# C∗-Algebra-Valued Extended Rectangular Hb-Metric Spaces and Presić-Type Fixed Point Theorem with an Application

Authors: Madjid Kohi; Hasan Hosseinzadeh; Rasoul Abazari; Ali Bagheri; Vahid Parvaneh
Journal: Journal of Mathematics
 (2022)

---

## Abstract

In this article, we introduce the concept of C∗-algebra-valued extended rectangular Hb-metric spaces and use it to prove the Banach contraction principle (BCP). Also, we extend this concept to Presić type contractions and use a method to shorten the proof. We also take an application to show the usefulness of our main result.


---

## Body

## 1. Introduction

One of the most important topics in mathematics is metric spaces, which many authors have extended to different spaces (see [1–16] for most information). In [3], the concept of rectangular metric spaces was introduced in which the triangular inequality of metric spaces was replaced by another inequality called rectangular inequality. Later, George et al. in [5], introduced the concept of rectangular b-metric space, which generalizes the concepts of metric spaces, rectangular metric spaces, and b-metric spaces.In this paper, inspired by Branchiari’s work, we generalize the definitions ofG-metric [12] and extended rectangular b-metric space [2] to a metric, that we will call ℋb-metric space. Next, we introduce the concept of C∗-algebra-valued extended rectangular ℋb-metric spaces and use it to prove the Banach contraction principle. Also, in the next section, we generalize this concept to Presić type contractions [15] and use a method to shorten the proof.LetA be a unital C∗-algebra with the unity element I, then, we denote AI=ι∈A,ιȷ=ȷι,ι≥I,∀ȷ∈A, θ be the zero element, and ς=ς∗ς1/2. An involution on A is a conjugate linear map ς↦ς∗ on A such that(1)ς+τ∗=ς∗+τ∗,λς∗=λ¯ς∗,ς∗∗=ς,ςτ∗=τ∗ς∗,for allς,τ∈A. The pair A,∗ is called a ∗-algebra. A Banach ∗-algebra is a ∗-algebra A together with a complete submultiplicative norm such that ς∗=ς for all ς∈A. A C∗-algebra is a Banach ∗-algebra such that ς∗ς=ς2 for all ς∈A. Let H be a Hilbert space and BH the set of all bounded linear operators on H, then, BH is a C∗-algebra with the operator norm. An element ς in A is self-adjoint or hermitian if ς=ς∗. Let Asa be the set of all self-adjoint elements in A, and σς=λ∈C:λI−ςis not invertible, where σς is spectrum of ς∈A. An element ς∈A of a C∗-algebra is positive if ς is hermitian and σς⊆ℝ+. We write ς>0 to mean that ς is positive. Using the positive element, one can define a partial ordering ≤ on Asa as follows: ς≤τ if and only if τ−ς≥θ. It is clear that if ς,τ∈Asa and c∈A, then, ς≤τ⇒c∗ςc≤c∗τc, and if ς,τ∈A+ are invertible, then, ς≤τ⇒θ≤τ−1≤ς−1.Definition 1 (see [12]).
LetΩ be a nonempty set and G:Ω3⟶0,+∞ be a function satisfying the following properties:(G1)Gν,ω,υ=0 if ν=ω=υ;(G2)0<Gν,ν,ω for all ν,ω∈Ω with ν≠ω;(G3)Gν,ν,ω≤Gν,ω,υ for all ν,ω,υ∈Ω with υ≠ω;(G4)Gν,ω,υ=Gν,υ,ω=Gω,υ,ν=⋯ (symmetry in all three variables);(G5)Gν,ω,υ≤Gν,ι,ι+Gι,ω,υ for all ν,ω,υ,ι∈Ω (rectangle inequality).
Then, the functionG is called a G-metric on Ω and the pair Ω,G is called a G-metric space.Proposition 1 (see [12]).
EveryG-metric space Ω,G defines a metric space Ω,ΓG by ΓGν,ω=Gν,ω,ω+Gω,ν,ν, for all ν,ω∈Ω.Definition 2 (see [9]).
LetΩ be a nonempty set. Suppose the mapping Γ:Ω×Ω⟶A satisfies (for all ν,ω,υ∈Ω):(i)
θ≤Γν,ω and Γν,ω=θ if and only if ν=ω;(ii)
Γν,ω=Γω,ν;(iii)
Γν,ω≤Γν,υ+Γυ,ω.
Then,Γ is called a C∗-algebra-valued metric on Ω, and Ω,A,Γ is called a C∗-algebra-valued metric space.Definition 3 (see [16]).
LetΩ be a nonempty set and S3 be the permutation group on 1,2,3. If G:Ω3⟶A+ is a mapping satisfying the following properties:(1)
Gν,ω,υ=θ⇔ν=ω=υ;(2)
Gνσ1,ωσ2,υσ3=Gν,ω,υ for all ν,ω,υ∈Ω,σ∈S3;(3)
Gν,ν,υ≤Gν,ω,υ for all ν,ω,υ∈Ω with ω≠υ;(4)
Gν,ω,υ≤Gν,ι,ι+Gι,ω,υ for all ν,ω,υ,ι∈Ω.
Then,G is called a C∗-algebra-valued G-metric on Ω, and Ω,A,G is called a C∗-algebra-valued G-metric space.Proposition 2 (see [16]).
LetΩ,A,G be a C∗-algebra-valued G-metric space, νn⊆Ω, ν∈Ω. The following statements are equivalent:(1)
νn→Gν;(2)
Gνn,νn,ν⟶0 as n⟶+∞;(3)
Gνn,ν,ν⟶0 as n⟶+∞.Proposition 3 (see [16]).
LetΩ,A,G be a C∗-algebra-valued G-metric space. The following statements are equivalent:(1)
The sequenceνn is G-Cauchy;(2)
For everyε>0, there is N∈ℕ such that Gνn,νm,νm<ε, for all n,m≥N.Definition 4 (see [3]).
LetΩ be a nonempty set. A mapping r:Ω2⟶ℝ+ is called a rectangular metric on Ω, if r satisfies the following (for all ν,ω∈Ω and all distinct u,v∈Ω,ν,ω):(1)
rζν,ω=0; if and only if ν=ω;(2)
rζν,ω=rζω,ν;(3)
rζν,ω≤rȷν,e+rζe,c+rζc,ω.
Then, the pairΩ,r is called a rectangular metric space.Definition 5 (see [5]).
LetΩ be a nonempty set. A mapping rb:Ω2⟶ℝ+ is called a rectangular b-metric on Ω, if rb satisfies the following (for all ν,ω∈Ω and all distinct u,v∈Ω,ν,ω):(1)
rbν,ω=0; if and only if ν=ω;(2)
rbν,ω=rbω,ν;(3)
rbν,ω≤srbν,e+rbe,c+rbc,ω.
Then, the pairΩ,rζ is called a rectangular b-metric space with the coefficient s≥1.Definition 6 (see [2]).
LetΩ be a nonempty set and ζ:Ω2⟶1,∞. A mapping rζ:Ω2⟶ℝ+ is called an extended rectangular b-metric on Ω, if rζ satisfies the following for all ν,ω∈Ω and all distinct u,v∈Ω,ν,ω:(1)
rζν,ω=0; if and only if ν=ω;(2)
rζν,ω=rζω,ν;(3)
rζν,ω≤ζν,ωrȷν,e+rζe,c+rζc,ω
Then, the pairΩ,rζ is called an extended rectangular b-metric space.Definition 7 (see [1]).
LetΩ be a nonempty set and s≥1 be a given real number. Suppose that Gb:Ω3⟶0,+∞ satisfies(Gb 1) Gbν,ω,υ=0 if ν=ω=υ;(Gb 2) 0<Gbν,ν,ω for all ν,ω∈Ω with ν≠ω;(Gb 3) Gbν,ν,ω≤Gbν,ω,υ for all ν,ω,υ∈Ω with ω≠υ;(Gb 4) Gbν,ω,υ=Gbν,υ,ω=Gbω,υ,ν=… (symmetry in all three variables);(Gb 5) Gbν,ω,υ≤sGbν,ι,ι+Gbι,ω,υ for all ν,ω,υ,ι∈Ω (triangle inequality).
Then,Gb is called a generalized b-metric and the pair Ω,Gb is called a generalized b-metric space or Gb-metric space.
The aim of this paper is to prove the existence and uniqueness of fixed points for the sequence of self-mappingsTii∈ℕ0 on Ω, where Ω,A,ℋζ is a C∗-algebra-valued extended rectangular ℋb-metric space.


## 2. Main Results

Before starting the main theorem, we first introduce the following definitions.Definition 8.
LetΩ be a nonempty set, ζ:Ω3⟶AI and ζν,ω,υ∈AI such that ζν,ω,υ≥I. Suppose that the mapping ℋζ:Ω3⟶A+ satisfies the following properties:(G1)ℋζν,ω,υ=θ if ν=ω=υ;(G2)0<ℋζν,ν,ω for all ν,ω∈Ω with ν≠ω;(G3)ℋζν,ν,ω≤ℋζν,ω,υ for all ν,ω,υ∈Ω with υ≠ω;(G4)ℋζν,ω,υ=ℋζν,υ,ω=ℋζω,υ,ν=⋯ (symmetry in all three variables);(G5)ℋζν,ω,υ≤ζν,ω,υℋζν,e,e+ℋζe,c,c+ℋζc,ω,υ for all ν,ω,υ,e,c∈Ω, and for all distinct points e,c∈Ω∖ν,ω,υ (rectangle inequality).
Then, the functionℋζ is called a C∗-algebra-valued extended rectangular ℋb-metric on Ω, and Ω,A,ℋζ is called a C∗-algebra-valued extended rectangular ℋb-metric space with the coefficient ζν,ω,υ.Example 1.
LetΩ=ℝ. ℋζ is defined by(2)ℋζν,ω,υ=maxν,ω,υ2,
for allν,ω,υ∈Ω and e,c∈Ω,ν,ω,υ, we have(3)ℋζν,ω,υ=maxν,ω,υ2≤maxν,e,e2+maxe,c,c2+maxc,ω,υ2≤ℋζν,e,e+ℋζe,c,c+ℋζc,ω,υ≤ζν,ω,υℋζν,e,e+ℋζe,c,c+ℋζc,ω,υ.
In the last part of the above inequality, we use the fact that for allν,ω,υ∈Ω, we have ζν,ω,υ≥I. Hence, ℋζ is a C∗-algebra-valued extended rectangular ℋb-metric space with(4)ζν,ω,υ=maxν,ω,υ+1.Definition 9.
LetΩ,A,ℋζ be a C∗-algebra-valued extended rectangular ℋb-metric space. Then,(i)
A sequenceνn is ℋb-convergent to ν in Ω if(5)limm,n⟶+∞ℋζν,νn,νm=0.That is, for eachε>0, there exists N∈ℕ such that(6)ℋζν,νn,νm<ε.for allm,n≥N;(ii)
A sequenceνn is called ℋb-Cauchy if, for each ε>0, there exists N∈ℕ such that(7)ℋζνl,νn,νm<ε,for allm,n,l≥N.Definition 10.
The tripleΩ,A,ℋζ is called complete C∗-algebra-valued extended rectangular ℋb-metric space, if every ℋb-Cauchy sequence in Ω is ℋb-convergent in Ω.Definition 11.
LetΩ,ℋζ and Ω′,ℋζ′ be C∗-algebra-valued extended rectangular ℋb-metric spaces, and f:Ω,ℋζ⟶Ω′,ℋζ′ be a function. Then, f is called ℋb-continuous at a point ι∈Ω if for every ε>0, there is δ>0 such that ν,ω∈Ω and ℋζι,ν,ω<δ implies ℋζfι,fν,fω<ε. A function f is ℋb-continuous at Ω if and only if it is ℋb-continuous at all points of Ω.Definition 12.
Suppose thatΩ,A,ℋζ be a C∗-algebra-valued extended rectangular ℋb-metric space. The sequence Ti:Ω⟶Ω is called contractive, if there is an element ρ∈A with ρ<1 such that(8)ℋζTiν,Tjω,Tjυ≤ρ∗ℋζν,ω,υρ,for allν,ω,υ∈Ω.
Now, we establish the main result of this manuscript as follows.Theorem 1.
LetΩ,A,ℋζ be a complete C∗-algebra-valued extended rectangular ℋb-metric space, Tii∈ℕ be a sequence of contractive mappings on Ω, and limn,m⟶∞ζνn,νm,νmρ<1, then, Tii∈ℕ has a unique fixed point in Ω.Proof.
Chooseν0∈Ω, and set(9)νn+1=Tnνn,n=1,2,….
By condition (8), we get(10)ℋζνn,νn+1,νn+1=ℋζTn−1νn−1,Tnνn,Tnνn≤ρ∗ℋζνn−1,νn,νnρ⋯≤ρn∗ℋζν0,ν1,ν1ρn=ρn∗Fnρn,where Fn=ℋζν0,ν1,ν1. Now, we show that νn is an ℋb-Cauchy sequences in Ω,A,ℋζ, for this, we prove two cases as follows:
(1) Letm=2r+1 with r≥1. Using (G5) for any n∈ℕ, we have(11)ℋζνn,νn+2r+1,νn+2r+1≤ζνn,νn+2r+1,νn+2r+1ℋζνn,νn+1,νn+1+ℋζνn+1,νn+2,νn+2+ℋζνn+2,νn+2r+1,νn+2r+1≤ζνn,νn+2r+1,νn+2r+1ℋζνn,νn+1,νn+1+ζνn,νn+2r+1,νn+2r+1ℋζνn+1,νn+2,νn+2+ζνn,νn+2r+1,νn+2r+1×ζνn+2,νn+2r+1,νn+2r+1ℋζνn+2,νn+3,νn+3+ℋζνn+3,νn+2r+1,νn+2r+1≤ζνn,νn+2r+1,νn+2r+1ℋζνn,νn+1,νn+1+ζνn,νn+2r+1,νn+2r+1ℋζνn+1,νn+2,νn+2+ζνn,νn+2r+1,νn+2r+1×ζνn+2,νn+2r+1,νn+2r+1ℋζνn+2,νn+3,νn+3+⋯+ζνn,νn+2r+1,νn+2r+1×⋯×ζνn+2r−2,νn+2r+1,νn+2r+1ℋζνn+2r−2,νn+2r+1,νn+2r+1≤ζνn,νn+2r+1,νn+2r+1ρn∗Fnρn+ζνn,νn+2r+1,νn+2r+1ρn+1∗Fnρn+1+ζνn,νn+2r+1,νn+2r+1×ζνn+2,νn+2r+1,νn+2r+1ρn+2∗Fnρn+2+⋯+ζνn,νn+2r+1,νn+2r+1×⋯×ζνn+2r−2,νn+2r+1,νn+2r+1ρn+2r−2∗Fnρn+2r−2+ζνn,νn+2r+1,νn+2r+1×⋯×ζνn+2r−2,νn+2r+1,νn+2r+1ρn+2r∗Fnρn+2r=ζνn,νn+2r+1,νn+2r+1ρn∗Fnρn+ρn+1∗Fnρn+1+ζνn,νn+2r+1,νn+2r+1×ζνn+2,νn+2r+1,νn+2r+1ρn+2∗Fnρn+2+ρn+3∗Fnρn+3+⋯+ζνn,νn+2r+1,νn+2r+1×⋯×ζνn+2r−2,νn+2r+1,νn+2r+1ρn+2r−2∗Fnρn+2r−2+ρn+2r−1∗Fnρn+2r−1+ζνn,νn+2r+1,νn+2r+1×⋯×ζνn+2r−2,νn+2r+1,νn+2r+1ρn+2r∗Fnρn+2r=ζνn,νn+2r+1,νn+2r+1ρnI+ρ∗FnρnI+ρ+ζνn,νn+2r+1,νn+2r+1×ζνn+2,νn+2r+1,νn+2r+1ρn+2I+ρ∗Fnρn+2I+ρ+⋯+ζνn,νn+2r+1,νn+2r+1×⋯×ζνn+2r−2,νn+2r+1,νn+2r+1ρn+2r−2I+ρ∗Fnρn+2r−2I+ρ+ζνn,νn+2r+1,νn+2r+1×⋯×ζνn+2r−2,νn+2r+1,νn+2r+1ρn+2r∗Fnρn+2r.
So,(12)ℋζνn,νn+2r+1,νn+2r+1≤ρnI+ρ∗FnρnI+ρ∑i=0r−1ρ2i∏j=0iζνn+2j,νn+2r+1,νn+2r+1+ρn+2r∗Fnρn+2r∏j=0r−1ζνn+2j,νn+2r+1,νn+2r+1=Fn1/2ρnI+ρ2∑i=0r−1ρ2i∏j=0iζνn+2j,νn+2r+1,νn+2r+1+Fn1/2ρn+2r2∏j=0r−1ζνn+2j,νn+2r+1,νn+2r+1≤FnρnI+ρ2∑i=0r−1ρi2∏j=0iζνn+2j,νn+2r+1,νn+2r+1+Fnρn+2r2∏j=0r−1ζνn+2j,νn+2r+1,νn+2r+1.
Note that the above inequality is dominated by(13)∑i=0r−1ρi2∏j=0iζνn+2j,νn+2r+1,νn+2r+1≤∑i=0r−1ρi2∏j=0iζν2j,νn+2r+1,νn+2r+1.
From hypothesis, we havelimn,m⟶∞ζνn,νm,νmρ<1. So, by the ratio test, we have(14)ρi+12∏j=0i+1ζν2j,νn+2r+1,νn+2r+1ρi2∏j=0iζν2j,νn+2r+1,νn+2r+1≤limi⟶∞ρ2ζν2i,νn+2r+1,νn+2r+1<1.
That is, the series∑i=0∞ρi2∏j=0iζν2j,νn+2r+1,νn+2r+1 is convergent for every r∈ℕ. Let(15)S=∑i=0∞ρi2∏j=0iζν2j,νn+2r+1,νn+2r+1,Sn=∑i=0nρi2∏j=0iζν2j,νn+2r+1,νn+2r+1.
As a result, by the above inequality, we have(16)ℋζνn,νn+2r+1,νn+2r+1≤FnρnI+ρ2Sr−1−Sn−1+Fnρn+2r2∏j=0r−1ζνn+2j,νn+2r+1,νn+2r+1⟶0,asn⟶∞.
(2) Letm=2r with r≥1.(17)ℋζνn,νn+2r,νn+2r≤ζνn,νn+2r,νn+2rℋζνn,νn+1,νn+1+ℋζνn+1,νn+2,νn+2+ℋζνn+2,νn+2r,νn+2r≤ζνn,νn+2r,νn+2rℋζνn,νn+1,νn+1+ζνn,νn+2r,νn+2rℋζνn+1,νn+2,νn+2+ζνn,νn+2r,νn+2r×ζνn+2,νn+2r,νn+2rℋζνn+2,νn+3,νn+3+ℋζνn+3,νn+2r,νn+2r≤ζνn,νn+2r,νn+2rℋζνn,νn+1,νn+1+ζνn,νn+2r,νn+2rℋζνn+1,νn+2,νn+2+ζνn,νn+2r,νn+2r×ζνn+2,νn+2r,νn+2rℋζνn+2,νn+3,νn+3+⋯+ζνn,νn+2r,νn+2r×⋯×ζνn+2r−2,νn+2r,νn+2rℋζνn+2r−2,νn+2r,νn+2r≤ζνn,νn+2r,νn+2rρn∗Fnρn+ζνn,νn+2r,νn+2rρn+1∗Fnρn+1+ζνn,νn+2r,νn+2r×ζνn+2,νn+2r,νn+2rρn+2∗Fnρn+2+⋯+ζνn,νn+2r,νn+2r×⋯×ζνn+2r−2,νn+2r,νn+2rρn+2r−2∗Fnρn+2r−2+ζνn,νn+2r,νn+2r×⋯×ζνn+2r−2,νn+2r,νn+2rρn+2r∗Fnρn+2r=ζνn,νn+2r,νn+2rρn∗Fnρn+ρn+1∗Fnρn+1+ζνn,νn+2r,νn+2r×ζνn+2,νn+2r,νn+2rρn+2∗Fnρn+2+ρn+3∗Fnρn+3+⋯+ζνn,νn+2r,νn+2r×⋯×ζνn+2r−2,νn+2r,νn+2rρn+2r−2∗Fnρn+2r−2+ρn+2r−1∗Fnρn+2r−1+ζνn,νn+2r,νn+2r×⋯×ζνn+2r−2,νn+2r,νn+2rρn+2r∗Fnρn+2r=ζνn,νn+2r,νn+2rρnI+ρ∗FnρnI+ρ+ζνn,νn+2r,νn+2r×ζνn+2,νn+2r,νn+2rρn+2I+ρ∗Fnρn+2I+ρ+⋯+ζνn,νn+2r,νn+2r×⋯×ζνn+2r−2,νn+2r,νn+2rρn+2r−2I+ρ∗Fnρn+2r−2I+ρ+ζνn,νn+2r,νn+2r×⋯×ζνn+2r−2,νn+2r,νn+2rρn+2r∗Fnρn+2r.
So,(18)ℋζνn,νn+2r,νn+2r≤ρnI+ρ∗FnρnI+ρ∑i=0r−1ρ2i∏j=0iζνn+2j,νn+2r,νn+2r+ρn+2r∗Fnρn+2r∏j=0r−1ζνn+2j,νn+2r,νn+2r=Fn1/2ρnI+ρ2∑i=0r−1ρ2i∏j=0iζνn+2j,νn+2r,νn+2r+Fn1/2ρn+2r2∏j=0r−1ζνn+2j,νn+2r,νn+2r≤FnρnI+ρ2∑i=0r−1ρ2i∏j=0iζνn+2j,νn+2r,νn+2r+Fnρn+2r2∏j=0r−1ζνn+2j,νn+2r,νn+2r.
Therefore, again by the ratio test, we have(19)ℋζνn,νn+2r,νn+2r≤FnρnI+ρ2Sr−1−Sn−1+Fnρn+2r2∏j=0r−1ζνn+2j,νn+2r,νn+2r⟶0,asn⟶∞.
Hence, we will have(20)limn⟶∞ℋζνn,νn+p,νn+p=0.
Therefore, this indicates that the sequenceνn is ℋb-Cauchy in Ω,A,ℋζ. Since Ω is complete, then, ν∈Ω such that νn⟶ν, that is,(21)limn⟶∞νn+1=limn⟶∞Tnνn=ν,
From continuity ofTii∈ℕ0, we have(22)θ≤ℋζν,Tiν,Tiν≤ζν,Tiν,Tiνℋζν,νn+1,νn+1+ℋζνn+1,νn+2,νn+2+ℋζνn+2,Tiν,Tiν=ζν,Tiν,Tiνℋζν,νn+1,νn+1+ℋζνn+1,νn+2,νn+2+ℋζTn+1νn+1,Tiν,Tiν≤ζν,Tiν,Tiνℋζν,νn+1,νn+1+ℋζνn+1,νn+2,νn+2+ρ∗ℋζνn+1,ν,νρ⟶θasn⟶∞.
So,ℋζν,Tiν,Tiν=0, therefore, ν is a fixed point of Tii∈ℕ0. Now, suppose that ν′ is another fixed point of Tii∈ℕ0. Since(23)θ≤ℋζν,ν′,ν′=ℋζTiν,Tjν′,Tjν′≤ρ∗ℋζν,ν′,ν′ρ,
one has(24)θ≤ℋζν,ν′,ν′=ℋζTiν,Tjν′,Tjν′≤ρ∗ℋζν,ν′,ν′ρ≤ρ∗ℋζν,ν′,ν′ρ=ρ∗2ℋζν,ν′,ν′<ℋζν,ν′,ν′.
So, it is a contradiction. Hence,ℋζν,ν′,ν′=θ and ν=ν′.


## 3. Presić Fixed Point Theorems

In this section, we obtain some results for Presić contractive mappings on theC∗-algebra-valued extended rectangular ℋb-metric space.Theorem 2 (see [17]).
LetΩ,Γ be a complete metric space and let T:Ω⟶Ω so that(25)ΓTι,Tκ≤ϑΓι,κfor allι,κ∈Ω,where ϑ∈0,1. Then, there exists σ∈Ω such that σ=Tσ. Also, for every ∂0∈Ω, the sequence ∂n+1=T∂n converges to σ.The BCP has expanded and generalized in various ways (see [10, 14]). Presić in [15] reached the following result.Theorem 3 (see [15]).
LetΩ,Γ be a complete metric space and let T:Ωk⟶Ω, where k is a positive integer. Suppose that(26)ΓT∂1,…,∂k,T∂2,…,∂k+1≤∑i=1kρiΓ∂i,∂i+1,
for all∂1,…,∂k+1∈Ω, where ρi≥0 and ∑i=1kρi∈0,1. Then, T has a unique fixed point ∂∗, that is, T∂∗,…,∂∗=∂∗. In addition, for all arbitrary points ∂1,…,∂k+1∈Ω, the sequence ∂n defined by ∂n+k=T∂n,∂n+1,…,∂n+k−1, converges to ∂∗.It is obvious that fork=1, Theorem 3 coincides with the BCP. The above theorem has been generalized by Ćirić and Presić [4] as follows.Theorem 4 (see [4]).
LetΩ,Γ be a complete metric space and T:Ωk⟶Ω, where k is a positive integer. Suppose that(27)ΓT∂1,…,∂k,T∂2,…,∂k+1≤ρmaxΓ∂i,∂i+1:1≤i≤k,
for all∂1,…,∂k+1∈Ω, where ρ∈0,1. Then, T has a fixed point ∂∗∈Ω. Also, for all points ∂1,…,∂k+1∈Ω, the sequence ∂n defined by ∂n+k=T∂n,∂n+1,…,∂n+k−1, converges to ∂∗. The fixed point of T is unique if(28)ΓTp,…,p,Tϱ,…,ϱ<Γp,ϱ,
for allp,ϱ∈Ω with p≠ϱ.Now, we have the main result of this section, first, we prove the following lemma:Lemma 1.
Suppose thatℋζ1,ℋζ2,…,ℋζn are C∗-algebra-valued extended rectangular ℋb-metric space on nonempty sets Ω1,Ω2,…,Ωn, respectively, and let ϖ:0,∞n⟶0,∞ so that ϖσ1,…,σn=0 if and only if σi=0 for all i=1,2,…,n and(29)ϖζ1ν11,ν21,ν31ℋζ1ν11,e1,e1+ℋζ1e1,c1,c1+ℋζ1c1,ν21,ν31+ζ2ν12,ν22,ν32ℋζ2ν12,e2,e2+ℋζ2e2,c2,c2+ℋζ2c2,ν22,ν32⋮+ζnν1n,ν2n,ν3nℋζnν1n,en,en+ℋζnen,cn,cn+ℋζncn,ν2n,ν3n≤ζν11,ν12,…,ν1n,ν21,ν22,…,ν2n,ν31,ν32,…,ν3nϖℋζ1ν11,e1,e1,ℋζ2ν12,e2,e2,…,ℋζnν1n,en,en+ϖℋζ1e1,c1,c1,ℋζ2e2,c2,c2,…,ℋζnen,cn,cn+ϖℋζ1c1,ν21,ν31,ℋζ2c2,ν22,ν32,…,ℋζncn,ν2n,ν3n,
for allνij∈0,∞, where(30)ζν11,ν12,…,ν1n,ν21,ν22,…,ν2n,ν31,ν32,…,ν3n=ϖζ1ν11,ν21,ν31,ζ2ν12,ν22,ν32,…,ζnν1n,ν2n,ν3n.
Then,(31)ℋζ˜ν11,ν12,…,ν1n,ν21,ν22,…,ν2n,ν31,ν32,…,ν3n=ϖℋζ1ν11,ν21,ν31,ℋζ2ν12,ν22,ν32,…,ℋζnν1n,ν2n,ν3n.
is aC∗-algebra-valued extended rectangular ℋb-metric in Ω1×Ω2×⋯×Ωn3.Proof.
We only show the triangular inequality in aC∗-algebra-valued extended rectangular ℋb-metric space. Let νij∈Ωj for all 1≤i≤3 and 1≤j≤n. So,(32)ℋ˜ζν11,ν12,…,ν1n,ν21,ν22,…,ν2n,ν31,ν32,…,ν3n=ϖℋζ1ν11,ν21,ν31,ℋζ2ν12,ν22,ν32,…,ℋζnν1n,ν2n,ν3n≤ϖζ1ν11,ν21,ν31ℋζ1ν11,e1,e1+ℋζ1e1,c1,c1+ℋζ1c1,ν21,ν31+ζ2ν12,ν22,ν32ℋζ2ν12,e2,e2+ℋζ2e2,c2,c2+ℋζ2c2,ν22,ν32⋮+ζnν1n,ν2n,ν3nℋζnν1n,en,en+ℋζnen,cn,cn+ℋζncn,ν2n,ν3n≤ζν11,ν12,…,ν1n,ν21,ν22,…,ν2n,ν31,ν32,…,ν3nϖℋζ1ν11,e1,e1,ℋζ2ν12,e2,e2,…,ℋζnν1n,en,en+ϖℋζ1e1,c1,c1,ℋζ2e2,c2,c2,…,ℋζnen,cn,cn+ϖℋζ1c1,ν21,ν31,ℋζ2c2,ν22,ν32,…,ℋζncn,ν2n,ν3n≤ζν11,ν12,…,ν1n,ν21,ν22,…,ν2n,ν31,ν32,…,ν3nℋ˜bν11,ν12,…,ν1n,e1,e2,…,en,e1,e2,…,en+ℋ˜ζe1,e2,…,en,c1,c2,…,cn,c1,c2,…,cn+ℋ˜ζc1,c2,…,cn,ν21,ν22,…,ν2n,ν31,ν32,…,ν3n.Theorem 5.
LetΩ,A,ℋζ be a C∗-algebra-valued extended rectangular ℋb-metric space and Ti:Ωn⟶Ω as a sequence of continuous functions such that(33)ℋζTiν11,ν12,…,ν1n,Tjν21,ν22,…,ν2n,Tjν31,ν32,…,ν3n≤ρ∗∑i=1nℋζν1i,ν2i,ν3iρ,
for allνij⊆Ω, where ρ∈A with ρ<1. Then, Ti has at least a Presić type fixed point.Proof.
We define the sequence of mappingsT˜i:Ωn⟶Ωn by(34)T˜iσ1,…,σn=Tiσ1,…,σn,…,Tiσ1,…,σn.
Clearly,T˜i is continuous. We show that T˜i satisfies all the conditions of Theorem 1. We know that(35)ℋζ˜ν11,ν12,…,ν1n,ν21,ν22,…,ν2n,ν31,ν32,…,ν3n≔ℋζν11,ν21,ν31+ℋζν12,ν22,ν32+⋯+ℋζν1n,ν2n,ν3nn.
is aC∗-algebra-valued extended rectangular ℋb-metric space. From (33), we have(36)ℋζ˜T˜iν11,ν12,…,ν1n,T˜jν21,ν22,…,ν2n,T˜jν31,ν32,…,ν3n=ℋζ˜Tiν11,ν12,…,ν1n,Tiν11,ν12,…,ν1n,…,Tiν11,ν12,…,ν1nTjν21,ν22,…,ν2n,Tjν21,ν22,…,ν2n,…,Tjν21,ν22,…,ν2nTjν31,ν32,…,ν3n,Tjν31,ν32,…,ν3n,…,Tjν31,ν32,…,ν3n=ℋζTiν11,ν12,…,ν1n,Tjν21,ν22,…,ν2n,Tjν31,ν32,…,ν3n≤ρ∗∑i=1nℋζν1i,ν2i,ν3iρ=ρ∗ℋζ˜ν11,ν12,…,ν1n,ν21,ν22,…,ν2n,ν31,ν32,…,ν3nρ.
Now, according to Theorem1, we infer that T˜i accepts at least one fixed point, which implies that there exist σ1,…,σn such that(37)Tiσ1,…,σn=σ1=⋯=σn.
That is,Ti has at least one fixed point of type Presić.Theorem 6.
LetΩ,A,ℋζ be a C∗-algebra-valued extended rectangular ℋb-metric space and Ti:Ωn⟶Ω be a sequence of continuous functions so that(38)ℋζTiν11,ν12,…,ν1n,Tjν21,ν22,…,ν2n,Tjν31,ν32,…,ν3n≤ρ∗maxℋζν1i,ν2i,ν3i:1≤i≤nρ,
for allνij∈Ω, where ρ∈A with ρ<1. Then, Ti has at least a Presić type fixed point.Proof.
LetT˜i:Ωn⟶Ωn be defined by(39)T˜iσ1,…,σn=Tiσ1,…,σn,…,Tiσ1,…,σn.
Evidently,T˜i is continuous. It will show that T˜i satisfies all the conditions of Theorem 1. We know that(40)ℋζ˜ν11,ν12,…,ν1n,ν21,ν22,…,ν2n,ν31,ν32,…,ν3n≔maxℋζν11,ν21,ν31,ℋζν12,ν22,ν32,…,ℋζν1n,ν2n,ν3n
is aC∗-algebra-valued extended rectangular ℋb-metric space. According to (38), we have(41)ℋ˜ζT˜iν11,ν12,…,ν1n,T˜jν21,ν22,…,ν2n,T˜jν31,ν32,…,ν3n=ℋ˜ζTiν11,ν12,…,ν1n,Tiν11,ν12,…,ν1n,…,Tiν11,ν12,…,ν1n,Tjν21,ν22,…,ν2n,Tjν21,ν22,…,ν2n,…,Tjν21,ν22,…,ν2nTjν31,ν32,…,ν3n,Tjν31,ν32,…,ν3n,…,Tjν31,ν32,…,ν3n=maxℋζTiν11,ν12,…,ν1n,Tjν21,ν22,…,ν2n,Tjν31,ν32,…,ν3n,ℋζTiν11,ν12,…,ν1n,Tjν21,ν22,…,ν2n,Tjν31,ν32,…,ν3n,…,ℋζTiν11,ν12,…,ν1n,Tjν21,ν22,…,ν2n,Tjν31,ν32,…,ν3n=ℋζTiν11,ν12,…,ν1n,Tjν21,ν22,…,ν2n,Tjν31,ν32,…,ν3n≤ρ∗maxℋζν1i,ν2i,ν3i:1≤i≤nρ=ρ∗ℋ˜ζν11,ν12,…,ν1n,ν21,ν22,…,ν2n,ν31,ν32,…,ν3nρ.
So, from Theorem1, we get T˜i has at least a fixed point which implies that Ti has at least a Presić-type fixed point.


## 4. Application

Consider the following Fredholm integral equation:(42)νs=∫EΛs,t,νtdt+hs,for alls,t∈E, where E is a measurable set, Λ:E2×ℝ⟶ℝ and h∈L∞E. Let Ω=L∞E, H=L2E and LH=A. Define ℋζ:Ω3⟶AI by(43)ℋζν,ω,υ=ℳν−ωp+ω−υp+υ−νp,for allν,ω,υ∈Ω, p≥1, where ℳ:H⟶H is the multiplicative operator defined by(44)ℳφψ=φ⋅ψ.Then,Ω,A,ℋζ is a C∗-algebra-valued extended rectangular ℋb-metric space. Now, define a mapping ζ:Ω3⟶AI by(45)ζν,ω,υ=ℳν−ωp+ω−υp+υ−νp,ifν≠ω≠υ,I2×2,ifν=ω=υ,where I2×2 is an square identity matrix in Ω. Now, we consider the following assumptions:(1)
For allν,ω,υ∈Ω there is a continuous function ψ:E2⟶ℝ and k∈0,1 such that(46)Λs,t,νt−Λs,t,ωt≤kψs,tνt−ωt.(2)
sups∈E∫Eψs,tdt≤1.Theorem 7.
Suppose that assumptions (1) and (2) hold. Then, the integral equation (42) has a unique solution in Ω.Proof.
We defineTi:Ω⟶Ω by(47)Tiνs=∫EΛs,t,νtdt+hs,∀s,t∈E.
Setρ=kI, then, ρ∈LH+ and ρ=k<1. For every φ∈H and p≥1, we have(48)ℋζTiν,Tiω,Tiυ=ℳTiν−Tiωp+Tiω−Tiυp+Tiυ−Tiνp=supφ=1ℳTiν−Tiωp+Tiω−Tiυp+Tiυ−Tiνpφ,φ=supφ=1∫E∫EΛs,t,νt−Λs,t,ωtdtpφsφs¯ds+supφ=1∫E∫EΛs,t,ωt−Λs,t,υtdtpφsφs¯ds+supφ=1∫E∫EΛs,t,υt−Λs,t,νtdtpφsφs¯ds≤supφ=1∫E∫EΛs,t,νt−Λs,t,ωtdtpφs2ds+supφ=1∫E∫EΛs,t,ωt−Λs,t,υtdtpφs2ds+supφ=1∫E∫EΛs,t,υt−Λs,t,νtdtpφs2ds≤kpsupφ=1∫E∫Eψs,tdtpφs2dsν−ω∞p+ω−υ∞p+υ−ν∞p≤ksups∈E∫Eψs,tdtsupφ=1∫Eφs2dsν−ω∞p+ω−υ∞p+υ−ν∞p≤kν−ω∞p+ω−υ∞p+υ−ν∞p=ρℋζν,ω,υ.
Sinceρ<1, therefore, it is obvious that the sequence of mappings Tii∈ℕ0 estimates all the conditions of Theorem 1. Hence, the Fredholm integral equation (42) has a unique solution, that is, Tii∈ℕ0 has a unique fixed point.



---

*Source: 7347385-2022-05-26.xml*