# Fixed Points of Monotone Total Asymptotically Nonexpansive Mapping in Hyperbolic Space via New Algorithm

Authors: Amna Kalsoom; Maliha Rashid; Tian-Chuan Sun; Amna Bibi; Abdul Ghaffar; Mustafa Inc; Ayman A. Aly
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, we consider an extensive class of monotone nonexpansive mappings and introduce a new iteration algorithm to approximate the fixed point for monotone total asymptotically nonexpansive mappings in the framework of hyperbolic space. Faster convergence and stability results are proved for that iteration; also, fixed point is approximated numerically in a nontrivial example by using MATLAB.


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## Body

## 1. Introduction

The concept of hyperbolic space was given by Reich and Shafrir [1] in 1990, which is defined as a metric space Σ,σ that has a family Fourier of metric lines; for any two unique endpoints l,m∈Σ, there is a unique metric line in Fourier. This metric line works out a unique metric segment symbolize by l,m, which is an isometric image of 0,σl,m. A unique point z∈l.m is denoted by αl⊕1−αm which satisfies(1)σl,z=1−ασl,m,σz,m=ασl,m,where α∈0,1. Such a metric space is called a convex metric space, such that(2)σαl⊕1−αm,αp⊕1−αq≤ασl,p+1−ασm,q,for all l,m,p,q∈Σ, then Σ is called hyperbolic metric space (abbreviated as H.M.S).The class of hyperbolic spaces contains the normed spaces, CAT(0) spaces, and many others. There are many examples in literature which show that hyperbolic spaces are more general than Banach spaces; for details, see [2] and Example 1.1 of [3].Recently, a new direction has been discovered dealing with the extension of the Banach Contraction Principle [4] to partially ordered metric spaces. The case of monotone nonexpansive mappings was first considered in [5]. After that, Dehaish and Khamsi [6] gave an analogue to Browder [7] and Göhde [8] fixed point theorems for monotone nonexpansive mappings. In 2018, Alfuraidan and Khamsi [9] extended Goebel and Kirk’s fixed point theorem [10] for asymptotically nonexpansive mappings to the case of monotone mappings. Multiple articles [11–14] can be found in the literature on fixed point of asymptotically nonexpansive mapping using multistep iterations and strong convergence analysis.In 2016, Alber et al. [15] introduced the concept of total asymptotically nonexpansive mappings that generalizes the family of mapping that are the extension of asymptotically nonexpansive mappings. Example 1 of [16] and Example 3.1 of [4] show that total asymptotically nonexpansive mappings properly contain the asymptotically nonexpansive mappings.In this article, we define monotone total asymptotically nonexpansive mappings (abbreviated as M.T.A.N.M) and also extend Alber’s fixed point theorem [10] for the respective class. In addition, this result also generalizes the results of Alfuraidan and Khamsi in hyperbolic space [9].In Section4, we introduce a new iteration scheme, prove fast convergence and stability results, and also provide comparison with some famous iterations listed as Banach [17], Mann [18], Ishikawa [19], Agarwal et al. [20], Noor [21], Abbas and Nazir [22], Vatan Two-step [23], an accelerated iteration [24] (by Chen and Wen [24]), and Thakur New [25]. Numerically, we compare the convergence of new iteration with these iterations in a nontrivial example.


## 2. Preliminaries

LetΣ,σ,≺¯ be a partially ordered (abbreviated as P.O) metric space, any two points a,b∈Σ are comparable whenever a≺¯b or b≺¯a.Definition 1.
ConsiderΣ,σ,≺¯ be a partially ordered space and Y be a self map of Σ, which is said to be(i)
monotone or order preserving [5] if(3)a≺¯b⇒Ya≺¯Yb(ii)
monotone Lipschitzian mapping [5] if Y is order preserving and there exist L≥0 such that(4)σYa,Yb≤Hσa,b.
IfH=1, the mapping Y is said to be order preserving nonexpansive mapping(iii)
monotone asymptotically nonexpansive mapping [9] if there exists a sequence Hυ for υ∈ℕ such that(5)limυ⟶∞Hυ=1,σYυa,Yυb≤Hυσa,b,for every a,b∈Σ, such that a and b are comparableNow, we will define M.T.A.N.M in hyperbolic space.Definition 2.
LetΣ be a hyperbolic metric space having a nonempty subset K. A self map Y is monotone total asymptotically nonexpansive mapping on K if there exist nonnegative sequences μυ and ξυ with μυ⟶0,ξυ⟶0, as υ⟶∞, a strictly increasing continuous function(6)ϑ:0,∞⟶0,∞withϑ0=0,such that(7)σYυs,Yυt≤σs,t+μυϑσs,t+ξυforallυ≥1,and there exists a constant R∗>0 such that ϑλ≤R∗λ for λ>0, then(8)σYυs,Yυt≤1+R∗μυσs,t+ξυ,for every comparable elements s,t∈K.Example 1.
Consider the real lineℝ as a hyperbolic metric space and K be the subset of ℝ, K=0,π/2, and T:K⟶K be a mapping defined by Tx=sinx. Suppose that there exist two nonnegative sequences xn and ξn with xn⟶0 and ξn⟶0 as n⟶0 and a strictly increasing continuous function θ:0,∞⟶0,∞ with θ0=0,θλ=λ+1,λ∈0,∞. Then, T is M.T.A.N.M.Example 2.
Consider the real planeℝ2 as a hyperbolic metric space. Let σ:Σ×Σ⟶ℝ be defined as(9)σw,h=w1−h1+w1w2−h1h2wherew=w1,w2,h=h1,h2.
LetK=0,1×0,1⊂Σ and Y:K⟶K be a mapping defined by(10)Yw,h=w10,h10ifw,h≠12,12;0,0ifw,h=12,12.
Thus,0,0 is the fixed point of Y. Suppose ∃ two nonnegative sequences μυ and ξυ with μυ⟶0,ξυ⟶0, as υ⟶∞, and a strictly increasing continuous function(11)ϑ:0,∞⟶0,∞withϑ0=0.Now, we consider the following cases with assumptionsw1,w2≺¯h1,h2:Case 1.
Ifw1,w2=1/2,1/2=h1,h2, then Yw1,w2=0,0=Yh1,h2. In this case, Y is monotone and satisfies all the conditions of total asymptotically nonexpansive mapping.Case 2.
Ifw1,w2=1/2,1/2,h1,h2≠1/2,1/2, then Yw1,w2=0,0, Yh1,h2=h1/10,h2/10, and Yυw1,w2=0,0,Yυh1,h2=h1/10υ,h2/10υ. Also,(12)σYυw1,w2,Yυh1,h2=h110υ+h1h2102υ≤12−h1+122−h1h2≤σw1,w2,h1,h2+μυϑσw1,w2,h1,h2+ξυimplies that Y is a M.T.A.N.M.Case 3.
Ifw1,w2≠1/2,1/2≠h1,h2, then Yw1,w2=w1/10,w2/10,Yh1,h2=h1/10,h2/10, and Yυw1,w2=w1/10υ,w2/10υ,Yυh1,h2=h1/10υ,h2/10υ. Now,(13)σYυw1,w2,Yυh1,h2=w1−h110υ+w1w2−h1h2102υ≤w1−h1+w1w2−h1h2≤σw1,w2,h1,h2+μυϑσw1,w2,h1,h2+ξυ.Hence,Y is M.T.A.N.M.Next, we have some definitions and lemmas that will be useful in the proof of the main result.Definition 3 (see [26]).
A hyperbolic spaceΣ with metric σ is said to be uniformly convex if for any w∈Σ, for every z>0, and for each ε>0(14)δz,ε=inf1−1zσ12s⊕12t,w;σs,w≤z,σt,w≤z,σs,t≥zε>0.The functionδ is called the modulus of uniform convexity of Σ.A hyperbolic spaceΣ,σ satisfies the propertyR [26]. If Kυ is nonincreasing sequence of nonempty, bounded, closed, and convex subset of Σ, ∩υ=1∞Kυ≠ϑ.Definition 4 (see [27]).
A bounded sequencerυ∈Σ is Δ-convege tor∈Σ, if r is the unique asymptotic centre of every subsequence rυkofrυ.Definition 5 (see [9]).
A partially ordered hyperbolic metric spaceΣ satisfies the monotone weak Opial condition if any sequence in Σ which is monotone and weakly converges to s, then the following(15)limsupυ⟶∞σsυ,s≤limsupυ⟶∞σsυ,t,for every t∈Σ such that s≺¯t or t≺¯s.Throughout in article, the order intervals are assumed to be closed and convex and any of the subsets(16)s,⟶=y∈Σ;s≺¯y,⟵,t=y∈Σ;y≺¯t.for every s,t∈Σ.Lemma 6 [9].
SupposeΣ,σ be uniformly convex H.M.S, and K be a subset of Σ which is closed nonempty and convex. Let τ:K⟶0,∞ be a type function if ∃ a bounded sequence sυ∈Σ such that(17)τs=limsupυ⟶∞σsυ,s,for any s∈K. Since Σ is hyperbolic space, τ is convex and continuous with distinctive minimum point u∈K such that(18)τu=infτs;s∈K=τ0.Moreover, ifYυu in K is the minimizing sequence of τ, i.e.,(19)limυ⟶∞τYυu=τ0,then, Yυu strongly converges to u.


## 3. Main Result

Theorem 7.
Let a uniformly convex P.O H.M.S beΣ,σ,≼ with nonempty convex closed bounded subset K. Let Y be a continuous M.T.A.N.M on K. Assume ∃s0∈K, such that s0≼Ys0. Then, Y has a fixed point.Proof.
Lets0∈K be such that(20)s0≼Ys0.
By the monotonicity ofY, we get(21)Yυs0≼Yυ+1s0,for each υ∈ℕ, and Yυs0 is a monotone increasing sequence. Also, the order intervals are closed and convex. So, we have(22)K∞=⋂υ≥0Yυs0,⟶∩K=∩s∈K;Yυs0≼s≠ϕ.
Lets∈K∞, then(23)Yυs0≼s,and the monotonicity of Y implies(24)YYυs0=Yυ+1s0≼Ys,for every υ∈ℕ, i.e., YK∞⊂K∞. Consider the type function τ:K∞⟶0,+∞ produced by Yυs0 given as τs=limsupυ⟶∞σYυs0,s for any s∈K∞. Above lemma shows the occurrence of a unique a∈K∞ such that τa=infτs;s∈K∞=τ0. Since a∈K∞, we have Yda∈K∞, for every d∈ℕ, which implies(25)τYda=limsupυ⟶∞σYυs0,Yda≤limsupυ⟶∞σYυs0,a+μpϑσYυs0,a+ξd=τa+μdlimsupυ⟶∞ϑσYυs0,a+ξd.
AsY is total asymptotically nonexpansive mapping, so μd⟶0, ξd⟶0, when d⟶∞. Hence,(26)limd⟶∞τYda=τ0,hence, Yda is a minimizing sequence of τ. Using Lemma 6, Yda converges to a. Since Y is continous, we have(27)limυ⟶∞YYυa=Ya=a,
i.e.,a is a fixed point of Y.☐The following corollary is the conclusion of Theorem 3.3 of [9].Corollary 8.
Let a uniformly convex P.O H.M.S beΣ,σ,≼ with nonempty convex closed bounded subset K. Let Y be a continuous M.T.A.N.M on K. Assume ∃s0∈K, such that s0≼Ys0. Then, Y has a fixed point.The corollary given below is the consequence of Theorem7 by replacing the continuity condition with weak Opial condition.Corollary 9.
LetΣ,σ′ be a uniformly convex P.O H.M.S, satisfying monotone weak Opial condition with a nonempty convex closed bounded subset K. Let Y be a M.T.A.N.M on K. Assume ∃s0∈K, such that s0≼Ys0. Then, Y has a fixed point.


## 4. Convergence Theorem and Stability Results

We introduce the new iteration scheme given below, letK be a nonempty convex subset of a hyperbolic space Σ, for υ≥0, where αυ, βυ, and γυ are sequences in 0,1, such that(28)s0∈K,ηυ=1−γυsυ⊕γυYυsυ,θυ=1−βυηυ⊕βυYυηυ,sυ+1=Yυ1−αυYυsυ⊕αυYυθυ.Fastness and stability play an important role for an iteration process to be preferred on another iteration process, so now, we prove that new iteration is stable and has good speed of convergence than others. For faster convergence and new class of mapping on metric spaceΣ,σ introduced by Berinde [28], satisfying(29)σYs,Yt≤aσs,t+Lσs,Ys,here, we will modify this mapping as(30)σYυs,Yυt≤aσs,t+Lσs,Yυs,for all s,t∈Σ, where a∈0,1 and L≥0.The following definitions and lemma will be helpful for faster convergence results given in [29].Definition 10.
Letaυ and bυ be two sequences, having convergent points a and b, respectively, then aυ converges faster than bυ if(31)limυ⟶∞σaυ,aσbυ,b=0.Definition 11.
Letkυ and lυ be two fixed point schemes that converge to the same fixed point s. If(32)σkυ,s≤sυandσlυ,s≤tυforallυ≥0,where sυ and tυ are two sequences that converge to 0. If sυ converges faster than tυ, then kυ converges faster than lυ to s.Lemma 12.
Ifb is a real number such that 0≤b<1 and aυυ=0∞ be a sequence such that(33)limυ⟶∞aυ=0,then for any positive sequence sυυ=0∞ satisfying(34)sυ+1≤bsυ+aυ⇒limυ⟶∞sυ=0.Lemma 13 (see [4]).
Supposelυ, mυ, and δυ be sequences of nonnegative satisfying(35)lυ+1≤1+δυlυ+mυ∀υ≥1.IfΣδυ<∞ and Σmυ<∞, then limυ⟶∞lυ+1 exists.Lemma 14 (see [4]).
SupposeΣ be a uniformly convex H.M.S. Let P∈0,∞ be such that(36)limsupυ⟶∞σsυ,a≤P,limsupυ⟶∞σθυ,a≤P,limυ⟶∞σa,αυsυ⊕1−αυθυ=P,where αυ∈a,b with 0<a≤b<1. Then, we get(37)limυ⟶∞σsυ,θυ=0.Lemma 15 (see [26]).
LetΣ,σ′ be a P.O hyperbolic space. Let K be a nonempty convex and closed subset of Σ. Let Y:K⟶K be a monotone mapping. Let s1∈K, such that s1≺¯Ys1 or Yt1≺¯t1. Then, the sequence sυ in 1 then(a)
sυ≺¯Ysυ≺¯sυ+1 or sυ+1≺¯Ysυ≺¯sυ(b)
sυ≺¯s or s≺¯sυ, provided that sυΔ-converge to s∈K,∀υ∈ℕLemma 16.
Let a uniformly convex P.O H.M.S beΣ,σ,≼ with nonempty convex closed bounded subset K. Let Y:K⟶K be a M.T.A.N.M with FY≠ϑ. If the sequence sυ is defined by 1 with s1≺¯Ys1 or Ys1≺¯s1. Then, the following holds(38)alimυ⟶∞σsυ,sexistfors∈FY,blimυ⟶∞σYυsυ,sυ=0.Proof.
Lets∈FY⇒Ys=s. By the above lemma s≺¯sυ, as Y is monotone(39)Ys≺¯Ysυ⇒s≺¯Yυsυ.
Now, using Definition2 and after simplification, we get(40)σηυ,s≤1+R∗μυσsυ,s+ξυ.
Again using Definition2 and (40), we get(41)σθυ,s≤1+R∗μυ2σsυ,s+2+R∗μυξυ.
Consider(42)σsυ+1,s=σYυ1−αυYυηυ⊕αυYυθυ,s.
Using Definition2, (40) and (41), we get(43)σsυ+1,s≤1+δυσsυ,s+bυ,where δυ=4+6R∗μυ+4R∗μυ2+R∗μυ3R∗μυ and bυ=4+6R∗μυ+4R∗μυ2+R∗μυ3ξυ. Using Lemma 13, limυ⟶∞σsυ,s exist for s∈FY.
For partb, we have to show that(44)limυ⟶∞σYυsυ,sυ=0,the proof resembles to Theorem 2.1 of [4].☐☐Theorem 17.
Let a uniformly convex P.O H.M.S beΣ,σ,≼ with convex closed bounded and nonempty subset Y. Let Y be a continuous M.T.A.N.M on Y, with FY≠ϑ. If sυ is defined by 1 with s1≺¯Ys1 or Ys1≺¯s1. If s≺¯s1 or s1≺¯s for s∈FY, then sυΔ-converges to a fixed point of Y.Proof.
By Lemma16,(45)limυ⟶∞σsυ,sexist∀s∈FY,sequence is bounded and limυ⟶∞σsυ,Yυsυ=0. Let sυk be any subsequence of sυ for k∈ℕ, such that sυkΔ-converges to p∈Y. By Lemma 15, we have(46)s1≺¯sυk≺¯p.
Now, we have to show that everyΔ-convergent subsequence of sυ has a unique Δ-limit in FY. Let sυk and sυr be two subsequences of sυΔ-converging to w and h, respectively. By Lemma 16, sυk is bounded and(47)limυ⟶∞σsυk,Yυsυk=0.
We claim thatw∈FY, and τ produced by sυk is(48)τw=limsupk⟶∞σsυk,w.
From Theorem7, Yw=w, same for Yh=h. By the definition of Δ-convergence and Lemma 6, we have(49)limsupk⟶∞σsυ,w=limsupk⟶∞σsυk,w<limsupk⟶∞σsυk,h=limsupυ⟶∞σsυ,h=limsupr⟶∞σsυr,h<limsupr⟶∞σsυr,w=limsupυ⟶∞σsυ,w,which is contradiction, unless w=h.☐Theorem 18.
Let a uniformly convex P.O H.M.S beΣ,σ,≼ with nonempty convex closed bounded subset K. Let Y be a continuous M.T.A.N.M on K with FY≠ϑ. If sυ is defined by 1 with s1≺¯Ys1 or Ys1≺¯s1. If s≺¯s1 or s1≺¯s for s∈FY, then sυ converges to a fixed point of Y if and only if(50)liminfυ⟶∞σsυ,FY=0.Proof.
Ifsυ converges to a fixed point of Y, then(51)liminfυ⟶∞σsυ,FY=0forallυ∈ℕ.
Conversely, considerliminfυ⟶∞σsυ,FY=0. From Lemma 16(52)limυ⟶∞σsυ,sexistforeachs∈FY,therefore, limυ⟶∞σsυ,F(Y exists. As liminfυ⟶∞σsυ,FY=0, so we get(53)limυ⟶∞σsυ,FY=0.
Now, we prove thatsυ is a Cauchy sequence in K. For ε>0,∃υ0∈ℕ, such that for all υ≥υ0(54)σsυ,FY<ε2,in particular infσsυ0,s:s∈FY<ε/2, so ∃ a fixed point p∈FY such that(55)σsυ0,s<ε2.
Form,υ≥υ0,(56)σsυ+m,sυ≤σsυ+m,s+σs,sυ<2σsυ0,s<ε.
Hence,sυ is a Cauchy sequence in closed subset K of Σ; therefore, it converges in K such that(57)limυ⟶∞sυ=qforq∈K.
As we havelimυ⟶∞σsυ,FY=0⇒σq,FY=0, since FY is closed so q∈FY.
Now, we prove that newly proposed iteration scheme1 is faster than Thakur New [25] for a mapping defined in 2 in hyperbolic metric space.☐Theorem 19.
LetΣ,σ,≼ be a P.O H.M.S. Let K be a nonempty convex closed bounded subset of Σ, and Y be a mapping satisfying (30) with FY≠ϑ. Let sυ be defined by (28), and uυ defined in [25], then sυ converges faster than uυ.Proof.
Lets∈FY⇒Yυs=s. Now, using (28) and (30), we have(58)σηυ,s≤1−γυσsυ,s+γυaσsυ,s+Lσs,Yυs=1−γυ1−aσsυ,s.
Again, using (28), (30), and (58), we have(59)σθυ,s≤1−βυ1−a1−γυ1−aσsυ,s.
Further, consider(60)σsυ+1,s=σYυ1−αυYυηυ⊕αυYυθυ,s.
Now, using (30) and then (28), we get(61)σsυ+1,s≤aa1−γυ1−a1−1−aαυβυσsυ,s.
Let(62)Kυ=aυa1−γ1−a1−1−aαβυσs1,s,and συ=aυ1−γ1−a1−1−aαβυσu1,s calculated in Theorem 3.1 of Thakur New [25]. Then,(63)Kυσυ=aυa1−γ1−a1−1−aαβυσs1,saυ1−γ1−a1−1−aαβυσu1,s=a1−γ1−a1−1−aαβυσs1,s1−γ1−a1−1−aαβυσu1,s⟶0asυ⟶∞.
Hence,sυ converges faster than uυ.☐Now, we will prove the stability result for this we have the following definition by [30].Definition 20.
Letpυυ=0∞⊂K be any arbitrary sequence, then iteration sequence sυ+1 converging to unique fixed point s is said to be Y-stable if for ευ=σpυ,sυ+1υ≥0, we have(64)limυ⟶∞ευ=0⇔limυ⟶∞pυ=s.Theorem 21.
LetΣ,σ′ be a P.O H.M.S. Let K be a nonempty convex closed bounded subset of Σ, and Y be a mapping satisfying 2 with FY≠ϑ. Let sυ be defined by 1, satisfying Σαυ=∞, then the iteration 1 is Y-stable.Proof.
Letpυυ=0∞⊂K be any arbitrary sequence, the sequence defined by (28) converging to unique fixed point s, and(65)ευ=σpυ,sυ+1.
We have to prove that(66)limυ⟶∞ευ=0⇔limυ⟶∞pυ=s.
Letlimυ⟶∞ευ=0, then(67)σpυ,s≤σpυ,sυ+1+σsυ+1,s=ευ+σYυ1−αυYυsυ⊕αυYυθυ,s≤ευ+aσ1−αυYυηυ⊕αυYυθυ,s+Lσs,Yυs=ευ+aσ1−αυYυηυ⊕αυYυθυ,s≤ευ+aa1−γυ1−a1−1−aαυβυσsυ,s,
0≤aa1−γυ1−a1−1−aαυβυ<1, applying limυ⟶∞, we get(68)limυ⟶∞σpυ,s=0⇒limυ⟶∞pυ=s.
Conversely, letlimυ⟶∞pυ=s, we have(69)ευ=σpυ,sυ+1≤σpυ,s+σsυ+1,s≤σpυ,s+aa1−γυ1−a1−1−aαυβυσsυ,s⇒limυ⟶∞ευ=0.
Hence, it isY-stable.☐We have a nontrivial example for M.T.A.N.M, and fixed point is numerically approximated by using MATLAB.Example 3.
LetΣ=ℝ2 be a hyperbolic space. Define a relation as(70)r1,t1′r2,t2⇔r1≤r2andt1≤t2.
Letσ:Σ×Σ⟶ℝ be defined as(71)σr,t=r1−r2+r1t1−r2t2wherer=r1,t1,y=r2,t2.
LetK=0,3×0,3⊂Σ and Y:K⟶K be a mapping defined by(72)Yr,t=1−cosr2,expt/2−12;r,t∈K.
As0,0 is the fixed point of Y, we have to show that Y is monotone for this consider(73)r1,t1′r2,t2⇔r1≤r2andt1≤t2,then(74)1−cosr12≤1−cosr22andexpt1/2−12≤expt2/2−12,that is(75)Yr1,t1′Yr2,t2.
Next,(76)Yυr1,t1=0,0,Yυr2,t2=0,0,forlargeυ,⇒σYυr1,t1,Yυr2,t2=0≤σr1,t1,r2,t2.
Hence,Y is M.T.A.N.M with μυ=ξυ=0.The rate of convergence of Mann, Ishikawa, Agarwal, Noor, Abbas, Thakur, and accelerated and new iterations for the mapping defined in Example2 is given below. Table 1 shows the convergence behavior of Mann, Ishikawa, Agarwal, and Noor iterations with new iteration for the parameters α=0.6,β=0.3, and γ=0.5, with the initial values x0=0,y0=3, and tolerance=10−6. New iteration requires less number of iterations for convergence.Table 1
The convergence behavior of Mann, Ishikawa, Agarwal, and Noor iterations with new iteration for the parametersα=0:6,β=0:3,andγ=0:5, with the initial values x0=0,y0=3, and tolerance=10-6.
StepsMannIshikawaAgarwalNoorNew0(0, 3)(0, 3)(0, 3)(0, 3)(0, 3)1(0, 2.244506)(0, 2.013105)(0, 1.509443)(0, 1.916515)(0, 0.311990)2(0, 1.519333)(0, 1.196547)(0, 0.479098)(0, 1.097796)(0, 0.011353)3(0, 0.949002)(0, 0.663545)(0, 0.116180)(0, 0.594503)(0, 0.000384)4(0, 0.561764)(0, 0.355178)(0, 0.025817)(0, 0.313100)(0, 0.000012)5(0, 0.321995)(0, 0.186769)(0, 0.005616)(0, 0.162620)(0, 0.000000)6(0, 0.181202)(0, 0.097333)(0, 0.001216)(0, 0.083873)(0, 0.000000)7(0, 0.100930)(0, 0.050492)(0, 0.000263)(0, 0.043105)(0, 0.000000)8(0, 0.055900)(0, 0.026132)(0, 0.000056)(0, 0.022113)(0, 0.000000)9(0, 0.030863)(0, 0.013508)(0, 0.000012)(0, 0.011333)(0, 0.000000)10(0, 0.017010)(0, 0.006978)(0, 0.000002)(0, 0.005806)(0, 0.000000)11(0, 0.009366)(0, 0.003603)(0, 0.000000)(0, 0.002973)(0, 0.000000)12(0, 0.005155)(0, 0.001860)(0, 0.000000)(0, 0.001522)(0, 0.000000)13(0, 0.002836)(0, 0.000960)(0, 0.000000)(0, 0.000779)(0, 0.000000)14(0, 0.001560)(0, 0.000495)(0, 0.000000)(0, 0.000399)(0, 0.000000)15(0, 0.000858)(0, 0.000256)(0, 0.000000)(0, 0.000204)(0, 0.000000)16(0, 0.000472)(0, 0.000132)(0, 0.000000)(0, 0.000104)(0, 0.000000)17(0, 0.000259)(0, 0.000068)(0, 0.000000)(0, 0.000053)(0, 0.000000)18(0, 0.000142)(0, 0.000035)(0, 0.000000)(0, 0.000027)(0, 0.000000)19(0, 0.000078)(0, 0.000018)(0, 0.000000)(0, 0.000014)(0, 0.000000)20(0, 0.000043)(0, 0.000009)(0, 0.000000)(0, 0.000007)(0, 0.000000)21(0, 0.000023)(0, 0.000004)(0, 0.000000)(0, 0.000003)(0, 0.000000)22(0, 0.000013)(0, 0.000002)(0, 0.000000)(0, 0.000001)(0, 0.000000)23(0, 0.000007)(0, 0.000001)(0, 0.000000)(0, 0.000000)(0, 0.000000)24(0, 0.000003)(0, 0.000000)(0, 0.000000)(0, 0.000000)(0, 0.000000)25(0, 0.000002)(0, 0.000000)(0, 0.000000)(0, 0.000000)(0, 0.000000)26(0, 0.000001)(0, 0.000000)(0, 0.000000)(0, 0.000000)(0, 0.000000)27(0, 0.000000)(0, 0.000000)(0, 0.000000)(0, 0.000000)(0, 0.000000)Table2 shows the convergence of Abbas, Thakur, and accelerated iteration with new iteration for the same initial values, parameters, and tolerance.Table 2
The convergence of Abbas, Thakur, and accelerated iteration with new iteration for the same initial values, parameters, and tolerance 10-6.
StepsAbbasAn acceleratedThakurNew0(0, 3)(0, 3)(0, 3)(0, 3)1(0, 0.917554)(0, 0.890744)(0, 0.969760)(0, 0.311990)2(0, 0.133217)(0, 0.126269)(0, 0.161770)(0, 0.011353)3(0, 0.015917)(0, 0.015024)(0, 0.022578)(0, 0.000384)4(0, 0.001852)(0, 0.001747)(0, 0.003065)(0, 0.000012)5(0, 0.000214)(0, 0.000202)(0, 0.000414)(0, 0.000000)6(0, 0.000024)(0, 0.000023)(0, 0.000056)(0, 0.000000)7(0, 0.000002)(0, 0.000002)(0, 0.000007)(0, 0.000000)8(0, 0.000000)(0, 0.000000)(0, 0.000001)(0, 0.000000)9(0, 0.000000)(0, 0.000000)(0, 0.000000)(0, 0.000000)The following figures show the convergence behavior of different iterations with new iteration in Figures1–3.Figure 1
Convergence behavior of Mann, Ishikawa, and Abbas with new iteration for the initial values (0,3) with parametersα=0.6,β=0.3,γ=0.5.Figure 2
Convergence behavior of Agarwal, accelerated with new iteration for the initial values (0,3) with parametersα=0.6,β=0.3,γ=0.5.Figure 3
Convergence behavior of Noor and Thakur with new iteration for the initial values (0,3) with parametersα=0.6,β=0.3,γ=0.5.All iterations converges tox,y=0,0. Comparison shows that new iteration requires the least number of iterations for convergence. Table 3 shows that different parameters have an effect on iterations and by changing the initial values, new iteration not only converges faster but also stable than other iterationsTable 3
Influence of parameters and initial values by setting stopping parameter at 10-15.
Forα=0.7, β=γ=0.1 with x0,y0=0,3Iteration processMannIshikawaAgarwalNoorAbbasAn acceleratedThakurNewNumber of iterations5048264822222513Forα=0.2=β, γ=0.3 with x0,y0=0,2.5Iteration processMannIshikawaAgarwalNoorAbbasAn acceleratedThakurNewNumber of iterations2212022620315152212Forα=0.3=β=0.7, γ=0.2 with x0,y0=0,2Iteration processMannIshikawaAgarwalNoorAbbasAn acceleratedThakurNewNumber of iterations1401162411516162112Forα=0.5=β=0.9, γ=0.2 with x0,y0=0,1Iteration processMannIshikawaAgarwalNoorAbbasAn acceleratedThakurNewNumber of iterations7457205618181811Forα=0.3=β=0.9, γ=0.5 with x0,y0=0,1.5Iteration processMannIshikawaAgarwalNoorAbbasAn acceleratedThakurNewNumber of iterations1381092310614141711


## 5. Conclusions

In the present article, the concept of monotone asymptotically nonexpansive mapping has been generalized to monotone total asymptotically nonexpansive mapping in the framework of hyperbolic space. New iteration has been introduced to approximate the fixed point for that mapping. We proved the existence of fixed point, faster convergence, and stability results for new iteration. We also constructed a nontrivial example to approximate the fixed point numerically and compare the convergence result of new iteration with some well-known iterations by using MATLAB.By relaxing the condition of monotonicity, we can also achieve some similar results presented in recent articles [31, 32] by using the proposed iteration.



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