Fixed Point and Homotopy Methods in Cone A-Metric Spaces and Application to the Existence of Solutions to Urysohn Integral Equation

Abstract

The purpose of this article is to introduce an ordered implicit relation that can be used for the existence of fixed points of new contractions defined in cone A-metric spaces. We investigate a fixed-point method for proving the existence of Urysohn integral equation solutions. We prove an homotopy result by the application of obtained fixed-point theorem. The hypothesis is demonstrated with examples.

1. Introduction

Ran and Reuring [1] studied in metric spaces with partial ordering of elements and thus generalised the Banach Contraction Principle [2]. Many authors afterward expanded on this concept (see [3,4,5,6]). Popa [6] studied implicit contractions that obeyed an implicit relation established on the metric spaces and introduced sufficient requirements for the existence of fixed points for such self-mappings. Beg et al. [7,8] published several fixed-point theorems for contractions in an ordered metric space after that. In addition, Berinde et al. [9,10] and Sedghi et al. [11] contributed some noteworthy common fixed results to implicit contractions in an ordered metric space. The fixed-point theorems provided in [1,3,4,6] were generalised by Altun and Simsek and stated as follows:

Theorem 1

([12]). Let $(P,d_p,⪯)$ be a partially ordered metric space and $K:P\to P$ be a nondecreasing self-mapping that satisfies (1) for all $\mathsf{x},\mathsf{y}\in P$ with $\mathsf{x}⪯\mathsf{y}$:

$$\mathcal{T}\left(d_p(K\mathsf{x},K\mathsf{y}),d_p(\mathsf{x},\mathsf{y}),d_p(\mathsf{x},K\mathsf{x}),d_p(\mathsf{y},K\mathsf{y}),d_p(\mathsf{x},K\mathsf{y}),d_p(\mathsf{y},K\mathsf{x})\right)\le \mathbf{0},$$

(1)

where $\mathcal{T}:{[0,\infty )}^6\to (-\infty ,\infty )$ is a mapping that satisfies an implicit relation on P. If either K is continuous or $(P,d_p,⪯)$ is regular, then K possesses a fixed point provided that there exists $l_0\in P$ such that $l_0⪯K\left(l_0\right)$.

We can obtain many contractive conditions depending on the definitions of the mapping $\mathcal{T}$ in (1). For instance, if we define $\mathcal{T}:{[0,\infty )}^6\to (-\infty ,\infty )$ by

$$\mathcal{T}({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6)={\ell }_1-\psi \left(max\left\{{\ell }_2,{\ell }_3,{\ell }_4,\frac{1}{2}({\ell }_5+{\ell }_6)\right\}\right),$$

then we have a main theorem presented in [3]. If we define $\mathcal{T}$ by

$$\mathcal{T}({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6)={\ell }_1-k{\ell }_2;0\le k<1,$$

then we have the results that appeared in [1].

In the study of fixed-point theory, metric structures are particularly essential (see [13,14,15,16,17]). Huang and Zhang [18] developed the concept of a cone metric and used the normal cone concept to expand many earlier results. Rezapour and Hamlbarani [19] conducted research on non-normal cones, which improved several of Huang’s findings. To generalise the implicit relation stated in [20,21], vector spaces can be used. In this paper, we construct an implicit relation in a cone $\mathbf{A}$-metric space, which we use to prove several fixed-point theorems that generalise the results of [7,8,12,18,19]. As an application, we prove an homotopy outcome and show the existence of solution to a UIE by using the the obtained fixed point result.

The obtained results are independent of Ercan’s [22] findings and are a true generalisation of other findings in the literature. In addition, during the past several decades, fixed-point theory has played a key role in solving many problems arising in nonlinear analysis and optimization theory, such as differential hemivariational inequalities systems [23], monotone bilevel equilibrium problems [24], generalized global fractional-order composite dynamical systems [25] and generalized time-dependent hemivariational inequalities systems [26].

Throughout, in this paper, we make use of the following notation:

$${\zeta }_{{\sigma }_1}^{{\sigma }_2}=(\underset{n-1}{\underbrace{{\sigma }_1,{\sigma }_1,....{\sigma }_1}},{\sigma }_2).$$

2. Basic Notions

Definition 1.

A binary relation $\mathcal{R}$ over a set $X\ne Ø$ defines a partial order if it satisfies the following axioms:

(1) 

$\mathcal{R}$ is reflexive.

(2) 

$\mathcal{R}$ is antisymmetric.

(3) 

$\mathcal{R}$ is transitive.

A set with partial order $\mathcal{R}$ is known as a partially ordered set denoted by $(X,\mathcal{R})$.

Let $(E,∥.∥)$ be a real Banach space, and $\mathcal{C}\subseteq E$ satisfies the following axioms:

(1)

$\mathcal{C}$ is non-empty closed set such that $\mathcal{C}\ne \left\{\mathbf{0}\right\}.$

(2)

∀${\ell }_1,{\ell }_2\in \mathbb{R}$ such that ${\ell }_1,{\ell }_2\ge 0$ along with ${\theta }_1,{\theta }_2\in \mathcal{C}$, we have ${\ell }_1{\theta }_1+{\ell }_2{\theta }_2\in \mathcal{C}$.

(3)

$\mathcal{C}\cap (-\mathcal{C})=\left\{\mathbf{0}\right\}$.

Then, $\mathcal{C}$ defines a cone in E. Given $\mathcal{C}\subset E$, we define the partial order ${⪯}_{ϵ}$ on $\mathcal{C}$ as follows:

$$\eta {⪯}_{ϵ}\varrho \iff \varrho -\eta \in \mathcal{C}\mathrm{for}\mathrm{all}\eta ,\xi \in E.$$

Note that $\eta {\prec }_{ϵ}\xi$ represents $\eta {⪯}_{ϵ}\xi$ but $\eta \ne \xi$ and $\eta \ll \xi$ shows that $\xi -\eta \in {\mathcal{C}}^{\circ }$ (the interior of $\mathcal{C}$).

Definition 2

([18]). The cone $\aleph \subseteq E$ is called normal if, for all $\chi ,\varsigma \in \aleph$, there exists a number $\mathcal{K}>0$ so that,

$$0{⪯}_{ϵ}\chi {⪯}_{ϵ}\varsigma \Rightarrow \parallel \chi \parallel \le \mathcal{K}\parallel \varsigma \parallel .$$

Throughout this paper, we use the notation ℜ as a partial order in any ordinary set P and ${⪯}_{ϵ}$ as a partial order in $\mathcal{C}\subseteq E$. If $P\subseteq E$, then ℜ and ${⪯}_{ϵ}$ would be considered as identical.

Definition 3

([18]). Let $P\ne Ø$, and if the mapping $d_c:P\times P\mapsto E$ satisfies the following axioms:

(1) 

$\mathbf{0}{⪯}_{ϵ}{d}_c({\ell }_1,{\ell }_2)$ and $d_c({\ell }_1,{\ell }_2)=\mathbf{0}$ if and only if ${\ell }_1={\ell }_2$;

(2) 

$d_c({\ell }_1,{\ell }_2)=d_c({\ell }_2,{\ell }_1)$; and

(3) 

$d_c({\ell }_1,{\ell }_3){⪯}_{ϵ}{d}_c({\ell }_1,{\ell }_2)+d_c({\ell }_2,{\ell }_3)$, ∀${\ell }_1,{\ell }_2,{\ell }_3\in P$,

then P is called a cone metric, and the pair $(P,d_c)$ represents a cone metric space.

The generalizations of a metric space enriched the fixed-point theory and its applications in various contemporary fields. Gahler [27,28,29] (1963), coined the idea of a 2-metric space as a generalization of a metric space.

Definition 4

([27]). Let $P\ne Ø$, and if a mapping $d_2:P\times P\times P\mapsto [0,\infty )$ satisfies the following axioms:

(d1) 

for distinct points ${\ell }_1,{\ell }_2\in P$, ∃${\ell }_3\in P$ such that $d_2({\ell }_1,{\ell }_2,{\ell }_3)\ne \mathbf{0}$;

(d2) 

$d_2({\ell }_1,{\ell }_2,{\ell }_3)=\mathbf{0}$, if any two elements of the set $\{{\ell }_1,{\ell }_2,{\ell }_3\}$ in P are equal;

(d3) 

$d_2({\ell }_1,{\ell }_2,{\ell }_3)=d_2({\ell }_1,{\ell }_3,{\ell }_2)=d_2({\ell }_2,{\ell }_1,{\ell }_3)$

$=d_2({\ell }_3,{\ell }_1,{\ell }_2)=d_2({\ell }_2,{\ell }_3,{\ell }_1)=d_2({\ell }_3,{\ell }_2,{\ell }_1)$; and

(d4) 

$d_2({\ell }_1,{\ell }_2,{\ell }_3)\le d_2({\ell }_1,{\ell }_2,\ell )+d_2({\ell }_1,\ell ,{\ell }_3)+d_2(\ell ,{\ell }_2,{\ell }_3)$, for all ${\ell }_1,{\ell }_2,{\ell }_3,\ell \in P$,

then $d_2$ is called a 2-metric, and $(P,d_2)$ is then called a 2-metric space.

The distance $d_2({\ell }_1,{\ell }_2,{\ell }_3)$ represents the area of a triangle formed by the points ${\ell }_1,{\ell }_2,{\ell }_3\in P$. A 2-metric, in contrast to an ordinary metric, is not a continuous function of its variables. As a result, Dhage introduced the concept of a D-metric space in [30] as follows:

Definition 5

([30]). Let $P\ne Ø$, and if a mapping $D:P\times P\times P\mapsto [0,\infty )$ satisfies the following axioms:

(D1) 

$D({\ell }_1,{\ell }_2,{\ell }_3)\ge \mathbf{0}$ and $D({\ell }_1,{\ell }_2,{\ell }_3)=\mathbf{0}$ if and only if ${\ell }_1={\ell }_2={\ell }_3$;

(D2) 

$D({\ell }_1,{\ell }_2,{\ell }_3)=D({\ell }_1,{\ell }_3,{\ell }_2)=D({\ell }_2,{\ell }_1,{\ell }_3)=D({\ell }_3,{\ell }_1,{\ell }_2)=D({\ell }_2,{\ell }_3,{\ell }_1)=D({\ell }_3,{\ell }_2,{\ell }_1)$; and

(D3) 

$D({\ell }_1,{\ell }_2,{\ell }_3)\le D({\ell }_1,{\ell }_2,\ell )+D({\ell }_1,\ell ,{\ell }_3)+D(\ell ,{\ell }_2,{\ell }_3)$, for all ${\ell }_1,{\ell }_2,{\ell }_3,\ell \in P$,

then $(P,D)$ denotes the D-metric space.

Mustafa and Sims [31,32] (2006) introduced G-metric space as a generalization of D-metric space. The deficiency of Dhage’s theory of D-metric is thus corrected. Moreover, Sedghi et al. [33] (2007) observed that the condition $\left(D1\right)$ can be modified as follows:

Definition 6

([33]). Let $P\ne Ø$, and if a mapping $D^{*}:P\times P\times P\mapsto [0,\infty )$ satisfies the following axioms:

(D’1) 

$D^{*}({\ell }_1,{\ell }_2,{\ell }_3)=\mathbf{0}$ if and only if ${\ell }_1={\ell }_2={\ell }_3$;

(D’2) 

$D^{*}({\ell }_1,{\ell }_2,{\ell }_2)\ge \mathbf{0}$;

(D’3) 

$D^{*}({\ell }_1,{\ell }_2,{\ell }_3)=D^{*}({\ell }_1,{\ell }_3,{\ell }_2)=D^{*}({\ell }_2,{\ell }_1,{\ell }_3)=D^{*}({\ell }_3,{\ell }_1,{\ell }_2)=D^{*}({\ell }_2,{\ell }_3,{\ell }_1)=D^{*}({\ell }_3,{\ell }_2,{\ell }_1)$; and

(D’4) 

$D^{*}({\ell }_1,{\ell }_2,{\ell }_3)\le D^{*}({\ell }_1,{\ell }_2,\ell )+D^{*}(\ell ,{\ell }_3,{\ell }_3)$,

then $(P,D^{*})$ is called a $D^{*}$-metric space. $D^{*}$-metric space can be viewed as an improved version of D-metric space.

Later, Sedghi et al. [34] (2012) introduced another generalization of a metric space known as an S-metric space. Abbas et al. [35] (2015), introduced the idea of $\mathbf{A}$-metric spaces as follows:

Definition 7

([35]). Let $P\ne Ø$, and if a mapping ${\mathit{d}}_P:P^n\mapsto [0,\infty )$ satisfies the following axioms: (for all ${\ell }_i,\ell \in P$, $i=1,2,\dots \dots .n$)

(A1) 

${\mathit{d}}_P({\ell }_1,{\ell }_2,\dots \dots ,{\ell }_{n-1},{\ell }_n)\ge 0$;

(A2) 

${\mathit{d}}_P({\ell }_1,{\ell }_2,\dots \dots ,{\ell }_{n-1},{\ell }_n)=0$ if and only if ${\ell }_1={\ell }_2=\dots \dots ={\ell }_{n-1}={\ell }_n$; and

(A3) 

${\mathit{d}}_P({\ell }_1,{\ell }_2,\dots \dots ,{\ell }_{n-1},{\ell }_n)\le {\mathit{d}}_P\left({\zeta }_{{\ell }_1}^{\ell }\right)+{\mathit{d}}_P\left({\zeta }_{{\ell }_2}^{\ell }\right)+\dots \dots +{\mathit{d}}_P\left({\zeta }_{{\ell }_{n-1}}^{\ell }\right)+{\mathit{d}}_P\left({\zeta }_{{\ell }_n}^{\ell }\right)$,

then $(P,{\mathit{d}}_P)$ is called a $\mathit{A}$-metric space.

One can check that $\mathbf{A}$-metric space is an n-dimensional S-metric space. We can have a metric d and a S-metric by assuming $n=2$ and $n=3$, respectively, in an $\mathbf{A}$-metric space.

Example 1

([35]). Let $P=R$, and define ${\mathit{d}}_P:P^n\mapsto [0,\infty )$ by

$${\mathit{d}}_P({\ell }_1,{\ell }_2,\cdots ,{\ell }_{n-1},{\ell }_n)=\sum _{\varpi =1}^n\sum _{\varpi <\ell }|{\ell }_{\varpi }-{\ell }_{\ell }|.$$

Then, $(P,{\mathit{d}}_P)$ is an $\mathit{A}$-metric space.

Example 2.

Let $P=\mathbb{R}$, and define ${\mathit{d}}_P:P^n\mapsto [0,\infty )$ by

$$\begin{array}{ccc}\hfill {\mathit{d}}_P({\ell }_1,{\ell }_2,\dots \dots ,{\ell }_{n-1},{\ell }_n)& =& |{\ell }_n+{\ell }_{n-1}+\dots \dots +{\ell }_2-(n-1){\ell }_1|\hfill \\ & +& |{\ell }_n+{\ell }_{n-1}+\dots \dots +{\ell }_3-(n-2){\ell }_2|\hfill \\ & ⋮& \\ & +& |{\ell }_n+{\ell }_{n-1}+{\ell }_{n-2}-3{\ell }_{n-3}|\hfill \\ & +& |{\ell }_n+{\ell }_{n-1}-2{\ell }_{n-2}|\hfill \\ & +& |{\ell }_n-{\ell }_{n-1}|.\hfill \end{array}$$

Then, $(P,{\mathit{d}}_A)$ is an $\mathit{A}$-metric space. For more details on $\mathit{A}$-metric spaces, see [35].

Fernandez et al. [36] (2017) introduced the concept of a cone $\mathbf{A}$-metric space. They also established some new results in a cone $\mathbf{A}$-metric space.

Definition 8

([36]). Let $P\ne Ø$, and if an operator ${\mathit{d}}_{cA}:P^n\mapsto E$ satisfies the following axioms: (for all ${\ell }_i,\ell \in P$, $i=1,2,\dots \dots .n$)

(d1) 

$\mathbf{0}{⪯}_{ϵ}{\mathit{d}}_{cA}({\ell }_1,{\ell }_2,\dots \dots ,{\ell }_{n-1},{\ell }_n)$;

(d2) 

${\mathit{d}}_{cA}({\ell }_1,{\ell }_2,\dots \dots ,{\ell }_{n-1},{\ell }_n)=\mathbf{0}$ if and only if ${\ell }_1={\ell }_2=\dots \dots ={\ell }_{n-1}={\ell }_n$; and

(d3) 

${\mathit{d}}_{cA}({\ell }_1,{\ell }_2,\dots \dots ,{\ell }_{n-1},{\ell }_n){⪯}_{ϵ}{\mathit{d}}_{cA}\left({\zeta }_{{\ell }_1}^{\ell }\right)+{\mathit{d}}_{cA}\left({\zeta }_{{\ell }_2}^{\ell }\right)+\dots \dots \dots +{\mathit{d}}_{cA}\left({\zeta }_{{\ell }_{n-1}}^{\ell }\right)+{\mathit{d}}_{cA}\left({\zeta }_{{\ell }_n}^{\ell }\right)$.

then $(P,{\mathit{d}}_{cA})$ is called a cone $\mathit{A}$-metric space.

A cone metric space is a special case of a cone $\mathbf{A}$-metric space with $n=2$. One can easily check that ${\mathit{d}}_{cA}\left({\zeta }_x^y\right)={\mathit{d}}_{cA}\left({\zeta }_y^x\right)$ for a cone $\mathbf{A}$-metric space ([35,36]).

Example 3

([36]). Let $E=C[a,b]$ be the set of continuous functions on the interval $[a,b]$ with the supremum norm. Define $P=\mathbb{R}$ and $\mathcal{C}=\{\ell \in E:\ell (t)\ge 0,t\in [a,b\left]\right\}$ and ${\mathit{d}}_{cA}:P^n\mapsto E$ by

$$\begin{array}{c}\hfill {\mathit{d}}_{cA}({\ell }_1,{\ell }_2,\cdots ,{\ell }_{n-1},{\ell }_n)\left(t\right)=\sum _{\alpha =1}^n\sum _{\alpha <\varpi }|{\ell }_{\alpha }-{\ell }_{\varpi }|e^t.\end{array}$$

Then, $(P,{\mathit{d}}_{cA})$ is a cone $\mathit{A}$-metric space.

Definition 9.

Let E be a real Banach space, and let $(P,{\mathit{d}}_{cA})$ be a cone $\mathbf{A}$-metric space along with ${\epsilon }_A\in E$ such that $0{\ll }_{ϵ}{\epsilon }_A$. A sequence $\left\{{\ell }_n\right\}$ is a Cauchy sequence if we can find a $n_1\in \mathbb{N}$ so that ${\mathit{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right){\ll }_{ϵ}{\epsilon }_A$ ∀ $n,m>n_1$. A sequence $\left\{{\ell }_n\right\}$ is convergent if we can find a $n_1\in \mathbb{N}$ so that ${\mathit{d}}_{cA}\left({\zeta }_{{\ell }_n}^{\ell }\right){\ll }_{ϵ}{\epsilon }_A$ ∀ $n\ge n_1$ and $\ell \in P$. A cone $\mathbf{A}$-metric $(P,{\mathit{d}}_{cA})$ is complete if every Cauchy sequence converges in P.

3. Ordered Implicit Relations

In the domain of implicit relation introduced by Popa [20,21], all the coordinates of the 6-tuple are ordered vectors, and then such kinds of relations will be known as ordered implicit relations. In this article, motivated by [7,8,9,10,11], we introduce an implicit relation in a cone $\mathbf{A}$-metric space. This relation is an ordered implicit relation and will be applied as a mathematical tool to obtain the proofs of stated fixed-point theorems. The obtained fixed-point results are independent of the results proved by Ercan [22].

Assume that $(E,∥.∥)$ is a real Banach space and $B(E,E)$ denotes the space of all bounded linear operators $\mathcal{S}:E\to E$ such that ${∥\mathcal{S}∥}_1<1$, where ${∥.∥}_1$ is a usual norm in $B(E,E)$.

Motivated by the research work presented in [7,8,9,10,37,38], we introduce a new ordered implicit relation as follows:

Definition 10.

Let E be a real Banach space, and suppose that the operator $ϝ:E^6\to E$ satisfies the following axioms:

$\left({ϝ}_1\right)$

if ${\eta }_5{⪯}_{ϵ}{\ell }_5$, ${\eta }_6{⪯}_{ϵ}{\ell }_6$

then $ϝ({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6){⪯}_{ϵ}ϝ({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\eta }_5,{\eta }_6)$,

$\left({ϝ}_2\right)$

if either

$$ϝ(\eta ,\ell ,\ell ,\eta ,\eta +(n-1)\ell ,\mathbf{0}){⪯}_{ϵ}\mathbf{0}$$

or

$$ϝ(\eta ,\ell ,\eta ,\ell ,\mathbf{0},\eta +(n-1)\ell ){⪯}_{ϵ}\mathbf{0},$$

then there exists $\mathcal{S}\in B(E,E)$ such that $\eta {⪯}_{ϵ}\frac{1}{n-1}\mathcal{S}(\ell )$, for all $\eta ,\ell \in E$.

$\left({ϝ}_3\right)$

$ϝ(\eta ,\mathbf{0},\mathbf{0},\eta ,\eta ,\mathbf{0})\succ \mathbf{0}$ whenever $∥\eta ∥>0$.

Then, ϝ is called an ordered implicit relation.

Remark 1.

It is remarked that the composition of finite number of bounded operators is bounded—in particular, if $\mathcal{S}\in B(E,E)$, then ${\mathcal{S}}^n\in B(E,E)$. It is known from operator theory that, if $\mathcal{S}\in B(E,E)$ verifying ${∥\mathcal{S}∥}_1<1$, then ${(I-\mathcal{S})}^{-1}$ can be represented by a convergent series. As a result, the operator ${\mathcal{S}}^n{(I-\mathcal{S})}^{-1}$ is bounded.

Let $\mathcal{H}=\{ϝ:E^6\to E|ϝ\mathrm{satisfies}\mathrm{the}\mathrm{conditions}{ϝ}_1,{ϝ}_2,{ϝ}_3\}$.

Example 4.

Let ${⪯}_{ϵ}$ be a partial ordering on a cone $\mathcal{C}$. Assume that the pair $(E,∥.∥)$ is a real Banach space and for $n>1$, $ϝ:E^6\to E$ be defined by $ϝ({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6)={\ell }_1-\frac{\varpi }{n-1}max\{{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6\}\mathrm{for}\mathrm{all}{\ell }_i\in E(i=1\mathrm{to}6)\mathrm{and}0\le \varpi <\frac{1}{2}.$ Then, the operator $ϝ\in \mathcal{H}$:

$\left({ϝ}_1\right).$ Let ${\ell }_5{⪯}_{ϵ}{\gamma }_5$ and ${\ell }_6{⪯}_{ϵ}{\gamma }_6$, then ${\gamma }_5-{\ell }_5\in \mathcal{C}$ and ${\gamma }_6-{\ell }_6\in \mathcal{C}$. Now, we show that $ϝ({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6)-ϝ({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\gamma }_5,{\gamma }_6)\in \mathcal{C}$. Consider,

$$\begin{array}{ccc}& & ϝ({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6)-ϝ({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\gamma }_5,{\gamma }_6)\hfill \\ & =& {\ell }_1-\frac{\alpha }{n-1}max\{{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6\}-\left({\ell }_1-\frac{\varpi }{n-1}max\{{\ell }_2,{\ell }_3,{\ell }_4,{\gamma }_5,{\gamma }_6\}\right)\hfill \\ & =& \frac{\varpi }{n-1}max\{\mathbf{0},\mathbf{0},\mathbf{0},{\gamma }_5-{\ell }_5,{\gamma }_6-{\ell }_6\}\in \mathcal{C}.\hfill \end{array}$$

Thus, $ϝ({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\gamma }_5,{\gamma }_6){⪯}_{ϵ}ϝ({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6)$.

$\left({ϝ}_2\right).$ Let $\ell ,\gamma \in E$ along with $\mathbf{0}{⪯}_{ϵ}\ell ,\mathbf{0}{⪯}_{ϵ}\gamma$. Let $ϝ(\gamma ,\ell ,\ell ,\gamma ,\gamma +(n-1)\ell ,\mathbf{0}){⪯}_{ϵ}\mathbf{0}$, we obtain

$$-\gamma +\frac{\varpi }{n-1}max\{\ell ,\ell ,\gamma ,\gamma +(n-1)\ell ,\mathbf{0}\}\in \mathcal{C}.$$

Take $\gamma =\mathbf{0}$, and then $\varpi \ell \in \mathcal{C}$. Thus, ∃$\mathcal{S}:E\to E$ taken as $\mathcal{S}(\ell )=\eta \ell$ ($0\le \eta <\frac{1}{2}$ and $\eta =\varpi$ is a constant) so that $\gamma {⪯}_{ϵ}\mathcal{S}(\ell )$. If $\gamma \ne \mathbf{0}$, we obtain, $-\gamma +\frac{\varpi }{n-1}max\{\ell ,\ell ,\gamma ,\gamma +(n-1)\ell ,\mathbf{0}\}\in \mathcal{C}$ ⇒ $\varpi (\gamma +\ell )-\gamma \in \mathcal{C}$ further implies $\varpi \ell -(1-\varpi )\gamma \in \mathcal{C}$. Hence, $(1-\varpi )\gamma {⪯}_{ϵ}\varpi \ell$; thus, ∃$\mathcal{S}:E\to E$ taken as $\mathcal{S}(\ell )=\eta \ell$ ($\eta =\frac{\varpi }{1-\varpi }$ represent constant) so that $\gamma {⪯}_{ϵ}\mathcal{S}(\ell )$.

$\left({ϝ}_3\right).$ Consider $\ell \in E$ be such that $∥\ell ∥>0$ and $\mathbf{0}{⪯}_{ϵ}ϝ(\ell ,\mathbf{0},\mathbf{0},\ell ,\ell ,\mathbf{0})$, and then

$$\ell -\frac{\varpi }{n-1}max\{\mathbf{0},\mathbf{0},\ell ,\ell ,\mathbf{0}\}\in \mathcal{C}.$$

As a result, we obtain $\frac{\varpi }{n-1}\ell {⪯}_{ϵ}\mathbf{0}$, which is true when $∥\ell ∥>0$.

Let ${ϝ}_i:E^6\to E$ $(i=1,2,3)$ be defined by

(i)

${ϝ}_1({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6)={\ell }_1-\frac{\varpi }{n-1}{\ell }_2;0\le \varpi <1.$

(ii)

${ϝ}_2({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6)={\ell }_1-\frac{{\varpi }_1}{n-1}{\ell }_2-\frac{{\varpi }_2}{n-1}{\ell }_3-\frac{{\varpi }_3}{n-1}{\ell }_4,{\varpi }_1,{\varpi }_2,{\varpi }_3\ge 0:\frac{{\varpi }_1+{\varpi }_2+{\varpi }_3}{n-1}<1$.

(iii)

${ϝ}_3({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6)=\varpi {\ell }_1-{\ell }_2;\varpi >1.$

Then, each ${ϝ}_i\in \mathcal{H}$.

We will apply the ordered implicit relation along with implicit contraction satisfying certain conditions to solve the following fixed-point problem.

“find $j^{*}\in (P,{\mathit{d}}_{cA})$ so that $g\left(j^{*}\right)=j^{*}$” where $g:P\to P$ is a self-mapping satisfying (2), for all comparable elements $\eta ,\kappa \in P$

$$(I-\mathsf{S})({\mathbf{d}}_{cA}\left({\zeta }_{\eta }^{g\left(\eta \right)}\right){⪯}_{ϵ}{\mathbf{d}}_{cA}\left({\zeta }_{\eta }^{\kappa }\right)\mathrm{implies}$$

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{g\left(\eta \right)}^{g\left(\kappa \right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{\eta }^{\kappa }\right),{\mathbf{d}}_{cA}\left({\zeta }_{\eta }^{g\left(\eta \right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{\kappa }^{g\left(\kappa \right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{\eta }^{g\left(\kappa \right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{\kappa }^{g\left(\eta \right)}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0},$$

(2)

and $\mathsf{S}\in B(E,E)$, $I:E\to E$ an identity operator, $ϝ\in \mathcal{H}$.

4. Main Results

Popa [6] derived new results by introducing new contractive conditions. Later, this idea was generalized to partially ordered metric spaces [12]. Nazam et al. [39] extended the idea presented in [12] to cone metric space. Mujahid Abbas introduced the idea of $\mathbf{A}$-metric space as a generalization of a metric space and defined some topological structure on it [35]. Fernandez introduced the cone $\mathbf{A}$-metric space [36]. This section deals with suggested fixed-point problems that extend the corresponding ones in [6,12,39]. We present the following theorems in this regard.

4.1. Result for Increasing Self-Mapping

Theorem 2.

Let $(P,{\mathit{d}}_{cA})$ be a complete cone $\mathit{A}$-metric space along with a cone $\mathcal{C}\subset E$ and $g:P\to P$ be a self-mapping. Let $\mathsf{S}\in B(E,E)$ so that ${∥\mathsf{S}∥}_1<1$, $I:E\to E$ be an identity operator. If for all comparable elements $\ell ,\kappa \in P$, $ϝ\in \mathcal{H}$, g satisfies the following condtions:

$$(I-\mathsf{S}){\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right){⪯}_{ϵ}{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)implies$$

$$ϝ\left(\begin{array}{c}{\mathit{d}}_{cA}\left({\zeta }_{g(\ell )}^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right),{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g(\ell )}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0},$$

(3)

and

(1) 

there is ${\ell }_0\in P$ such that ${\ell }_0\mathcal{R}g\left({\ell }_0\right)$;

(2) 

for all $\ell ,\kappa \in P$, $\ell \mathcal{R}\kappa$ implies $g(\ell )\mathcal{R}g\left(\kappa \right)$; and

(3) 

for each sequence $\left\{{\ell }_n\right\}$, with comparable sequential terms, converging to $c^{*}$, we have ${\ell }_n\mathcal{R}{c}^{*}$ for all $n\in \mathbb{N}$.

Then, there exists $c^{*}\in P$ such that $c^{*}=g\left(c^{*}\right)$.

Proof. 

We construct a sequence $\left\{{\ell }_n\right\}$ such that $g\left({\ell }_{n-1}\right)={\ell }_n$. By (1), there exists ${\ell }_0\in P$ such that ${\ell }_0\mathcal{R}g\left({\ell }_0\right)$, so, taking $\ell ={\ell }_0$ in (3), we have

$$(I-\mathsf{S}){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{g\left({\ell }_0\right)}\right)=(I-\mathsf{S}){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right){⪯}_{ϵ}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)\mathrm{implies}$$

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{g\left({\ell }_0\right)}^{g\left({\ell }_1\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{g\left({\ell }_0\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{g\left({\ell }_1\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{g\left({\ell }_1\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{g\left({\ell }_0\right)}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0},$$

that is,

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_1}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0}.$$

(4)

By (d3), we have

$${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_2}\right){⪯}_{ϵ}(n-1){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)+{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),$$

and by rewriting (4) and applying (${ϝ}_1$), we obtain

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),(n-1){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)+{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),\mathbf{0}\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0}.$$

(5)

By $\left({ϝ}_2\right)$, there exists $\mathsf{S}\in B(E,E)$ verifying ${∥\mathsf{S}∥}_1<1$ and

$${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right){⪯}_{ϵ}\frac{1}{n-1}\mathsf{S}\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)\right).$$

By (2), we have $g\left({\ell }_0\right)\mathcal{R}g\left({\ell }_1\right)$, that is ${\ell }_1\mathcal{R}{\ell }_2$, so, putting $\ell ={\ell }_1$ in (3), we have

$$(I-\mathsf{S}){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{g\left({\ell }_1\right)}\right)=(I-\mathsf{S}){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right){⪯}_{ϵ}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right).$$

This implies that

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{g\left({\ell }_1\right)}^{g\left({\ell }_2\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{g\left({\ell }_1\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{g\left({\ell }_2\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{g\left({\ell }_2\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{g\left({\ell }_1\right)}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0}.$$

By (d3), we have

$${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_3}\right){⪯}_{ϵ}(n-1){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right)+{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{{\ell }_3}\right),$$

and $\left({ϝ}_1\right)$ implies

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{{\ell }_3}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{{\ell }_3}\right),(n-1){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right)+{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{{\ell }_3}\right),\mathbf{0}\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0}.$$

(6)

By $\left({ϝ}_2\right)$, we have

$${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{{\ell }_3}\right){⪯}_{ϵ}\frac{1}{n-1}\mathsf{S}\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right)\right){⪯}_{ϵ}\frac{1}{{(n-1)}^2}{\mathsf{S}}^2\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)\right).$$

The same reasoning leads to a sequence $\left\{{\ell }_n\right\}$ such that

${\ell }_n\mathcal{R}{\ell }_{n+1}$ and ${\ell }_{n+1}=g\left({\ell }_n\right)$. Furthermore, observing that

$$(I-\mathsf{S}){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{n-1}}^{g\left({\ell }_{n-1}\right)}\right)=(I-\mathsf{S}){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{n-1}}^{{\ell }_n}\right){⪯}_{ϵ}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{n-1}}^{{\ell }_n}\right),$$

we have

$${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_{n+1}}\right){⪯}_{ϵ}\frac{1}{n-1}\mathsf{S}\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{n-1}}^{{\ell }_n}\right)\right){⪯}_{ϵ}\frac{1}{{(n-1)}^2}{\mathsf{S}}^2\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{n-2}}^{{\ell }_{n-1}}\right)\right){⪯}_{ϵ}\cdots {⪯}_{ϵ}\frac{1}{{(n-1)}^n}{\mathsf{S}}^n\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)\right).$$

For $m,n\in \mathbb{N}$ such that $m>n$, we note that

$$\begin{array}{ccc}\hfill {\mathit{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right)& {⪯}_{ϵ}& (n-1)[{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_{n+1}}\right)+{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{n+1}}^{{\ell }_{n+2}}\right)+\cdots +{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{m-1}}^{{\ell }_m}\right)]\hfill \\ & {⪯}_{ϵ}& \frac{1}{{(n-1)}^{n-1}}[{\mathsf{S}}^n+\frac{1}{(n-1)}{\mathsf{S}}^{n+1}+\dots \dots +\frac{1}{{(n-1)}^{m-n-2}}{\mathsf{S}}^{m-1}+..]\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)\right)\hfill \\ & =& \frac{1}{{(n-1)}^{n-1}}\left\{{\mathsf{S}}^n{(I-\frac{1}{n-1}\mathsf{S})}^{-1}\right\}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right).\left(\mathrm{By}\mathrm{Remark}1\right)\hfill \end{array}$$

We deduce the following information from the last inequality.

$$∥{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right)∥\le \frac{1}{{(n-1)}^{n-1}}sup\frac{∥{\mathsf{S}}^n{(I-\frac{1}{n-1}\mathsf{S})}^{-1}(\ell )∥}{∥\ell ∥}∥{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)∥.$$

By Remark 1 and the assumption that ${∥\mathsf{S}∥}_1<1$, we infer (using the facts given in [18,19]) that ${lim}_{n\to \infty }{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right)=\mathbf{0}$, or equivalently, ${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right){\ll }_{ϵ}{C}_A$ for some $\mathbf{0}{\ll }_{ϵ}{C}_A$. By using the technique appeared in [19], we can obtain that ${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right){\ll }_{ϵ}{C}_A$. Thus, $\left\{{\ell }_n\right\}$ is a Cauchy sequence in P. $(P,{\mathbf{d}}_{cA})$ is a complete cone $\mathbf{A}$-metric space, which assures the existence of $c^{*}\in P$ such that ${\ell }_n\to c^{*}$ as $n\to \infty$, or equivalently, there exists $n_2\in N$ satisfying

$${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{c^{*}}\right){\ll }_{ϵ}c\mathrm{for}\mathrm{all}n\ge n_2.$$

Our concern is to show that

$$(I-\mathsf{S}){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{g\left({\ell }_n\right)}\right){⪯}_{ϵ}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{c^{*}}\right).$$

We assume on the contrary that

$$(I-\mathsf{S}){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{g\left({\ell }_n\right)}\right)\succ {\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{c^{*}}\right)$$

and

$$(I-\mathsf{S}){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{n+1}}^{g\left({\ell }_{n+1}\right)}\right)\succ {\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{n+1}}^{c^{*}}\right)\mathrm{for}\mathrm{some}n\in \mathbb{N}.$$

By (d3) and then by (3), we have the following information:

$$\begin{array}{ccc}\hfill {\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{g\left({\ell }_n\right)}\right)& {⪯}_{ϵ}& {\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{c^{*}}\right)+(n-1){\mathbf{d}}_{cA}\left({\zeta }_{c^{*}}^{g\left({\ell }_n\right)}\right)\hfill \\ & \prec & (I-\mathsf{S}){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{g\left({\ell }_n\right)}\right)+(n-1)(I-\mathsf{S}){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{n+1}}^{g\left({\ell }_{n+1}\right)}\right)\hfill \\ & \prec & (I-\mathsf{S}){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{g\left({\ell }_n\right)}\right)+(n-1)\frac{1}{n-1}\mathsf{S}(I-\mathsf{S}){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{g\left({\ell }_n\right)}\right)\hfill \\ & =& (I-{\mathsf{S}}^2)\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{g\left({\ell }_n\right)}\right)\right).\hfill \end{array}$$

Thus,

$${\mathsf{S}}^2\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{g\left({\ell }_n\right)}\right)\right)\prec \mathbf{0},$$

which contradicts the definition of cone A-metric; therefore, for each $n\ge 1$, we have

$$(I-\mathsf{S})\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{g\left({\ell }_n\right)}\right)\right){⪯}_{ϵ}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{c^{*}}\right).$$

By (3), it implies that

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{g\left({\ell }_n\right)}^{g\left(c^{*}\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{c^{*}}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{g\left({\ell }_n\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{c^{*}}^{g\left(c^{*}\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{g\left(c^{*}\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{c^{*}}^{g\left({\ell }_n\right)}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0}$$

or

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{n+1}}^{g\left(c^{*}\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{c^{*}}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_{n+1}}\right),{\mathbf{d}}_{cA}\left({\zeta }_{c^{*}}^{g\left(c^{*}\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{g\left(c^{*}\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{c^{*}}^{{\ell }_{n+1}}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0}.$$

(7)

We claim that $∥{\mathbf{d}}_{cA}\left({\zeta }_{c^{*}}^{g\left(c^{*}\right)}\right)∥=0$. On the contrary, suppose that $∥{\mathbf{d}}_{cA}\left({\zeta }_{c^{*}}^{g\left(c^{*}\right)}\right)∥>0$, and letting $n\to \infty$ in (7), we have

$$ϝ\left({\mathbf{d}}_{cA}\left({\zeta }_{c^{*}}^{g\left(c^{*}\right)}\right),\mathbf{0},\mathbf{0},{\mathbf{d}}_{cA}\left({\zeta }_{c^{*}}^{g\left(c^{*}\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{c^{*}}^{g\left(c^{*}\right)}\right),\mathbf{0}\right){⪯}_{ϵ}\mathbf{0}.$$

This is a contradiction to $\left({ϝ}_3\right)$. Thus,$∥{\mathbf{d}}_{cA}\left({\zeta }_{c^{*}}^{g\left(c^{*}\right)}\right)∥=0$. Hence ${\mathbf{d}}_{cA}\left({\zeta }_{c^{*}}^{g\left(c^{*}\right)}\right)=\mathbf{0}$. By (d1), we have $c^{*}=g\left(c^{*}\right)$. This completes the proof.  □

Remark 2.

If the operator $ϝ:E^6\to E$ is taken as

$$ϝ({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6)={\ell }_1-\phi \left(max\left\{{\ell }_2,{\ell }_3,{\ell }_4,\frac{1}{2}({\ell }_5+{\ell }_6)\right\}\right),forall{\ell }_i\in E$$

Here, $\phi :E\to E$ is taken as non-decreasing mapping with ${lim}_{n\to \infty }{\phi }^n\left(v\right)={\mathbf{0}}_{\mathbf{E}}$. Then, Theorem (2), generalizes the corresponding result in [3]. Taking $ϝ:E^6\to E$ by

$$ϝ({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6)={\ell }_1-\frac{k}{n-1}{\ell }_2;k\in [0,1),n>1$$

in Theorem 2 leads to the generalization of [1]. Furthermore, Theorem 2 extends the corresponding theorems in [7,9,10,20,21].

We will take decreasing self-mapping for the theorem given below.

4.2. Result for Decreasing Self-Mapping

Theorem 3.

Let $(P,{\mathbf{d}}_{cA})$ be a complete cone $\mathit{A}$-metric space, and let $\mathcal{C}\subset E$ be a cone. Let $\mathsf{S}\in B(E,E)$ so that ${∥\mathsf{S}∥}_1<1$, and let $I:E\to E$ be an identity operator. If the mapping $g:P\to P$, for all comparable elements $\ell ,\kappa \in P$, and $ϝ\in \mathcal{H}$ satisfies the following condition

$$(I-\mathsf{S})({\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right){⪯}_{ϵ}{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)implies$$

$$ϝ\left(\begin{array}{c}{\mathit{d}}_{cA}\left({\zeta }_{g(\ell )}^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right),{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g(\ell )}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0},$$

(8)

and

(1) 

there exists ${\ell }_0\in P$ such that $g\left({\ell }_0\right)\mathcal{R}{\ell }_0$;

(2) 

for all $\ell ,\kappa \in P$, $\ell \mathcal{R}\kappa$ we find $g\left(\kappa \right)\mathcal{R}g(\ell )$; and

(3) 

for a sequence $\left\{{\ell }_n\right\}$ with all comparable sequential terms such that ${\ell }_n\to c^{*}$, we have ${\ell }_n\mathcal{R}{c}^{*}$ ∀ $n\in \mathbb{N}$.

Then, there exists $c^{*}\in P$ such that $c^{*}=g\left(c^{*}\right)$.

Proof. 

Let ${\ell }_0\in P$ and construct a sequence $\left\{{\ell }_n\right\}$ by ${\ell }_n=g\left({\ell }_{n-1}\right)$ for all $n\in \mathbb{N}$. Using condition (1), we have ${\ell }_1=g\left({\ell }_0\right)\mathcal{R}{\ell }_0$. By (2), we have $g\left({\ell }_0\right)\mathcal{R}g\left({\ell }_1\right)$, i.e., ${\ell }_1\mathcal{R}{\ell }_2$. Using (8), we obtain

$$(I-\mathsf{S})\left({\mathbf{d}}_{cA}\left({\zeta }_{g\left({\ell }_0\right)}^{{\ell }_0}\right)\right)=(I-\mathsf{S})\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_0}\right)\right){⪯}_{ϵ}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)\mathrm{implies}$$

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{g\left({\ell }_1\right)}^{g\left({\ell }_0\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_0}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{g\left({\ell }_1\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{g\left({\ell }_0\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{g\left({\ell }_0\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{g\left({\ell }_1\right)}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0},$$

that is,

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{{\ell }_1}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_0}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_1}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_2}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0}.$$

(9)

Now, (d3) implies

$${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_2}\right){⪯}_{ϵ}(n-1){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right)+{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)$$

and then using $\left({ϝ}_1\right)$, we obtain

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{{\ell }_1}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_0}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right),0,(n-1){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right)+{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0}.$$

(10)

By $\left({ϝ}_2\right)$, we have

$${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right){⪯}_{ϵ}\frac{1}{n-1}\mathsf{S}\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)\right).$$

As ${\ell }_1\mathcal{R}{\ell }_2$; thus, the use of (2) implies ${\ell }_3\mathcal{R}{\ell }_2$. Applying (8)

$$(I-\mathsf{S})\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{g\left({\ell }_1\right)}\right)\right)=(I-\mathsf{S})\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right)\right){⪯}_{ϵ}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right)\mathrm{implies}$$

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{g\left({\ell }_1\right)}^{g\left({\ell }_2\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{g\left({\ell }_1\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{g\left({\ell }_2\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{g\left({\ell }_2\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{g\left({\ell }_1\right)}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0},$$

By (d3), $\left({ϝ}_1\right)$ and $\left({ϝ}_2\right)$, we obtain

$${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{{\ell }_3}\right){⪯}_{ϵ}\frac{1}{(n-1)}\mathsf{S}\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right)\right).$$

By following above procedure, we have a sequence $\left\{{\ell }_n\right\}$ so that

$${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_{n+1}}\right){⪯}_{ϵ}\frac{1}{n-1}\mathsf{S}\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{n-1}}^{{\ell }_n}\right)\right){⪯}_{ϵ}\frac{1}{{(n-1)}^2}{\mathsf{S}}^2\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{n-2}}^{{\ell }_{n-1}}\right)\right){⪯}_{ϵ}\cdots {⪯}_{ϵ}\frac{1}{{(n-1)}^n}{\mathsf{S}}^n\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)\right).$$

Using the same reasoning as in Theorem 2 leads to $c^{*}=g\left(c^{*}\right)$.  □

4.3. Result for Monotone Self-Mapping

The following result encapsulates the statements of Theorems 2 and 3.

Theorem 4.

Let $(P,{\mathbf{d}}_{cA})$ be a complete cone $\mathbf{A}$-metric space and $\mathcal{C}\subset E$ be a cone. Let $\mathsf{S}\in B(E,E)$ so that ${∥\mathsf{S}∥}_1<1$, and $I:E\to E$ is an identity operator. If the mapping $g:P\to P$, for all comparable elements $\ell ,\kappa \in P$, and $ϝ\in \mathcal{H}$ satisfies the following condition

$$(I-\mathsf{S})({\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right){⪯}_{ϵ}{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)implies$$

$$ϝ\left(\begin{array}{c}{\mathit{d}}_{cA}\left({\zeta }_{g(\ell )}^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right),{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g(\ell )}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0}$$

(11)

and

(1) 

there exists ${\ell }_0\in P$ such that ${\ell }_0\mathcal{R}g\left({\ell }_0\right)$ or $g\left({\ell }_0\right)\mathcal{R}{\ell }_0$;

(2) 

the mapping g is monotone; and

(3) 

for a sequence $\left\{{\ell }_n\right\}$ with all comparable sequential terms such that ${\ell }_n\to c^{*}$, we have ${\ell }_n\mathcal{R}{c}^{*}$ ∀ $n\in \mathbb{N}$.

Then, there exists $c^{*}\in P$ such that $c^{*}=g\left(c^{*}\right)$.

Proof. 

Proceeding with an initial guess ${\ell }_0\in P$, we construct a sequence $\left\{{\ell }_n\right\}$ so that ${\ell }_n=g\left({\ell }_{n-1}\right)$ ∀ $n\in \mathbb{N}$. By assumption (1), we have ${\ell }_0\mathcal{R}g\left({\ell }_0\right)={\ell }_1$ and by (2), ${\ell }_2\mathcal{R}{\ell }_1$. Using (11), we obtain

$$(I-\mathsf{S})\left({\mathbf{d}}_{cA}\left({\zeta }_{g\left({\ell }_0\right)}^{{\ell }_0}\right)\right)=(I-\mathsf{S})\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_0}\right)\right){⪯}_{ϵ}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_0}\right)\mathrm{implies}$$

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{g\left({\ell }_0\right)}^{g\left({\ell }_1\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{g\left({\ell }_0\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{g\left({\ell }_1\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{g\left({\ell }_1\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{g\left({\ell }_0\right)}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0},$$

that is,

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_1}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0}.$$

(12)

By (d3) and $\left({ϝ}_1\right)$,

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),(n-1){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)+{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),0)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0}.$$

(13)

Using $\left({ϝ}_2\right)$, we have

$${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right){⪯}_{ϵ}\frac{1}{n-1}\mathsf{S}\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)\right).$$

Since ${\ell }_2\mathcal{R}{\ell }_1$, by (11), we have

$$(I-\mathsf{S})\left({\mathbf{d}}_{cA}\left({\zeta }_{g\left({\ell }_0\right)}^{g\left({\ell }_1\right)}\right)\right)=(I-\mathsf{S})\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right)\right){⪯}_{ϵ}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right)\mathrm{implies}$$

$$ϝ\left(\begin{array}{c}{\mathbf{d}}_{cA}\left({\zeta }_{g\left({\ell }_1\right)}^{g\left({\ell }_2\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{g\left({\ell }_1\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{g\left({\ell }_2\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{g\left({\ell }_2\right)}\right),{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{g\left({\ell }_1\right)}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0}.$$

Using (d3), we have

$${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_3}\right){⪯}_{ϵ}(n-1){\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right)+{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{{\ell }_3}\right).$$

By $\left({ϝ}_1\right)$ and $\left({ϝ}_2\right)$, we have

$${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_2}^{{\ell }_3}\right){⪯}_{ϵ}\frac{1}{n-1}\mathsf{S}\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{{\ell }_2}\right)\right){⪯}_{ϵ}\frac{1}{{(n-1)}^2}{\mathsf{S}}^2({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right).$$

By following same way, we construct a sequence $\left\{{\ell }_n\right\}$ such that

$${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_{n+1}}\right){⪯}_{ϵ}\frac{1}{n-1}\mathsf{S}\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{n-1}}^{{\ell }_n}\right)\right){⪯}_{ϵ}\frac{1}{{(n-1)}^2}{\mathsf{S}}^2\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_{n-2}}^{{\ell }_{n-1}}\right)\right){⪯}_{ϵ}\cdots {⪯}_{ϵ}\frac{1}{{(n-1)}^n}{\mathsf{S}}^n\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_0}^{{\ell }_1}\right)\right).$$

Analysis similar to that in the proof of Theorem 2 will lead to the desired conclusion.  □

Remark 3.

(1). To obtain a unique fixed point by using Theorems 2–4, we take an upper bound or lower bound for every pair $\ell ,\kappa \in P$.

(2). For a normal cone, one can replace ${ϝ}_3$ by

$ϝ(\ell ,{\ell }_1,{\ell }_2,\varpi ,\varpi +(n-1){\ell }_1,\varpi +(n-1){\ell }_2){⪯}_{ϵ}\mathbf{0}$∀, $\ell {\ll }_{ϵ}{\epsilon }_A$, ${\ell }_1{\ll }_{ϵ}{\epsilon }_A$, ${\ell }_2{\ll }_{ϵ}{\epsilon }_A$ and $\varpi {\ll }_{ϵ}{\epsilon }_A$.

5. Examples and Consequences

We elaborate our results through the following examples.

Example 5.

Let $E=({\mathbb{R}}^{,}\parallel ·\parallel )$ as a real Banach space, along with a cone $\mathcal{C}=\{\ell \in \mathbb{R}:\ell \ge 0\}$ in E. Define the cone $\mathbf{A}$-metric by ${\mathit{d}}_{cA}({\ell }_1,{\ell }_2,\dots \dots ,{\ell }_n)={\sum }_{i=1}^n{\sum }_{i<j}|{\ell }_i-{\ell }_j|$. Let $P=\{0,\frac{1}{6}\}\subset E$ and $g:P\to P$ be given by

$$g\left(0\right)=\frac{1}{6},g\left(\frac{1}{6}\right)=\frac{1}{6}.$$

Then, the mapping g is monotone with respect to partial order ${⪯}_{ϵ}$. Let $\mathsf{S}(\ell )=\frac{\ell }{3}$, then $\parallel \mathsf{S}\parallel <1$ and hence $\mathsf{S}\in B(E,E)$. Furthermore, $\mathsf{S}\left(\mathcal{C}\right)\subset \mathcal{C}$. Now, if $\ell =0$ and $\kappa =\frac{1}{6}$, then $\ell {⪯}_{ϵ}\kappa$.

$${\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right)=\frac{(n-1)}{6},{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g\left(\kappa \right)}\right)=0,{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g(\ell )}\right)=0,{\mathit{d}}_{cA}\left({\zeta }_{g(\ell )}^{g\left(\kappa \right)}\right)=0.$$

$$\begin{array}{ccc}\hfill \mathsf{S}({\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right)& =& \frac{{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right)}{3}=\frac{(n-1)}{18},\hfill \\ \hfill (I-\mathsf{S})({\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right)& =& \frac{(n-1)}{9}.\hfill \end{array}$$

$$\begin{array}{ccc}& & \frac{\alpha }{n-1}max\left\{{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)\right),{\mathit{d}}_{cA}({\zeta }_{\ell }^{g(\ell )},{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g(\ell )}\right)\}\hfill \\ & =& \frac{\alpha }{n-1}max\left\{\frac{(n-1)}{6},\frac{(n-1)}{6},0\right\}\hfill \\ & =& \frac{\alpha }{6}.\hfill \end{array}$$

Thus, for every $\alpha \in \left[0,\frac{1}{2}\right)$, $(I-\mathsf{S})({\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right){⪯}_{ϵ}{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)$ implies

$${\mathit{d}}_{cA}({\zeta }_{g(\ell )}^{g\left(\kappa \right)}{⪯}_{ϵ}\frac{\alpha }{n-1}max\left\{{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)\right),{\mathit{d}}_{cA}({\zeta }_{\ell }^{g(\ell )},{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g(\ell )}\right)\}.$$

Now,

$$\begin{array}{ccc}& & ϝ\left(\begin{array}{c}{\mathit{d}}_{cA}\left({\zeta }_{g(\ell )}^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)),{\mathit{d}}_{cA}({\zeta }_{\ell }^{g(\ell )},{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g(\ell )}\right)\hfill \end{array}\right)\hfill \\ & =& {\mathit{d}}_{cA}\left({\zeta }_{g(\ell )}^{g\left(\kappa \right)}\right)-\frac{\alpha }{n-1}max\left\{{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)\right),{\mathit{d}}_{cA}({\zeta }_{\ell }^{g(\ell )},{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g(\ell )}\right)\}.\hfill \end{array}$$

Then, $(I-\mathsf{S})\left(\mathit{A}\left({\zeta }_{\ell }^{g(\ell )}\right)\right){⪯}_{ϵ}\mathit{A}\left({\zeta }_{\ell }^{\kappa }\right)$ implies

$$ϝ\left(\begin{array}{c}{\mathit{d}}_{cA}\left({\zeta }_{g(\ell )}^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)),{\mathit{d}}_{cA}({\zeta }_{\ell }^{g(\ell )},{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g\left(\kappa \right)}\right),{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g(\ell )}\right)\hfill \end{array}\right){⪯}_{ϵ}\mathbf{0}.$$

Use of Theorem 2 leads to have a fixed point of g, such that $g\left(\frac{1}{6}\right)=\frac{1}{6}$.

Corollary 1.

Let $(P,{\mathit{d}}_{cA})$ be a complete cone $\mathbf{A}$-metric space. Let $\mathsf{S}\in B(E,E)$ with ${∥\mathsf{S}∥}_1<1$ and $I:E\to E$ be an identity operator. If the mapping $g:P\to P$, for all comparable elements $\ell ,\kappa \in P$, and $ϝ\in \mathcal{H}$ satisfies the following condition

$$(I-\mathsf{S})\left({\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right)\right){⪯}_{ϵ}{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)\Rightarrow {\mathit{d}}_{cA}\left({\zeta }_{g(\ell )}^{g\left(\kappa \right)}\right){⪯}_{ϵ}\mathsf{S}\left({\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)\right),$$

and

(1) 

there exists ${\ell }_0\in P$ such that either ${\ell }_0\mathcal{R}g\left({\ell }_0\right)$ or $g\left({\ell }_0\right)\mathcal{R}{\ell }_0$;

(2) 

for all $\ell ,\kappa \in P$, $\ell \mathcal{R}\kappa$ we have $g(\ell )\mathcal{R}g\left(\kappa \right)$ or $g\left(\kappa \right)\mathcal{R}g(\ell )$; and

(3) 

for a sequence $\left\{{\ell }_n\right\}$ with ${\ell }_n\to c^{*}$ whose all sequential terms are comparable, we have ${\ell }_n\mathcal{R}{c}^{*}$ for all $n\in \mathbb{N}$.

Then, there exists $c^{*}\in P$ such that $c^{*}=g\left(c^{*}\right)$.

Proof. 

Define $ϝ:E^6\to E$ by $ϝ({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4,{\ell }_5,{\ell }_6)=\varpi {\ell }_1-{\ell }_2;\varpi >1,$ and the operator $\mathsf{S}:E\to E$ by $\mathsf{S}(\ell )=h\ell$ for all $\ell \in E$; $h<1$, so that $\mathsf{S}\in B(E,E)$. Now, by Theorem 4, there exists $c^{*}\in P$ such that $c^{*}=g\left(c^{*}\right)$.  □

Corollary 2.

Let $(P,{\mathit{d}}_{cA})$ be a complete cone $\mathbf{A}$-metric space. Let $\mathsf{S}\in B(E,E)$ with ${∥\mathsf{S}∥}_1<1$ and $I:E\to E$ be an identity operator. If the mapping $g:P\to P$, for all comparable elements $\ell ,\kappa \in P$, and $ϝ\in \mathcal{H}$ satisfies the following condition

$$(I-\mathsf{S})\left({\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right)\right){⪯}_{ϵ}{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)\Rightarrow {\mathit{d}}_{cA}\left({\zeta }_{g(\ell )}^{g\left(\kappa \right)}\right){⪯}_{ϵ}\frac{1}{n-1}\mathsf{S}\left({\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)\right),$$

and

(1) 

there exists ${\ell }_0\in P$ verifying either ${\ell }_0\mathcal{R}g\left({\ell }_0\right)$ or $g\left({\ell }_0\right)\mathcal{R}{\ell }_0$;

(2) 

for all $\ell ,\kappa \in P$, $\ell \mathcal{R}\kappa$ implies either $g(\ell )\mathcal{R}g\left(\kappa \right)$ or $g\left(\kappa \right)\mathcal{R}g(\ell )$; and

(3) 

for a sequence $\left\{{\ell }_n\right\}$ converging to $c^{*}$ and ${\ell }_n\mathcal{R}{\ell }_{n-1}$, we have ${\ell }_n\mathcal{R}{c}^{*}$ for all $n\in \mathbb{N}$.

Then, there exists a vector $c^{*}\in P$ such that $c^{*}=g\left(c^{*}\right)$.

Proof. 

Taking $\mathsf{S}:E\to E$ such that $\mathsf{S}\left(v\right)=v$, ∀$v\in E$ and following the proof of Corollary 1, we obtain the result.  □

Example 6.

Let $E=({\mathbb{R}}^3,\parallel ·\parallel )$ with $\parallel \ell \parallel =max\left\{\left(\right|{\ell }_1|,|{\ell }_2|,|{\ell }_3\left|\right)\right\}$, then $(E,∥.∥)$ is a real Banach space. Take a cone in E as $\mathcal{C}=\{(\ell ,\xi ,\nu )\in {\mathbb{R}}^3:\ell ,\xi ,\nu \ge 0\}$. Define the cone $\mathbf{A}$-metric by ${\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\xi }\right)=(n-1)\left\{\right|{\ell }_1-{\xi }_1|,|{\ell }_2-{\xi }_2|,|{\ell }_3-{\xi }_3\left|\right\}$, where $\ell =({\ell }_1,{\ell }_2,{\ell }_3)$ and $\xi =({\xi }_1,{\xi }_2,{\xi }_3)$. Let $P=\{(0,0,0),(0,0,\frac{1}{4}),(0,\frac{1}{4},0)\}\subset E$ and define g by

$$g(0,0,0)=\left(0,0,\frac{1}{4}\right),g\left(0,0,\frac{1}{4}\right)=\left(0,0,\frac{1}{4}\right),g\left(0,\frac{1}{4},0\right)=(0,0,0),$$

where g is monotonic with respect to partial order ${⪯}_{ϵ}$. Take

$$\mathsf{S}=\left(\begin{array}{ccc}\frac{1}{3}& 0& 0\\ 0& \frac{1}{3}& 0\\ 0& 0& \frac{1}{3}\end{array}\right).$$

Define $\mathsf{S}(\ell )=\frac{\ell }{3}$ and

$$\begin{array}{ccc}\hfill \parallel \mathsf{S}(\ell )\parallel & =& max\left\{\frac{|{\ell }_1|}{3},\frac{|{\ell }_2|}{3},\frac{|{\ell }_3|}{3}\right\}\hfill \\ & =& \frac{1}{3}max\left\{\right|{\ell }_1|,|{\ell }_2|,|{\ell }_3\left|\right\}\hfill \\ & =& \frac{1}{3}\parallel \ell \parallel .\hfill \end{array}$$

Then, $\parallel \mathsf{S}\parallel =\frac{1}{3}<1$ and hence $\mathsf{S}\in B(E,E)$. Furthermore, $\mathsf{S}\left(\mathcal{C}\right)\subset \mathcal{C}$. Now, if $\ell =(0,0,0)$ and $\kappa =(0,0,\frac{1}{4})$, then $\ell {⪯}_{ϵ}\kappa$. Additionally,

$${\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right)=(n-1)\left(0,0,\frac{1}{4}\right),{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g\left(\kappa \right)}\right)=\left(0,0,0\right),{\mathit{d}}_{cA}\left({\zeta }_{\kappa }^{g(\ell )}\right)=\left(0,0,0\right),{\mathit{d}}_{cA}\left({\zeta }_{g(\ell )}^{g\left(\kappa \right)}\right)=\left(0,0,0\right).$$

$$\begin{array}{ccc}\hfill \mathsf{S}\left({\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right)\right)& =& \frac{{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right)}{3}=(n-1)\left(0,0,\frac{1}{12}\right),\hfill \\ \hfill (I-\mathsf{S})\left({\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right)\right)& =& (n-1)\left(0,0,\frac{1}{4}\right)-(n-1)\left(0,0,\frac{1}{12}\right)=(n-1)\left(0,0,\frac{1}{6}\right).\hfill \end{array}$$

$${\mathit{d}}_{cA}\left({\zeta }_{g(\ell )}^{g\left(\kappa \right)}\right){⪯}_{ϵ}\frac{1}{n-1}\mathsf{S}\left({\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)\right)$$

Then, $(I-\mathsf{S})\left({\mathit{d}}_{cA}\left({\zeta }_{\ell }^{g(\ell )}\right)\right){⪯}_{ϵ}{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)$ implies

$${\mathit{d}}_{cA}\left({\zeta }_{g(\ell )}^{g\left(\kappa \right)}\right){⪯}_{ϵ}\frac{1}{n-1}\mathsf{S}\left({\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)\right)$$

Applying Corollary 1, we have $(0,0,\frac{1}{4})\in P$ such that $g(0,0,\frac{1}{4})=(0,0,\frac{1}{4})$.

6. A Homotopy Result

We establish a homotopy result as an application of Corollary 1.

Definition 11

([36]). Let $(X,{\mathit{d}}_{cA})$ be a cone $\mathit{A}$-metric space over Banach algebra E. Then, for a $\ell \in P$ and $0{\ll }_{ϵ}{\epsilon }_A$, the open ball with center ℓ and radius ${\epsilon }_A$ is

$$B\left({\zeta }_{\ell }^{{\epsilon }_A}\right)=\{{\ell }_1\in P:{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{{\ell }_1}\right){\ll }_{ϵ}{\epsilon }_A\}.$$

The closed ball with center ℓ and radius ${\epsilon }_A$ is

$$\overline{B\left({\zeta }_{\ell }^c\right)}=\{{\ell }_1\in P:{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{{\ell }_1}\right){⪯}_{ϵ}{\epsilon }_A\}$$

Definition 12.

Let $(X,{\mathit{d}}_{cA})$ be a cone $\mathit{A}$-metric space. Then, for any set $A\subset X$ and $\ell \in X$, define

$${\mathit{d}}_{cA}\left({\zeta }_{\ell }^A\right)=inf\{{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{{\ell }_1}\right):{\ell }_1\in A\}.$$

Theorem 5.

Let $(E,∥.∥)$ be a real Banach space and $\mathcal{C}\subset E$ be a cone. Let $(P,{\mathit{d}}_{cA})$ be a complete cone $\mathit{A}$-metric space and $V\subset P$ be an open set. Let the operator $\mathsf{S}\in B(E,E)$ be such that ${∥\mathsf{S}∥}_1<1$ satisfying $\mathsf{S}\left(\mathcal{C}\right)\subset \mathcal{C}$. Suppose that the mapping $l:{\overline{V}}^{n-1}\times [0,1]\to P$ satisfies the axiom (1) of Corollary 2 in the first variable and

(1) 

$\ell \ne l\left({\zeta }_{\ell }^{{\ell }_1}\right)$ for each $\ell \in \partial V$ ($\partial V$ represents the boundary of V in P);

(2) 

there exists $Y\ge 0$ such that

$$∥{\mathit{d}}_{cA}\left({\zeta }_{l\left({\zeta }_{\ell }^{{\ell }_1}\right)}^{l\left({\zeta }_{\ell }^{\varsigma }\right)}\right)∥\le Y|{\ell }_1-\varsigma |,$$

where $\ell \in \overline{V}$ and ${\ell }_1,\varsigma \in [0,1]$; and

(3) 

if r is a radius of an open ball V satisfying $∥{\mathit{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)∥\le r$, then $\ell \mathcal{R}\kappa$ for any $\ell ,\kappa \in V$.

Then, whenever $l\left({\zeta }_{(·)}^0\right)$ possesses a fixed point in V, $l\left({\zeta }_{(·)}^1\right)$ possesses a fixed point in V.

Proof. 

Let

$$G=\{o_1\in [0,1]|\ell =l\left({\zeta }_{\ell }^{o_1}\right);\mathrm{when}\ell \in V\}.$$

Define the partial order ${⪯}_{ϵ}$ on E by $e{⪯}_{ϵ}w$ if and only if $\parallel e\parallel \le \parallel w\parallel$, for all $e,w\in E$. As $l\left({\zeta }_{.}^0\right)$ possesses a fixed point in V, so $0\in B_G$. Thus, $G\ne \varphi$, also ${\mathbf{d}}_{cA}\left({\zeta }_{\ell }^{l\left({\zeta }_{\ell }^{o_1}\right)}\right)={\mathbf{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)$, $(I-\mathsf{S})\left({\mathbf{d}}_{cA}\left({\zeta }_{\ell }^{l\left({\zeta }_{\ell }^{o_1}\right)}\right)\right){⪯}_{ϵ}{\mathbf{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)$ for all $\ell \mathcal{R}\kappa$, then by Corollary 1, we obtain

$${\mathbf{d}}_{cA}\left({\zeta }_{l\left({\zeta }_{\ell }^{o_1}\right)}^{l\left({\zeta }_{\kappa }^{o_1}\right)}\right){⪯}_{ϵ}\frac{1}{n-1}\mathsf{S}\left({\mathbf{d}}_{cA}\left({\zeta }_{\ell }^{\kappa }\right)\right).$$

Take ${\left\{o_n\right\}}_{n=1}^{\infty }\subseteq G$ such that $o_n\to o_1\in [0,1]$ whenever $n\to \infty$. As $o_n\in G$ for $n\in \mathbb{N}$, we have ${\ell }_n\in V$ so that ${\ell }_n=l\left({\zeta }_{{\ell }_n}^{o_n}\right)$. Since, $l\left({\zeta }_{\ell }^{(·)}\right)$ is monotone, so, for $n,m\in \mathbb{N}$, we have ${\ell }_m\mathcal{R}{\ell }_n$. Since

$$(I-\mathsf{S})\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{l\left({\zeta }_{{\ell }_m}^{o_m}\right)}\right)\right)=(I-\mathsf{S})\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right)\right){⪯}_{ϵ}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right),$$

we have

$${\mathbf{d}}_{cA}\left({\zeta }_{l\left({\zeta }_{{\ell }_n}^{o_m}\right)}^{l\left({\zeta }_{{\ell }_m}^{o_m}\right)}\right){⪯}_{ϵ}\frac{1}{n-1}\mathsf{S}\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right)\right)$$

and

$$\begin{array}{ccc}\hfill {\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right)& =& {\mathbf{d}}_{cA}\left({\zeta }_{l\left({\zeta }_{{\ell }_n}^{o_n}\right)}^{l\left({\zeta }_{{\ell }_m}^{o_m}\right)}\right)\hfill \\ & {⪯}_{ϵ}& {\mathit{d}}_{cA}\left({\zeta }_{l\left({\zeta }_{{\ell }_n}^{o_n}\right)}^{l\left({\zeta }_{{\ell }_n}^{o_m}\right)}\right)+(n-1){\mathbf{d}}_{cA}\left({\zeta }_{l\left({\zeta }_{{\ell }_n}^{o_m}\right)}^{l\left({\zeta }_{{\ell }_m}^{{\theta }_m}\right)}\right)\hfill \\ \hfill ∥{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right)∥& \le & Y|o_n-o_m|+∥\mathsf{S}\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right)\right)∥\hfill \\ \hfill ∥{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right)∥& \le & \frac{Y}{1-∥\mathsf{S}∥}|o_n-o_m|.\hfill \end{array}$$

Since ${\left\{o_n\right\}}_{n=1}^{\infty }$ is a Cauchy sequence in $[0,1]$, we have

$$\underset{n,m\to \infty }{lim}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right)=\mathbf{0},$$

and thus ${\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{{\ell }_m}\right){\ll }_{ϵ}{\epsilon }_A$, as $n,m\to \infty$. This implies that $\left\{{\ell }_n\right\}$ is a Cauchy sequence in P. As P is a complete cone $\mathbf{A}$-metric space, we can obtain an $\ell \in \overline{V}$ such that ${lim}_{n\to \infty }{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{\ell }\right){\ll }_{ϵ}{\epsilon }_A$. Thus, ${\ell }_n\mathcal{R}\ell$ ∀ $n\in \mathbb{N}$. Take

$$\begin{array}{ccc}\hfill {\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{l\left({\zeta }_{\ell }^{o_1}\right)}\right)& =& {\mathbf{d}}_{cA}\left({\zeta }_{l\left({\zeta }_{{\ell }_n}^{o_n}\right)}^{l\left({\zeta }_{\ell }^{o_1}\right)}\right)\hfill \\ & {⪯}_{ϵ}& {\mathbf{d}}_{cA}\left({\zeta }_{l\left({\zeta }_{{\ell }_n}^{o_n}\right)}^{l\left({\zeta }_{{\ell }_n}^{o_1}\right)}\right)+(n-1){\mathbf{d}}_{cA}\left({\zeta }_{l\left({\zeta }_{{\ell }_n}^{o_1}\right)}^{l\left({\zeta }_{\ell }^{o_1}\right)}\right)\hfill \\ \hfill ∥{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{l\left({\zeta }_{\ell }^{o_1}\right)}\right)∥& \le & Y|o_n-o_1|+∥\mathsf{S}\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{\ell }\right)\right)∥.\hfill \end{array}$$

This implies

$$\underset{n\to \infty }{lim}{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_n}^{l\left({\zeta }_{\ell }^{o_1}\right)}\right)=\mathbf{0}.$$

Thus, ${\mathbf{d}}_{cA}\left({\zeta }_{\ell }^{l\left({\zeta }_{\ell }^{o_1}\right)}\right)=\mathbf{0}$. Consequently $o_1\in G$, and hence G is closed in $[0,1]$.

Now, to show is that G is open in $[0,1]$. For this, take ${\vartheta }_1\in G$, we have ${\ell }_1\in V$ such that $l\left({\zeta }_{{\vartheta }_1}^{{\ell }_1}\right)={\ell }_1$. As V is open; therefore, we can have $r>0$ with $G\left({\zeta }_{{\ell }_1}^r\right)\subseteq V$. Consider

$$p={\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{\partial U}\right)=inf\{{\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{\xi }\right):\xi \in \partial V\}.$$

Take $r=p>0$ by fixing $ϵ>0$ so that $ϵ<\frac{(1-∥\mathsf{S}∥)p}{Y}$. Consider $o_1\in ({\vartheta }_1-ϵ,{\vartheta }_1+ϵ)$. Then,

$$\ell \in \overline{B\left({\zeta }_{{\ell }_1}^r\right)}=\{\ell \in P:∥{\mathbf{d}}_{cA}\left({\zeta }_{\ell }^{{\ell }_1}\right)∥\le r\},\mathrm{as}\ell \mathcal{R}{\ell }_1.$$

Consider

$$\begin{array}{ccc}\hfill {\mathbf{d}}_{cA}\left({\zeta }_{l\left({\zeta }_{\ell }^{o_1}\right)}^{{\ell }_1}\right)& =& {\mathbf{d}}_{cA}\left({\zeta }_{l\left({\zeta }_{\ell }^{o_1}\right)}^{l\left({\zeta }_{{\ell }_1}^{{\vartheta }_1}\right)}\right)\hfill \\ & {⪯}_{ϵ}& {\mathbf{d}}_{cA}\left({\zeta }_{l\left({\zeta }_{\ell }^{o_1}\right)}^{l\left({\zeta }_{\ell }^{{\vartheta }_1}\right)}\right)+(n-1){\mathbf{d}}_{cA}\left({\zeta }_{l\left({\zeta }_{\ell }^{{\vartheta }_1}\right)}^{l\left({\zeta }_{{\ell }_1}^{{\vartheta }_1}\right)}\right)\hfill \\ \hfill ∥{\mathbf{d}}_{cA}\left({\zeta }_{l\left({\zeta }_{\ell }^{o_1}\right)}^{{\ell }_1}\right)∥& \le & Y|{\vartheta }_1-o_1|+∥\mathsf{S}\left({\mathbf{d}}_{cA}\left({\zeta }_{{\ell }_1}^{\ell }\right)\right)∥\hfill \\ & \le & Yϵ+∥\mathsf{S}∥p<p.\hfill \end{array}$$

Hence, $l\left({\zeta }_{·}^t\right):\overline{G\left({\zeta }_{\ell }^r\right)}\to \overline{G\left({\zeta }_{\ell }^r\right)}$ possesses a fixed point in $\overline{V}$ for every fixed $o_1\in ({\vartheta }_1-ϵ,{\vartheta }_1+ϵ)$, application of Corollary 1 leads to the proof. The obtained fixed point will necessarily be in V as in the previous situation. Thus, ${\vartheta }_1\in G$ for each ${\vartheta }_1\in (o_1-ϵ,o_1+ϵ)$, and therefore G is open in $[0,1]$. Hence, G is open as well as closed in $[0,1]$ and by connectedness, $G=[0,1]$. Hence, $l\left({\zeta }_{·}^1\right)$ possesses a fixed point in V.  □

7. Application to the Existence of the Solution to Urysohn Integral Equation (UIE)

In this section, our concern is to obtain a unique converging point for UIE:

$$\ell \left(v\right)=\mathsf{b}\left(v\right)+\underset{\mathrm{IR}}{\int }{L}_1(v,t,\ell \left(t\right))dt.$$

(14)

This above Equation (14) depending on the range of integration (IR), is a generalization of many integral equations in literature (see [40,41,42]). In the present article, we use a fixed point method to calculate a single converging point to Urysohn Integral Equation, which also leads to convergence of different mathematical structures. Assume IR as the set of finite measure and ${\mathcal{J}}_{\mathrm{IR}}^2=\left\{\ell |{\int }_{\mathrm{IR}}{|\ell \left(t\right)|}^{2}dt<\infty \right\}$. Assume $∥.∥:{\mathcal{J}}_{\mathrm{IR}}^2\to [0,\infty )$ with

$${∥\ell ∥}_2=\sqrt{{\int }_{\mathrm{IR}}{|\ell \left(t\right)|}^{2}dt},\mathrm{for}\mathrm{all}\ell ,j\in {\mathcal{J}}_{\mathrm{IR}}^2.$$

Similarly, take:

$${∥\ell ∥}_{2,\nu }=\sqrt{sup\left\{e^{-\nu {\int }_{\mathrm{IR}}\alpha \left(t\right)dt}{\int }_{\mathrm{IR}}\right|\ell \left(t\right){|}^{2}dt\}},\mathrm{for}\mathrm{all}\ell \in {\mathcal{J}}_{\mathrm{IR}}^2,\nu >1.$$

One can observe $E=({\mathcal{J}}_{\mathrm{IR}}^2,{∥.∥}_{2,\nu })$ represents a Banach space. Assume a cone $\mathcal{Z}=\{\ell \in {\mathcal{J}}_{\mathrm{IR}}^2:\ell \left(t\right)>0\mathrm{for}\mathrm{almost}\mathrm{every}\mathrm{s}\}$ in E. Define ${\mathsf{c}}_{\nu }$ as ${\mathsf{c}}_{\nu }(x_1,x_2)=x_1{∥x_1-x_2∥}_{2,\nu }^2$ for all $x_1,x_2\in \mathcal{Z}$. Then, ${\mathsf{c}}_{\nu }$ represents a cone $\mathbf{A}$-metric. Define partial order ${⪯}_{ϵ}$ on E, such that

$$\alpha {⪯}_{ϵ}\varpi \mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\alpha \left(s\right)\varpi \left(s\right)\ge \varpi \left(s\right),\mathrm{for}\mathrm{all}a,\varpi \in E.$$

Then, $(E,{⪯}_{ϵ},{\mathsf{c}}_{\nu })$ is a complete cone $\mathbf{A}$-metric space. To show the existence of solutions to UIE, we need the following conditions:

(C1)

The kernel $L_1:\mathrm{IR}\times \mathrm{IR}\times \mathbb{R}\to \mathbb{R}$ fulfils the Carathéodory axiom along with

$$\left|L_1(v,t,\ell \left(t\right))\right|\le w(v,t)+e(v,t)\left|\ell \left(t\right)\right|;w,e\in {\mathcal{J}}^2(\mathrm{IR}\times \mathrm{IR}),e(v,t)>0.$$

(C2)

Assume a continuous and bounded function $\mathsf{b}:\mathrm{IR}\to [1,\infty )$ over IR.

(C3)

There exists a constant $C>0$ such that

$$\underset{v\in \mathrm{IR}}{sup}\underset{\mathrm{IR}}{\int }\left|L_1(v,t)\right|dt\le C.$$

(C4)

For any ${\ell }_0\in {\mathcal{J}}_{\mathrm{IR}}^2$, there exists ${\ell }_1=K\left({\ell }_0\right)$ such that ${\ell }_1{⪯}_{ϵ}{\ell }_0$ or ${\ell }_0{⪯}_{ϵ}{\ell }_1$.

(C4’)

Take a sequence $\left\{{\ell }_n\right\}$ satisfying ${\ell }_{n-1}{⪯}_{ϵ}{\ell }_n$ such that ${\ell }_n\to p$, which further implies to ${\ell }_n{⪯}_{ϵ}p$ for all $n\in \mathbb{N}$.

(C5)

We can find a non-negative and measurable function $q:\mathrm{IR}\times \mathrm{IR}\to \mathbb{R}$ such that

$$\alpha \left(v\right):={\int }_{\mathrm{IR}}{q}^2(v,s)dt\le \frac{1}{\nu }$$

also integrable over $IR$ such that

$$\left|L_1(v,s,\ell \left(s\right))-L_1(v,s,j\left(s\right))\right|\le q(v,s)|\ell \left(s\right)-j\left(s\right)|$$

for all $v,s\in \mathrm{IR}$ and $\ell ,j\in E$ such that $\ell {⪯}_{ϵ}j$.

Theorem 6.

Assume that $L_1$ satisfies all axioms (C1)–(C5), and then we obtain a single converging point for UIE.

Proof. 

Defining a function $K:E\to E$, using defined notations by

$$\left(K\ell \right)\left(v\right)=b\left(v\right)+{\int }_{\mathrm{IR}}{L}_1(v,t,\ell \left(t\right))dt,$$

K is ⪯-preserving:

Let $\ell ,j\in E$ along with $\ell ⪯j$, so that $\ell \left(s\right)j\left(s\right)\ge \ell \left(s\right)$. For almost every $v\in IR$, we have

$$\left(K\ell \right)\left(v\right)=b\left(v\right)+{\int }_{\mathrm{IR}}{L}_1(v,t,\ell \left(t\right))dt\ge 1,$$

which shows $\left(K\ell \right)\left(v\right)\left(Kj\right)\left(v\right)\ge \left(K\ell \right)\left(v\right)$. Hence, $\left(K\ell \right)⪯\left(K\ell \right)$.

Self-operator:

The use of (C1) and (C3) leads to a continuous and compact function $K:\mathcal{Z}\to \mathcal{Z}$(see [40], Lemma 3).

Using (C4), we have ${\ell }_1=K\left({\ell }_0\right)$ with ${\ell }_1⪯{\ell }_0$ or ${\ell }_0⪯{\ell }_1$, for each ${\ell }_0\in \mathcal{Z}$ and K taken as ⪯-preserving. Hence, ${\ell }_n=R^n\left({\ell }_0\right))$ with ${\ell }_n⪯{\ell }_{n+1}$ or ${\ell }_{n+1}⪯{\ell }_n$ for all $n\ge 0$.

Applying (C5) along with Holder inequality, we find a contractive condition of Theorem 2.

$$\begin{array}{ccc}\hfill \ell {\left|(K\ell )\left(v\right)-(Kj)\left(v\right)\right|}^2& =& \ell {\left|\underset{\mathrm{IR}}{\int }{L}_1(v,s,\ell \left(s\right))ds-\underset{\mathrm{IR}}{\int }{L}_1(v,s,j\left(s\right))ds\right|}^2\hfill \\ & ⪯& \ell {\left(\underset{\mathrm{IR}}{\int }\left|L_1(v,s,\ell \left(s\right))-L_1(v,s,j\left(s\right))\right|ds\right)}^2\hfill \\ & ⪯& \ell {\left(\underset{\mathrm{IR}}{\int }q(v,s)|\ell \left(s\right)-j\left(s\right)|ds\right)}^2\hfill \\ & ⪯& \ell \underset{IR}{\int }{q}^2(v,s)ds·\underset{IR}{\int }{|\ell \left(s\right)-j\left(s\right)|}^{2}ds\hfill \\ & =& \ell \alpha \left(v\right)\underset{\mathrm{IR}}{\int }{|\ell \left(s\right)-j\left(s\right)|}^{2}ds.\hfill \end{array}$$

Now,

$$\begin{array}{ccc}\hfill \ell \underset{IR}{\int }{\left|(K\ell )\left(v\right)-(Kj)\left(v\right)\right|}^{2}dv& ⪯& \ell \underset{\mathrm{IR}}{\int }\left(\alpha \left(v\right)\underset{\mathrm{IR}}{\int }{|\ell \left(s\right)-j\left(s\right)|}^{2}ds\right)dv\hfill \\ & =& \ell \underset{\mathrm{IR}}{\int }\left(\alpha \left(v\right)e^{\nu {\int }_{\mathrm{IR}}\alpha \left(s\right)ds}·e^{-\nu {\int }_{\mathrm{IR}}\alpha \left(s\right)ds}\underset{\mathrm{IR}}{\int }{|\ell \left(s\right)-j\left(s\right)|}^{2}ds\right)dv\hfill \\ & ⪯& \ell {∥\ell -j∥}_{2,\nu }^2\underset{\mathrm{IR}}{\int }\alpha \left(v\right)e^{\nu {\int }_{\mathrm{IR}}\alpha \left(s\right)ds}dv\hfill \\ & ⪯& \ell \frac{1}{\nu }{∥\ell -j∥}_{2,\nu }^2{e}^{\nu {\int }_{\mathrm{IR}}\alpha \left(s\right)ds}.\hfill \end{array}$$

Thus,

$$\ell e^{-\nu {\int }_{\mathrm{IR}}\alpha \left(s\right)ds}\underset{\mathrm{IR}}{\int }{\left|(K\ell )\left(v\right)-(Kj)\left(v\right)\right|}^{2}dv⪯\ell \frac{1}{\nu }{∥\ell -j∥}_{2,\nu }^2.$$

This implies that

$$\ell {∥K\ell -Kj∥}_{2,\nu }^2⪯\ell \frac{1}{\nu }{∥\ell -j∥}_{2,\nu }^2.$$

$$c_{\nu }(K\ell ,Kj)⪯\frac{1}{\nu }{c}_{\nu }(\ell ,j)$$

$$\nu c_{\nu }(K\ell ,Kj)⪯c_{\nu }(\ell ,j).$$

Define $\mathcal{J}:E^6\to E$ by

$$\mathcal{J}(p_1,p_2,p_3,p_4,p_5,p_6)=k{p}_1-p_2;k>1.$$

We have

$$\mathcal{J}\left(c(K{x}_1,K{x}_2),c(x_1,x_2),c(x_1,K{x}_2),c(x_2,K{x}_2),c(x_1,K{x}_2),c(x_2,K{x}_1)\right){⪯}_{ϵ}{\mathbf{0}}_{\mathbf{E}}.$$

By using (C1)–(C5) and 2, we have a unique fixed point for K, which implies that UIE (14) has a unique converging point.  □

8. Conclusions

This paper contains various fixed-point results in the cone A-metric spaces. The fixed-point results extend the findings that appeared in [1,6,18]. These fixed-point results were presented successfully by using two different partial orders. The significance and validity of these results was shown by demonstrating various examples and applications. We suggest the readers and interested researchers to compare the results presented in this paper with the results appearing in [43] for further study.