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

Anam Arif

¹,

Muhammad Nazam

^2,*

Hamed H. Al-Sulami

³,

Aftab Hussain

and

Hasan Mahmood

Department of Mathematics, Government College University, Lahore 54000, Pakistan

Department of Mathematics, Allama Iqbal Open University, Islamabad 44000, Pakistan

Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

Author to whom correspondence should be addressed.

Symmetry2022,14(7), 1328;https://doi.org/10.3390/sym14071328

Submission received: 26 May 2022 /Revised: 16 June 2022 /Accepted: 18 June 2022 /Published: 27 June 2022

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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 coneA-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.

Keywords:

fixed point;implicit order relation;cone

A

-metric space;implicit contraction

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

x, y \in P

with

x ⪯ y

T (d_{p} (K x, K y), d_{p} (x, y), d_{p} (x, K x), d_{p} (y, K y), d_{p} (x, K y), d_{p} (y, K x)) \leq 0,

(1)

where

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 (l_{0})

We can obtain many contractive conditions depending on the definitions of the mapping

T

in (1). For instance, if we define

T : {[0, \infty)}^{6} \to (- \infty, \infty)

T (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}) = ℓ_{1} - ψ (max \{ℓ_{2}, ℓ_{3}, ℓ_{4}, \frac{1}{2} (ℓ_{5} + ℓ_{6})\}),

then we have a main theorem presented in [3]. If we define

T

T (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}) = ℓ_{1} - k ℓ_{2}; 0 \leq 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

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:

ζ_{σ_{1}}^{σ_{2}} = (\underset{n - 1}{\underset{︸}{σ_{1}, σ_{1}, . . . . σ_{1}}}, σ_{2}) .

2.Basic Notions

Definition 1.

A binary relation

R

over a set

X \neq Ø

defines a partial order if it satisfies the following axioms:

(1): $R$ is reflexive.
(2): $R$ is antisymmetric.
(3): $R$ is transitive.

A set with partial order

R

is known as a partially ordered set denoted by

(X, R)

Let

(E, ∥.∥)

be a real Banach space, and

C \subseteq E

satisfies the following axioms:

(1): $C$ is non-empty closed set such that $C \neq {0} .$
(2): ∀ $ℓ_{1}, ℓ_{2} \in R$ such that $ℓ_{1}, ℓ_{2} \geq 0$ along with $θ_{1}, θ_{2} \in C$ , we have $ℓ_{1} θ_{1} + ℓ_{2} θ_{2} \in C$ .
(3): $C \cap (- C) = {0}$ .

Then,

C

defines a cone inE. Given

C \subset E

, we define the partial order

⪯_{ϵ}

C

as follows:

η ⪯_{ϵ} ϱ \Leftrightarrow ϱ - η \in C for all η, ξ \in E .

Note that

η ≺_{ϵ} ξ

represents

η ⪯_{ϵ} ξ

but

η \neq ξ

and

η ≪ ξ

shows that

ξ - η \in C^{\circ}

(the interior of

C

Definition 2

([18]).The cone

ℵ \subseteq E

is called normal if, for all

χ, ς \in ℵ

, there exists a number

K > 0

so that,

0 ⪯_{ϵ} χ ⪯_{ϵ} ς \Rightarrow ∥ χ ∥ \leq K ∥ ς ∥ .

Throughout this paper, we use the notationℜ as a partial order in any ordinary setP and

⪯_{ϵ}

as a partial order in

C \subseteq E

. If

P \subseteq E

, thenℜ and

⪯_{ϵ}

would be considered as identical.

Definition 3

([18]).Let

P \neq Ø

, and if the mapping

d_{c} : P \times P \mapsto E

satisfies the following axioms:

(1): $0 ⪯_{ϵ} d_{c} (ℓ_{1}, ℓ_{2})$ and $d_{c} (ℓ_{1}, ℓ_{2}) = 0$ if and only if $ℓ_{1} = ℓ_{2}$ ;
(2): $d_{c} (ℓ_{1}, ℓ_{2}) = d_{c} (ℓ_{2}, ℓ_{1})$ ; and
(3): $d_{c} (ℓ_{1}, ℓ_{3}) ⪯_{ϵ} d_{c} (ℓ_{1}, ℓ_{2}) + d_{c} (ℓ_{2}, ℓ_{3})$ , ∀ $ℓ_{1}, ℓ_{2}, ℓ_{3} \in P$ ,

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 \neq Ø

, and if a mapping

d_{2} : P \times P \times P \mapsto [0, \infty)

satisfies the following axioms:

(d1): for distinct points $ℓ_{1}, ℓ_{2} \in P$ , ∃ $ℓ_{3} \in P$ such that $d_{2} (ℓ_{1}, ℓ_{2}, ℓ_{3}) \neq 0$ ;
(d2): $d_{2} (ℓ_{1}, ℓ_{2}, ℓ_{3}) = 0$ , if any two elements of the set ${ℓ_{1}, ℓ_{2}, ℓ_{3}}$ in P are equal;
(d3): $d_{2} (ℓ_{1}, ℓ_{2}, ℓ_{3}) = d_{2} (ℓ_{1}, ℓ_{3}, ℓ_{2}) = d_{2} (ℓ_{2}, ℓ_{1}, ℓ_{3})$
$= d_{2} (ℓ_{3}, ℓ_{1}, ℓ_{2}) = d_{2} (ℓ_{2}, ℓ_{3}, ℓ_{1}) = d_{2} (ℓ_{3}, ℓ_{2}, ℓ_{1})$ ; and
(d4): $d_{2} (ℓ_{1}, ℓ_{2}, ℓ_{3}) \leq d_{2} (ℓ_{1}, ℓ_{2}, ℓ) + d_{2} (ℓ_{1}, ℓ, ℓ_{3}) + d_{2} (ℓ, ℓ_{2}, ℓ_{3})$ , for all $ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ \in P$ ,

then

d_{2}

is called a 2-metric, and

(P, d_{2})

is then called a 2-metric space.

The distance

d_{2} (ℓ_{1}, ℓ_{2}, ℓ_{3})

represents the area of a triangle formed by the points

ℓ_{1}, ℓ_{2}, ℓ_{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 aD-metric space in [30] as follows:

Definition 5

([30]).Let

P \neq Ø

, and if a mapping

D : P \times P \times P \mapsto [0, \infty)

satisfies the following axioms:

(D1): $D (ℓ_{1}, ℓ_{2}, ℓ_{3}) \geq 0$ and $D (ℓ_{1}, ℓ_{2}, ℓ_{3}) = 0$ if and only if $ℓ_{1} = ℓ_{2} = ℓ_{3}$ ;
(D2): $D (ℓ_{1}, ℓ_{2}, ℓ_{3}) = D (ℓ_{1}, ℓ_{3}, ℓ_{2}) = D (ℓ_{2}, ℓ_{1}, ℓ_{3}) = D (ℓ_{3}, ℓ_{1}, ℓ_{2}) = D (ℓ_{2}, ℓ_{3}, ℓ_{1}) = D (ℓ_{3}, ℓ_{2}, ℓ_{1})$ ; and
(D3): $D (ℓ_{1}, ℓ_{2}, ℓ_{3}) \leq D (ℓ_{1}, ℓ_{2}, ℓ) + D (ℓ_{1}, ℓ, ℓ_{3}) + D (ℓ, ℓ_{2}, ℓ_{3})$ , for all $ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ \in P$ ,

then

(P, D)

denotes the D-metric space.

Mustafa and Sims [31,32] (2006) introducedG-metric space as a generalization ofD-metric space. The deficiency of Dhage’s theory ofD-metric is thus corrected. Moreover, Sedghi et al. [33] (2007) observed that the condition

(D 1)

can be modified as follows:

Definition 6

([33]).Let

P \neq Ø

, and if a mapping

D^{*} : P \times P \times P \mapsto [0, \infty)

satisfies the following axioms:

(D’1): $D^{*} (ℓ_{1}, ℓ_{2}, ℓ_{3}) = 0$ if and only if $ℓ_{1} = ℓ_{2} = ℓ_{3}$ ;
(D’2): $D^{*} (ℓ_{1}, ℓ_{2}, ℓ_{2}) \geq 0$ ;
(D’3): $D^{*} (ℓ_{1}, ℓ_{2}, ℓ_{3}) = D^{*} (ℓ_{1}, ℓ_{3}, ℓ_{2}) = D^{*} (ℓ_{2}, ℓ_{1}, ℓ_{3}) = D^{*} (ℓ_{3}, ℓ_{1}, ℓ_{2}) = D^{*} (ℓ_{2}, ℓ_{3}, ℓ_{1}) = D^{*} (ℓ_{3}, ℓ_{2}, ℓ_{1})$ ; and
(D’4): $D^{*} (ℓ_{1}, ℓ_{2}, ℓ_{3}) \leq D^{*} (ℓ_{1}, ℓ_{2}, ℓ) + D^{*} (ℓ, ℓ_{3}, ℓ_{3})$ ,

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 anS-metric space. Abbas et al. [35] (2015), introduced the idea of

A

-metric spaces as follows:

Definition 7

([35]).Let

P \neq Ø

, and if a mapping

d_{P} : P^{n} \mapsto [0, \infty)

satisfies the following axioms: (for all

ℓ_{i}, ℓ \in P

i = 1, 2, \dots \dots . n

)

(A1): $d_{P} (ℓ_{1}, ℓ_{2}, \dots \dots, ℓ_{n - 1}, ℓ_{n}) \geq 0$ ;
(A2): $d_{P} (ℓ_{1}, ℓ_{2}, \dots \dots, ℓ_{n - 1}, ℓ_{n}) = 0$ if and only if $ℓ_{1} = ℓ_{2} = \dots \dots = ℓ_{n - 1} = ℓ_{n}$ ; and
(A3): $d_{P} (ℓ_{1}, ℓ_{2}, \dots \dots, ℓ_{n - 1}, ℓ_{n}) \leq d_{P} (ζ_{ℓ_{1}}^{ℓ}) + d_{P} (ζ_{ℓ_{2}}^{ℓ}) + \dots \dots + d_{P} (ζ_{ℓ_{n - 1}}^{ℓ}) + d_{P} (ζ_{ℓ_{n}}^{ℓ})$ ,

then

(P, d_{P})

is called a

A

-metric space.

One can check that

A

-metric space is ann-dimensionalS-metric space. We can have a metricd and aS-metric by assuming

n = 2

and

n = 3

, respectively, in an

A

-metric space.

Example 1

([35]).Let

P = R

, and define

d_{P} : P^{n} \mapsto [0, \infty)

d_{P} (ℓ_{1}, ℓ_{2}, \dots, ℓ_{n - 1}, ℓ_{n}) = \sum_{ϖ = 1}^{n} \sum_{ϖ < ℓ} | ℓ_{ϖ} - ℓ_{ℓ} | .

Then,

(P, d_{P})

is an

A

-metric space.

Example 2.

Let

P = R

, and define

d_{P} : P^{n} \mapsto [0, \infty)

\begin{matrix} d_{P} (ℓ_{1}, ℓ_{2}, \dots \dots, ℓ_{n - 1}, ℓ_{n}) & = & | ℓ_{n} + ℓ_{n - 1} + \dots \dots + ℓ_{2} - (n - 1) ℓ_{1} | \\ + & | ℓ_{n} + ℓ_{n - 1} + \dots \dots + ℓ_{3} - (n - 2) ℓ_{2} | \\ ⋮ \\ + & | ℓ_{n} + ℓ_{n - 1} + ℓ_{n - 2} - 3 ℓ_{n - 3} | \\ + & | ℓ_{n} + ℓ_{n - 1} - 2 ℓ_{n - 2} | \\ + & | ℓ_{n} - ℓ_{n - 1} | . \end{matrix}

Then,

(P, d_{A})

is an

A

-metric space. For more details on

A

-metric spaces, see [35].

Fernandez et al. [36] (2017) introduced the concept of a cone

A

-metric space. They also established some new results in a cone

A

-metric space.

Definition 8

([36]).Let

P \neq Ø

, and if an operator

d_{c A} : P^{n} \mapsto E

satisfies the following axioms: (for all

ℓ_{i}, ℓ \in P

i = 1, 2, \dots \dots . n

)

(d1): $0 ⪯_{ϵ} d_{c A} (ℓ_{1}, ℓ_{2}, \dots \dots, ℓ_{n - 1}, ℓ_{n})$ ;
(d2): $d_{c A} (ℓ_{1}, ℓ_{2}, \dots \dots, ℓ_{n - 1}, ℓ_{n}) = 0$ if and only if $ℓ_{1} = ℓ_{2} = \dots \dots = ℓ_{n - 1} = ℓ_{n}$ ; and
(d3): $d_{c A} (ℓ_{1}, ℓ_{2}, \dots \dots, ℓ_{n - 1}, ℓ_{n}) ⪯_{ϵ} d_{c A} (ζ_{ℓ_{1}}^{ℓ}) + d_{c A} (ζ_{ℓ_{2}}^{ℓ}) + \dots \dots \dots + d_{c A} (ζ_{ℓ_{n - 1}}^{ℓ}) + d_{c A} (ζ_{ℓ_{n}}^{ℓ})$ .

then

(P, d_{c A})

is called a cone

A

-metric space.

A cone metric space is a special case of a cone

A

-metric space with

n = 2

. One can easily check that

d_{c A} (ζ_{x}^{y}) = d_{c A} (ζ_{y}^{x})

for a cone

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 = R

and

C = {ℓ \in E : ℓ (t) \geq 0, t \in [a, b]}

and

d_{c A} : P^{n} \mapsto E

\begin{matrix} d_{c A} (ℓ_{1}, ℓ_{2}, \dots, ℓ_{n - 1}, ℓ_{n}) (t) = \sum_{α = 1}^{n} \sum_{α < ϖ} | ℓ_{α} - ℓ_{ϖ} | e^{t} . \end{matrix}

Then,

(P, d_{c A})

is a cone

A

-metric space.

Definition 9.

Let E be a real Banach space, and let

(P, d_{c A})

be a cone

A

-metric space along with

ε_{A} \in E

such that

0 ≪_{ϵ} ε_{A}

. A sequence

{ℓ_{n}}

is a Cauchy sequence if we can find a

n_{1} \in N

so that

d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}}) ≪_{ϵ} ε_{A}

∀

n, m > n_{1}

. A sequence

{ℓ_{n}}

is convergent if we can find a

n_{1} \in N

so that

d_{c A} (ζ_{ℓ_{n}}^{ℓ}) ≪_{ϵ} ε_{A}

∀

n \geq n_{1}

and

ℓ \in P

. A cone

A

-metric

(P, d_{c A})

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

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

S : E \to E

such that

{∥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:

$(ϝ_{1})$: if $η_{5} ⪯_{ϵ} ℓ_{5}$ , $η_{6} ⪯_{ϵ} ℓ_{6}$
then $ϝ (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}) ⪯_{ϵ} ϝ (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, η_{5}, η_{6})$ ,
$(ϝ_{2})$: if either
$ϝ (η, ℓ, ℓ, η, η + (n - 1) ℓ, 0) ⪯_{ϵ} 0$
or
$ϝ (η, ℓ, η, ℓ, 0, η + (n - 1) ℓ) ⪯_{ϵ} 0,$
then there exists $S \in B (E, E)$ such that $η ⪯_{ϵ} \frac{1}{n - 1} S (ℓ)$ , for all $η, ℓ \in E$ .
$(ϝ_{3})$: $ϝ (η, 0, 0, η, η, 0) ≻ 0$ whenever $∥η∥ > 0$ .

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

S \in B (E, E)

, then

S^{n} \in B (E, E)

. It is known from operator theory that, if

S \in B (E, E)

verifying

{∥S∥}_{1} < 1

, then

{(I - S)}^{- 1}

can be represented by a convergent series. As a result, the operator

S^{n} {(I - S)}^{- 1}

is bounded.

Let

H = {ϝ : E^{6} \to E | ϝ satisfies the conditions ϝ_{1}, ϝ_{2}, ϝ_{3}}

Example 4.

Let

⪯_{ϵ}

be a partial ordering on a cone

C

. Assume that the pair

(E, ∥.∥)

is a real Banach space and for

n > 1

ϝ : E^{6} \to E

be defined by

ϝ (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}) = ℓ_{1} - \frac{ϖ}{n - 1} max {ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}} for all ℓ_{i} \in E (i = 1 to 6) and 0 \leq ϖ < \frac{1}{2} .

Then, the operator

ϝ \in H

(ϝ_{1}) .

Let

ℓ_{5} ⪯_{ϵ} γ_{5}

and

ℓ_{6} ⪯_{ϵ} γ_{6}

, then

γ_{5} - ℓ_{5} \in C

and

γ_{6} - ℓ_{6} \in C

. Now, we show that

ϝ (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}) - ϝ (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, γ_{5}, γ_{6}) \in C

. Consider,

\begin{matrix} ϝ (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}) - ϝ (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, γ_{5}, γ_{6}) \\ = & ℓ_{1} - \frac{α}{n - 1} max {ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}} - (ℓ_{1} - \frac{ϖ}{n - 1} max {ℓ_{2}, ℓ_{3}, ℓ_{4}, γ_{5}, γ_{6}}) \\ = & \frac{ϖ}{n - 1} max {0, 0, 0, γ_{5} - ℓ_{5}, γ_{6} - ℓ_{6}} \in C . \end{matrix}

Thus,

ϝ (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, γ_{5}, γ_{6}) ⪯_{ϵ} ϝ (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6})

(ϝ_{2}) .

Let

ℓ, γ \in E

along with

0 ⪯_{ϵ} ℓ, 0 ⪯_{ϵ} γ

. Let

ϝ (γ, ℓ, ℓ, γ, γ + (n - 1) ℓ, 0) ⪯_{ϵ} 0

, we obtain

- γ + \frac{ϖ}{n - 1} max {ℓ, ℓ, γ, γ + (n - 1) ℓ, 0} \in C .

Take

γ = 0

, and then

ϖ ℓ \in C

. Thus, ∃

S : E \to E

taken as

S (ℓ) = η ℓ

(

0 \leq η < \frac{1}{2}

and

η = ϖ

is a constant) so that

γ ⪯_{ϵ} S (ℓ)

. If

γ \neq 0

, we obtain,

- γ + \frac{ϖ}{n - 1} max {ℓ, ℓ, γ, γ + (n - 1) ℓ, 0} \in C

⇒

ϖ (γ + ℓ) - γ \in C

further implies

ϖ ℓ - (1 - ϖ) γ \in C

. Hence,

(1 - ϖ) γ ⪯_{ϵ} ϖ ℓ

; thus, ∃

S : E \to E

taken as

S (ℓ) = η ℓ

(

η = \frac{ϖ}{1 - ϖ}

represent constant) so that

γ ⪯_{ϵ} S (ℓ)

(ϝ_{3}) .

Consider

ℓ \in E

be such that

∥ℓ∥ > 0

and

0 ⪯_{ϵ} ϝ (ℓ, 0, 0, ℓ, ℓ, 0)

, and then

ℓ - \frac{ϖ}{n - 1} max {0, 0, ℓ, ℓ, 0} \in C .

As a result, we obtain

\frac{ϖ}{n - 1} ℓ ⪯_{ϵ} 0

, which is true when

∥ℓ∥ > 0

Let

ϝ_{i} : E^{6} \to E

(i = 1, 2, 3)

be defined by

(i): $ϝ_{1} (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}) = ℓ_{1} - \frac{ϖ}{n - 1} ℓ_{2}; 0 \leq ϖ < 1 .$
(ii): $ϝ_{2} (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}) = ℓ_{1} - \frac{ϖ_{1}}{n - 1} ℓ_{2} - \frac{ϖ_{2}}{n - 1} ℓ_{3} - \frac{ϖ_{3}}{n - 1} ℓ_{4}, ϖ_{1}, ϖ_{2}, ϖ_{3} \geq 0 : \frac{ϖ_{1} + ϖ_{2} + ϖ_{3}}{n - 1} < 1$ .
(iii): $ϝ_{3} (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}) = ϖ ℓ_{1} - ℓ_{2}; ϖ > 1 .$

Then, each

ϝ_{i} \in 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, d_{c A})

so that

g (j^{*}) = j^{*}

” where

g : P \to P

is a self-mapping satisfying (2), for all comparable elements

η, κ \in P

(I - S) (d_{c A} (ζ_{η}^{g (η)}) ⪯_{ϵ} d_{c A} (ζ_{η}^{κ}) implies

ϝ (\begin{matrix} d_{c A} (ζ_{g (η)}^{g (κ)}), d_{c A} (ζ_{η}^{κ}), d_{c A} (ζ_{η}^{g (η)}), d_{c A} (ζ_{κ}^{g (κ)}), d_{c A} (ζ_{η}^{g (κ)}), d_{c A} (ζ_{κ}^{g (η)}) \end{matrix}) ⪯_{ϵ} 0,

(2)

and

S \in B (E, E)

I : E \to E

an identity operator,

ϝ \in 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

A

-metric space as a generalization of a metric space and defined some topological structure on it [35]. Fernandez introduced the cone

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, d_{c A})

be a complete cone

A

-metric space along with a cone

C \subset E

and

g : P \to P

be a self-mapping. Let

S \in B (E, E)

so that

{∥S∥}_{1} < 1

I : E \to E

be an identity operator. If for all comparable elements

ℓ, κ \in P

ϝ \in H

, g satisfies the following condtions:

(I - S) d_{c A} (ζ_{ℓ}^{g (ℓ)}) ⪯_{ϵ} d_{c A} (ζ_{ℓ}^{κ}) i m p l i e s

ϝ (\begin{matrix} d_{c A} (ζ_{g (ℓ)}^{g (κ)}), d_{c A} (ζ_{ℓ}^{κ}), d_{c A} (ζ_{ℓ}^{g (ℓ)}), d_{c A} (ζ_{κ}^{g (κ)}), d_{c A} (ζ_{ℓ}^{g (κ)}), d_{c A} (ζ_{κ}^{g (ℓ)}) \end{matrix}) ⪯_{ϵ} 0,

(3)

and

(1): there is $ℓ_{0} \in P$ such that $ℓ_{0} R g (ℓ_{0})$ ;
(2): for all $ℓ, κ \in P$ , $ℓ R κ$ implies $g (ℓ) R g (κ)$ ; and
(3): for each sequence ${ℓ_{n}}$ , with comparable sequential terms, converging to $c^{*}$ , we have $ℓ_{n} R c^{*}$ for all $n \in N$ .

Then, there exists

c^{*} \in P

such that

c^{*} = g (c^{*})

Proof.

We construct a sequence

{ℓ_{n}}

such that

g (ℓ_{n - 1}) = ℓ_{n}

. By (1), there exists

ℓ_{0} \in P

such that

ℓ_{0} R g (ℓ_{0})

, so, taking

ℓ = ℓ_{0}

in (3), we have

(I - S) d_{c A} (ζ_{ℓ_{0}}^{g (ℓ_{0})}) = (I - S) d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}) ⪯_{ϵ} d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}) implies

ϝ (\begin{matrix} d_{c A} (ζ_{g (ℓ_{0})}^{g (ℓ_{1})}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}), d_{c A} (ζ_{ℓ_{0}}^{g (ℓ_{0})}), d_{c A} (ζ_{ℓ_{1}}^{g (ℓ_{1})}), d_{c A} (ζ_{ℓ_{0}}^{g (ℓ_{1})}), d_{c A} (ζ_{ℓ_{1}}^{g (ℓ_{0})}) \end{matrix}) ⪯_{ϵ} 0,

that is,

ϝ (\begin{matrix} d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{1}}) \end{matrix}) ⪯_{ϵ} 0 .

(4)

By (d3), we have

d_{c A} (ζ_{ℓ_{0}}^{ℓ_{2}}) ⪯_{ϵ} (n - 1) d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}) + d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}),

and by rewriting (4) and applying (

ϝ_{1}

), we obtain

ϝ (\begin{matrix} d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), (n - 1) d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}) + d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), 0 \end{matrix}) ⪯_{ϵ} 0 .

(5)

(ϝ_{2})

, there exists

S \in B (E, E)

verifying

{∥S∥}_{1} < 1

and

d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}) ⪯_{ϵ} \frac{1}{n - 1} S (d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}})) .

By (2), we have

g (ℓ_{0}) R g (ℓ_{1})

, that is

ℓ_{1} R ℓ_{2}

, so, putting

ℓ = ℓ_{1}

in (3), we have

(I - S) d_{c A} (ζ_{ℓ_{1}}^{g (ℓ_{1})}) = (I - S) d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}) ⪯_{ϵ} d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}) .

This implies that

ϝ (\begin{matrix} d_{c A} (ζ_{g (ℓ_{1})}^{g (ℓ_{2})}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{1}}^{g (ℓ_{1})}), d_{c A} (ζ_{ℓ_{2}}^{g (ℓ_{2})}), d_{c A} (ζ_{ℓ_{1}}^{g (ℓ_{2})}), d_{c A} (ζ_{ℓ_{2}}^{g (ℓ_{1})}) \end{matrix}) ⪯_{ϵ} 0 .

By (d3), we have

d_{c A} (ζ_{ℓ_{1}}^{ℓ_{3}}) ⪯_{ϵ} (n - 1) d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}) + d_{c A} (ζ_{ℓ_{2}}^{ℓ_{3}}),

and

(ϝ_{1})

implies

ϝ (\begin{matrix} d_{c A} (ζ_{ℓ_{2}}^{ℓ_{3}}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{2}}^{ℓ_{3}}), (n - 1) d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}) + d_{c A} (ζ_{ℓ_{2}}^{ℓ_{3}}), 0 \end{matrix}) ⪯_{ϵ} 0 .

(6)

(ϝ_{2})

, we have

d_{c A} (ζ_{ℓ_{2}}^{ℓ_{3}}) ⪯_{ϵ} \frac{1}{n - 1} S (d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}})) ⪯_{ϵ} \frac{1}{{(n - 1)}^{2}} S^{2} (d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}})) .

The same reasoning leads to a sequence

{ℓ_{n}}

such that

ℓ_{n} R ℓ_{n + 1}

and

ℓ_{n + 1} = g (ℓ_{n})

. Furthermore, observing that

(I - S) d_{c A} (ζ_{ℓ_{n - 1}}^{g (ℓ_{n - 1})}) = (I - S) d_{c A} (ζ_{ℓ_{n - 1}}^{ℓ_{n}}) ⪯_{ϵ} d_{c A} (ζ_{ℓ_{n - 1}}^{ℓ_{n}}),

we have

d_{c A} (ζ_{ℓ_{n}}^{ℓ_{n + 1}}) ⪯_{ϵ} \frac{1}{n - 1} S (d_{c A} (ζ_{ℓ_{n - 1}}^{ℓ_{n}})) ⪯_{ϵ} \frac{1}{{(n - 1)}^{2}} S^{2} (d_{c A} (ζ_{ℓ_{n - 2}}^{ℓ_{n - 1}})) ⪯_{ϵ} \dots ⪯_{ϵ} \frac{1}{{(n - 1)}^{n}} S^{n} (d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}})) .

For

m, n \in N

such that

m > n

, we note that

\begin{matrix} d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}}) & ⪯_{ϵ} & (n - 1) [d_{c A} (ζ_{ℓ_{n}}^{ℓ_{n + 1}}) + d_{c A} (ζ_{ℓ_{n + 1}}^{ℓ_{n + 2}}) + \dots + d_{c A} (ζ_{ℓ_{m - 1}}^{ℓ_{m}})] \\ ⪯_{ϵ} & \frac{1}{{(n - 1)}^{n - 1}} [S^{n} + \frac{1}{(n - 1)} S^{n + 1} + \dots \dots + \frac{1}{{(n - 1)}^{m - n - 2}} S^{m - 1} + . .] (d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}})) \\ = & \frac{1}{{(n - 1)}^{n - 1}} {S^{n} {(I - \frac{1}{n - 1} S)}^{- 1}} d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}) . (By Remark 1) \end{matrix}

We deduce the following information from the last inequality.

∥d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}})∥ \leq \frac{1}{{(n - 1)}^{n - 1}} sup \frac{∥S^{n} {(I - \frac{1}{n - 1} S)}^{- 1} (ℓ)∥}{∥ℓ∥} ∥d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}})∥ .

By Remark 1 and the assumption that

{∥S∥}_{1} < 1

, we infer (using the facts given in [18,19]) that

{lim}_{n \to \infty} d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}}) = 0

, or equivalently,

d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}}) ≪_{ϵ} C_{A}

for some

0 ≪_{ϵ} C_{A}

. By using the technique appeared in [19], we can obtain that

d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}}) ≪_{ϵ} C_{A}

. Thus,

{ℓ_{n}}

is a Cauchy sequence inP.

(P, d_{c A})

is a complete cone

A

-metric space, which assures the existence of

c^{*} \in P

such that

ℓ_{n} \to c^{*}

n \to \infty

, or equivalently, there exists

n_{2} \in N

satisfying

d_{c A} (ζ_{ℓ_{n}}^{c^{*}}) ≪_{ϵ} c for all n \geq n_{2} .

Our concern is to show that

(I - S) d_{c A} (ζ_{ℓ_{n}}^{g (ℓ_{n})}) ⪯_{ϵ} d_{c A} (ζ_{ℓ_{n}}^{c^{*}}) .

We assume on the contrary that

(I - S) d_{c A} (ζ_{ℓ_{n}}^{g (ℓ_{n})}) ≻ d_{c A} (ζ_{ℓ_{n}}^{c^{*}})

and

(I - S) d_{c A} (ζ_{ℓ_{n + 1}}^{g (ℓ_{n + 1})}) ≻ d_{c A} (ζ_{ℓ_{n + 1}}^{c^{*}}) for some n \in N .

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

\begin{matrix} d_{c A} (ζ_{ℓ_{n}}^{g (ℓ_{n})}) & ⪯_{ϵ} & d_{c A} (ζ_{ℓ_{n}}^{c^{*}}) + (n - 1) d_{c A} (ζ_{c^{*}}^{g (ℓ_{n})}) \\ ≺ & (I - S) d_{c A} (ζ_{ℓ_{n}}^{g (ℓ_{n})}) + (n - 1) (I - S) d_{c A} (ζ_{ℓ_{n + 1}}^{g (ℓ_{n + 1})}) \\ ≺ & (I - S) d_{c A} (ζ_{ℓ_{n}}^{g (ℓ_{n})}) + (n - 1) \frac{1}{n - 1} S (I - S) d_{c A} (ζ_{ℓ_{n}}^{g (ℓ_{n})}) \\ = & (I - S^{2}) (d_{c A} (ζ_{ℓ_{n}}^{g (ℓ_{n})})) . \end{matrix}

Thus,

S^{2} (d_{c A} (ζ_{ℓ_{n}}^{g (ℓ_{n})})) ≺ 0,

which contradicts the definition of coneA-metric; therefore, for each

n \geq 1

, we have

(I - S) (d_{c A} (ζ_{ℓ_{n}}^{g (ℓ_{n})})) ⪯_{ϵ} d_{c A} (ζ_{ℓ_{n}}^{c^{*}}) .

By (3), it implies that

ϝ (\begin{matrix} d_{c A} (ζ_{g (ℓ_{n})}^{g (c^{*})}), d_{c A} (ζ_{ℓ_{n}}^{c^{*}}), d_{c A} (ζ_{ℓ_{n}}^{g (ℓ_{n})}), d_{c A} (ζ_{c^{*}}^{g (c^{*})}), d_{c A} (ζ_{ℓ_{n}}^{g (c^{*})}), d_{c A} (ζ_{c^{*}}^{g (ℓ_{n})}) \end{matrix}) ⪯_{ϵ} 0

ϝ (\begin{matrix} d_{c A} (ζ_{ℓ_{n + 1}}^{g (c^{*})}), d_{c A} (ζ_{ℓ_{n}}^{c^{*}}), d_{c A} (ζ_{ℓ_{n}}^{ℓ_{n + 1}}), d_{c A} (ζ_{c^{*}}^{g (c^{*})}), d_{c A} (ζ_{ℓ_{n}}^{g (c^{*})}), d_{c A} (ζ_{c^{*}}^{ℓ_{n + 1}}) \end{matrix}) ⪯_{ϵ} 0 .

(7)

We claim that

∥d_{c A} (ζ_{c^{*}}^{g (c^{*})})∥ = 0

. On the contrary, suppose that

∥d_{c A} (ζ_{c^{*}}^{g (c^{*})})∥ > 0

, and letting

n \to \infty

in (7), we have

ϝ (d_{c A} (ζ_{c^{*}}^{g (c^{*})}), 0, 0, d_{c A} (ζ_{c^{*}}^{g (c^{*})}), d_{c A} (ζ_{c^{*}}^{g (c^{*})}), 0) ⪯_{ϵ} 0 .

This is a contradiction to

(ϝ_{3})

. Thus,

∥d_{c A} (ζ_{c^{*}}^{g (c^{*})})∥ = 0

. Hence

d_{c A} (ζ_{c^{*}}^{g (c^{*})}) = 0

. By (d1), we have

c^{*} = g (c^{*})

. This completes the proof. □

Remark 2.

If the operator

ϝ : E^{6} \to E

is taken as

ϝ (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}) = ℓ_{1} - φ (max \{ℓ_{2}, ℓ_{3}, ℓ_{4}, \frac{1}{2} (ℓ_{5} + ℓ_{6})\}), f o r a l l ℓ_{i} \in E

Here,

φ : E \to E

is taken as non-decreasing mapping with

{lim}_{n \to \infty} φ^{n} (v) = 0_{E}

. Then, Theorem (2), generalizes the corresponding result in [3]. Taking

ϝ : E^{6} \to E

ϝ (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}) = ℓ_{1} - \frac{k}{n - 1} ℓ_{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, d_{c A})

be a complete cone

A

-metric space, and let

C \subset E

be a cone. Let

S \in B (E, E)

so that

{∥S∥}_{1} < 1

, and let

I : E \to E

be an identity operator. If the mapping

g : P \to P

, for all comparable elements

ℓ, κ \in P

, and

ϝ \in H

satisfies the following condition

(I - S) (d_{c A} (ζ_{ℓ}^{g (ℓ)}) ⪯_{ϵ} d_{c A} (ζ_{ℓ}^{κ}) i m p l i e s

ϝ (\begin{matrix} d_{c A} (ζ_{g (ℓ)}^{g (κ)}), d_{c A} (ζ_{ℓ}^{κ}), d_{c A} (ζ_{ℓ}^{g (ℓ)}), d_{c A} (ζ_{κ}^{g (κ)}), d_{c A} (ζ_{ℓ}^{g (κ)}), d_{c A} (ζ_{κ}^{g (ℓ)}) \end{matrix}) ⪯_{ϵ} 0,

(8)

and

(1): there exists $ℓ_{0} \in P$ such that $g (ℓ_{0}) R ℓ_{0}$ ;
(2): for all $ℓ, κ \in P$ , $ℓ R κ$ we find $g (κ) R g (ℓ)$ ; and
(3): for a sequence ${ℓ_{n}}$ with all comparable sequential terms such that $ℓ_{n} \to c^{*}$ , we have $ℓ_{n} R c^{*}$ ∀ $n \in N$ .

Then, there exists

c^{*} \in P

such that

c^{*} = g (c^{*})

Proof.

Let

ℓ_{0} \in P

and construct a sequence

{ℓ_{n}}

ℓ_{n} = g (ℓ_{n - 1})

for all

n \in N

. Using condition (1), we have

ℓ_{1} = g (ℓ_{0}) R ℓ_{0}

. By (2), we have

g (ℓ_{0}) R g (ℓ_{1})

, i.e.,

ℓ_{1} R ℓ_{2}

. Using (8), we obtain

(I - S) (d_{c A} (ζ_{g (ℓ_{0})}^{ℓ_{0}})) = (I - S) (d_{c A} (ζ_{ℓ_{1}}^{ℓ_{0}})) ⪯_{ϵ} d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}) implies

ϝ (\begin{matrix} d_{c A} (ζ_{g (ℓ_{1})}^{g (ℓ_{0})}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{0}}), d_{c A} (ζ_{ℓ_{1}}^{g (ℓ_{1})}), d_{c A} (ζ_{ℓ_{0}}^{g (ℓ_{0})}), d_{c A} (ζ_{ℓ_{1}}^{g (ℓ_{0})}), d_{c A} (ζ_{ℓ_{0}}^{g (ℓ_{1})}) \end{matrix}) ⪯_{ϵ} 0,

that is,

ϝ (\begin{matrix} d_{c A} (ζ_{ℓ_{2}}^{ℓ_{1}}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{0}}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{1}}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{2}}) \end{matrix}) ⪯_{ϵ} 0 .

(9)

Now, (d3) implies

d_{c A} (ζ_{ℓ_{0}}^{ℓ_{2}}) ⪯_{ϵ} (n - 1) d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}) + d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}})

and then using

(ϝ_{1})

, we obtain

ϝ (\begin{matrix} d_{c A} (ζ_{ℓ_{2}}^{ℓ_{1}}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{0}}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}), 0, (n - 1) d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}) + d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}) \end{matrix}) ⪯_{ϵ} 0 .

(10)

(ϝ_{2})

, we have

d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}) ⪯_{ϵ} \frac{1}{n - 1} S (d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}})) .

ℓ_{1} R ℓ_{2}

; thus, the use of (2) implies

ℓ_{3} R ℓ_{2}

. Applying (8)

(I - S) (d_{c A} (ζ_{ℓ_{1}}^{g (ℓ_{1})})) = (I - S) (d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}})) ⪯_{ϵ} d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}) implies

ϝ (\begin{matrix} d_{c A} (ζ_{g (ℓ_{1})}^{g (ℓ_{2})}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{1}}^{g (ℓ_{1})}), d_{c A} (ζ_{ℓ_{2}}^{g (ℓ_{2})}), d_{c A} (ζ_{ℓ_{1}}^{g (ℓ_{2})}), d_{c A} (ζ_{ℓ_{2}}^{g (ℓ_{1})}) \end{matrix}) ⪯_{ϵ} 0,

By (d3),

(ϝ_{1})

and

(ϝ_{2})

, we obtain

d_{c A} (ζ_{ℓ_{2}}^{ℓ_{3}}) ⪯_{ϵ} \frac{1}{(n - 1)} S (d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}})) .

By following above procedure, we have a sequence

{ℓ_{n}}

so that

d_{c A} (ζ_{ℓ_{n}}^{ℓ_{n + 1}}) ⪯_{ϵ} \frac{1}{n - 1} S (d_{c A} (ζ_{ℓ_{n - 1}}^{ℓ_{n}})) ⪯_{ϵ} \frac{1}{{(n - 1)}^{2}} S^{2} (d_{c A} (ζ_{ℓ_{n - 2}}^{ℓ_{n - 1}})) ⪯_{ϵ} \dots ⪯_{ϵ} \frac{1}{{(n - 1)}^{n}} S^{n} (d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}})) .

Using the same reasoning as in Theorem 2 leads to

c^{*} = g (c^{*})

. □

4.3. Result for Monotone Self-Mapping

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

Theorem 4.

Let

(P, d_{c A})

be a complete cone

A

-metric space and

C \subset E

be a cone. Let

S \in B (E, E)

so that

{∥S∥}_{1} < 1

, and

I : E \to E

is an identity operator. If the mapping

g : P \to P

, for all comparable elements

ℓ, κ \in P

, and

ϝ \in H

satisfies the following condition

(I - S) (d_{c A} (ζ_{ℓ}^{g (ℓ)}) ⪯_{ϵ} d_{c A} (ζ_{ℓ}^{κ}) i m p l i e s

ϝ (\begin{matrix} d_{c A} (ζ_{g (ℓ)}^{g (κ)}), d_{c A} (ζ_{ℓ}^{κ}), d_{c A} (ζ_{ℓ}^{g (ℓ)}), d_{c A} (ζ_{κ}^{g (κ)}), d_{c A} (ζ_{ℓ}^{g (κ)}), d_{c A} (ζ_{κ}^{g (ℓ)}) \end{matrix}) ⪯_{ϵ} 0

(11)

and

(1): there exists $ℓ_{0} \in P$ such that $ℓ_{0} R g (ℓ_{0})$ or $g (ℓ_{0}) R ℓ_{0}$ ;
(2): the mapping g is monotone; and
(3): for a sequence ${ℓ_{n}}$ with all comparable sequential terms such that $ℓ_{n} \to c^{*}$ , we have $ℓ_{n} R c^{*}$ ∀ $n \in N$ .

Then, there exists

c^{*} \in P

such that

c^{*} = g (c^{*})

Proof.

Proceeding with an initial guess

ℓ_{0} \in P

, we construct a sequence

{ℓ_{n}}

so that

ℓ_{n} = g (ℓ_{n - 1})

∀

n \in N

. By assumption (1), we have

ℓ_{0} R g (ℓ_{0}) = ℓ_{1}

and by (2),

ℓ_{2} R ℓ_{1}

. Using (11), we obtain

(I - S) (d_{c A} (ζ_{g (ℓ_{0})}^{ℓ_{0}})) = (I - S) (d_{c A} (ζ_{ℓ_{1}}^{ℓ_{0}})) ⪯_{ϵ} d_{c A} (ζ_{ℓ_{1}}^{ℓ_{0}}) implies

ϝ (\begin{matrix} d_{c A} (ζ_{g (ℓ_{0})}^{g (ℓ_{1})}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}), d_{c A} (ζ_{ℓ_{0}}^{g (ℓ_{0})}), d_{c A} (ζ_{ℓ_{1}}^{g (ℓ_{1})}), d_{c A} (ζ_{ℓ_{0}}^{g (ℓ_{1})}), d_{c A} (ζ_{ℓ_{1}}^{g (ℓ_{0})}) \end{matrix}) ⪯_{ϵ} 0,

that is,

ϝ (\begin{matrix} d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{1}}) \end{matrix}) ⪯_{ϵ} 0 .

(12)

By (d3) and

(ϝ_{1})

ϝ (\begin{matrix} d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}), d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), (n - 1) d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}) + d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), 0) \end{matrix}) ⪯_{ϵ} 0 .

(13)

Using

(ϝ_{2})

, we have

d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}) ⪯_{ϵ} \frac{1}{n - 1} S (d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}})) .

Since

ℓ_{2} R ℓ_{1}

, by (11), we have

(I - S) (d_{c A} (ζ_{g (ℓ_{0})}^{g (ℓ_{1})})) = (I - S) (d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}})) ⪯_{ϵ} d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}) implies

ϝ (\begin{matrix} d_{c A} (ζ_{g (ℓ_{1})}^{g (ℓ_{2})}), d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}), d_{c A} (ζ_{ℓ_{1}}^{g (ℓ_{1})}), d_{c A} (ζ_{ℓ_{2}}^{g (ℓ_{2})}), d_{c A} (ζ_{ℓ_{1}}^{g (ℓ_{2})}), d_{c A} (ζ_{ℓ_{2}}^{g (ℓ_{1})}) \end{matrix}) ⪯_{ϵ} 0 .

Using (d3), we have

d_{c A} (ζ_{ℓ_{1}}^{ℓ_{3}}) ⪯_{ϵ} (n - 1) d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}}) + d_{c A} (ζ_{ℓ_{2}}^{ℓ_{3}}) .

(ϝ_{1})

and

(ϝ_{2})

, we have

d_{c A} (ζ_{ℓ_{2}}^{ℓ_{3}}) ⪯_{ϵ} \frac{1}{n - 1} S (d_{c A} (ζ_{ℓ_{1}}^{ℓ_{2}})) ⪯_{ϵ} \frac{1}{{(n - 1)}^{2}} S^{2} (d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}}) .

By following same way, we construct a sequence

{ℓ_{n}}

such that

d_{c A} (ζ_{ℓ_{n}}^{ℓ_{n + 1}}) ⪯_{ϵ} \frac{1}{n - 1} S (d_{c A} (ζ_{ℓ_{n - 1}}^{ℓ_{n}})) ⪯_{ϵ} \frac{1}{{(n - 1)}^{2}} S^{2} (d_{c A} (ζ_{ℓ_{n - 2}}^{ℓ_{n - 1}})) ⪯_{ϵ} \dots ⪯_{ϵ} \frac{1}{{(n - 1)}^{n}} S^{n} (d_{c A} (ζ_{ℓ_{0}}^{ℓ_{1}})) .

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

ℓ, κ \in P

(2). For a normal cone, one can replace

ϝ_{3}

ϝ (ℓ, ℓ_{1}, ℓ_{2}, ϖ, ϖ + (n - 1) ℓ_{1}, ϖ + (n - 1) ℓ_{2}) ⪯_{ϵ} 0

∀,

ℓ ≪_{ϵ} ε_{A}

ℓ_{1} ≪_{ϵ} ε_{A}

ℓ_{2} ≪_{ϵ} ε_{A}

and

ϖ ≪_{ϵ} ε_{A}

5. Examples and Consequences

We elaborate our results through the following examples.

Example 5.

Let

E = (R^{,} ∥ \cdot ∥)

as a real Banach space, along with a cone

C = {ℓ \in R : ℓ \geq 0}

in E. Define the cone

A

-metric by

d_{c A} (ℓ_{1}, ℓ_{2}, \dots \dots, ℓ_{n}) = \sum_{i = 1}^{n} \sum_{i < j} | ℓ_{i} - ℓ_{j} |

. Let

P = {0, \frac{1}{6}} \subset E

and

g : P \to P

be given by

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

Then, the mapping g is monotone with respect to partial order

⪯_{ϵ}

. Let

S (ℓ) = \frac{ℓ}{3}

, then

∥ S ∥ < 1

and hence

S \in B (E, E)

. Furthermore,

S (C) \subset C

. Now, if

ℓ = 0

and

κ = \frac{1}{6}

, then

ℓ ⪯_{ϵ} κ

d_{c A} (ζ_{ℓ}^{g (ℓ)}) = \frac{(n - 1)}{6}, d_{c A} (ζ_{κ}^{g (κ)}) = 0, d_{c A} (ζ_{κ}^{g (ℓ)}) = 0, d_{c A} (ζ_{g (ℓ)}^{g (κ)}) = 0 .

\begin{matrix} S (d_{c A} (ζ_{ℓ}^{g (ℓ)}) & = & \frac{d_{c A} (ζ_{ℓ}^{g (ℓ)})}{3} = \frac{(n - 1)}{18}, \\ (I - S) (d_{c A} (ζ_{ℓ}^{g (ℓ)}) & = & \frac{(n - 1)}{9} . \end{matrix}

\begin{matrix} \frac{α}{n - 1} max {d_{c A} (ζ_{ℓ}^{κ})), d_{c A} (ζ_{ℓ}^{g (ℓ)}, d_{c A} (ζ_{κ}^{g (κ)}), d_{c A} (ζ_{ℓ}^{g (κ)}), d_{c A} (ζ_{κ}^{g (ℓ)})} \\ = & \frac{α}{n - 1} max \{\frac{(n - 1)}{6}, \frac{(n - 1)}{6}, 0\} \\ = & \frac{α}{6} . \end{matrix}

Thus, for every

α \in [0, \frac{1}{2})

(I - S) (d_{c A} (ζ_{ℓ}^{g (ℓ)}) ⪯_{ϵ} d_{c A} (ζ_{ℓ}^{κ})

implies

d_{c A} (ζ_{g (ℓ)}^{g (κ)} ⪯_{ϵ} \frac{α}{n - 1} max {d_{c A} (ζ_{ℓ}^{κ})), d_{c A} (ζ_{ℓ}^{g (ℓ)}, d_{c A} (ζ_{κ}^{g (κ)}), d_{c A} (ζ_{ℓ}^{g (κ)}), d_{c A} (ζ_{κ}^{g (ℓ)})} .

Now,

\begin{matrix} ϝ (\begin{matrix} d_{c A} (ζ_{g (ℓ)}^{g (κ)}), d_{c A} (ζ_{ℓ}^{κ})), d_{c A} (ζ_{ℓ}^{g (ℓ)}, d_{c A} (ζ_{κ}^{g (κ)}), d_{c A} (ζ_{ℓ}^{g (κ)}), d_{c A} (ζ_{κ}^{g (ℓ)}) \end{matrix}) \\ = & d_{c A} (ζ_{g (ℓ)}^{g (κ)}) - \frac{α}{n - 1} max {d_{c A} (ζ_{ℓ}^{κ})), d_{c A} (ζ_{ℓ}^{g (ℓ)}, d_{c A} (ζ_{κ}^{g (κ)}), d_{c A} (ζ_{ℓ}^{g (κ)}), d_{c A} (ζ_{κ}^{g (ℓ)})} . \end{matrix}

Then,

(I - S) (A (ζ_{ℓ}^{g (ℓ)})) ⪯_{ϵ} A (ζ_{ℓ}^{κ})

implies

ϝ (\begin{matrix} d_{c A} (ζ_{g (ℓ)}^{g (κ)}), d_{c A} (ζ_{ℓ}^{κ})), d_{c A} (ζ_{ℓ}^{g (ℓ)}, d_{c A} (ζ_{κ}^{g (κ)}), d_{c A} (ζ_{ℓ}^{g (κ)}), d_{c A} (ζ_{κ}^{g (ℓ)}) \end{matrix}) ⪯_{ϵ} 0 .

Use of Theorem 2 leads to have a fixed point of g, such that

g (\frac{1}{6}) = \frac{1}{6}

Corollary 1.

Let

(P, d_{c A})

be a complete cone

A

-metric space. Let

S \in B (E, E)

with

{∥S∥}_{1} < 1

and

I : E \to E

be an identity operator. If the mapping

g : P \to P

, for all comparable elements

ℓ, κ \in P

, and

ϝ \in H

satisfies the following condition

(I - S) (d_{c A} (ζ_{ℓ}^{g (ℓ)})) ⪯_{ϵ} d_{c A} (ζ_{ℓ}^{κ}) \Rightarrow d_{c A} (ζ_{g (ℓ)}^{g (κ)}) ⪯_{ϵ} S (d_{c A} (ζ_{ℓ}^{κ})),

and

(1): there exists $ℓ_{0} \in P$ such that either $ℓ_{0} R g (ℓ_{0})$ or $g (ℓ_{0}) R ℓ_{0}$ ;
(2): for all $ℓ, κ \in P$ , $ℓ R κ$ we have $g (ℓ) R g (κ)$ or $g (κ) R g (ℓ)$ ; and
(3): for a sequence ${ℓ_{n}}$ with $ℓ_{n} \to c^{*}$ whose all sequential terms are comparable, we have $ℓ_{n} R c^{*}$ for all $n \in N$ .

Then, there exists

c^{*} \in P

such that

c^{*} = g (c^{*})

Proof.

Define

ϝ : E^{6} \to E

ϝ (ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}) = ϖ ℓ_{1} - ℓ_{2}; ϖ > 1,

and the operator

S : E \to E

S (ℓ) = h ℓ

for all

ℓ \in E

;

h < 1

, so that

S \in B (E, E)

. Now, by Theorem 4, there exists

c^{*} \in P

such that

c^{*} = g (c^{*})

. □

Corollary 2.

Let

(P, d_{c A})

be a complete cone

A

-metric space. Let

S \in B (E, E)

with

{∥S∥}_{1} < 1

and

I : E \to E

be an identity operator. If the mapping

g : P \to P

, for all comparable elements

ℓ, κ \in P

, and

ϝ \in H

satisfies the following condition

(I - S) (d_{c A} (ζ_{ℓ}^{g (ℓ)})) ⪯_{ϵ} d_{c A} (ζ_{ℓ}^{κ}) \Rightarrow d_{c A} (ζ_{g (ℓ)}^{g (κ)}) ⪯_{ϵ} \frac{1}{n - 1} S (d_{c A} (ζ_{ℓ}^{κ})),

and

(1): there exists $ℓ_{0} \in P$ verifying either $ℓ_{0} R g (ℓ_{0})$ or $g (ℓ_{0}) R ℓ_{0}$ ;
(2): for all $ℓ, κ \in P$ , $ℓ R κ$ implies either $g (ℓ) R g (κ)$ or $g (κ) R g (ℓ)$ ; and
(3): for a sequence ${ℓ_{n}}$ converging to $c^{*}$ and $ℓ_{n} R ℓ_{n - 1}$ , we have $ℓ_{n} R c^{*}$ for all $n \in N$ .

Then, there exists a vector

c^{*} \in P

such that

c^{*} = g (c^{*})

Proof.

Taking

S : E \to E

such that

S (v) = v

, ∀

v \in E

and following the proof of Corollary 1, we obtain the result. □

Example 6.

Let

E = (R^{3}, ∥ \cdot ∥)

with

∥ ℓ ∥ = max \{(| ℓ_{1} |, | ℓ_{2} |, | ℓ_{3} |)\}

, then

(E, ∥.∥)

is a real Banach space. Take a cone in E as

C = {(ℓ, ξ, ν) \in R^{3} : ℓ, ξ, ν \geq 0}

. Define the cone

A

-metric by

d_{c A} (ζ_{ℓ}^{ξ}) = (n - 1) {| ℓ_{1} - ξ_{1} |, | ℓ_{2} - ξ_{2} |, | ℓ_{3} - ξ_{3} |}

, where

ℓ = (ℓ_{1}, ℓ_{2}, ℓ_{3})

and

ξ = (ξ_{1}, ξ_{2}, ξ_{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) = (0, 0, \frac{1}{4}), g (0, 0, \frac{1}{4}) = (0, 0, \frac{1}{4}), g (0, \frac{1}{4}, 0) = (0, 0, 0),

where g is monotonic with respect to partial order

⪯_{ϵ}

. Take

S = (\begin{matrix} \frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{3} \end{matrix}) .

Define

S (ℓ) = \frac{ℓ}{3}

and

\begin{matrix} ∥ S (ℓ) ∥ & = & max \{\frac{| ℓ_{1} |}{3}, \frac{| ℓ_{2} |}{3}, \frac{| ℓ_{3} |}{3}\} \\ = & \frac{1}{3} max {| ℓ_{1} |, | ℓ_{2} |, | ℓ_{3} |} \\ = & \frac{1}{3} ∥ ℓ ∥ . \end{matrix}

Then,

∥ S ∥ = \frac{1}{3} < 1

and hence

S \in B (E, E)

. Furthermore,

S (C) \subset C

. Now, if

ℓ = (0, 0, 0)

and

κ = (0, 0, \frac{1}{4})

, then

ℓ ⪯_{ϵ} κ

. Additionally,

d_{c A} (ζ_{ℓ}^{g (ℓ)}) = (n - 1) (0, 0, \frac{1}{4}), d_{c A} (ζ_{κ}^{g (κ)}) = (0, 0, 0), d_{c A} (ζ_{κ}^{g (ℓ)}) = (0, 0, 0), d_{c A} (ζ_{g (ℓ)}^{g (κ)}) = (0, 0, 0) .

\begin{matrix} S (d_{c A} (ζ_{ℓ}^{g (ℓ)})) & = & \frac{d_{c A} (ζ_{ℓ}^{g (ℓ)})}{3} = (n - 1) (0, 0, \frac{1}{12}), \\ (I - S) (d_{c A} (ζ_{ℓ}^{g (ℓ)})) & = & (n - 1) (0, 0, \frac{1}{4}) - (n - 1) (0, 0, \frac{1}{12}) = (n - 1) (0, 0, \frac{1}{6}) . \end{matrix}

d_{c A} (ζ_{g (ℓ)}^{g (κ)}) ⪯_{ϵ} \frac{1}{n - 1} S (d_{c A} (ζ_{ℓ}^{κ}))

Then,

(I - S) (d_{c A} (ζ_{ℓ}^{g (ℓ)})) ⪯_{ϵ} d_{c A} (ζ_{ℓ}^{κ})

implies

d_{c A} (ζ_{g (ℓ)}^{g (κ)}) ⪯_{ϵ} \frac{1}{n - 1} S (d_{c A} (ζ_{ℓ}^{κ}))

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, d_{c A})

be a cone

A

-metric space over Banach algebra E. Then, for a

ℓ \in P

and

0 ≪_{ϵ} ε_{A}

, the open ball with center ℓ and radius

ε_{A}

B (ζ_{ℓ}^{ε_{A}}) = {ℓ_{1} \in P : d_{c A} (ζ_{ℓ}^{ℓ_{1}}) ≪_{ϵ} ε_{A}} .

The closed ball with center ℓ and radius

ε_{A}

\bar{B (ζ_{ℓ}^{c})} = {ℓ_{1} \in P : d_{c A} (ζ_{ℓ}^{ℓ_{1}}) ⪯_{ϵ} ε_{A}}

Definition 12.

Let

(X, d_{c A})

be a cone

A

-metric space. Then, for any set

A \subset X

and

ℓ \in X

, define

d_{c A} (ζ_{ℓ}^{A}) = inf {d_{c A} (ζ_{ℓ}^{ℓ_{1}}) : ℓ_{1} \in A} .

Theorem 5.

Let

(E, ∥.∥)

be a real Banach space and

C \subset E

be a cone. Let

(P, d_{c A})

be a complete cone

A

-metric space and

V \subset P

be an open set. Let the operator

S \in B (E, E)

be such that

{∥S∥}_{1} < 1

satisfying

S (C) \subset C

. Suppose that the mapping

l : {\bar{V}}^{n - 1} \times [0, 1] \to P

satisfies the axiom (1) of Corollary 2 in the first variable and

(1): $ℓ \neq l (ζ_{ℓ}^{ℓ_{1}})$ for each $ℓ \in \partial V$ ( $\partial V$ represents the boundary of V in P);
(2): there exists $Y \geq 0$ such that
$∥d_{c A} (ζ_{l (ζ_{ℓ}^{ℓ_{1}})}^{l (ζ_{ℓ}^{ς})})∥ \leq Y | ℓ_{1} - ς |,$
where $ℓ \in \bar{V}$ and $ℓ_{1}, ς \in [0, 1]$ ; and
(3): if r is a radius of an open ball V satisfying $∥d_{c A} (ζ_{ℓ}^{κ})∥ \leq r$ , then $ℓ R κ$ for any $ℓ, κ \in V$ .

Then, whenever

l (ζ_{(\cdot)}^{0})

possesses a fixed point in V,

l (ζ_{(\cdot)}^{1})

possesses a fixed point in V.

Proof.

Let

G = {o_{1} \in [0, 1] | ℓ = l (ζ_{ℓ}^{o_{1}}); when ℓ \in V} .

Define the partial order

⪯_{ϵ}

onE by

e ⪯_{ϵ} w

if and only if

∥ e ∥ \leq ∥ w ∥

, for all

e, w \in E

. As

l (ζ_{.}^{0})

possesses a fixed point inV, so

0 \in B_{G}

. Thus,

G \neq ϕ

, also

d_{c A} (ζ_{ℓ}^{l (ζ_{ℓ}^{o_{1}})}) = d_{c A} (ζ_{ℓ}^{κ})

(I - S) (d_{c A} (ζ_{ℓ}^{l (ζ_{ℓ}^{o_{1}})})) ⪯_{ϵ} d_{c A} (ζ_{ℓ}^{κ})

for all

ℓ R κ

, then by Corollary 1, we obtain

d_{c A} (ζ_{l (ζ_{ℓ}^{o_{1}})}^{l (ζ_{κ}^{o_{1}})}) ⪯_{ϵ} \frac{1}{n - 1} S (d_{c A} (ζ_{ℓ}^{κ})) .

Take

{o_{n}}_{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 N

, we have

ℓ_{n} \in V

so that

ℓ_{n} = l (ζ_{ℓ_{n}}^{o_{n}})

. Since,

l (ζ_{ℓ}^{(\cdot)})

is monotone, so, for

n, m \in N

, we have

ℓ_{m} R ℓ_{n}

. Since

(I - S) (d_{c A} (ζ_{ℓ_{n}}^{l (ζ_{ℓ_{m}}^{o_{m}})})) = (I - S) (d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}})) ⪯_{ϵ} d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}}),

we have

d_{c A} (ζ_{l (ζ_{ℓ_{n}}^{o_{m}})}^{l (ζ_{ℓ_{m}}^{o_{m}})}) ⪯_{ϵ} \frac{1}{n - 1} S (d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}}))

and

\begin{matrix} d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}}) & = & d_{c A} (ζ_{l (ζ_{ℓ_{n}}^{o_{n}})}^{l (ζ_{ℓ_{m}}^{o_{m}})}) \\ ⪯_{ϵ} & d_{c A} (ζ_{l (ζ_{ℓ_{n}}^{o_{n}})}^{l (ζ_{ℓ_{n}}^{o_{m}})}) + (n - 1) d_{c A} (ζ_{l (ζ_{ℓ_{n}}^{o_{m}})}^{l (ζ_{ℓ_{m}}^{θ_{m}})}) \\ ∥d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}})∥ & \leq & Y | o_{n} - o_{m} | + ∥S (d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}}))∥ \\ ∥d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}})∥ & \leq & \frac{Y}{1 - ∥S∥} | o_{n} - o_{m} | . \end{matrix}

Since

{o_{n}}_{n = 1}^{\infty}

is a Cauchy sequence in

[0, 1]

, we have

lim_{n, m \to \infty} d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}}) = 0,

and thus

d_{c A} (ζ_{ℓ_{n}}^{ℓ_{m}}) ≪_{ϵ} ε_{A}

, as

n, m \to \infty

. This implies that

{ℓ_{n}}

is a Cauchy sequence inP. AsP is a complete cone

A

-metric space, we can obtain an

ℓ \in \bar{V}

such that

{lim}_{n \to \infty} d_{c A} (ζ_{ℓ_{n}}^{ℓ}) ≪_{ϵ} ε_{A}

. Thus,

ℓ_{n} R ℓ

∀

n \in N

. Take

\begin{matrix} d_{c A} (ζ_{ℓ_{n}}^{l (ζ_{ℓ}^{o_{1}})}) & = & d_{c A} (ζ_{l (ζ_{ℓ_{n}}^{o_{n}})}^{l (ζ_{ℓ}^{o_{1}})}) \\ ⪯_{ϵ} & d_{c A} (ζ_{l (ζ_{ℓ_{n}}^{o_{n}})}^{l (ζ_{ℓ_{n}}^{o_{1}})}) + (n - 1) d_{c A} (ζ_{l (ζ_{ℓ_{n}}^{o_{1}})}^{l (ζ_{ℓ}^{o_{1}})}) \\ ∥d_{c A} (ζ_{ℓ_{n}}^{l (ζ_{ℓ}^{o_{1}})})∥ & \leq & Y | o_{n} - o_{1} | + ∥S (d_{c A} (ζ_{ℓ_{n}}^{ℓ}))∥ . \end{matrix}

This implies

lim_{n \to \infty} d_{c A} (ζ_{ℓ_{n}}^{l (ζ_{ℓ}^{o_{1}})}) = 0 .

Thus,

d_{c A} (ζ_{ℓ}^{l (ζ_{ℓ}^{o_{1}})}) = 0

. Consequently

o_{1} \in G

, and henceG is closed in

[0, 1]

Now, to show is thatG is open in

[0, 1]

. For this, take

ϑ_{1} \in G

, we have

ℓ_{1} \in V

such that

l (ζ_{ϑ_{1}}^{ℓ_{1}}) = ℓ_{1}

. AsV is open; therefore, we can have

r > 0

with

G (ζ_{ℓ_{1}}^{r}) \subseteq V

. Consider

p = d_{c A} (ζ_{ℓ_{1}}^{\partial U}) = inf {d_{c A} (ζ_{ℓ_{1}}^{ξ}) : ξ \in \partial V} .

Take

r = p > 0

by fixing

ϵ > 0

so that

ϵ < \frac{(1 - ∥S∥) p}{Y}

. Consider

o_{1} \in (ϑ_{1} - ϵ, ϑ_{1} + ϵ)

. Then,

ℓ \in \bar{B (ζ_{ℓ_{1}}^{r})} = {ℓ \in P : ∥d_{c A} (ζ_{ℓ}^{ℓ_{1}})∥ \leq r}, as ℓ R ℓ_{1} .

Consider

\begin{matrix} d_{c A} (ζ_{l (ζ_{ℓ}^{o_{1}})}^{ℓ_{1}}) & = & d_{c A} (ζ_{l (ζ_{ℓ}^{o_{1}})}^{l (ζ_{ℓ_{1}}^{ϑ_{1}})}) \\ ⪯_{ϵ} & d_{c A} (ζ_{l (ζ_{ℓ}^{o_{1}})}^{l (ζ_{ℓ}^{ϑ_{1}})}) + (n - 1) d_{c A} (ζ_{l (ζ_{ℓ}^{ϑ_{1}})}^{l (ζ_{ℓ_{1}}^{ϑ_{1}})}) \\ ∥d_{c A} (ζ_{l (ζ_{ℓ}^{o_{1}})}^{ℓ_{1}})∥ & \leq & Y | ϑ_{1} - o_{1} | + ∥S (d_{c A} (ζ_{ℓ_{1}}^{ℓ}))∥ \\ \leq & Y ϵ + ∥S∥ p < p . \end{matrix}

Hence,

l (ζ_{\cdot}^{t}) : \bar{G (ζ_{ℓ}^{r})} \to \bar{G (ζ_{ℓ}^{r})}

possesses a fixed point in

\bar{V}

for every fixed

o_{1} \in (ϑ_{1} - ϵ, ϑ_{1} + ϵ)

, application of Corollary 1 leads to the proof. The obtained fixed point will necessarily be inV as in the previous situation. Thus,

ϑ_{1} \in G

for each

ϑ_{1} \in (o_{1} - ϵ, o_{1} + ϵ)

, and thereforeG is open in

[0, 1]

. Hence,G is open as well as closed in

[0, 1]

and by connectedness,

G = [0, 1]

. Hence,

l (ζ_{\cdot}^{1})

possesses a fixed point inV. □

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:

ℓ (v) = b (v) + \int_{IR} L_{1} (v, t, ℓ (t)) d t .

(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

J_{IR}^{2} = \{ℓ | \int_{IR} {| ℓ (t) |}^{2} d t < \infty\}

. Assume

∥.∥ : J_{IR}^{2} \to [0, \infty)

with

{∥ℓ∥}_{2} = \sqrt{\int_{IR} {| ℓ (t) |}^{2} d t}, for all ℓ, j \in J_{IR}^{2} .

Similarly, take:

{∥ℓ∥}_{2, ν} = \sqrt{sup {e^{- ν \int_{IR} α (t) d t} \int_{IR} | ℓ (t) |^{2} d t}}, for all ℓ \in J_{IR}^{2}, ν > 1 .

One can observe

E = (J_{IR}^{2}, {∥.∥}_{2, ν})

represents a Banach space. Assume a cone

Z = {ℓ \in J_{IR}^{2} : ℓ (t) > 0 for almost every s}

inE. Define

c_{ν}

c_{ν} (x_{1}, x_{2}) = x_{1} {∥x_{1} - x_{2}∥}_{2, ν}^{2}

for all

x_{1}, x_{2} \in Z

. Then,

c_{ν}

represents a cone

A

-metric. Define partial order

⪯_{ϵ}

onE, such that

α ⪯_{ϵ} ϖ if and only if α (s) ϖ (s) \geq ϖ (s), for all a, ϖ \in E .

Then,

(E, ⪯_{ϵ}, c_{ν})

is a complete cone

A

-metric space. To show the existence of solutions to UIE, we need the following conditions:

(C1): The kernel $L_{1} : IR \times IR \times R \to R$ fulfils the Carathéodory axiom along with
$|L_{1} (v, t, ℓ (t))| \leq w (v, t) + e (v, t) |ℓ (t)|; w, e \in J^{2} (IR \times IR), e (v, t) > 0 .$
(C2): Assume a continuous and bounded function $b : IR \to [1, \infty)$ over IR.
(C3): There exists a constant $C > 0$ such that
$sup_{v \in IR} \int_{IR} |L_{1} (v, t)| d t \leq C .$
(C4): For any $ℓ_{0} \in J_{IR}^{2}$ , there exists $ℓ_{1} = K (ℓ_{0})$ such that $ℓ_{1} ⪯_{ϵ} ℓ_{0}$ or $ℓ_{0} ⪯_{ϵ} ℓ_{1}$ .
(C4’): Take a sequence ${ℓ_{n}}$ satisfying $ℓ_{n - 1} ⪯_{ϵ} ℓ_{n}$ such that $ℓ_{n} \to p$ , which further implies to $ℓ_{n} ⪯_{ϵ} p$ for all $n \in N$ .
(C5): We can find a non-negative and measurable function $q : IR \times IR \to R$ such that
$α (v) : = \int_{IR} q^{2} (v, s) d t \leq \frac{1}{ν}$
also integrable over $I R$ such that
$|L_{1} (v, s, ℓ (s)) - L_{1} (v, s, j (s))| \leq q (v, s) | ℓ (s) - j (s) |$

for all

v, s \in IR

and

ℓ, j \in E

such that

ℓ ⪯_{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

(K ℓ) (v) = b (v) + \int_{IR} L_{1} (v, t, ℓ (t)) d t,

K is ⪯-preserving:

Let

ℓ, j \in E

along with

ℓ ⪯ j

, so that

ℓ (s) j (s) \geq ℓ (s)

. For almost every

v \in I R

, we have

(K ℓ) (v) = b (v) + \int_{IR} L_{1} (v, t, ℓ (t)) d t \geq 1,

which shows

(K ℓ) (v) (K j) (v) \geq (K ℓ) (v)

. Hence,

(K ℓ) ⪯ (K ℓ)

Self-operator:

The use of (C1) and (C3) leads to a continuous and compact function

K : Z \to Z

(see [40], Lemma 3).

Using (C4), we have

ℓ_{1} = K (ℓ_{0})

with

ℓ_{1} ⪯ ℓ_{0}

ℓ_{0} ⪯ ℓ_{1}

, for each

ℓ_{0} \in Z

andK taken as ⪯-preserving. Hence,

ℓ_{n} = R^{n} (ℓ_{0}))

with

ℓ_{n} ⪯ ℓ_{n + 1}

ℓ_{n + 1} ⪯ ℓ_{n}

for all

n \geq 0

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

\begin{matrix} ℓ {|(K ℓ) (v) - (K j) (v)|}^{2} & = & ℓ {|\int_{IR} L_{1} (v, s, ℓ (s)) d s - \int_{IR} L_{1} (v, s, j (s)) d s|}^{2} \\ ⪯ & ℓ {(\int_{IR} |L_{1} (v, s, ℓ (s)) - L_{1} (v, s, j (s))| d s)}^{2} \\ ⪯ & ℓ {(\int_{IR} q (v, s) | ℓ (s) - j (s) | d s)}^{2} \\ ⪯ & ℓ \int_{I R} q^{2} (v, s) d s \cdot \int_{I R} {| ℓ (s) - j (s) |}^{2} d s \\ = & ℓ α (v) \int_{IR} {| ℓ (s) - j (s) |}^{2} d s . \end{matrix}

Now,

\begin{matrix} ℓ \int_{I R} {|(K ℓ) (v) - (K j) (v)|}^{2} d v & ⪯ & ℓ \int_{IR} (α (v) \int_{IR} {| ℓ (s) - j (s) |}^{2} d s) d v \\ = & ℓ \int_{IR} (α (v) e^{ν \int_{IR} α (s) d s} \cdot e^{- ν \int_{IR} α (s) d s} \int_{IR} {| ℓ (s) - j (s) |}^{2} d s) d v \\ ⪯ & ℓ {∥ℓ - j∥}_{2, ν}^{2} \int_{IR} α (v) e^{ν \int_{IR} α (s) d s} d v \\ ⪯ & ℓ \frac{1}{ν} {∥ℓ - j∥}_{2, ν}^{2} e^{ν \int_{IR} α (s) d s} . \end{matrix}

Thus,

ℓ e^{- ν \int_{IR} α (s) d s} \int_{IR} {|(K ℓ) (v) - (K j) (v)|}^{2} d v ⪯ ℓ \frac{1}{ν} {∥ℓ - j∥}_{2, ν}^{2} .

This implies that

ℓ {∥K ℓ - K j∥}_{2, ν}^{2} ⪯ ℓ \frac{1}{ν} {∥ℓ - j∥}_{2, ν}^{2} .

c_{ν} (K ℓ, K j) ⪯ \frac{1}{ν} c_{ν} (ℓ, j)

ν c_{ν} (K ℓ, K j) ⪯ c_{ν} (ℓ, j) .

Define

J : E^{6} \to E

J (p_{1}, p_{2}, p_{3}, p_{4}, p_{5}, p_{6}) = k p_{1} - p_{2}; k > 1 .

We have

J (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})) ⪯_{ϵ} 0_{E} .

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

8. Conclusions

This paper contains various fixed-point results in the coneA-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.

Author Contributions

Conceptualization, M.N. and A.A.; methodology, A.A.; software, A.A.; validation, M.N., H.M. and A.H.; formal analysis, M.N.; investigation, A.A; resources, A.A., H.H.A.-S.; writing—original draft preparation, A.A.; writing—review and editing, M.N.; visualization, M.N.; supervision, M.N. and H.M.; project administration, A.H. and H.H.A.-S.; funding acquisition, H.H.A.-S. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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Arif, A.; Nazam, M.; Al-Sulami, H.H.; Hussain, A.; Mahmood, H. Fixed Point and Homotopy Methods in ConeA-Metric Spaces and Application to the Existence of Solutions to Urysohn Integral Equation.Symmetry2022,14, 1328. https://doi.org/10.3390/sym14071328

AMA Style

Arif A, Nazam M, Al-Sulami HH, Hussain A, Mahmood H. Fixed Point and Homotopy Methods in ConeA-Metric Spaces and Application to the Existence of Solutions to Urysohn Integral Equation.Symmetry. 2022; 14(7):1328. https://doi.org/10.3390/sym14071328

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Arif, Anam, Muhammad Nazam, Hamed H. Al-Sulami, Aftab Hussain, and Hasan Mahmood. 2022. "Fixed Point and Homotopy Methods in ConeA-Metric Spaces and Application to the Existence of Solutions to Urysohn Integral Equation"Symmetry 14, no. 7: 1328. https://doi.org/10.3390/sym14071328

APA Style

Arif, A., Nazam, M., Al-Sulami, H. H., Hussain, A., & Mahmood, H. (2022). Fixed Point and Homotopy Methods in ConeA-Metric Spaces and Application to the Existence of Solutions to Urysohn Integral Equation.Symmetry,14(7), 1328. https://doi.org/10.3390/sym14071328

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

Abstract

1. Introduction

2.Basic Notions

3.Ordered Implicit Relations

4. Main Results

4.1. Result for Increasing Self-Mapping

4.2. Result for Decreasing Self-Mapping

4.3. Result for Monotone Self-Mapping

5. Examples and Consequences

6. A Homotopy Result

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

8. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

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