_{m+1}\left(w\right))\right)+q^{2}M\left(w\right)·\mathcal{D}(z_n\left(w\right),z_m\left(w\right)).\hfill \end{array}$$

On simplification yields

$$\begin{array}{cc}\hfill (1-q^{2}M\left(w\right))\mathcal{D}(z_n\left(w\right),z_m\left(w\right))& \le (q+q^{2}K\left(w\right))\mathcal{D}(z_n\left(w\right),z_{n+1}\left(w\right))\hfill \\ \hfill & +q^2(1+K\left(w\right))\mathcal{D}(z_m\left(w\right),z_{m+1}\left(w\right)).\hfill \end{array}$$

Utilizing the hypothesis that $\mathcal{S}$ is asymptotically regular (all terms in the RHS vanish on letting $n\to \infty$), we obtain the sequence ${\left(z_n\left(w\right)\right)}_{n\in \mathbb{N}}$ a Cauchy sequence for each fixed $w\in \nabla .$ The completeness of the b-metric space ensures the existence of some $z^{+}\left(w\right)\in \mathcal{Z}(w\in \nabla )$ such that it is the limit of the sequenece, that is, $z^{+}\left(w\right)={\displaystyle \underset{n\to \infty }{lim}}{z}_n\left(w\right)(w\in \nabla ).$ Since $z_0(·)$ is measurable, $z_1(·)$ is measurable. Inductively, the mapping $w\to z_n\left(w\right)$ is measurable for each $n\in \mathbb{N}.$ The limit function $z^{+}:\nabla \to \mathcal{Z}$ is measurable, since the pointwise limit of measurable mappings is again measurable. Now, we establish that the limit is a random fixed point of $\mathcal{S},$ that is, $z^{+}\left(w\right)=\mathcal{S}(w,z^{+}\left(w\right))$ for all $w\in \nabla .$ Estimate the b-metric distance between the limit $z^{+}\left(w\right)$ and its image under the mapping $\mathcal{S}$ as follows:

$$\begin{array}{cc}\hfill \mathcal{D}(z^{+}\left(w\right),\mathcal{S}(w,z^{+}\left(w\right))& ={\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}(z_n\left(w\right),\mathcal{S}(w,z_n\left(w\right)))=0.\hfill \end{array}$$

Therefore, $z^{+}\left(w\right)=\mathcal{S}(w,z^{+}\left(w\right))$ for each $w\in \nabla .$ Let us show that $\mathcal{S}$ has a unique random fixed point and it is $z^{+}.$ If we take any random variable ${\tilde{u}}_0:\nabla \to \mathcal{Z}$ and we define the sequence

$$\begin{array}{c}\hfill {\tilde{z}}_0\left(w\right)={\tilde{u}}_0\left(w\right)\mathrm{and}{\tilde{z}}_n\left(w\right)=\mathcal{S}(w,{\tilde{z}}_{n-1}\left(w\right))={\mathcal{S}}^n(w,{\tilde{u}}_0\left(w\right)),\end{array}$$

for all $w\in \nabla$, $n\in \mathbb{N},$ we get that the sequence ${\left({\tilde{z}}_n\left(w\right)\right)}_{n\in \mathbb{N}}$ converges to $z^{+}\left(w\right)$ as $n\to \infty$ for each fixed $w\in \nabla ,$ where $z^{+}$ is the random fixed point of $\mathcal{S}$. Estimate the b-metric distance between $z_n\left(w\right)$ and ${\tilde{z}}_n\left(w\right)$ as follows:

$$\begin{array}{cc}\hfill \mathcal{D}(z_n\left(w\right),{\tilde{z}}_n\left(w\right))& =\mathcal{D}(\mathcal{S}(w,z_{n-1}\left(w\right)),\mathcal{S}(w,{\tilde{z}}_{n-1}\left(w\right)))\hfill \\ \hfill & \le M\left(w\right)\mathcal{D}(z_{n-1}\left(w\right),{\tilde{z}}_{n-1}\left(w\right))+K\left(w\right)[\mathcal{D}(z_n\left(w\right),z_{n-1}\left(w\right))\hfill \\ \hfill & +\mathcal{D}({\tilde{z}}_n\left(w\right),{\tilde{z}}_{n-1}\left(w\right))].\hfill \end{array}$$

As $n\to \infty ,$ the second and third terms (in the RHS) tend to zero because of the asymptotic regularity of the mapping $\mathcal{S}$, as a consequence the sequence becomes Cauchy using Lemma 2. This proves that ${\tilde{z}}_n\left(w\right)\stackrel{n\to \infty }{\to }{z}^{+}\left(w\right)$ for each $w\in \nabla .$

Now, let us consider another scenario. For an arbitrary random variable ${\tilde{u}}_0:\nabla \to \mathcal{Z}$. Then, for each $w\in \nabla ,$ there exists some $g\left(w\right)\in \mathcal{Z}$ that is comparable to $u_0\left(w\right)$ and ${\tilde{u}}_0\left(w\right)$ simultaneously. Thus, if we construct the sequence

$$\begin{array}{c}\hfill g_0\left(w\right)=g\left(w\right)\mathrm{and}{g}_n\left(w\right)=\mathcal{S}(w,g_{n-1}\left(w\right))={\mathcal{S}}^n(w,g\left(w\right))\end{array}$$

for all $w\in \nabla$ and $n\in \mathbb{N},$ then $z_n\left(w\right)$ is comparable to $g_n\left(w\right)$ for each $w\in \nabla$, and ${\tilde{z}}_n\left(w\right)$ is comparable to $g_n\left(w\right)$ for each $w\in \nabla .$ By estimating the b-metric distance in this situation, we have

$$\begin{array}{cc}\hfill \mathcal{D}(z_n\left(w\right),\widetilde{z_n}\left(w\right))& \le \mathcal{D}(\mathcal{S}(w,z_{n-1}\left(w\right)),\mathcal{S}(w,g_{n-1}\left(w\right)))+\mathcal{D}(\mathcal{S}(w,g_{n-1}\left(w\right),\mathcal{S}(w,{\tilde{z}}_{n-1}\left(w\right)))\hfill \\ \hfill & \le M\left(w\right)\mathcal{D}(z_{n-1}\left(w\right),g_{n-1}\left(w\right))+K\left(w\right)[\mathcal{D}(z_{n-1}\left(w\right),z_n\left(w\right))\hfill \\ \hfill & +\mathcal{D}(g_{n-1}\left(w\right),g_n\left(w\right))]+M\left(w\right)\mathcal{D}(g_{n-1}\left(w\right),{\tilde{z}}_{n-1}\left(w\right))\hfill \\ \hfill & +K\left(w\right)[\mathcal{D}(g_{n-1}\left(w\right),g_n\left(w\right))+\mathcal{D}({\tilde{z}}_{n-1}\left(w\right),{\tilde{z}}_n\left(w\right))].\hfill \end{array}$$

We get ${\displaystyle \underset{n\to \infty }{lim}}\mathcal{D}(z_n\left(w\right),\widetilde{z_n}\left(w\right))=0$, which in turn establishes that ${\tilde{z}}_n\left(w\right)\stackrel{n\to \infty }{\to }{z}^{+}\left(w\right)$ for each $w\in \nabla .$ □

Remark 12.

By setting $K=0$ and $M=0$ in (30), we obtain the stochastic versions of Banach’s contraction principle and Kannan’s theorem, respectively, within a preordered b-metric space.

6. Conclusions

This paper offers a thorough exploration of fixed-point and common fixed-point theorems in different contexts, including b-metric spaces, preordered b-metric spaces, and random b-metric spaces. We have visually illustrated the discussed examples through various graphical plots. Additionally, several well-known fixed-point theorems emerge as corollaries of our proven results (see, for instance, [8,9,12,13,14,16,17,19,23,25,32,34,35,36]). Our proposed results on fixed points offer valuable insights, such as solving nonlinear integral equations, stability of dynamical systems, etc. Endowing a b-metric space with a preordering structure helps us in situations when we are looking for a positive or negative solution.