Some Coincidence and Common Fixed-Point Results on Cone b2-Metric Spaces over Banach Algebras with Applications to the Infinite System of Integral Equations

Abstract

In this article, common fixed-point theorems for self-mappings under different types of generalized contractions in the context of the cone $b_2$-metric space over the Banach algebra are discussed. The existence results obtained strengthen the ones mentioned previously in the literature. An example and an application to the infinite system of integral equations are also presented to validate the main results.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and Preliminaries

Gähler [1] proposed the definition of 2-metric spaces as a generalization of an ordinary metric space. He defined that $ds,m,z$ geometrically represents the area of a triangle with vertices $s,m,z\in \aleph$. 2-metric is not a continuous function of its variables. This was one of the key drawbacks of the 2-metric space while an ordinary metric is a continuous function.

Keeping these drawbacks in mind, Dhage [2], in his PhD thesis, proposed a concept of the $D$-metric space as a generalized version of the 2-metric space. He also defined an open ball in such spaces and studied other topological properties of the mentioned structure. According to him, $Ds,m,z$ represented the perimeter of a triangle. He stated that the $D$-metric induced a Hausdorff topology and in the $D$-metric space, the family of all open balls forms a basis for such topology.

Later, Mustafa and Sims [3] illustrate that the topological structure of Dhage’s $D$-metric is invalid. Then, they revised the $D$-metric and expanded the notion of a metric in which each triplet of an arbitrary set is given a real number called as the $G$-metric space [4].

In addition, the definition of the $D^{\ast }$-metric space is proposed by Sedghi et al. [5] as an updated version of Dhage’s $D$-metric space. Later, they analyzed and found that $G$-metric and $D^{\ast }$-metric have shortcomings. Later, they proposed a new simplified sturcture called the $S$-metric space [6].

On the other hand, by swapping the real numbers with the ordered Banach space and established cone metric space, Huang and Zhang [7] generalized the notion of a metric space and demonstrated some fixed-point results of contractive maps using the normality condition in such spaces. Rezapour and Hamlbarani [8] subsequently ignored the normality assumption and obtained some generalizations of the Huang and Zhang [7] results. However, it should be noted that the equivalence between cone metric spaces and metric spaces has been developed in recent studies by some scholars in the context of the presence of fixed points in the mapping involved. Liu and Xu [9] proposed the concept of a cone metric space over the Banach algebra in order to solve these shortcomings by replacing the Banach space with the Banach algebra. This became an interesting discovery in the study of fixed-point theory since it can be shown that cone metric spaces over the Banach algebra are not equal to metric spaces in terms of the presence of the fixed points of mappings. Among these generalizations, by generalizing the cone 2-metric spaces [10] over the Banach algebra and $b_2$-metric spaces [11], Fernandez et al. [12] examined cone $b_2$-metric spaces over the Banach algebra with the constant $b\ge 1$. In the setting of the new structure, they proved some fixed-point theorems under different types of contractive mappings and showed the existence and uniqueness of a solution to a class of system of integral equations as an application.

Recently, in 2020, Islam et al. [13] initiated the notion of the cone $b_2$-metric space over the Banach algebra with constant $b\succcurlyeq e$ which is a generalization of the definition of Fernandez et al. [12]. They proved some fixed-point theorems under $\alpha$-admissible Hardy-Rogers contractions which generalize many of the results from the existence literature, and as an application, they proved results which guarantee the existence of solution of an infinite system of integral equations.

In 1973, Hardy and Rogers [14] proposed a new definition of mappings called the contraction of Hardy-Rogers that generalizes the theory of the Banach contraction and the theorem of Reich [15] in a metric space setting. For other related work about the concept of Hardy-Rogers contractions, see, for instance, [16, 17] and the references therein.

We recollect certain essential notes, definitions required, and primary results consistent with the literature.

Definition 1 (see [18]).

Consider $\widehat{\mathcal{U}}$ the Banach algebra which is real, and the multiplication operation is defined under the below properties (for all $s,m,z\in \widehat{\mathcal{U}},\rho \in ℝ$):

(a₁) $smz=smz$

(a₂) $sm+z=sm+sz$ and $s+mz=sz+mz$

(a₃) $\rho sm=\rho sm=s\rho m$

(a₄) $sm\le sm$

Unless otherwise stated, we will assume in this article that $\widehat{\mathcal{U}}$ is a real Banach algebra. If $s\in \widehat{\mathcal{U}}$ occurs, we call $e$ the unit of $\widehat{\mathcal{U}}$, so that $es=se=s$. We call $\widehat{\mathcal{U}}$ a unital in this case. If an inverse element $m\in \widehat{\mathcal{U}}$ exists, the element $s\in \widehat{\mathcal{U}}$ is said to be invertible, so that $sm=ms=e$. The inverse of $s$ in such case is unique and is denoted by $s^{-1}$. We require the following propositions in the sequel.

Proposition 2 (see [18]).

Consider the unital Banach algebra $\widehat{\mathcal{U}}$ with unit $e$, and let $s\in \widehat{\mathcal{U}}$ be the arbitrary element. If the spectral radius $rs<1$, i.e., $$\begin{array}{}\text{(1)}& rs=\underset{n⟶\infty }{\lim }{s^n}^{1/n}=\inf {s^n}^{1/n}<1,\end{array}$$

then $e-s$ is invertible. In fact, $$\begin{array}{}\text{(2)}& {e-s}^{-1}=\sum _{k=1}^{\infty }{s}^k.\end{array}$$

Remark 3.

We see from [18] that $rs\le s$ for all $s\in \widehat{\mathcal{U}}$ with unit $e$.

Remark 4 (see [19]).

In Proposition 2, by replacing “$rs<1$” by $s\le 1$, then the conclusion remains the same.

Remark 5 (see [19]).

If $rs<1$, then $s^n⟶0$ as $n⟶\infty$.

Definition 6.

Consider the Banach algebra $\widehat{\mathcal{U}}$ with unit element $e$, zero element ${\theta }_{\widehat{\mathcal{U}}}$, and ${\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$. Then, ${\mathcal{C}}_{\widehat{\mathcal{U}}}\subset \widehat{\mathcal{U}}$ is a cone in $\widehat{\mathcal{U}}$ if:

(b₁) $e\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$

(b₂) ${\mathcal{C}}_{\widehat{\mathcal{U}}}+{\mathcal{C}}_{\widehat{\mathcal{U}}}\subset {\mathcal{C}}_{\widehat{\mathcal{U}}}$

(b₃) $\lambda {\mathcal{C}}_{\widehat{\mathcal{U}}}\subset {\mathcal{C}}_{\widehat{\mathcal{U}}}$ for all $\lambda \ge 0$

(b₄) ${\mathcal{C}}_{\widehat{\mathcal{U}}}·{\mathcal{C}}_{\widehat{\mathcal{U}}}\subset {\mathcal{C}}_{\widehat{\mathcal{U}}}$

(b₅) ${\mathcal{C}}_{\widehat{\mathcal{U}}}\cap -{\mathcal{C}}_{\widehat{\mathcal{U}}}={\theta }_{\widehat{\mathcal{U}}}$

Define the partial order relation $\preccurlyeq$ in $\widehat{\mathcal{U}}$ w.r.t ${\mathcal{C}}_{\widehat{\mathcal{U}}}$ by $s\preccurlyeq m$ if and only if $m-s\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$; also, $s\prec m$ if $s\preccurlyeq m$ but $s\ne m$ while $s\ll m$ stands for $m-s\in \mathrm{int}{\mathcal{C}}_{\widehat{\mathcal{U}}}$, where $\mathrm{int}{\mathcal{C}}_{\widehat{\mathcal{U}}}$ is the interior of ${\mathcal{C}}_{\widehat{\mathcal{U}}}$. ${\mathcal{C}}_{\widehat{\mathcal{U}}}$ is solid if $\mathrm{int}{\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$.

If there is $M>0$ such that for all $s,m\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$, we have $$\begin{array}{}\text{(3)}& {\theta }_{\widehat{\mathcal{U}}}\preccurlyeq s\preccurlyeq m\text{implies}s\le Mm,\end{array}$$

then ${\mathcal{C}}_{\widehat{\mathcal{U}}}$ is normal. If $M$ is least and positive in the above, then it is the normal constant of ${\mathcal{C}}_{\widehat{\mathcal{U}}}$ [7].

Definition 7 (see [7, 9]).

Let $d:\aleph \times \aleph ⟶\widehat{\mathcal{U}}$ and $\aleph \ne \varnothing$ be the mapping:

(c₁) For all $s,m\in \aleph$, $ds,m\succcurlyeq {\theta }_{\widehat{\mathcal{U}}}$ and $ds,m={\theta }_{\widehat{\mathcal{U}}}$ if and only if $s=m$

(c₂) For all $s,m\in \aleph$, $ds,m=dm,s$

(c₃) For all $s,m,z\in \aleph$, $ds,z\preccurlyeq ds,m+dm,z$

Then, $\aleph ,d$ over the Banach algebra $\widehat{\mathcal{U}}$ with cone metric $d$ is a cone metric space.

In [20], over the Banach algebra with constant $b\ge 1$, the cone $b$-metric space is introduced as a generalization of the cone metric space over the Banach algebra while in Mitrovic and Hussain [16], over the Banach algebra with parameter $b\succcurlyeq e$, the concept of cone $b$-metric spaces is introduced.

Definition 8 (see [16]).

Let $d:\aleph \times \aleph ⟶\widehat{\mathcal{U}}$ and $\aleph \ne \varnothing$ be the mapping:

(e₁) For all $s,m\in \aleph$, ${\theta }_{\widehat{\mathcal{U}}}\preccurlyeq ds,m$ and $ds,m={\theta }_{\widehat{\mathcal{U}}}$ if and only if $s=m$

(e₂) For all $s,m\in \aleph$, $ds,m=dm,s$

(e₃) There is $b\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$, $b\succcurlyeq e$, and for all $s,m,z\in \aleph$, $ds,z\preccurlyeq bds,m+dm,z$

Then, $\aleph ,d$ over the Banach algebra $\widehat{\mathcal{U}}$ with cone $b$-metric $d$ is a cone $b$-metric space. Note that if we take $b=e$, then it reduces to the cone metric space over the Banach algebra $\widehat{\mathcal{U}}$.

Definition 9 (see [1]).

Let $d:\aleph \times \aleph \times \aleph ⟶{ℝ}^{+}$ and $\aleph \ne \varnothing$:

(f₁) There is a point $z\in \aleph$ for $s,m\in \aleph$ such that $ds,m,z\ne 0$, if at least two of $s,m,z$ are not equal

(f₂) $ds,m,z=0$ if and only if at least two of $s,m,z$ are equal

(f₃) $ds,m,z=dPs,m,z$ for all $s,m,z\in \aleph$, where $Ps,m,z$ stands for all permutations of $s,m,z$

(f₄) $ds,m,z\le ds,m,t+ds,z,t+dm,z,t$ for all $s,m,z,t\in \aleph$

Then, $\aleph ,d$ is a $2$-metric space with $2$-metric $d$.

Definition 10 (see [12]).

Let $d:\aleph \times \aleph \times \aleph ⟶\widehat{\mathcal{U}}$, $\aleph \ne \varnothing$, and $b\ge 1$ be a real number:

(g₁) There is a point $z\in \aleph$ for $s,m\in \aleph$ such that $ds,m,z\ne {\theta }_{\widehat{\mathcal{U}}}$ if at least two of $s,m,z$ are not equal

(g₂) $ds,m,z={\theta }_{\widehat{\mathcal{U}}}$ if and only if at least two of $s,m,z$ are equal

(g₃) $ds,m,z=dPs,m,z$ for all $s,m,z\in \aleph$, where $Ps,m,z$ stands for all permutations of $s,m,z$

(g₄) $ds,m,z\preccurlyeq bds,m,t+ds,z,t+dm,z,t$ for all $s,m,z,t\in \aleph$

Then, $\aleph ,d$ over the Banach algebra $\widehat{\mathcal{U}}$ with parameter $b\ge 1$ is a cone $b_2$-metric space. By taking $b=1$, it became a cone 2-metric space. We refer the reader to [21] for other details about the cone 2-metric space over the Banach algebra $\widehat{\mathcal{U}}$.

Islam et al. [13] initiated the concept of the cone $b_2$-metric space over the Banach algebra with parameter $b\succcurlyeq e$.

Definition 11 (see [13]).

Let $d:\aleph \times \aleph \times \aleph ⟶\widehat{\mathcal{U}}$ and $\aleph \ne \varnothing$:

(h₁) There is a point $z\in \aleph$ for $s,m\in \aleph$ such that $ds,m,z\ne {\theta }_{\widehat{\mathcal{U}}}$ if at least two of $s,m,z$ are not equal

(h₂) $ds,m,z={\theta }_{\widehat{\mathcal{U}}}$ iff at least two of $s,m,z$ are equal

(h₃) $ds,m,z=dPs,m,z$ for all $s,m,z\in \aleph$, where $Ps,m,z$ stands for all permutations of $s,m,z$

(h₄) $ds,m,z\preccurlyeq bds,m,t+ds,z,t+dm,z,t$ for all $s,m,z,t\in \aleph$ with $b\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$, $b\succcurlyeq e$

Then, $\aleph ,d$ over the Banach algebra with parameter $b\succcurlyeq e$ is a cone $b_2$-metric space. By taking $b=e$, it reduces to a cone 2-metric space.

Example 12 (see [13]).

Let $\widehat{\mathcal{U}}=C_{ℝ}^{1}0,1$. For each $ht\in \widehat{\mathcal{U}}$, $ht={ht}_{\infty }+{h^{\prime }t}_{\infty }$. Then, $\widehat{\mathcal{U}}$ is a Banach algebra with unit $e=1$ as a constant function, and multiplication is defined pointwise. Let ${\mathcal{C}}_{\widehat{\mathcal{U}}}=ht\in \widehat{\mathcal{U}}\mid ht\ge 0,t\in 0,1$. Then, ${\mathcal{C}}_{\widehat{\mathcal{U}}}$ is a cone in $\widehat{\mathcal{U}}$. Let $\aleph =k,0\in {ℝ}^2\mid \mid 0\le k\le 1\cup 0,1$. For all $S,M,Z\in \aleph$, define $d:\aleph \times \aleph \times \aleph ⟶\widehat{\mathcal{U}}$ as $$\begin{array}{}\text{(4)}& dS,M,Z=\begin{array}{cc}dPS,M,Z,\hfill & P\text{denotes}\text{permutations},\hfill \\ \Delta ·ht,\hfill & \text{otherwise},\hfill \end{array}\end{array}$$where $\Delta$ is the square of the area of the triangle formed by $S,M,Z$ and $h:0,1⟶ℝ$ defined by $ht=e^t$. Consider $$\begin{array}{}\text{(5)}& ds,0,m,0,0,1·e^t\preccurlyeq ds,0,m,0,z,0·e^t+ds,0,z,0,0,1·e^t+dz,0,m,0,0,1·e^t.\end{array}$$

That is, $1/4{s-m}^2·e^t\preccurlyeq 1/4{s-z}^2+{z-m}^2·e^t$, showing that $d$ is not a cone 2-metric, because $-3/16e^t\cancel{\in }{\mathcal{C}}_{\widehat{\mathcal{U}}}$ for $0\le s,m,z\le 1$ with $z=1/2$, $m=0$, and $s=1$. But for $b\ge 2\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$ is a cone $b_2$-metric space over the Banach algebra $\widehat{\mathcal{U}}$.

Definition 13 (see [13]).

Consider that $\aleph ,d$ is a cone $b_2$-metric space over the Banach algebra $\widehat{\mathcal{U}}$ with $b\succcurlyeq e$, and let $s_n$ be a sequence in $\aleph ,d$; then,

(i₁) $s_n$ is said to converge to $s\in \aleph$ if for every $c\gg {\theta }_{\widehat{\mathcal{U}}}$ there is $N\in \mathbb{N}$ such that $d{s}_n,s,a\ll c$ for all $n\ge N$. That is, $$\begin{array}{}\text{(6)}& \underset{n⟶\infty }{\lim }{s}_n=s\mathit{\text{or}}{s}_n⟶sn⟶\infty .\end{array}$$

(i₂) If for every $c\gg {\theta }_{\widehat{\mathcal{U}}}$ there exists $N\in \mathbb{N}$ such that $d{s}_n,s_m,a\ll c$ for all $m,n\ge N$, then we say that $s_n$ is a Cauchy sequence

(i₃) $\aleph ,d$ is complete if every Cauchy sequence is convergent in $\aleph$

Definition 14 (see [22]).

Let a sequence $s_n$ be in $\widehat{\mathcal{U}}$; then, sequence $s_n$ is a $c$-sequence, if for each $c\gg {\theta }_{\widehat{\mathcal{U}}}$ there is $N\in \mathbb{N}$ such that $s_n\ll c$ for all $n>N$.

Lemma 15 (see [23]).

Consider the Banach algebra $\widehat{\mathcal{U}}$ and $\mathrm{int}{\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$. Also, consider $s_n$ a $c$-sequence in $\widehat{\mathcal{U}}$ and $k\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$ where $k$ is arbitrary; then, $k{s}_n$ is a $c$-sequence.

Lemma 16 (see [23]).

Consider the Banach algebra $\widehat{\mathcal{U}}$ and $\mathrm{int}{\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$. Let $s_n$ and $z_n$ be $c$-sequences in $\widehat{\mathcal{U}}$. Then, for arbitrary $\eta ,\zeta \in {\mathcal{C}}_{\widehat{\mathcal{U}}}$, we have $\eta s_n+\zeta z_n$ which is also a $c$-sequence.

Lemma 17 (see [23]).

Consider the Banach algebra $\widehat{\mathcal{U}}$ and $\mathrm{int}{\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$. Let $s_n\subset {\mathcal{C}}_{\widehat{\mathcal{U}}}$ such that $s_n⟶0$ as $n⟶\infty$. Then, $s_n$ is a $c$-sequence.

Lemma 18 (see [19]).

Let $\widehat{\mathcal{U}}$ be the Banach algebra, $e$ be their unit element, and $m,s\in \widehat{\mathcal{U}}$. If $m,s$ commutes, then

(k₁) $rs+m\le rs+rm$

(k₂) $rsm\le rsrm$

Lemma 19 (see [19]).

Consider the Banach algebra $\widehat{\mathcal{U}}$ and $s\in \widehat{\mathcal{U}}$. If $0\le rs<1$, then $$\begin{array}{}\text{(7)}& r{e-s}^{-1}\le {1-rs}^{-1}.\end{array}$$

Lemma 20 (see [24]).

Consider the Banach algebra $\widehat{\mathcal{U}}$, $e$ is their unit element, and ${\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$. Let $L\in \widehat{\mathcal{U}}$ and $s_n=L^n$. If $rL<1$, then $s_n$ is a $c$-sequence.

Lemma 21 (see [25]).

Let ${\mathcal{C}}_{\widehat{\mathcal{U}}}\subset \widehat{\mathcal{U}}$ be a cone.

(l₁) If $s,m\in \widehat{\mathcal{U}}$, $k\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$, and $m\preccurlyeq s$, then $km\preccurlyeq ks$

(l₂) If $m,k\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$, $rk<1$, and $m\preccurlyeq km$, then $m=0$

(l₃) For any $n\in \mathbb{N}$, $r{k}^n<1$ with $k\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$ and $rk<1$

Lemma 22 (see [26]).

Consider the Banach algebra $\widehat{\mathcal{U}}$ and $\mathrm{int}{\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$.

(n₁) If $m,s,z\in \widehat{\mathcal{U}}$ and $m\preccurlyeq s\ll z$, then $m\ll s$

(n²) If $m\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$ and $m\ll c$ for $c\gg {\theta }_{\widehat{\mathcal{U}}}$, then $m={\theta }_{\widehat{\mathcal{U}}}$

Lemma 23 (see [21]).

Consider the Banach algebra $\widehat{\mathcal{U}}$ and $\mathrm{int}{\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$. Let $m\in \widehat{\mathcal{U}}$, and suppose that $k\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$ is an arbitrary given vector such that $m\ll c$ for any ${\theta }_{\widehat{\mathcal{U}}}\ll c$, then $km\ll c$ for any ${\theta }_{\widehat{\mathcal{U}}}\ll c$.

Lemma 24 (see [27]).

Consider the Banach algebra $\widehat{\mathcal{U}}$ and $\mathrm{int}{\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$. Let ${\theta }_{\widehat{\mathcal{U}}}\preccurlyeq m\ll c$ for each ${\theta }_{\widehat{\mathcal{U}}}\ll c$; then, $m={\theta }_{\widehat{\mathcal{U}}}$.

Lemma 25 (see [27]).

Consider the Banach algebra $\widehat{\mathcal{U}}$ and $\mathrm{int}{\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$. If $s_n⟶0$ as $n⟶\infty$, then for each $c\gg {\theta }_{\widehat{\mathcal{U}}}$, there is $N\in \mathbb{N}$ with $n>N$, such that $s_n\ll c$.

Definition 26 (see [28]).

Let $g$ and $g$ be self-maps of a set $\aleph$. If $m=gs=fs$ for some $s\in \aleph$, then for $g$ and $f$, $s$ is known as a coincidence point and $m$ is known as a point of coincidence of $g$ and $f$.

Definition 27 (see [29]).

The mappings $g,f:\aleph ⟶\aleph$ are said to be weakly compatible, whenever $fs=gs$ and $fgs=gfs$ for any $s\in \aleph$.

Lemma 28 (see [28]).

Let the mappings $g$ and $f$ be weakly compatible self-maps of a set $\aleph$. If $g$ and $f$ have a unique point of coincidence $m=fs=gs$, then $m$ is the unique common fixed point of $g$ and $f$.

2. Main Results

In this section, in the framework of the cone $b_2$-metric space over Banach algebras with parameter $b\succcurlyeq e$, we prove some common fixed-point results.

Proposition 29.

Let $\aleph ,d$ over the Banach algebra $\widehat{\mathcal{U}}$ be the complete cone $b_2$-metric space and ${\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$ be a cone in $\widehat{\mathcal{U}}$. If a sequence $s_n$ in $\aleph$ converges to $s\in \aleph$, then we have the following:

(i) $d{s}_n,s,a$ is a $c$-sequence for all $a\in \aleph$

(ii) For any $\alpha \in \mathbb{N}$, $d{s}_n,s_{n+\alpha },a$ is a $c$-sequence for all $a\in \aleph$

Proof.

Since the proof is easy, so we left it.

Now, we here state and prove our first main results which generalize and extend many of the conclusions from the existence literature.☐

Theorem 30.

Let $\aleph ,d$ over the Banach algebra $\widehat{\mathcal{U}}$ be a cone $b_2$-metric space with $b\succcurlyeq e$ and ${\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$ be a cone in $\widehat{\mathcal{U}}$. Let ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ be four families of self-mappings on $\aleph$. For all $i,j,k,l\in \mathbb{N}$, if a sequence ${{\eta }_n}_{n=1}^{\infty }$ exists of nonnegative integers, such that for all $s,m,z\in \aleph$, $$\begin{array}{}\text{(8)}& d{E}_i^{{\eta }_i}s,F_j^{{\eta }_j}m,a\preccurlyeq {\vartheta }_{1}d{G}_k^{{\eta }_k}s,H_l^{{\eta }_l}m,a+{\vartheta }_{2}d{G}_k^{{\eta }_k}s,E_i^{{\eta }_i}s,a+{\vartheta }_{3}d{H}_l^{{\eta }_l}m,F_j^{{\eta }_j}m,a+{\vartheta }_{4}d{G}_k^{{\eta }_k}s,F_j^{{\eta }_j}m,a+{\vartheta }_{5}d{H}_l^{{\eta }_l}m,E_i^{{\eta }_i}s,a,\end{array}$$where ${\vartheta }_1,{\vartheta }_2,{\vartheta }_3,{\vartheta }_4,{\vartheta }_5\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$ with ${\sum }_{w=1}^{3}r{\vartheta }_w+2r{\vartheta }_{4}rb+2r{\vartheta }_{5}rb<1$, $r{\vartheta }_{2}rb+r{\vartheta }_{5}r{b}^2<1$, $r{\vartheta }_{3}rb+r{\vartheta }_{4}r{b}^2<1$, and ${\vartheta }_1,{\vartheta }_2,{\vartheta }_3,{\vartheta }_4,{\vartheta }_5,b$ commute. If $E_i\aleph \subseteq H_l\aleph$, $F_j\aleph \subseteq G_k\aleph$, and one of $E_i\aleph$, $G_k\aleph$, $H_l\aleph$, and $F_j\aleph$ are a complete subspace of $\aleph$ for each $i,j,k,l\ge 1$, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique point of coincidence in $\aleph$. Moreover, if $F_j,H_l$ and $E_i,G_k$ are weakly compatible, respectively, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique common fixed point.

Proof.

Set $E_i^{{\eta }_i}=S_{2i+1}$, $F_j^{{\eta }_j}=T_{2j+2}$, $G_k^{{\eta }_k}=I_{2k+3}$, and $H_l^{{\eta }_l}=J_{2l+2}$, $i,j,k,l\ge 1$. Then, by (8), we have $$\begin{array}{}\text{(9)}& d{S}_{2i+1}s,T_{2j+2}m,a\preccurlyeq {\vartheta }_{1}d{I}_{2k+3}s,J_{2l+2}m,a+{\vartheta }_{2}d{I}_{2k+3}s,S_{2i+1}s,a+{\vartheta }_{3}d{J}_{2l+2}m,T_{2j+2}m,a+{\vartheta }_{4}d{I}_{2k+3}s,T_{2j+2}m,a+{\vartheta }_{5}d{J}_{2l+2}m,S_{2i+1}s,a.\end{array}$$

Choose $s_0\in \aleph$ to be arbitrary. Since $E_i\aleph \subseteq H_l\aleph$ and $F_j\aleph \subseteq G_k\aleph$ for each $i,j,k,l\ge 1$, there exists $s_1,s_2\in \aleph$ such that $S_1{s}_0=J_2{s}_1$ and $T_2{s}_1=I_3{s}_2$. Continuing this process, we can define $s_n$ by $S_{2n+1}{s}_{2n}=J_{2n+2}{s}_{2n+1}$ and $T_{2n+2}{s}_{2n+1}=I_{2n+3}{s}_{2n+2}$.

Denote ${\mu }_{2n}=J_{2n+2}{s}_{2n+1}=S_{2n+1}{s}_{2n}$ and ${\mu }_{2n+1}=I_{2n+3}{s}_{2n+2}=T_{2n+2}{s}_{2n+1}$ for $n=0,1,2,\cdots$. Now, we show that ${\mu }_n$ is a Cauchy sequence.

From (9), we know that $$\begin{array}{}\text{(10)}& d{\mu }_{2n},{\mu }_{2n+1},a=d{S}_{2n+1}{s}_{2n},T_{2n+2}{s}_{2n+1},a\preccurlyeq {\vartheta }_{1}d{I}_{2n+1}{s}_{2n},J_{2n+2}{s}_{2n+1},a+{\vartheta }_{2}d{I}_{2n+1}{s}_{2n},S_{2n+1}{s}_{2n},a+{\vartheta }_{3}d{J}_{2n+2}{s}_{2n+1},T_{2n+2}{s}_{2n+1},a+{\vartheta }_{4}d{I}_{2n+1}{s}_{2n},T_{2n+2}{s}_{2n+1},a+{\vartheta }_{5}d{J}_{2n+2}{s}_{2n+1},S_{2n+1}{s}_{2n},a={\vartheta }_{1}d{\mu }_{2n-1},{\mu }_{2n},a+{\vartheta }_{2}d{\mu }_{2n-1},{\mu }_{2n},a+{\vartheta }_{3}d{\mu }_{2n},{\mu }_{2n+1},a+{\vartheta }_{4}d{\mu }_{2n-1},{\mu }_{2n+1},a+{\vartheta }_{5}d{\mu }_{2n},{\mu }_{2n},a\preccurlyeq {\vartheta }_{1}d{\mu }_{2n-1},{\mu }_{2n},a+{\vartheta }_{2}d{\mu }_{2n-1},{\mu }_{2n},a+{\vartheta }_{3}d{\mu }_{2n},{\mu }_{2n+1},a+{\vartheta }_{4}bd{\mu }_{2n-1},{\mu }_{2n+1},{\mu }_{2n}+{\vartheta }_{4}bd{\mu }_{2n-1},{\mu }_{2n},a+{\vartheta }_{4}bd{\mu }_{2n},{\mu }_{2n+1},a.\text{(11)}& d{\mu }_{2n},{\mu }_{2n+1},{\mu }_{2n-1}=d{S}_{2n+1}{s}_{2n},T_{2n+2}{s}_{2n+1},{\mu }_{2n-1}\le {\vartheta }_{1}d{I}_{2n+1}{s}_{2n},J_{2n+2}{s}_{2n+1},{\mu }_{2n-1}+{\vartheta }_{2}d{I}_{2n+1}{s}_{2n},S_{2n+1}{s}_{2n},{\mu }_{2n-1}+{\vartheta }_{3}d{J}_{2n+2}{s}_{2n+1},T_{2n+2}{s}_{2n+1},{\mu }_{2n-1}+{\vartheta }_{4}d{I}_{2n+1}{s}_{2n},T_{2n+2}{s}_{2n+1},{\mu }_{2n-1}+{\vartheta }_{5}d{J}_{2n+2}{s}_{2n+1},S_{2n+1}{s}_{2n},{\mu }_{2n-1}={\vartheta }_{1}d{\mu }_{2n-1},{\mu }_{2n},{\mu }_{2n-1}+{\vartheta }_{2}d{\mu }_{2n-1},{\mu }_{2n},{\mu }_{2n-1}+{\vartheta }_{3}d{\mu }_{2n},{\mu }_{2n+1},{\mu }_{2n-1}+{\vartheta }_{4}d{\mu }_{2n-1},{\mu }_{2n+1},{\mu }_{2n-1}+{\vartheta }_5\text{d}{\mu }_{2n},{\mu }_{2n},{\mu }_{2n-1}\preccurlyeq {\vartheta }_{3}d{\mu }_{2n},{\mu }_{2n+1},{\mu }_{2n-1}.\end{array}$$

It means that $d{\mu }_{2n},{\mu }_{2n+1},{\mu }_{2n-1}={\theta }_{\widehat{\mathcal{U}}}$. Therefore, (10) becomes $$\begin{array}{}\text{(12)}& e-{\vartheta }_3-{\vartheta }_{4}bd{\mu }_{2n},{\mu }_{2n+1},a\preccurlyeq {\vartheta }_1+{\vartheta }_2+{\vartheta }_{4}bd{\mu }_{2n-1},{\mu }_{2n},a,\end{array}$$that is, $$\begin{array}{}\text{(13)}& d{\mu }_{2n},{\mu }_{2n+1},a\preccurlyeq L_{1}d{\mu }_{2n-1},{\mu }_{2n},a,\end{array}$$where $L_1={e-{\vartheta }_3-{\vartheta }_{4}b}^{-1}{\vartheta }_1+{\vartheta }_2+{\vartheta }_{4}b$. Similarly, it is not difficult to show that $$\begin{array}{}\text{(14)}& d{\mu }_{2n+2},{\mu }_{2n+1},a=d{S}_{2n+3}{s}_{2n+2},T_{2n+2}{s}_{2n+1},a\preccurlyeq {\vartheta }_{1}d{I}_{2n+3}{s}_{2n+2},J_{2n+2}{s}_{2n+1},a+{\vartheta }_{2}d{I}_{2n+3}{s}_{2n+2},S_{2n+3}{s}_{2n+2},a+{\vartheta }_{3}d{J}_{2n+2}{s}_{2n+1},T_{2n+2}{s}_{2n+1},a+{\vartheta }_{4}d{I}_{2n+3}{s}_{2n+2},T_{2n+2}{s}_{2n+1},a+{\vartheta }_{5}d{J}_{2n+2}{s}_{2n+1},S_{2n+3}{s}_{2n+2},a={\vartheta }_{1}d{\mu }_{2n+1},{\mu }_{2n},a+{\vartheta }_{2}d{\mu }_{2n+1},{\mu }_{2n+2},a+{\vartheta }_{3}d{\mu }_{2n},{\mu }_{2n+1},a+{\vartheta }_{4}d{\mu }_{2n+1},{\mu }_{2n+1},a+{\vartheta }_{5}d{\mu }_{2n},{\mu }_{2n+2},a\preccurlyeq {\vartheta }_{1}d{\mu }_{2n+1},{\mu }_{2n},a+{\vartheta }_{2}d{\mu }_{2n+1},{\mu }_{2n+2},a+{\vartheta }_{3}d{\mu }_{2n},{\mu }_{2n+1},a+{\vartheta }_{5}bd{\mu }_{2n},{\mu }_{2n+2},{\mu }_{2n+1}+{\vartheta }_{5}bd{\mu }_{2n},{\mu }_{2n+1},a+{\vartheta }_{5}bd{\mu }_{2n+1},{\mu }_{2n+2},a.\end{array}$$

As $d{\mu }_{2n},{\mu }_{2n+2},{\mu }_{2n+1}={\theta }_{\widehat{\mathcal{U}}}$, therefore (14) becomes $$\begin{array}{}\text{(15)}& e-{\vartheta }_2-{\vartheta }_{5}bd{\mu }_{2n+2},{\mu }_{2n+1},a\preccurlyeq {\vartheta }_1+{\vartheta }_3+{\vartheta }_{5}bd{\mu }_{2n},{\mu }_{2n+1},a,\end{array}$$that is, $$\begin{array}{}\text{(16)}& d{\mu }_{2n+2},{\mu }_{2n+1},a\preccurlyeq L_{2}d{\mu }_{2n},{\mu }_{2n+1},a,\end{array}$$where $L_2={e-{\vartheta }_2-{\vartheta }_{5}b}^{-1}{\vartheta }_1+{\vartheta }_3+{\vartheta }_{5}b$. Set $K=L_1{L}_2$, and using inequalities (13) and (16), we deduce that $$\begin{array}{}\text{(17)}& d{\mu }_{2n+2},{\mu }_{2n+1},a\preccurlyeq L_{2}d{\mu }_{2\text{n}+1},{\mu }_{2n},a\preccurlyeq L_2{L}_{1}d{\mu }_{2n},{\mu }_{2n-1},a\preccurlyeq L_2{L}_1{L}_{2}d{\mu }_{2n-1},{\mu }_{2n-2},a=L_{2}Kd{\mu }_{2n-1},{\mu }_{2n-2},a\preccurlyeq \cdots \preccurlyeq L_2{K}^{n}d{\mu }_1,{\mu }_0,a,\text{(18)}& d{\mu }_{2n+3},{\mu }_{2n+2},a\preccurlyeq L_{1}d{\mu }_{2n+2},{\mu }_{2n+1},a\preccurlyeq L_1{L}_2{K}^{n}d{\mu }_1,{\mu }_0,a=K^{n+1}d{\mu }_1,{\mu }_0,a.\end{array}$$

In this case, for all $t<n$, similar to (17) and (18), we have $$\begin{array}{c}\text{(19)}& d{\mu }_{2n+1},{\mu }_{2n},{\mu }_{2t+1}=L_2{K}^{n-t-1}d{\mu }_{2t+1},{\mu }_{2t+2},{\mu }_{2t+1}={\theta }_{\widehat{\mathcal{U}}},d{\mu }_{2n-1},{\mu }_{2n},{\mu }_{2t+1}=K^{n-t-1}d{\mu }_{2t+1},{\mu }_{2t+2},{\mu }_{2t+1}={\theta }_{\widehat{\mathcal{U}}},\\ d{\mu }_{2n+1},{\mu }_{2n},{\mu }_{2t}=K^{n-t}d{\mu }_{2t+1},{\mu }_{2t},{\mu }_{2t}={\theta }_{\widehat{\mathcal{U}}},\\ d{\mu }_{2n-1},{\mu }_{2n},{\mu }_{2t}=L_2{K}^{n-t-1}d{\mu }_{2t+1},{\mu }_{2t},{\mu }_{2t}={\theta }_{\widehat{\mathcal{U}}}.\end{array}$$

Therefore, for all $m<n$, $a\in \aleph$, and using the above inequalities, we have $$\begin{array}{}\text{(20)}& d{\mu }_{2n+1},{\mu }_{2m+1},a\preccurlyeq bd{\mu }_{2n+1},{\mu }_{2m+1},{\mu }_{2n}+bd{\mu }_{2n+1},{\mu }_{2n},a+bd{\mu }_{2n},{\mu }_{2m+1},a=bd{\mu }_{2n+1},{\mu }_{2n},a+bd{\mu }_{2n},{\mu }_{2m+1},a\preccurlyeq bd{\mu }_{2n+1},{\mu }_{2n},a+b^{2}d{\mu }_{2n},{\mu }_{2m+1},{\mu }_{2n-1}+b^{2}d{\mu }_{2n},{\mu }_{2n-1},a+b^{2}d{\mu }_{2n-1},{\mu }_{2m+1},a=bd{\mu }_{2n+1},{\mu }_{2n},a+b^{2}d{\mu }_{2n},{\mu }_{2n-1},a+b^{2}d{\mu }_{2n-1},{\mu }_{2m+1},a.\end{array}$$

Continuing this process, we get $$\begin{array}{c}\text{(21)}& d{\mu }_{2n+1},{\mu }_{2m+1},a\le bd{\mu }_{2n+1},{\mu }_{2n},a+b^{2}d{\mu }_{2n},{\mu }_{2n-1},a+b^{3}d{\mu }_{2n-1},{\mu }_{2n-2},a+b^{4}d{\mu }_{2n-2},{\mu }_{2n-3},a+\cdots +b^{n-m-4}d{\mu }_{2m+5},{\mu }_{2m+4},a+b^{n-m-3}d{\mu }_{2m+4},{\mu }_{2m+3},a+b^{n-m-2}d{\mu }_{2m+3},{\mu }_{2m+2},a+b^{n-m-1}d{\mu }_{2m+2},{\mu }_{2m+1},a\le b{K}^{n}d{\mu }_1,{\mu }_0,a+b^2{L}_2{K}^{n-1}d{\mu }_1,{\mu }_0,a+b^3{K}^{n-1}d{\mu }_1,{\mu }_0,a+b^4{L}_2{K}^{n-2}d{\mu }_1,{\mu }_0,a+\cdots +b^{n-m-4}{K}^{m+2}d{\mu }_1,{\mu }_0,a+b^{n-m-3}{L}_2{K}^{m+1}d{\mu }_1,{\mu }_0,a+b^{n-m-2}{K}^{m+1}d{\mu }_1,{\mu }_0,a+b^{n-m-1}{L}_2{K}^{m}d{\mu }_1,{\mu }_0,a\le b{K}^n+b^2{L}_2{K}^{n-1}+b^3{K}^{n-1}+b^4{L}_2{K}^{\text{n}-2}+\cdots +b^{n-m-4}{K}^{m+2}+b^{n-m-3}{L}_2{K}^{m+1}+b^{n-m-2}{K}^{m+1}+b^{n-m-1}{L}_2{K}^{m}d{\mu }_1,{\mu }_0,a,d{\mu }_{2n+1},{\mu }_{2m+1},a=b{K}^n+b^3{K}^{n-1}+\cdots +b^{n-m-4}{K}^{m+2}+b^{n-m-2}{K}^{m+1}+b^2{L}_2{K}^{n-1}+b^4{L}_2{K}^{n-2}+\cdots +b^{n-m-3}{L}_2{K}^{m+1}+b^{n-m-1}{L}_2{K}^{m}d{\mu }_1,{\mu }_0,a=K^{m+1}{b}^{n-m-2}e+b^{-2}K+\cdots +b^{m-n+3}{K}^{n-m-1}d{\mu }_1,{\mu }_0,a+L_2{K}^m{b}^{n-m-1}e+b^{-2}K+\cdots +b^{m-n+3}{K}^{n-m-1}d{\mu }_1,{\mu }_0,a=K^{m+1}{b}^{n-m-2}\sum _{r=0}^{\infty }{b^{-2}K}^r+L_2{K}^m{b}^{n-m-1}\sum _{r=0}^{\infty }{b^{-2}K}^{r}d{\mu }_1,{\mu }_0,a.\end{array}$$

That is, $$\begin{array}{}\text{(22)}& d{\mu }_{2n+1},{\mu }_{2m+1},a\preccurlyeq \sum _{r=0}^{\infty }{b^{-2}K}^r{K}^{m+1}{b}^{n-m-2}+L_2{K}^m{b}^{n-m-1}d{\mu }_1,{\mu }_0,a\preccurlyeq {e-b^{-2}K}^{-1}{K}^{m}K{b}^{n-m-2}+L_2{b}^{n-m-1}d{\mu }_1,{\mu }_0,a=M{K}^{m}d{\mu }_1,{\mu }_0,a,\end{array}$$where $M={e-b^{-2}K}^{-1}K{b}^{n-m-2}+L_2{b}^{n-m-1}$.

Similarly, we have $$\begin{array}{}\text{(23)}& d{\mu }_{2n},{\mu }_{2m+1},a\le bd{\mu }_{2n},{\mu }_{2n-1},a+b^{2}d{\mu }_{2n-1},{\mu }_{2n-2},a+b^{3}d{\mu }_{2n-2},{\mu }_{2n-3},a+b^{4}d{\mu }_{2n-3},{\mu }_{2n-4},a+\cdots +b^{n-m-4}d{\mu }_{2m+5},{\mu }_{2m+4},a+b^{n-m-3}d{\mu }_{2m+4},{\mu }_{2m+3},a+b^{n-m-2}d{\mu }_{2m+3},{\mu }_{2m+2},a+b^{n-m-1}d{\mu }_{2m+2},{\mu }_{2m+1},a\le b{L}_2{K}^{n-1}+b^3{L}_2{K}^{n-2}+\cdots +b^{n-m-4}{L}_2{K}^{m+2}+b^{n-m-2}{L}_2{K}^{m+1}+b^2{K}^{n-1}+b^4{K}^{n-2}+\cdots +b^{n-m-3}{K}^{m+2}+b^{n-m-1}{K}^{m}d{\mu }_1,{\mu }_0,a\le b^{n-m-2}{L}_2{K}^{m+1}e+b^{-2}K+\cdots +b^{m-n+3}{K}^{n-m-1}d{\mu }_1,{\mu }_0,a+b^{n-m-1}{K}^{m}e+b^{-2}K+\cdots +b^{m-n+3}{K}^{n-m-1}d{\mu }_1,{\mu }_0,a\le b^{n-m-2}{L}_2{K}^{m+1}\sum _{r=0}^{\infty }{b^{-2}K}^r+b^{n-m-1}{K}^m\sum _{r=0}^{\infty }{b^{-2}K}^{r}d{\mu }_1,{\mu }_0,a\le \sum _{r=0}^{\infty }{b^{-2}K}^r{b}^{n-m-2}{L}_2{K}^{m+1}+b^{n-m-1}{K}^{m}d{\mu }_1,{\mu }_0,a\le {e-b^{-2}K}^{-1}{K}^m{b}^{n-m-2}{L}_{2}K+b^{n-m-1}d{\mu }_1,{\mu }_0,a=N{K}^{m}d{\mu }_1,{\mu }_0,a,\end{array}$$where $N={e-b^{-2}K}^{-1}{b}^{n-m-2}{L}_{2}K+b^{n-m-1}$.

Similar to the above, one can easily get that $$\begin{array}{c}\text{(24)}& d{\mu }_{2n},{\mu }_{2m},a\preccurlyeq O{K}^{m}d{\mu }_1,{\mu }_0,a,d{\mu }_{2n+1},{\mu }_{2m},a\preccurlyeq P{K}^{m}d{\mu }_1,{\mu }_0,a,\end{array}$$where $O={e-b^{-2}K}^{-1}{b}^{n-m}{L}_2+b^{n-m-1}K$ and $P={e-b^{-2}K}^{-1}{b}^{n-m-1}+b^{n-m-2}{L}_2$.

From Lemmas 18 and 19, we have that $$\begin{array}{}\text{(25)}& rK=r{L}_1{L}_2\le r{L}_1\times r{L}_2=r{e-{\vartheta }_3-{\vartheta }_{4}b}^{-1}{\vartheta }_1+{\vartheta }_2+{\vartheta }_{4}b\times r{e-{\vartheta }_2-{\vartheta }_{5}b}^{-1}{\vartheta }_1+{\vartheta }_3+{\vartheta }_{5}b\le r{e-{\vartheta }_3-{\vartheta }_{4}b}^{-1}\times r{\vartheta }_1+{\vartheta }_2+{\vartheta }_{4}b\times r{e-{\vartheta }_2-{\vartheta }_{5}b}^{-1}\times r{\vartheta }_1+{\vartheta }_3+{\vartheta }_{5}b\le \frac{r{\vartheta }_1+r{\vartheta }_2+r{\vartheta }_{4}rb}{1-r{\vartheta }_3-r{\vartheta }_{4}rb}\times \frac{r{\vartheta }_1+r{\vartheta }_3+r{\vartheta }_{5}rb}{1-r{\vartheta }_2-r{\vartheta }_{5}rb}<1.\end{array}$$

Since $rK<1$, therefore in the light of Remark 5 and Lemma 25, $K^m⟶0$ as $m⟶\infty$, and so, for every $c\in \mathrm{int}{\mathcal{C}}_{\widehat{\mathcal{U}}}$, there exists $n_0\in \mathbb{N}$ such that $K^m\ll c$ for all $n>n_0$; that is, the sequence $K^m$ is a $c$-sequence. By Lemma 15, the sequences $M{K}^{m}d{\mu }_1,{\mu }_0,a$, $N{K}^{m}d{\mu }_1,{\mu }_0,a$, $O{K}^{m}d{\mu }_1,{\mu }_0,a$, and $P{K}^{m}d{\mu }_1,{\mu }_0,a$ are also $c$-sequences. Therefore, for any $c\in \widehat{\mathcal{U}}$ with ${\theta }_{\widehat{\mathcal{U}}}\ll c$, there exists $n_1\in \mathbb{N}$ such that, for any $n>m>n_1$, we have $d{\mu }_n,{\mu }_m,a\ll c$ for all $n>n_1$ and for all $a\in \aleph$. Thus, from Lemma 24, it means that $d{\mu }_n,{\mu }_m,a={\theta }_{\widehat{\mathcal{U}}}$. This implies that ${\mu }_n$ is a Cauchy sequence in $\aleph$.

If $H_l\aleph$ is complete for each $l=1,2,3,\cdots$, there exists $q\in H_l\aleph$ such that $$\begin{array}{}\text{(26)}& {\mu }_{2n}=J_{2n+2}{s}_{2n+1}=S_{2n+1}{s}_{2n}⟶qn⟶\infty .\end{array}$$

So we can find a $p\in \aleph$ such that $J_{2n+2}p=q$ (if $E_i\aleph$ is complete for each $i=1,2,3,\cdots$, there exists $q\in E_i\aleph \subseteq H_l\aleph$; then, the conclusion remains the same). Now, we show that $T_{2n+2}p=q$. By (9), we have $$\begin{array}{c}\text{(27)}& d{T}_{2n+2}p,q,a\preccurlyeq bd{T}_{2n+2}p,q,S_{2n+1}{s}_{2n}+bd{T}_{2n+2}p,S_{2n+1}{s}_{2n},a+bd{S}_{2n+1}{s}_{2n},q,a\preccurlyeq bd{T}_{2n+2}p,q,S_{2n+1}{s}_{2n}+bd{S}_{2n+1}{s}_{2n},q,a+b{\vartheta }_{1}d{I}_{2n+1}{s}_{2n},{\text{J}}_{2n+2}{s}_{2n+1},a+b{\vartheta }_{2}d{I}_{2n+1}{s}_{2n},S_{2n+1}{s}_{2n},a+b{\vartheta }_{3}d{J}_{2n+2}{s}_{2n+1},T_{2n+2}p,a+b{\vartheta }_{4}d{I}_{2n+1}{s}_{2n},T_{2n+2}p,a+b{\vartheta }_{5}d{J}_{2n+2}{s}_{2n+1},S_{2n+1}{s}_{2n},a,d{T}_{2n+2}p,q,a=bd{T}_{2n+2}p,q,{\mu }_{2n}+bd{\mu }_{2n},q,a+b{\vartheta }_{1}d{\mu }_{2n-1},q,a+b{\vartheta }_{2}d{\mu }_{2n-1},{\mu }_{2n},a+b{\vartheta }_{3}dq,T_{2n+2}p,a+b{\vartheta }_{4}d{\mu }_{2n-1},T_{2n+2}p,a+b{\vartheta }_{5}dq,{\mu }_{2n},a\preccurlyeq bd{T}_{2n+2}p,q,{\mu }_{2n}+bd{\mu }_{2n},q,a+b{\vartheta }_{1}d{\mu }_{2n-1},q,a+b{\vartheta }_{2}d{\mu }_{2n-1},{\mu }_{2n},a+b{\vartheta }_{3}dq,T_{2n+2}p,a+b^2{\vartheta }_{4}d{\mu }_{2n-1},T_{2n+2}p,q+b^2{\vartheta }_{4}d{\mu }_{2n-1},q,a+b^2{\vartheta }_{4}dq,T_{2n+2}p,a+b{\vartheta }_{5}dq,{\mu }_{2n},a.\end{array}$$

That is, $$\begin{array}{}\text{(28)}& e-b{\vartheta }_3-b^2{\vartheta }_{4}d{T}_{2n+2}p,q,a\preccurlyeq bd{\mu }_{2n},T_{2n+2}p,a+b^2{\vartheta }_{4}d{\mu }_{2n-1},T_{2n+2}p,q+b+b{\vartheta }_{5}d{\mu }_{2n},q,a+b{\vartheta }_1+b^2{\vartheta }_{4}d{\mu }_{2n-1},q,a+b{\vartheta }_{2}d{\mu }_{2n-1},{\mu }_{2n},a.\end{array}$$

Therefore, it follows from Proposition 29 and Lemmas 15 and 16 that $$\begin{array}{}\text{(29)}& e-b{\vartheta }_3-b^2{\vartheta }_{4}d{T}_{2n+2}p,q,a\preccurlyeq z_n,\end{array}$$where $z_n$ is a $c$-sequence in ${\mathcal{C}}_{\widehat{\mathcal{U}}}$. In addition, from Proposition 2 and $$\begin{array}{}\text{(30)}& rb{\vartheta }_3+b^2{\vartheta }_4\le rbr{\vartheta }_3+r{b}^{2}r{\vartheta }_4<1,\end{array}$$

it means that $e-b{\vartheta }_3+b^2{\vartheta }_4$ is invertible. In this case, we have $$\begin{array}{}\text{(31)}& e-b{\vartheta }_3-b^2{\vartheta }_{4}d{T}_{2n+2}p,q,a\ll c,\end{array}$$

for any $c\gg {\theta }_{\widehat{\mathcal{U}}}$, which together with Lemma 23 implies that ${\theta }_{\widehat{\mathcal{U}}}\preccurlyeq d{T}_{2n+2}p,q,a\ll c$, for any $a\in \aleph$, $n\in \mathbb{N}$, and $c\gg {\theta }_{\widehat{\mathcal{U}}}$ as $e-b{\vartheta }_3+b^2{\vartheta }_4$ is invertible. Therefore, by Lemma 24, we have $d{T}_{2n+2}p,q,a={\theta }_{\widehat{\mathcal{U}}}$ for any $n\in \mathbb{N}$. Namely, $T_{2n+2}p=q$ for any $n\in \mathbb{N}$. That is, $T_{2n+2}p=q=J_{2n+2}p$.

At the same time, as $q=T_{2n+2}p\in F_j\aleph \subseteq G_k\aleph$, there exists $u\in \aleph$ such that $I_{2n+3}u=q$.

Now, we show that $S_{2n+1}u=q$. From (9), we have that $$\begin{array}{}\text{(32)}& d{S}_{2n+1}u,q,a=d{S}_{2n+1}u,T_{2n+2}p,a\preccurlyeq {\vartheta }_{1}d{I}_{2n+3}u,J_{2n+2}p,a+{\vartheta }_{2}d{I}_{2n+3}u,S_{2n+1}u,a+{\vartheta }_{3}d{J}_{2n+2}p,T_{2n+2}p,a+{\vartheta }_{4}d{I}_{2n+3}u,T_{2n+2}p,a+{\vartheta }_{5}d{J}_{2n+2}p,S_{2n+1}u,a={\vartheta }_{1}dq,q,a+{\vartheta }_{2}dq,S_{2n+1}u,a+{\vartheta }_{3}dq,q,a+{\vartheta }_{4}dq,q,a+{\vartheta }_{5}dq,S_{2n+1}u,a.\end{array}$$

That is, $$\begin{array}{}\text{(33)}& d{S}_{2n+1}u,q,a\preccurlyeq {\vartheta }_2+{\vartheta }_{5}d{S}_{2n+1}u,q,a.\end{array}$$

Hence, by Lemma 20, we know that $d{S}_{2n+1}u,q,a={\theta }_{\widehat{\mathcal{U}}}$, and so $S_{2n+1}u=q$. Therefore, $S_{2n+1}u=I_{2n+3}u=q$ and $T_{2n+2}p=J_{2n+2}p=q$.

Next, if we assume $G_k\aleph$ is complete for each $k=1,2,3,\cdots$, there exists $q\in G_k\aleph$ such that $$\begin{array}{}\text{(34)}& {\mu }_{2n+1}=I_{2n+3}{s}_{2n+2}=T_{2n+2}{s}_{2n+1}⟶q\text{as}n⟶\infty .\end{array}$$

So, we can find $u\in \aleph$ such that $I_{2n+3}u=q$ (if $F_j\aleph$ is complete for each $j=1,2,3,\cdots$, there exists $q\in F_j\aleph \subseteq G_k\aleph$; then, the conclusion remains the same).

Now, we show that $S_{2n+1}u=q$. By (9), we get that $$\begin{array}{}\text{(35)}& d{S}_{2n+1}u,q,a\preccurlyeq bd{S}_{2n+1}u,q,T_{2n+2}{s}_{2n+1}+bd{S}_{2n+1}u,T_{2n+2}{s}_{2n+1},a+bd{T}_{2n+2}{s}_{2n+1},q,a\preccurlyeq bd{S}_{2n+1}u,q,T_{2n+2}{s}_{2n+1}+bd{T}_{2n+2}{s}_{2n+1},q,a+b{\vartheta }_{1}d{I}_{2n+3}u,J_{2n+2}{s}_{2n+1},a+b{\vartheta }_{2}d{I}_{2n+3}u,S_{2n+1}u,a+b{\vartheta }_{3}d{J}_{2n+2}{s}_{2n+1},T_{2n+2}{s}_{2n+1},a+b{\vartheta }_{4}d{I}_{2n+3}u,T_{2n+2}{s}_{2n+1},a+b{\vartheta }_{5}d{J}_{2n+2}{s}_{2n+1},S_{2n+1}u,a\preccurlyeq bd{S}_{2n+1}u,q,{\mu }_{2n+1}+bd{\mu }_{2n+1},q,a+b{\vartheta }_{1}dq,{\mu }_{2n},a+b{\vartheta }_{2}dq,S_{2n+1}u,a+b{\vartheta }_{3}d{\mu }_{2n},{\mu }_{2n+1},a+b{\vartheta }_{4}dq,{\mu }_{2n+1},a+b^2{\vartheta }_{5}d{\mu }_{2n},S_{2n+1}u,q+b^2{\vartheta }_{5}d{\mu }_{2n},q,a+b^2{\vartheta }_{5}dq,S_{2n+1}u,a.\end{array}$$

That is, $$\begin{array}{}\text{(36)}& e-b{\vartheta }_3-b^2{\vartheta }_{4}d{T}_{2n+2}p,q,a\le bd{\mu }_{2n},T_{2n+2}p,a+b^2{\vartheta }_{4}d{\mu }_{2n-1},T_{2n+2}p,q+b+b{\vartheta }_{5}d{\mu }_{2n},q,a+b{\vartheta }_1+b^2{\vartheta }_{4}d{\mu }_{2n-1},q,a+b{\vartheta }_{2}d{\mu }_{2n-1},{\mu }_{2n},a.\end{array}$$

Therefore, it follows from Proposition 29 and Lemmas 15 and 16 that $$\begin{array}{}\text{(37)}& e-b{\vartheta }_2-b^2{\vartheta }_{5}d{S}_{2n+1}u,q,a\preccurlyeq z_n^{\ast },\end{array}$$where $z_n^{\ast }$ is a $c$-sequence in ${\mathcal{C}}_{\widehat{\mathcal{U}}}$. In addition, from Proposition 2 and $$\begin{array}{}\text{(38)}& rb{\vartheta }_3+b^2{\vartheta }_4\le rbr{\vartheta }_3+r{b}^{2}r{\vartheta }_4<1,\end{array}$$

it means that $e-b{\vartheta }_2+b^2{\vartheta }_5$ is invertible. In this case, we have $$\begin{array}{}\text{(39)}& e-b{\vartheta }_2-b^2{\vartheta }_{5}d{S}_{2n+1}u,q,a\ll c,\end{array}$$

for any $c\gg {\theta }_{\widehat{\mathcal{U}}}$, which together with Lemma 23 implies that ${\theta }_{\widehat{\mathcal{U}}}\preccurlyeq d{S}_{2n+1}u,q,a\ll c$ for any $a\in \aleph$, $n\in \mathbb{N}$, and $c\gg {\theta }_{\widehat{\mathcal{U}}}$ as $e-b{\vartheta }_2+b^2{\vartheta }_5$ is invertible. Therefore, by Lemma 24, we have $d{S}_{2n+1}u,q,a={\theta }_{\widehat{\mathcal{U}}}$ for any $n\in \mathbb{N}$. Namely, $S_{2n+1}u=q$ for any $n\in \mathbb{N}$. That is, $S_{2n+1}u=q=I_{2n+3}u$.

At the same time, as $q=S_{2n+1}u\in E_i\aleph \subseteq H_l\aleph$, there exists $p\in \aleph$ such that $J_{2n+2}p=q$. Now, we show that $T_{2n+2}p=q$. From (9), we have $$\begin{array}{}\text{(40)}& d{T}_{2n+2}p,q,a=d{S}_{2n+1}u,T_{2n+2}p,a\preccurlyeq {\vartheta }_{1}d{I}_{2n+3}u,J_{2n+2}p,a+{\vartheta }_{2}d{I}_{2n+3}u,S_{2n+1}u,a+{\vartheta }_{3}d{J}_{2n+2}p,T_{2n+2}p,a+{\vartheta }_{4}d{I}_{2n+3}u,T_{2n+2}p,a+{\vartheta }_{5}d{J}_{2n+2}p,S_{2n+1}u,a={\vartheta }_{1}dq,q,a+{\vartheta }_{2}dq,q,a+{\vartheta }_{3}dq,T_{2n+2}p,a+{\vartheta }_{4}dq,T_{2n+2}p,a+{\vartheta }_{5}dq,q,a.\end{array}$$

That is, $$\begin{array}{}\text{(41)}& d{T}_{2n+2}p,q,a\preccurlyeq {\vartheta }_3+{\vartheta }_{4}d{T}_{2n+2}p,q,a.\end{array}$$

Hence, by Lemma 20, we know that $d{T}_{2n+2}p,q,a={\theta }_{\widehat{\mathcal{U}}}$, and so $T_{2n+2}p=q$. Therefore, $T_{2n+2}p=J_{2n+2}p=q$ and $S_{2n+1}u=I_{2n+3}u=q$.

Finally, we show that $S_{2i+1}$ and $I_{2k+3}$, $T_{2j+2}$, and $J_{2l+2}$ have a unique point of coincidence in $\aleph$. Assume that there is another point $z\in \aleph$ such that $T_{2n+2}x=J_{2n+2}=z$; then, $$\begin{array}{}\text{(42)}& dq,z,a=d{S}_{2n+1}u,T_{2n+2}x,a\preccurlyeq {\vartheta }_{1}d{I}_{2n+3}u,J_{2n+2}x,a+{\vartheta }_{2}d{I}_{2n+3}u,S_{2n+1}u,a+{\vartheta }_{3}d{J}_{2n+2}x,T_{2n+2}x,a+{\vartheta }_{4}d{I}_{2n+3}u,T_{2n+2}x,a+{\vartheta }_{5}d{J}_{2n+2}x,S_{2n+1}u,a={\vartheta }_{1}dq,z,a+{\vartheta }_{2}dq,q,a+{\vartheta }_{3}dz,z,a+{\vartheta }_{4}dq,z,a+{\vartheta }_{5}dz,q,a.\end{array}$$

That is, $$\begin{array}{}\text{(43)}& dq,z,a\preccurlyeq {\vartheta }_1+{\vartheta }_4+{\vartheta }_{5}dq,z,a.\end{array}$$

Hence, by Lemma 20, we have that $dq,z,a={\theta }_{\widehat{\mathcal{U}}}$, and so $q=z$; that is, $q$ is the unique point of coincidence of $T_{2j+2}$ and $J_{2l+2}$.

Similarly, we also have $q$ which is the unique point of coincidence of $S_{2i+1}$ and $I_{2k+3}$ by induction.

So, according to Lemma 28, $q$ is the unique common fixed point of $S_{2i+1},I_{2k+3}$ and $T_{2j+2},J_{2l+2}$ for each $i,j,k,l=1,2,3,\cdots$. Therefore, $q$ is the unique common fixed point of $S_{2i+1},I_{2k+3},T_{2j+2}$, and $J_{2l+2}$.

Now, it is left to show that $q$ is the unique common fixed point of ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{\text{k}=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$.

As $q=S_{2n+1}q=E_n^{{\eta }_n}q$, so we have $E_{n}q=E_n{E}_n^{{\eta }_n}q=E_n^{{\eta }_n}{E}_{n}q=S_{2n+1}q$, that is, $S_{2n+1}{E}_{n}q=E_{n}q$. But $S_{2n+1}q=q$ is unique; therefore, $E_{n}q=q$ for $n=1,2,3,\cdots$.

Also, as $q=T_{2n+2}q=F_n^{{\eta }_n}q$, so we have $F_{n}q=F_n{F}_n^{{\eta }_n}q=F_n^{{\eta }_n}{F}_{n}q=T_{2n+2}q$, that is, $T_{2n+2}{F}_{n}q=F_{n}q$. But $T_{2n+2}q=q$ is unique; therefore, $F_{n}q=q$ for $n=1,2,3,\cdots$. Similarly, $G_{n}q=q$ and $H_{n}q=q$ for $n=1,2,3,\cdots$. Thus, the four families of mappings ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique common fixed point.☐

Remark 31.

Theorem 30 of this paper extends and improves Theorem 2.1 of [30] from cone metric spaces to cone $b_2$-metric spaces; also, it extends and improves Theorem 3.2 of [17] and Theorem 3.1 of [31] from one family and two families, respectively, to four families of mappings.

We obtain a series of new common fixed-point results using Theorem 30 for four families of mappings in the context of cone $b_2$-metric spaces over Banach algebras, which generalize and improve many known results from the existence literature.

Corollary 32.

Let $\aleph ,d$ over the Banach algebra $\widehat{\mathcal{U}}$ be a cone $b_2$-metric space with $b\pm e$ and ${\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$ be a cone in $\widehat{\mathcal{U}}$. Let ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ be four families of self-mappings on $\aleph$. For all $i,j,k,l\in \mathbb{N}$, if a sequence ${{\eta }_n}_{n=1}^{\infty }$ exists of nonnegative integers, such that for all $s,m,z\in \aleph$, $$\begin{array}{}\text{(44)}& d{E}_i^{{\eta }_i}s,F_j^{{\eta }_j}m,a\preccurlyeq \alpha d{G}_k^{{\eta }_k}s,H_l^{{\eta }_l}m,a+\beta d{G}_k^{{\eta }_k}s,E_i^{{\eta }_i}s,a+d{H}_l^{{\eta }_l}m,F_j^{{\eta }_j}m,a+\gamma d{G}_k^{{\eta }_k}s,F_j^{{\eta }_j}m,a+d{H}_l^{{\eta }_l}m,E_i^{{\eta }_i}s,a,\end{array}$$where $\alpha ,\beta ,\gamma \in {\mathcal{C}}_{\widehat{\mathcal{U}}}$ with $r\alpha +r\beta +2r\gamma rb<1$, $r\beta rb+r\gamma r{b}^2<1$, and $\alpha ,\beta ,\gamma ,b$ commute. If $E_i\aleph \subseteq H_l\aleph$, $F_j\aleph \subseteq G_k\aleph$, and one of $E_i\aleph$, $G_k\aleph$, $H_l\aleph$, and $F_j\aleph$ are a complete subspace of $\aleph$ for each $i,j,k,l\ge 1$, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique point of coincidence in $\aleph$. Moreover, if $F_j,H_l$ and $E_i,G_k$ are weakly compatible, respectively, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique common fixed point.

Proof.

Let ${\vartheta }_1=\alpha ,{\vartheta }_2={\vartheta }_3=\beta ,{\vartheta }_4={\vartheta }_5=\gamma$ in Theorem 30.☐

Corollary 33.

Let $\aleph ,d$ over the Banach algebra $\widehat{\mathcal{U}}$ be a cone $b_2$-metric space with $b\pm e$ and ${\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$ be a cone in $\widehat{\mathcal{U}}$. Let ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ be four families of self-mappings on $\aleph$. For all $i,j,k,l\in \mathbb{N}$, if a sequence ${{\eta }_n}_{n=1}^{\infty }$ exists of nonnegative integers, such that for all $s,m,z\in \aleph$, $$\begin{array}{}\text{(45)}& d{E}_i^{{\eta }_i}s,F_j^{{\eta }_j}m,a\preccurlyeq {\vartheta }_{1}d{G}_k^{{\eta }_k}s,H_l^{{\eta }_l}m,a+{\vartheta }_{2}d{G}_k^{{\eta }_k}s,E_i^{{\eta }_i}s,a+{\vartheta }_{3}d{H}_l^{{\eta }_l}m,F_j^{{\eta }_j}m,a,\end{array}$$where ${\vartheta }_1,{\vartheta }_2,{\vartheta }_3\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$ with $r{\vartheta }_1+r{\vartheta }_2+r{\vartheta }_3<1$, $r{\vartheta }_{2}rb+r{\vartheta }_{3}rb<1$, and ${\vartheta }_1,{\vartheta }_2,{\vartheta }_3,b$ commute. If $E_i\aleph \subseteq H_l\aleph$, $F_j\aleph \subseteq G_k\aleph$, and one of $E_i\aleph$, $G_k\aleph$, $H_l\aleph$, and $F_j\aleph$ are a complete subspace of $\aleph$ for each $i,j,k,l\ge 1$, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique point of coincidence in $\aleph$. Moreover, if $F_j,H_l$ and $E_i,G_k$ are weakly compatible, respectively, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique common fixed point.

Proof.

Taking ${\vartheta }_4={\vartheta }_5={\theta }_{\widehat{\mathcal{U}}}$ in Theorem 30, one can get the desired result.☐

Remark 34.

We can have Theorem 3.1 in [21], when ${E_i}_{i=1}^{\infty }$ and ${F_j}_{j=1}^{\infty }$ are the same mapping and ${G_k}_{k=1}^{\infty }$ and ${H_l}_{l=1}^{\infty }$ are the identity mappings. Therefore, Theorem 3.1 of [21] is a special case of Corollary 33. Also, Corollary 33 of this paper generalizes Theorem 2.1 of [10] from the cone 2-metric space to the cone $b_2$-metric space and extends Theorem 6.1 in [12].

Corollary 35.

Let $\aleph ,d$ over the Banach algebra $\widehat{\mathcal{U}}$ be a cone $b_2$-metric space with $b\pm e$ and ${\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$ be a cone in $\widehat{\mathcal{U}}$. Let ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ be four families of self-mappings on $\aleph$. For all $i,j,k,l\in \mathbb{N}$, if a sequence ${{\eta }_n}_{n=1}^{\infty }$ exists of nonnegative integers, such that for all $s,m,z\in \aleph$, $$\begin{array}{}\text{(46)}& d{E}_i^{{\eta }_i}s,F_j^{{\eta }_j}m,a\preccurlyeq \alpha d{G}_k^{{\eta }_k}s,H_l^{{\eta }_l}m,a+\beta d{G}_k^{{\eta }_k}s,E_i^{{\eta }_i}s,a+d{H}_l^{{\eta }_l}m,F_j^{{\eta }_j}m,a,\end{array}$$where $\alpha ,\beta \in {\mathcal{C}}_{\widehat{\mathcal{U}}}$ with $r\alpha +2r\beta <1$, $2r\beta rb<1$, and $\alpha ,\beta ,b$ commute. If $E_i\aleph \subseteq H_l\aleph$, $F_j\aleph \subseteq G_k\aleph$, and one of $E_i\aleph$, $G_k\aleph$, $H_l\aleph$, and $F_j\aleph$ is a complete subspace of $\aleph$ for each $i,j,k,l\ge 1$, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique point of coincidence in $\aleph$. Moreover, if $F_j,H_l$ and $E_i,G_k$ are weakly compatible, respectively, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique common fixed point.

Proof.

One can the result taking ${\vartheta }_1=\alpha ,{\vartheta }_2={\vartheta }_3=\beta$, and ${\vartheta }_4={\vartheta }_5={\theta }_{\widehat{\mathcal{U}}}$ in Theorem 30.☐

Remark 36.

Corollary 35 of this paper extends Theorem 6 in [32]; therefore, Theorem 6 in [32] is a special case of Corollary 35.

Corollary 37.

Let $\aleph ,d$ over the Banach algebra $\widehat{\mathcal{U}}$ be a cone $b_2$-metric space with $b\succcurlyeq \text{e}$ and ${\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$ be a cone in $\widehat{\mathcal{U}}$. Let ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ be four families of self-mappings on $\aleph$. For all $i,j,k,l\in \mathbb{N}$, if a sequence ${{\eta }_n}_{n=1}^{\infty }$ exists of nonnegative integers, such that for all $s,m,z\in \aleph$, $$\begin{array}{}\text{(47)}& d{E}_i^{{\eta }_i}s,F_j^{{\eta }_j}m,a\preccurlyeq \alpha d{G}_k^{{\eta }_k}s,H_l^{{\eta }_l}m,a+\beta d{G}_k^{{\eta }_k}s,F_j^{{\eta }_j}m,a+\gamma d{H}_l^{{\eta }_l}m,E_i^{{\eta }_i}s,a,\end{array}$$where $\alpha ,\beta ,\gamma \in {\mathcal{C}}_{\widehat{\mathcal{U}}}$ with $r\alpha +2r\beta rb+2r\gamma rb<1$, $r\beta r{b}^2+r\gamma r{b}^2<1$, and $\alpha ,\beta ,\gamma ,b$ commute. If $E_i\aleph \subseteq H_l\aleph$, $F_j\aleph \subseteq G_k\aleph$, and one of $E_i\aleph$, $G_k\aleph$, $H_l\aleph$, and $F_j\aleph$ are a complete subspace of $\aleph$ for each $i,j,k,l\ge 1$, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique point of coincidence in $\aleph$. Moreover, if $F_j,H_l$ and $E_i,G_k$ are weakly compatible, respectively, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique common fixed point.

Proof.

One can get the result taking ${\vartheta }_1=\alpha$,${\vartheta }_2={\vartheta }_3={\theta }_{\widehat{\mathcal{U}}}$ and ${\vartheta }_4=\beta ,{\vartheta }_5=\gamma$ in Theorem 30.☐

Corollary 38.

Let $\aleph ,d$ over the Banach algebra $\widehat{\mathcal{U}}$ be a cone $b_2$-metric space with $b\succcurlyeq e$ and ${\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$ be a cone in $\widehat{\mathcal{U}}$. Let ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ be four families of self-mappings on $\aleph$. For all $i,j,k,l\in \mathbb{N}$, if a sequence ${{\eta }_n}_{n=1}^{\infty }$ exists of nonnegative integers, such that for all $s,m,z\in \aleph$, $$\begin{array}{}\text{(48)}& d{E}_i^{{\eta }_i}s,F_j^{{\eta }_j}m,a\preccurlyeq kd{G}_k^{{\eta }_k}s,E_i^{{\eta }_i}s,a+ld{H}_l^{{\eta }_l}m,F_j^{{\eta }_j}m,a,\end{array}$$where $k,l\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$ with $rk+rl<1$, $rkrb+rlrb<1$, and $k,l,b$ commute. If $E_i\aleph \subseteq H_l\aleph$, $F_j\aleph \subseteq G_k\aleph$, and one of $E_i\aleph$, $G_k\aleph$, $H_l\aleph$, and $F_j\aleph$ are a complete subspace of $\aleph$ for each $i,j,k,l\ge 1$, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique point of coincidence in $\aleph$. Moreover, if $F_j,H_l$ and $E_i,G_k$ are weakly compatible, respectively, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique common fixed point.

Proof.

Let ${\vartheta }_1={\vartheta }_4={\vartheta }_5={\theta }_{\widehat{\mathcal{U}}},{\vartheta }_2=k,{\vartheta }_3=l$ in Theorem 30.☐

Corollary 39.

Let $\aleph ,d$ over the Banach algebra $\widehat{\mathcal{U}}$ be a cone $b_2$-metric space with $b\succcurlyeq e$ and ${\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$ be a cone in $\widehat{\mathcal{U}}$. Let ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ be four families of self-mappings on $\aleph$. For all $i,j,k,l\in \mathbb{N}$, if a sequence ${{\eta }_n}_{n=1}^{\infty }$ exists of nonnegative integers, such that for all $s,m,z\in \aleph$, $$\begin{array}{}\text{(49)}& d{E}_i^{{\eta }_i}s,F_j^{{\eta }_j}m,a\preccurlyeq kd{G}_k^{{\eta }_k}s,F_j^{{\eta }_j}m,a+ld{H}_l^{{\eta }_l}m,E_i^{{\eta }_i}s,a,\end{array}$$where $k,l\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$ with $2rkrb+2rlrb<1$, $rkr{b}^2+rlr{b}^2<1$, and $k,l,b$ commute. If $E_i\aleph \subseteq H_l\aleph$, $F_j\aleph \subseteq G_k\aleph$, and one of $E_i\aleph$, $G_k\aleph$, $H_l\aleph$, and $F_j\aleph$ are a complete subspace of $\aleph$ for each $i,j,k,l\ge 1$, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique point of coincidence in $\aleph$. Moreover, if $F_j,H_l$ and $E_i,G_k$ are weakly compatible, respectively, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique common fixed point.

Proof.

Let ${\vartheta }_1={\vartheta }_2={\vartheta }_3={\theta }_{\widehat{\mathcal{U}}},{\vartheta }_4=k,{\vartheta }_5=l$ in Theorem 30.☐

Corollary 40.

Let $\aleph ,d$ over the Banach algebra $\widehat{\mathcal{U}}$ be a cone $b_2$-metric space with $b\succcurlyeq e$ and ${\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$ be a cone in $\widehat{\mathcal{U}}$. Let ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ be four families of self-mappings on $\aleph$. For all $i,j,k,l\in \mathbb{N}$, if a sequence ${{\eta }_n}_{n=1}^{\infty }$ exists of nonnegative integers, such that for all $s,m,z\in \aleph$, $$\begin{array}{}\text{(50)}& d{E}_i^{{\eta }_i}s,F_j^{{\eta }_j}m,a\preccurlyeq kd{G}_k^{{\eta }_k}s,H_l^{{\eta }_l}m,a,\end{array}$$where $k\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$ with $rk<1$ and $k,b$ commute. If $E_i\aleph \subseteq H_l\aleph$, $F_j\aleph \subseteq G_k\aleph$, and one of $E_i\aleph$, $G_k\aleph$, $H_l\aleph$, and $F_j\aleph$ are a complete subspace of $\aleph$ for each $i,j,k,l\ge 1$, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique point of coincidence in $\aleph$. Moreover, if $F_j,H_l$ and $E_i,G_k$ are weakly compatible, respectively, then ${E_i}_{i=1}^{\infty }$, ${F_j}_{j=1}^{\infty }$, ${G_k}_{k=1}^{\infty }$, and ${H_l}_{l=1}^{\infty }$ have a unique common fixed point.

Proof.

Let ${\vartheta }_1=k,{\vartheta }_2={\vartheta }_3={\vartheta }_4={\vartheta }_5={\theta }_{\widehat{\mathcal{U}}}$ in Theorem 30.☐

From the above corollary, we obtain the following.

Corollary 41.

Let $\aleph ,d$ over the Banach algebra $\widehat{\mathcal{U}}$ be a complete cone $b_2$-metric space with $b\succcurlyeq e$ and ${\mathcal{C}}_{\widehat{\mathcal{U}}}\ne \varnothing$ be a cone in $\widehat{\mathcal{U}}$. Let ${E_i}_{i=1}^{\infty }$ be the family of self-mapping on $\aleph$. For all $i\in \mathbb{N}$ and for all $s,m,z\in \aleph$, $$\begin{array}{}\text{(51)}& d{E}_{i}s,E_{i}m,a\preccurlyeq kds,m,a,\end{array}$$where $k\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$ with $rk<1$ and $k,b$ commute. Then, ${E_i}_{i=1}^{\infty }$ have a unique common fixed point.

Proof.

Taking ${\eta }_n=1$, $E_i=F_j$, and $G_k,H_l$ which are identity mappings in Corollary 40, then we can obtain the required result.☐

We finish this section with an example that will demonstrate the consequence of Theorem 30.

Example 42.

Let $\widehat{\mathcal{U}}={ℝ}^2$. For each $s_1,s_2\in \widehat{\mathcal{U}}$, $s_1,s_2=s_1+s_2$. The multiplication is defined by $sm=s_1,s_2{m}_1,m_2=s_1{m}_1,s_1{m}_2+s_2{m}_1$. Then, $\widehat{\mathcal{U}}$ is a Banach algebra with unit element $e=1,0$. Let ${\mathcal{C}}_{\widehat{\mathcal{U}}}=s_1,s_2\in {ℝ}^2\mid s_1,s_2\ge 0$. Then, ${\mathcal{C}}_{\widehat{\mathcal{U}}}$ is a cone in $\widehat{\mathcal{U}}$.

Let $\aleph =s,0\in {ℝ}^2\mid s\ge 0\cup 0,2\subset {ℝ}^2$ and define $d:\aleph \times \aleph \times \aleph ⟶\widehat{\mathcal{U}}$ as follows: $$\begin{array}{}\text{(52)}& dS,M,Z=\begin{array}{cc}0,0,\hfill & \text{if}\text{atleast}\text{two}\text{of}S,M,Z\text{are}\text{equal},\hfill \\ dPS,M,Z,\hfill & P\text{denotes}\text{permutations},\hfill \\ \Delta ,\Delta ,\hfill & \text{otherwise},\hfill \end{array}\end{array}$$where $\Delta$ is the square of the area of the triangle $S,M,Z$. We have $$\begin{array}{}\text{(53)}& ds,0,m,0,0,2\preccurlyeq ds,0,m,0,z,0+ds,0,z,0,0,2+dz,0,m,0,0,2.\end{array}$$

That is, ${s-m}^2\preccurlyeq {s-z}^2+{z-m}^2$, which shows that $d$ is not a cone 2-metric, because $-9/2,-9/2\cancel{\in }{\mathcal{C}}_{\widehat{\mathcal{U}}}$ for $s,m,z\ge 0$ with $s=5$, $m=0$, and $z=1/2$. But for the parameter $b=2,0\succcurlyeq e$ is a cone $b_2$-metric space over the Banach algebra $\widehat{\mathcal{U}}$.

Now, we define mappings $E_i:\aleph ⟶\aleph$$i=1,2,3,\cdots$ by $$\begin{array}{c}\text{(54)}& E_{i}s,0={\frac{1}{6}}^{1/2i-1}{\frac{3}{2}}^{1/2i-1}\frac{s}{s^{2i-2}},0,E_{i}0,2=0,0.\end{array}$$

We have $$\begin{array}{}\text{(55)}& E_i^{2i-1}s,0=E_i^{2i-2}{E}_{i}s,0=E_i^{2i-2}{\frac{1}{6}}^{1/2i-1}{\frac{3}{2}}^{1/2i-1}\frac{s}{s^{2i-2}},0=E_i^{2i-3}{\frac{1}{6}}^{2/2i-1}{\frac{3}{2}}^{2/2i-1}\frac{s^2}{s^{2i-2}},0=E_i^{2i-4}{\frac{1}{6}}^{3/2i-1}{\frac{3}{2}}^{3/2i-1}\frac{s^3}{s^{2i-2}},0=\cdots \cdots =E_i{\frac{1}{6}}^{2i-2/2i-1}{\frac{3}{2}}^{2i-2/2i-1}\frac{s^{2i-2}}{s^{2i-2}},0={\frac{1}{6}}^{2i-1/2i-1}{\frac{3}{2}}^{2i-1/2i-1}\frac{s^{2i-1}}{s^{2i-2}},0=\frac{1}{4}s,0.\end{array}$$

We define mappings $G_k:\aleph ⟶\aleph$ ($k=1,2,3,\cdots$) by $$\begin{array}{c}\text{(56)}& G_{k}s,0={\frac{1}{3}}^{1/2k-1}{\frac{1}{2}}^{-1/2k-1}\frac{s}{s^{2k-2}},0,G_{k}0,2=0,0.\end{array}$$

We have $$\begin{array}{c}\text{(57)}& G_k^{2k-1}s,0=G_k^{2k-2}{G}_{k}s,0=G_k^{2k-2}{\frac{1}{3}}^{1/2k-1}{\frac{1}{2}}^{-1/2k-1}\frac{s}{s^{2k-2}},0=G_k^{2k-3}{\frac{1}{3}}^{2/2i-1}{\frac{1}{2}}^{-2/2k-1}\frac{s^2}{s^{2k-2}},0=G_k^{2k-4}{\frac{1}{3}}^{3/2k-1}{\frac{1}{2}}^{-3/2k-1}\frac{s^3}{s^{2k-2}},0=\cdots \cdots =G_k{\frac{1}{3}}^{2k-2/2k-1}{\frac{1}{2}}^{-2k-2/2k-1}\frac{s^{2k-2}}{s^{2k-2}},0,G_k^{2k-1}s,0={\frac{1}{3}}^{2k-1/2k-1}{\frac{1}{2}}^{-2k+1/2k-1}\frac{s^{2k-1}}{s^{2k-2}},0=\frac{2}{3}s,0.\end{array}$$

Similarly, we define mappings $F_j,H_l:\aleph ⟶\aleph$ ($j,l=1,2,3,\cdots$) by $$\begin{array}{c}\text{(58)}& F_{j}s,0={\frac{2}{3}}^{1/2j-1}{\frac{3}{10}}^{1/2j-1}\frac{s}{s^{2j-2}},0,F_{j}0,2=0,0,\\ H_{l}s,0={\frac{2}{3}}^{1/2l-1}{\frac{1}{2}}^{-1/2l-1}\frac{s}{s^{2l-2}},0,\\ H_{l}0,2=0,0.\end{array}$$

Then, it is not difficult to show that $F_j^{2j-1}s,0=1/5s,0$ and $H_l^{2l-1}s,0=1/3s,0$. Choose ${\vartheta }_1=1/10,0$, ${\vartheta }_2={\vartheta }_3=1/8,0$, and ${\vartheta }_4={\vartheta }_5=1/16,0$. Clearly, $$\begin{array}{}\text{(59)}& \sum _{w=1}^{3}r{\vartheta }_w+2r{\vartheta }_{4}rb+2r{\vartheta }_{5}rb=\frac{1}{10}+\frac{1}{8}+\frac{1}{8}+2\frac{1}{16}2+2\frac{1}{16}2=\frac{34}{40}<1,\end{array}$$

also $r{\vartheta }_{2}rb+r{\vartheta }_{5}r{b}^2=21/8+41/16=1/2<1$ and $r{\vartheta }_{3}rb+r{\vartheta }_{4}r{b}^2=21/8+41/16=1/2<1$.

Now, considering the contractive condition (8), we have $$\begin{array}{}\text{(60)}& d\frac{s}{4},0,\frac{s}{5},0,0,2\preccurlyeq \frac{1}{10},0d\frac{2s}{3},0,\frac{s}{3},0,0,2+\frac{1}{8},0d\frac{2s}{3},0,\frac{s}{4},0,0,2+\frac{1}{8},0d\frac{s}{3},0,\frac{s}{5},0,0,2+\frac{1}{16},0d\frac{2s}{3},0,\frac{s}{5},0,0,2+\frac{1}{16},0d\frac{s}{3},0,\frac{s}{4},0,0,2,\end{array}$$that is, $$\begin{array}{}\text{(61)}& {\frac{s}{4}-\frac{s}{5}}^2,{\frac{s}{4}-\frac{s}{5}}^2\preccurlyeq \frac{1}{10},0{\frac{2s}{3}-\frac{s}{3}}^2,{\frac{2s}{3}-\frac{s}{3}}^2+\frac{1}{8},0{\frac{2s}{3}-\frac{s}{4}}^2,{\frac{2s}{3}-\frac{s}{4}}^2+\frac{1}{8},0{\frac{s}{3}-\frac{s}{5}}^2,{\frac{s}{3}-\frac{s}{5}}^2+\frac{1}{16},0{\frac{2s}{3}-\frac{s}{5}}^2,{\frac{2s}{3}-\frac{s}{5}}^2+\frac{1}{16},0+{\frac{s}{3}-\frac{s}{4}}^2,{\frac{s}{3}-\frac{s}{4}}^2=\frac{1}{10}{\frac{2s}{3}-\frac{s}{3}}^2,\frac{1}{10}{\frac{2s}{3}-\frac{s}{3}}^2+\frac{1}{8}{\frac{2s}{3}-\frac{s}{4}}^2,\frac{1}{8}{\frac{2s}{3}-\frac{s}{4}}^2+\frac{1}{8}{\frac{s}{3}-\frac{s}{5}}^2,\frac{1}{8}{\frac{s}{3}-\frac{s}{5}}^2+\frac{1}{16}{\frac{2s}{3}-\frac{s}{5}}^2,\frac{1}{16}{\frac{2s}{3}-\frac{s}{5}}^2+\frac{1}{16}{\frac{s}{3}-\frac{s}{4}}^2,\frac{1}{16}{\frac{s}{3}-\frac{s}{4}}^2,\end{array}$$which means that $$\begin{array}{}\text{(62)}& {\frac{s}{4}-\frac{s}{5}}^2\preccurlyeq \frac{1}{10}{\frac{2s}{3}-\frac{s}{3}}^2+\frac{1}{8}{\frac{2s}{3}-\frac{s}{4}}^2+\frac{1}{8}{\frac{s}{3}-\frac{s}{5}}^2+\frac{1}{16}{\frac{2s}{3}-\frac{s}{5}}^2+\frac{1}{16}{\frac{s}{3}-\frac{s}{4}}^2,{\frac{s}{20}}^{2°}\frac{1}{10}{\frac{s}{3}}^2+\frac{1}{8}{\frac{5s}{12}}^2+\frac{1}{8}{\frac{2s}{15}}^2+\frac{1}{16}{\frac{7s}{15}}^2+\frac{1}{16}{\frac{s}{12}}^2,\end{array}$$that is, $$\begin{array}{}\text{(63)}& \frac{s^2}{400}\preccurlyeq \frac{s^2}{90}+\frac{25s^2}{1152}+\frac{4s^2}{1800}+\frac{49s^2}{3600}+\frac{s^2}{2304},\end{array}$$

which shows that $s^2/400\preccurlyeq 2827s^2/57600$, and so $2827s^2/57600-s^2/400\in {\mathcal{C}}_{\widehat{\mathcal{U}}}$, which is true for all $s\ge 0$. Hence, condition (8) is true for all $s,m,a\in \aleph$ and $i,j,k,l\ge 1$, where ${\eta }_i=2i-1$, ${\eta }_j=2j-1$, ${\eta }_k=2k-1$, and ${\eta }_l=2l-1$. All other conditions of Theorem 30 are satisfied. By Theorem 30, $E_i$, $F_j$, $G_k$, and $H_l$ have a unique common fixed point $0,0$ for all $i,j,k,l\ge 1$.

3. Application to the Infinite System of Integral Equations

We give here a couple of auxiliary facts that will be needed in our further considerations. Let $\widehat{\mathcal{U}}={ℝ}^2$ with norm ${.}_{\widehat{\mathcal{U}}}$ be a real Banach algebra. Let $I=0,T$, and denote by $CI,\widehat{\mathcal{U}}$ the space consisting of all continuous functions defined on interval $I$ with values in the Banach algebra $\widehat{\mathcal{U}}$. The space $CI,\widehat{\mathcal{U}}$ will be equipped with $s=\max {sa}_{\widehat{\mathcal{U}}}:a\in I$.

Let $\aleph =CI,\widehat{\mathcal{U}}$ and define $d:{\aleph }^3⟶\widehat{\mathcal{U}}$ by $$\begin{array}{}\text{(64)}& dst,mt,z={\min st-mt,mt-z,st-z}^p,\end{array}$$where $p\ge 1$ and for all $st,mt,z\in \aleph$. Then, $\aleph ,d$ is a complete cone $b_2$-metric space over the Banach algebra.

We consider the infinite system of integral equations of the form $$\begin{array}{}\text{(65)}& s_{i}t=g_{i}t+{\int }_0^T{M}_{i}t,w{f}_{i}w,swdw,\end{array}$$where $i=1,2,3,\cdots$. Let $E_i:\aleph ⟶\aleph$. We redefine the above infinite system of integral equations as $$\begin{array}{}\text{(66)}& E_i{s}_{i}t=g_{i}t+{\int }_0^T{M}_{i}t,w{f}_{i}w,swdw,\end{array}$$

for all $s_{i}t\in \aleph$ and $t,w\in I$. Clearly, by using Corollary 41, the existence of solution to (65) is equivalent to the existence of a common fixed point of $E_i$.

We assume that

(i) $g_i:I⟶ℝ$ are continuous for each $i=1,2,3,\cdots$

(ii) $M_i:I\times ℝ⟶0,+\infty$ are continuous and ${\int }_0^T{M}_{i}t,wdw\le 1$ for each $i=1,2,3,\cdots$

(iii) $f_i:I\times ℝ⟶ℝ$ are continuous for each $i=1,2,3,\cdots$ such that

$$\begin{array}{}\text{(67)}& f_{i}w,sw-f_{i}w,mw\le v^{1/p}\min sw-mw,mw-z,sw-z,\end{array}$$

for all $sw,mw,z\in \aleph$ and $0\le v<1$.

Theorem 43.

Under the assumptions (i)–(iii), the infinite system of integral equations (65) has a unique solution in $\aleph$.

Proof.

Take $\widehat{\mathcal{U}}={ℝ}^2$ with norm $s=s_1,s_2=\mid s_1\mid +\mid s_2\mid$, and multiplication is defined by the following way: $$\begin{array}{}\text{(68)}& sm=s_1,s_2{m}_1,m_2=s_1{m}_1,s_1{m}_2+s_2{m}_1.\end{array}$$

Let ${\mathcal{C}}_{\widehat{\mathcal{U}}}=s_1,s_2\in \widehat{\mathcal{U}}:s_1,s_2\ge 0$. It is clear that ${\mathcal{C}}_{\widehat{\mathcal{U}}}$ is a normal cone and $\widehat{\mathcal{U}}$ is a Banach algebra with unit element $e=1,0$.

Consider the family of mapping $E_i:\aleph ⟶\aleph$ defined by (66). Let $s_{i}t,m_{i}t,z\in \aleph$. From (64), we deduce that $$\begin{array}{}\text{(69)}& d{E}_i{s}_{i}t,E_i{m}_{i}t,z=\underset{a\in 0,T}{\max }{\min E_i{s}_{i}t-E_i{m}_{i}t,E_i{s}_{i}t-z,E_i{m}_{i}t-z}^p\preccurlyeq {\underset{a\in 0,T}{\max }{E}_i{s}_{i}t-E_i{m}_{i}t}^p={\underset{a\in 0,T}{\max }{\int }_0^T{M}_{i}t,w{f}_{i}w,swdw-{\int }_0^T{M}_{i}t,w{f}_{i}w,mwdw}^p={\underset{a\in 0,T}{\max }{\int }_0^T{M}_{i}t,w{f}_{i}w,sw-f_{i}w,mwdw}^p\preccurlyeq {\underset{a\in 0,T}{\max }{\int }_0^T{M}_{i}t,w{f}_{i}w,sw-f_{i}w,mwdw}^p\preccurlyeq {\underset{a\in 0,T}{\max }{\int }_0^T{M}_{i}t,w{v}^{1/p}\min \text{s}w-mw,mw-z,sw-zdw}^p\preccurlyeq {{\int }_0^T\underset{a\in 0,T}{\max }{M}_{i}t,w{v}^{1/p}{\underset{a\in 0,T}{\max }{\min sw-mw,mw-z,sw-z}^p}^{1/p}dw}^p\le {{\int }_0^T\underset{a\in 0,T}{\max }{M}_{i}t,w{v}^{1/p}{d{s}_{i}t,m_{i}t,z}^{1/p}dw}^p.\end{array}$$

Therefore, $$\begin{array}{}\text{(70)}& d{E}_i{s}_{i}t,E_i{m}_{i}t,z\preccurlyeq vd{s}_{i}t,m_{i}t,z.\end{array}$$

Now, all the assumptions of Corollary 41 are fulfilled and the family of mapping $E_i$ has a unique common fixed point in $\aleph$, which means that the infinite system of integral equations (65) has a unique solution in $\aleph$.☐

Authors’ Contributions

All authors make equal contributions and read and supported the last original copy.

Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.

References

[1] S. Gähler, "2-metrische räume und ihre topologische struktur," Mathematische Nachrichten, vol. 26 no. 1-4, pp. 115-148, DOI: 10.1002/mana.19630260109, 1963.

[2] B. Dhage, A study of some fixed point theorems [Ph.D. thesis], 1984.

[3] Z. Mustafa, B. Sims, "Some remarks concerning d-metric spaces," Proceedings of the International Conference on Fixed Point Theory and Applications, pp. 189-198, .

[4] Z. Mustafa, B. Sims, "A new approach to generalized metric spaces," Journal of Nonlinear and convex Analysis, vol. 7 no. 2, 2006.

[5] S. Sedghi, N. Shobe, H. Zhou, "A common fixed point theorem in-metric spaces," Fixed point theory and Applications, vol. 2007 no. 1,DOI: 10.1155/2007/27906, 2007.

[6] S. Sedghi, N. Shobe, A. Aliouche, "A generalization of fixed point theorems in s-metric spaces," Matematiˇcki vesnik, vol. 64 no. 249, pp. 258-266, 2012.

[7] L.-G. Huang, X. Zhang, "Cone metric spaces and fixed point theorems of contractive mappings," Journal of mathematical Analysis and Applications, vol. 332 no. 2, pp. 1468-1476, DOI: 10.1016/j.jmaa.2005.03.087, 2007.

[8] S. Rezapour, R. Hamlbarani, "Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”," Journal of Mathematical Analysis and Applications, vol. 345 no. 2, pp. 719-724, DOI: 10.1016/j.jmaa.2008.04.049, 2008.

[9] H. Liu, S. Xu, "Cone metric spaces with banach algebras and fixed point theorems of generalized Lipschitz mappings," Fixed point theory and applications, vol. 2013 no. 1,DOI: 10.1186/1687-1812-2013-320, 2013.

[10] B. Singh, S. Jain, P. Bhagat, "Cone 2-metric space nad fixed point theorem of contractive mappings," Commentationes Mathematicae, vol. 52 no. 2, 2012.

[11] Z. Mustafa, V. Parvaneh, J. R. Roshan, Z. Kadelburg, "b2-metric spaces and some fixed point theorems," Fixed Point Theory and Applications, vol. 2014 no. 1, 2014.

[12] J. Fernandez, N. Malviya, K. Saxena, "Cone b2-metric spaces over Banach algebra with applications," São Paulo Journal of Mathematical Sciences, vol. 11 no. 1, pp. 221-239, DOI: 10.1007/s40863-017-0058-8, 2017.

[13] Z. Islam, M. Sarwar, M. de la Sen, "Fixed-point results for generalized a-admissible Hardy-Rogers’contractions in cone b2-metric spaces over Banach’s algebras with application," Advances in Mathematical Physics, vol. 2020,DOI: 10.1155/2020/8826060, 2020.

[14] G. E. Hardy, T. Rogers, "A generalization of a fixed point theorem of Reich," Canadian Mathematical Bulletin, vol. 16 no. 2, pp. 201-206, DOI: 10.4153/cmb-1973-036-0, 1973.

[15] S. Reich, "Fixed point theorem," Acts of the National Academy of Lincei Account-class of Physical-Mathematical & Natural Sciences, vol. 51 no. 1-2, 1971.

[16] Z. D. Mitrovic, N. Hussain, "On results of Hardy-Rogers and Reich in cone b-metric space over Banach algebra and applications," UPB Scientific Bulletin, Series A, vol. 81, pp. 147-154, 2019.

[17] M. Rangamma, P. R. B. Murthy, "Hardy and Rogers type contractive condition and common fixed point theorem in cone 2-metric space for a family of self-maps," Global Journal of Pure and Applied Mathematics, vol. 12 no. 3, pp. 2375-2383, 2016.

[18] W. Rudin, "Functional analysis 2nd ed.," International Series in Pure and Applied Mathematics, vol. 45 no. 3, 1991.

[19] S. Xu, S. Radenović, "Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality," Fixed point theory and applications, vol. 2014 no. 1,DOI: 10.1186/1687-1812-2014-102, 2014.

[20] H. Huang, S. Radenovi’c, "Some fixed point results of generalized Lipschitz mappings on cone b-metric spaces over Banach algebras," Journal of Computational Analysis & Applications, vol. 20 no. 3, 2016.

[21] T. Wang, J. Yin, Q. Yan, "Fixed point theorems on cone 2-metric spaces over Banach algebras and an application," Fixed point theory and applications, vol. 2015 no. 1,DOI: 10.1186/s13663-015-0459-2, 2015.

[22] H. Huang, G. Deng, S. Radenović, "Some topological properties and fixed point results in cone metric spaces over Banach algebras," Positivity, vol. 23 no. 1, pp. 21-34, DOI: 10.1007/s11117-018-0590-5, 2019.

[23] H. Huang, S. Radenovic, "Common fixed point theorems of generalized Lipschitz mappings in cone metric spaces over Banach algebras," Applied Mathematics & Information Sciences, vol. 9 no. 6, 2015.

[24] H. Huang, S. Radenovic, "Common fixed point theorems of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras and applications," Journal of Nonlinear Sciences and Applications, vol. 8 no. 5, pp. 787-799, DOI: 10.22436/jnsa.008.05.29, 2015.

[25] S. Shukla, S. Balasubramanian, M. Pavlović, "A generalized Banach fixed point theorem," Bulletin of the Malaysian Mathematical Sciences Society, vol. 39 no. 4, pp. 1529-1539, DOI: 10.1007/s40840-015-0255-5, 2016.

[26] S. Janković, Z. Kadelburg, S. Radenović, "On cone metric spaces: a survey," Nonlinear Analysis:Theory, Methods & Applications, vol. 74 no. 7, pp. 2591-2601, DOI: 10.1016/j.na.2010.12.014, 2011.

[27] S. Radenović, B. Rhoades, "Fixed point theorem for two non-self mappings in cone metric spaces," Computers & Mathematics with Applications, vol. 57 no. 10, pp. 1701-1707, DOI: 10.1016/j.camwa.2009.03.058, 2009.

[28] M. Abbas, G. Jungck, "Common fixed point results for noncommuting mappings without continuity in cone metric spaces," Journal of Mathematical Analysis and Applications, vol. 341 no. 1, pp. 416-420, DOI: 10.1016/j.jmaa.2007.09.070, 2008.

[29] C. Di Bari, P. Vetro, "ϕ -pairs and common fixed points in cone metric spaces," Rendiconti del circolo Matematico di Palermo, vol. 57 no. 2, pp. 279-285, DOI: 10.1007/s12215-008-0020-9, 2008.

[30] Y. Han, S.-Y. Xu, "New common fixed point results for four maps on cone metric spaces," Applied Mathematics, vol. 2 no. 9, pp. 1114-1118, DOI: 10.4236/am.2011.29153, 2011.

[31] G. Wu, L. Yang, "Some fixed point theorems on cone 2-metric spaces over Banach algebras," Journal of Fixed Point Theory and Applications, vol. 20 no. 3,DOI: 10.1007/s11784-018-0588-4, 2018.

[32] V. Badshah, P. Bhagat, S. Shukla, "Some common fixed point theorems in cone b2-metric spaces over Banach algebra," Acta Universitatis Apulensis, vol. 2019 no. 60, pp. 37-52, 2019.