# Convergence of Generalized Quasi-Nonexpansive Mappings in Hyperbolic Space

Authors: Naeem Saleem; Maliha Rashid; Fahd Jarad; Amna Kalsoom
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we consider a wider class of nonexpansive mappings (locally related quasi-nonexpansive) than monotone nonexpansive mappings. We obtained the convergence of fixed point for quasiϱ-preserving locally related quasi-nonexpansive mappings in hyperbolic space. An iterative process is also used to obtain the convergence results for this mapping. Fixed point is approximated numerically in a nontrivial example by using Matlab.


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## Body

## 1. Introduction

A first attempt to enrich metric spaces with convexity was essentially due to Takahashi [1] and is known as convex metric spaces. The class of convex metric spaces is general in nature and has constant curvature. Fixed point theory has many developments regarding convex metric spaces; for example, see [2–5].The concept of hyperbolic space was introduced by Kohlenbach [6] in 2005, which is more general than the concept of hyperbolic space in [7] and more restrictive than the hyperbolic space defined in [8]. This definition is different from Takahashi’s notion of convex metric space in the sense that every convex subset of a hyperbolic space is itself a hyperbolic space. These spaces are nonlinear in nature and more general than normed spaces. Fixed point theorems for nonexpansive mappings in hyperbolic space have been studied in [7, 9–16]. The existence of fixed point for nonexpansive mapping was initiated by Browder [17], Kirk [18], and Göhde [19] independently in 1965. In 1967, Diaz and Metcalf [20] gave an idea about quasi-nonexpansive mapping. Example of quasi-nonexpansive mappings was given by Doston [21] in 1972 which was not nonexpansive. Fixed point results for monotone nonexpansive mappings were presented by Bachar and Khamsi [22] in 2015. In 2019, the concept of ϱ-preserving was introduced by Al-Rawashdeh and Mehmood [23] which is generalized than the concept of monotone.In this article, we will generalize the results of [23] in hyperbolic space. A nontrivial example is also given in which Picard [24], Mann [23, 25], Ishikawa [26, 27], Agarwal [28], Abbas and Nazir [29], and Noor [30] iteration schemes are used to approximate the fixed point.


## 2. Preliminaries

A hyperbolic space [6] is a metric space X,d;X≠∅, together with a mapping
(1)H:X×X×0,1⟶X,which satisfies the following for all u,x,y,z∈X and α,β∈0,1(1)
du,Hx,y,α≤1−αdu,x+αdu,y,(2)
dHx,y,α,Hx,y,β=α−βdx,y,(3)
Hx,y,α=Hy,x,1−α,(4)
dHx,z,α,Hy,u,α≤1−αdx,y+αdz,u.If a space satisfies only Condition 1, it coincides with the convex metric space introduced by Takahashi [1].Throughout this article, we consider(2)Hx,y,α=αy⊕1−αx,for all x,y,z,H∈X and α,β∈0,1.LetX be a hyperbolic space and K be a nonempty subset of X and Γ:K⟶K be a mapping. According to [23], the underlying concepts are defined in hyperbolic space as follows: A mapping Γ is said to be nonexpansive if
(3)dΓx,Γy≤dx,y,forallx,y∈K,and quasi-nonexpansive provided FΓ is nonempty and for each y∈FΓ(4)dΓx,y≤dx,y,forallx∈K.Letϱ be a relation on X; a self-map Γ of X is said to be ϱ-preserving if
(5)xϱy⇒ΓxϱΓy,forallx,y∈X.LetX,d,≼ be a partially order hyperbolic space, and a self map Γ of X is said to be monotone nonexpansive if Γ is monotone and
(6)dΓx,Γy≤dx,ywheneverx≼y,forallx,y∈X.A mappingΓ:K⟶K is said to be
(i)
Locally related quasi-nonexpansive (abbreviated as L.R.Q.N) providedFΓ is nonempty, and for each y∈FΓ(7)dΓx,y≤dx,yandxϱy,forallx∈K,(ii)
Quasiϱ-preserving provided FΓ is nonempty, and for each y∈FΓ(8)Γxϱywheneverxϱy,forallx∈K.A quasiϱ-preserving mapping which is also L.R.Q.N is called quasiϱ-preserving L.R.Q.N.Condition (S) A hyperbolic space X having a relation ϱ on it satisfying condition S if every convergent sequence xn,xn⟶x where x∈X has a subsequence xnk such that xϱxnk for all k∈ℕ.LetX be a hyperbolic space and ϱ be the relation on X; ϱ is said to be compatible if for all x,y∈X,(a)
xϱy implies x⊕zϱy⊕z,(b)
xϱy implies αxϱαy for α∈0,1.Remark 1.
In the above definitions, we consider only a relation which needs not to be a partial order relation necessarily. The conditionS is utilized in Theorems 2, 5, 9, 13, 17, and 19 which is moderate than the conditions already been considered in the literature [31]. The condition
(9)liminfn⟶∞dxn,FΓ=0,mentioned in each Theorems 2, 5, 9, 13, 17, and 19 is based on the condition S (see the last paragraph on page 4 of [23]).


## 3. Main Results

LetX be a hyperbolic space and K be a nonempty subset of X, I be the identity map, and Γ:K⟶K be a mapping. For x0∈K, and α,β∈0,1, let xn be a sequence with iterations given as
(10)xn=Γxn−1=Γnx0,Picarditeration,(11)xn=Γαxn−1=Γαnx0,whereΓα=HΓ,I,α,Manniteration,(12)xn=Γα,βnxn−1=Γα,βnx0,whereΓα,β=HΓΓβ,I,α,Ishikawaiteration.Theorem 2.
LetX be hyperbolic space, having a compatible relation ϱ on it satisfying condition S. Let K be a closed and convex subset of X. Suppose Γ be a quasi ϱ-preserving L.R.Q.N self map of K. If there exists some c0∈K such that c0ϱy for all y∈FΓ, then, (10) converges to a fixed point of Γ in K if and only if
(13)liminfn⟶∞dxn,FΓ=0.Proof.
Ifxn converges to some fixed point of Γ, then liminfn⟶∞dxn,FΓ=0 holds. Conversely, suppose liminfn⟶∞dxn,FΓ=0 holds. As Γ is quasi ϱ-preserving and c0ϱy, we have Γc0ϱy. Similarly,
(14)Γn−1c0ϱy=xn−1ϱyforalln∈ℕ.SinceΓ is L.R.Q.N, so for all n∈ℕ and y∈FΓ,(15)dxn,y=dΓxn−1,y≤dxn−1,y,taking inf over y(16)dxn,FΓ≤dxn−1,FΓ,implies dxn,FΓ≥0 is a nonincreasing sequence and bounded as well. So
(17)limn⟶∞dxn,FΓ=0.Now, we prove thatxn is a Cauchy sequence. For a given ε>0, there exists k∈ℕ, such that for all n≥k,(18)dxn,FΓ<ε2.Forx∈FΓ and all l,m≥k, we have
(19)dxl,x=dΓlc0,x≤dΓkc0,x,dxm,x=dΓmc0,x≤dΓkc0,x,by adding
(20)dxl,x+dxm,x≤2dΓkc0,x.Takinginf over x(21)dxl,xm≤2dxn,FΓ<ε,implies xn is a Cauchy sequence. Since K is complete, so there exists x∈K, such that
(22)limn⟶∞xn=x.Next, we have to show thatFΓ is closed. Let x∈K be a limit point of FΓ; then, there exists a sequence xn⊆FΓ, and using condition S, a subsequence xnk of xn converges to x and
(23)xϱxnkforallk∈ℕ.Consider(24)dΓx,x≤dΓx,xnk+dx,xnk≤2dx,xnk⟶0ask⟶∞,so x∈FΓ.Example 1.
LetX=a1,b1∈R2;a1,b1>0 be a hyperbolic space and ϱ be the relation defined as
(25)a1,b1ϱa2,b2⇔a1≤a2andb1≥b2.Letd:X×X⟶ℝ be defined as
(26)da,b=a1−a2+a1b1−a2b2wherea=a1,b1,b=a2,b2.LetK=1,4×1,4⊂X and Γ:K⟶K be a mapping defined by
(27)Γa,b=1,2a,b=4,41,1a,b≠4,4.As1,1 is the fixed point of Γ, and a,bϱ1,1 implies a≤1 and b≥1 that is 1,b, where 1≤b≤4, we have
(28)dΓ1,b,1,1=0≤b−1=d1,b,1,1,which shows that Γ is L.R.Q.N.Next since1,4ϱ4,4 but Γ1,4ϱΓ4,4 does not hold, which shows that Γ is not ϱ-preserving.Also1,bϱ1,1 implies
(29)Γ1,b=1,1ϱ1,1=Γ1,1,so Γ is quasi ρ-preserving. As 3.9,4ϱ4,4(30)dΓ3.9,4,Γ4,4=1>0.5=d3.9,4,4,4,hence, Γ is not nonexpansive. This example shows that L.R.Q.N mapping is not necessarily ϱ-preserving or ϱ-preserving nonexpansive.Example 2.
LetX=ℝ2 be a hyperbolic space and ϱ be the relation defined as
(31)a1,b1ϱa2,b2⇔a1≤a2andb1=b2.Letd:X×X⟶ℝ be defined as
(32)da,b=a1−a2+a1b1−a2b2wherea=a1,b1,b=a2,b2.LetK=0.1,0.5×0.1,0.5⊂X and Γ:K⟶K be a mapping defined by
(33)Γa,b=cosa2,cosb2;a,b∈K.As0.450183611294874,0.450183611294874 is the fixed point of Γ, and a1,b1ϱ0.450183611294874,0.450183611294874 implies a1≤0.4501836112948740 and b1=0.450183611294874, this shows that Γ is L.R.Q.N.There are manyexamples in literature which shows that hyperbolic spaces are more general than Banach spaces for detail [1], so we have the following corollary which is Theorem 2.6 of [23].Corollary 3.
LetX be Banach space, having a compatible relation ϱ on it satisfying condition S. Let K be a closed and convex subset of X. Suppose Γ:K⟶K be a quasi ϱ-preserving L.R.Q.N mapping. If there exists some c0∈K such that c0ϱy for all y∈FΓ, then, the sequence (10) converges to a fixed point of Γ in K if and only if
(34)liminfn⟶∞dxn,FΓ=0.All the results of [23] are the consequences of Theorem 2.Following proposition from [1] will be helpful to proof the next results.Proposition 4.
LetH be a Takahashi convex structure on metric space X,d. If x,y∈X and t∈0,1, then
(1)
Hx,y,1=x and Hx,y,0=y,(2)
Hx,x,t=x,(3)
dx,Hx,y,t=1−tdx,y and dy,Hx,y,t=tdx,y.Theorem 5.
LetX be hyperbolic space, having a compatible relation ϱ on it satisfying condition S. Let K be a closed and convex subset of X. Suppose Γ:K⟶K be a quasi ϱ-preserving L.R.Q.N mapping. If there exists some c0∈K such that c0ϱy for all y∈FΓ, then sequence (11) converges to a fixed point of Γ in K if and only if
(35)limn⟶∞infdxn,FΓ=0.Proof.
Letq∈FΓ, then
(36)Γαq=HΓq,q,α=Hq,q,α=q,so FΓ⊂FΓα. Suppose y∈FΓα that is Γαy=y then from (11)
(37)dy,Γy=dHΓy,y,α,Γy≤1−αdΓy,Γy+αdy,Γy,implies dy,Γy=0 and y∈FΓ. Hence, FΓ=FΓα.Now, we show thatΓα is quasi ϱ-preserving. Suppose xϱy then Γxϱy and by the compatibility of ϱ,(38)αxϱαy,1−αΓxϱ1−αyandαx⊕1−αΓxϱαy⊕1−αΓx.Also(39)HΓx,x,α=αx⊕1−αΓxϱαy⊕1−αΓy=HΓy,y,α=y,implies
(40)Γαxϱy.Finally, we prove thatΓα is L.R.Q.N. For this let xϱy, then
(41)dΓαx,y=dHΓx,x,α,y≤αdΓx,y+1−αdx,y≤αdx,y+1−αdx,y=dx,y.Using Theorem2, we get the result.Remark 6.
IfΓ is ϱ-preserving, then Γα is also ϱ-preserving.Proof.
AsΓ is ρ-preserving, xϱy implies ΓxϱΓy. Using the compatibility of ϱ, we get
(42)αxϱαyand1−αΓxϱ1−αΓy,which implies
(43)HΓx,x,α=αx⊕1−αΓxϱαy⊕1−αΓy=HΓy,y,αΓαxϱΓαy.Now, we discuss the convergence of the iterative scheme (12) given as
(44)xn=Γα,βnxn−1=Γα,βnx0,whereΓα,β=HΓΓβ,I,α.Proposition 7.
Forα,β∈0,1, FΓα,β=FΓ, whenever Γ is ϱ-preserving nonexpansive and yϱΓβy, for y∈FΓα,β.Proof.
Letx∈FΓ that is Γx=x then
(45)Γα,βx=HΓΓβx,x,α=HΓHΓx,x,β,x,α=HΓx,x,α=Hx,x,α=x,so
(46)FΓ⊆FΓα,β.For the other inclusion, supposey∈FΓα,β that is Γα,βy=y, and consider
(47)dΓy,y=dΓy,Γα,βy=dΓy,HΓΓβy,y,α≤1−αdΓy,y+αdΓy,ΓΓβy≤1−αdΓy,y+αdy,ΓβyΓisnonexpansive=1−αdΓy,y+αdy,HΓy,y,β=1−αdΓy,y+αβdΓy,yusingdy,Hx,y,t=tdx,y=1−α+αβdΓy,y,which givesα1−βdΓy,y=0 and therefore y∈FΓ implies FΓα,β⊆FΓ. Hence,
(48)FΓα,β=FΓ.Proposition 8.
IfΓ is quasi ϱ-preserving L.R.Q.N mapping, then Γα,β is also quasi ϱ-preserving L.R.Q.N mapping.Proof.
Firstly we show that ifΓ is quasi ϱ-preserving, then Γα,β is quasi ϱ-preserving. For this, let x∈K and y∈FΓ⊆FΓα,β, such that xϱy implies Γxϱy. Also by Theorem 5, Γβ is quasi ϱ-preserving, y∈FΓβ; therefore, Γβxϱy. Further ΓΓβxϱy and by using the compatibility of ϱ, we get
(49)Γα,βx=HΓΓβx,x,α=αx⊕1−αΓΓβxϱαy⊕1−αy=Hy,y,α=y.Hence,Γα,β is quasi ϱ-preserving.Letx∈K and y∈FΓ, such that xϱy then ΓβxϱΓβy=y as Γβ is ϱ-preserving. Consider
(50)dΓα,βx,y=dHΓΓβx,x,α,y≤αdΓΓβx,y+1−αdx,y≤αdΓβx,y+1−αdx,yasΓisquasinonexpansive≤αdHΓx,x,β,y+1−αdx,y≤αβdΓx,y+α1−βdx,y+1−αdx,y≤αβdx,y+α1−βdx,y+1−αdx,y≤αβ+α−αβ+1−αdx,y≤dx,y.Theorem 9.
LetX be hyperbolic space, having a compatible relation ϱ on it satisfying condition S. Let K be a closed and convex subset of X. Suppose Γ:K⟶K be a quasi ϱ-preserving L.R.Q.N mapping. If there exists some c0∈K such that c0ϱy for all y∈FΓ, then the sequence (12) converges to a fixed point of Γ in K if and only if
(51)limn⟶∞infdxn,FΓ=0.Proof.
AsΓ is quasiϱ-preserving L.R.Q.N mapping, by the Proposition 8, Γα,β is also quasi ϱ-preserving L.R.Q.N mapping. For y∈FΓ,(52)Γα,βy=y,and Γα,βnc0ϱy for all n∈ℕ. By Theorem 2, we get conclusion.Next, we will discuss the convergence of the Agarwal iteration process defined in [28] as
(53)xn=Γγβnx0,where
(54)Γγβx=1−βΓx⊕βΓΓγx=HΓx,ΓΓγx,β,Γγx=1−γΓx⊕γx=HΓx,x,γ.Proposition 10.
IfΓ is quasi ϱ-preserving, then, Γγβ is also quasi ϱ-preserving.Proof.
SupposeΓ is quasi ρ-preservin,g then for x∈K and y∈FΓ, such that xϱy implies Γxϱy,Γγxϱy, and ΓΓγxϱy. Then, by using the compatibility of ϱ, we get
(55)Γγβx=1−βΓx⊕βΓΓγxϱ1−βy⊕βy=Hy,y,β=y.
Hence,Γγβ is quasi ρ-preserving.Proposition 11.
IfΓ is quasi ϱ-preserving L.R.Q.N mapping, then Γγβ is also quasi ϱ-preserving L.R.Q.N mapping.Proof.
Letx∈K and q∈FΓ, such that xϱq. Consider
(56)dΓγβx,q=dHΓx,ΓΓγx,β,q≤βdΓx,q+1−βdΓΓγx,q≤βdx,q+1−βdΓγx,qasΓisquasinonexpansive≤βdx,q+1−βdx,qasΓγisquasinonexpansive≤dx,q.
Hence,Γγβ is also quasi ϱ-preserving L.R.Q.N mapping.Proposition 12.
Forβ,γ∈0,1,FΓ⊆FΓγβ.Proof.
Letq∈FΓ that is Γq=q then
(57)Γγβq=HΓq,ΓΓγq,β=HΓq,ΓHΓq,q,γ,β=Hq,ΓHq,q,γ,β=Hq,Γq,β=Hq,q,β=q,so
(58)FΓ⊆FΓγβ.The following theorem describes the necessary and sufficient conditions for convergence of iterative sequence (53) of quasi ϱ-preserving L.R.Q.N mappings.Theorem 13.
LetX be hyperbolic space, having a compatible relation ϱ on it satisfying condition S. Let K be a closed and convex subset of X. Suppose Γ:K⟶K be a quasi ϱ-preserving L.R.Q.N mapping. If there exists some c0∈K such that c0ϱy for all y∈FΓ, then, the sequence (53) converges to a fixed point of Γ in K if and only if
(59)limn⟶∞infdxn,FΓ=0.Proof.
AsΓ is quasi ϱ-preserving L.R.Q.N mapping, by the Proposition 15, Γγβ is also quasi ϱ-preserving L.R.Q.N mapping. For y∈FΓ,(60)Γγβy=y,and Γγβnc0ϱy for all n∈ℕ. By Theorem 2, we get conclusion.Next, we will discuss the convergence of the iteration process (Abbas and Nazir) defined in [29] as
(61)xn=Γγα,βnx0,where
(62)Γγα,βx=1−αΓΓγβx⊕αΓΓγx=HΓΓγβx,ΓΓγx,α,Γγβx=1−βΓx⊕βΓΓγx=HΓx,ΓΓγx,β,Γγx=1−γΓx⊕γx=HΓx,x,γ.Proposition 14.
Forα,β,γ∈0,1,FΓ⊆FΓγα,β.Proof.
Letq∈FΓ that is Γq=q then
(63)Γγα,βq=HΓΓγβq,ΓΓγq,α=HΓHΓq,ΓΓγq,β,ΓHΓq,ΓΓγq,β,α=HΓHq,q,β,ΓHq,q,β,α=HΓq,Γq,α=Hq,q,α=q,so
(64)FΓ⊆FΓγα,β.Proposition 15.
IfΓ is quasiϱ-preserving, then Γγα,β is also quasi ϱ-preserving.Proof.
SupposeΓ is quasi ϱ-preserving, then, by Proposition 10 for β,γ∈0,1,Γγ and Γγβ are also quasi ϱ-preserving. Suppose for x∈K and y∈FΓ, such that xϱy implies Γxϱy,Γγxϱy,ΓΓγxϱy,Γγβxϱy, and ΓΓγβxϱy. Then, by using the compatibility of ϱ, we get
(65)Γγα,βx=1−αΓΓγβx⊕αΓΓγxϱ1−αy⊕αy=Hy,y,α=y.Hence,Γγα,β is quasi ϱ-preserving.Proposition 16.
IfΓ is quasi ϱ-preserving L.R.Q.N mapping, then Γγα,β is also quasi ϱ-preserving L.R.Q.N mapping.Proof.
Letx∈K and y∈FΓ, such that xϱy, then, Γγxϱy as Γγ is ϱ-preserving. Consider
(66)dΓγα,βx,y=dHΓΓγβx,ΓΓγx,α,≤αdΓΓγβx,y+1−αdΓΓγx,y≤αdΓγβx,y+1−αdΓγx,yasΓisquasinonexpansive≤αdx,y+1−αdx,y≤dx,y.Hence,Γγα,β is also quasi ρ-preserving L.R.Q.N mapping.The following theorem describes the necessary and sufficient conditions for convergence of iterative sequence (61) of quasi ϱ-preserving L.R.Q.N mappings.Theorem 17.
LetX be hyperbolic space, having a compatible relation ϱ on it satisfying condition S. Let K be a closed and convex subset of X. Suppose Γ:K⟶K be a quasi ϱ-preserving L.R.Q.N mapping. If there exists some c0∈K such that c0ϱy for all y∈FΓ, then, the sequence (61) converges to a fixed point of Γ in K if and only if
(67)limn⟶∞infdxn,FΓ=0.Proof.
AsΓ is quasi ϱ-preserving L.R.Q.N mapping, by the Proposition 15, Γγα,β is also quasi ϱ-preserving L.R.Q.N mapping. For y∈FΓ,(68)Γγα,βy=y,
andΓγα,βnc0ϱy for all n∈ℕ. By Theorem 2, we get conclusion.Next, we will discuss the convergence of the Noor iteration process defined in [30] as
(69)xn=Γα,β,γnx0,where
(70)Γα,β,γx=1−αΓΓβ,γx⊕αx=HΓΓβ,γx,x,α,Γβ,γx=1−βΓΓγx⊕βx=HΓΓγx,x,β,Γγx=1−γΓx⊕γx=HΓx,x,γ.Proposition 18.
Forα,β,γ∈0,1,FΓ⊆FΓβ,γ and FΓ⊆FΓα,β,γ.Proof.
Letq∈FΓ that is Γq=q then
(71)Γβ,γq=HΓΓγq,q,β=HΓHΓq,q,γ,q,β=HΓHq,q,γ,q,β=HΓq,q,β=Hq,q,β=q,so
(72)FΓ⊆FΓβ,γ.Now consider(73)Γα,β,γq=HΓΓβ,γq,q,α=HΓq,q,α=Hq,q,α=q,which implies
(74)FΓ⊆FΓα,β,γ.The following theorem describes the necessary and sufficient conditions for convergence of iterative sequence (53) of quasi ϱ-preserving L.R.Q.N mappings.Theorem 19.
LetX be hyperbolic space, having a compatible relation ϱ on it satisfying condition S. Let K be a closed and convex subset of X. Suppose Γ:K⟶K be a quasi ϱ-preserving L.R.Q.N mapping. If there exists some c0∈K such that c0ϱy for all y∈FΓ, then, the sequence (12) converges to a fixed point of Γ in K if and only if
(75)limn⟶∞infdxn,FΓ=0.Proof.
AsΓ is quasi ρ-preserving L.R.Q.N mapping, by the Proposition 18, Γα,β,γ is also quasi ρ-preserving L.R.Q.N mapping. For y∈FΓ,(76)Γα,β,γy=y,
andΓα,β,γnc0ϱy for all n∈ℕ. By Theorem 2, we get conclusion.All iterations converges tox,y=0.450183611294874,0.450183611294874.Table1 provides the rate of convergence of Mann, Ishikawa, Agarwal, Noor and Abbas iterations for the mapping given in Example 2. For α=γ=0.3,β=0.2, initial values are x0=0.1,y0=0.1, and tolerence=10−18.Table 1
Rate of convergence of Mann, Ishikawa, Agarwal, Noor, and Abbas iterations for the mapping given in Example2.
Iteration processMannIshikawaAgarwalNoorAbbasNumber of iterations1930133009Using different values forα,β and γ, we can see that in Table 2 Abbas iteration not only converges faster but also stable than other iterations.Table 2
Influence of parameters on convergence.
Forα=β=γ=0.1Iteration processMannIshikawaAgarwalNoorAbbasNumber of iterations151771411Forα=0.2=β,γ=0.3Iteration processMannIshikawaAgarwalNoorAbbasNumber of iterations112212209Forα=0.3,β=0.4,γ=0.5Iteration processMannIshikawaAgarwalNoorAbbasNumber of iterations192715265Forα=0.4,β=0.6,γ=0.8Iteration processMannIshikawaAgarwalNoorAbbasNumber of iterations2833193311Remark 20.
The numerical Example2 validates the existence of L.R.Q.N mappings for hyperbolic spaces in the perspective of different iterations with the help of Table 3 and Figure 1.Table 3
Table shows the behavior of different iterations of Example2 toward fixed point along the y-component (as the values of both components are same).
StepsMannIshikawaAgarwalNoorAbbas00.10.10.10.10.110.3782514578473090.3498656853580710.4691163101497750.3534881888569700.44805947672147720.4387345623397340.4207617666497370.4488587050723000.4236651399110460.45014584114059730.4484718471811850.4414970419949460.4502752524887840.4429261247836280.45018293730471940.4499303147350260.4476140306197770.4501772674302270.4481985674366770.45018359926709050.4501461882197050.4494230717790800.4501840504241170.4496407559961010.45018361108023060.4501780835288510.4499584706814490.4501835808977480.4496407559961010.45018361129104370.4501827948158040.4501169601880630.4501836133990040.4500351617371480.45018361129480580.4501834906973810.4501638794706420.4501836111492230.4501430166734080.45018361129487290.4501835934821150.4501777697328670.4501836113049560.4501725104342130.450183611294874100.4501836086638540.4501818819114160.4501836112941760.4501805756957780.450183611294874110.4501836109062610.4501830993139560.4501836112949220.4501827811917060.450183611294874120.4501836112374740.4501834597238100.4501836112948700.4501833842980930.450183611294874130.4501836112863950.4501835664225210.4501836112948740.4501835492212190.450183611294874140.4501836112936210.4501835980104910.4501836112948740.4501835943204540.450183611294874150.4501836112946890.4501836073620550.4501836112948740.4501836066531150.450183611294874160.4501836112948460.4501836101305690.4501836112948740.4501836100255560.450183611294874170.4501836112948700.4501836109501830.4501836112948740.4501836109477710.450183611294874180.4501836112948730.4501836111928290.4501836112948740.4501836111999560.450183611294874190.4501836112948740.4501836112646630.4501836112948740.4501836111999560.450183611294874200.4501836112948740.4501836112859300.4501836112948740.4501836112689180.450183611294874210.4501836112948740.4501836112922260.4501836112948740.4501836112877760.450183611294874220.4501836112948740.4501836112940900.4501836112948740.4501836112929330.450183611294874230.4501836112948740.4501836112946420.4501836112948740.4501836112943430.450183611294874240.4501836112948740.4501836112948050.4501836112948740.4501836112947280.450183611294874250.4501836112948740.4501836112948530.4501836112948740.4501836112948340.450183611294874260.4501836112948740.4501836112948680.4501836112948740.4501836112948630.450183611294874270.4501836112948740.4501836112948720.4501836112948740.4501836112948710.450183611294874280.4501836112948740.4501836112948730.4501836112948740.4501836112948730.450183611294874290.4501836112948740.4501836112948730.4501836112948740.4501836112948730.450183611294874300.4501836112948740.4501836112948740.4501836112948740.4501836112948740.450183611294874Figure 1
Convergence behavior of Mann, Ishikawa, Agarwal, Noor iterations with Abbas forα=γ=0.3,β=0.2, initial values are x0=0.1,y0=0.1, and tolerence=10−18.


## 4. Conclusion

In the present article, the concept of monotone has been generalized toϱ-preserving in the framework of hyperbolic space. We also constructed a nontrivial example to show that locally related quasi-nonexpansive mapping is not necessarily ϱ-preserving or ϱ-preserving nonexpansive and approximate the fixed point numerically and compare the convergence result of different iterations with Abbas iteration by using Matlab.



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*Source: 3785584-2022-06-14.xml*