# Fixed Point Approximation of Monotone Nonexpansive Mappings in Hyperbolic Spaces

Authors: Amna Kalsoom; Naeem Saleem; Hüseyin Işık; Tareq M. Al-Shami; Amna Bibi; Hafsa Khan
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Fixed points of monotoneα-nonexpansive and generalized β-nonexpansive mappings have been approximated in Banach space. Our purpose is to approximate the fixed points for the above mappings in hyperbolic space. We prove the existence and convergence results using some iteration processes.


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## Body

## 1. Introduction

In 1965, Browder [1], Göhde [2], and Kirk [3] started working in the approximation of fixed point for nonexpansive mappings. Firstly, Browder obtained fixed point theorem for nonexpansive mapping on a subset of a Hilbert space that is closed bounded and convex. Soon after, Browder [1] and Göhde [2] generalized the same result from a Hilbert space to a uniformly convex Banach space. Kirk [3] utilized normal structure property in a reflexive Banach space to sum up the similar results. Recently, Dehici and Najeh [4] and Tan and Cho [5] approximated fixed point result for nonexpansive mappings in Banach space and Hilbert space.Fixed point theory in partially ordered metric spaces has been initiated by Ran and Reurings [6] for finding application to matrix equation. Nieto and Lopez [7] extended their result for nondecreasing mapping and presented an application to differential equations. Recently, Song et al. [8] extended the notion of α-nonexpansive mapping to monotone α-nonexpansive mapping in order Banach spaces and obtained some existence and convergence theorem for the Mann iteration (see also [9] and the reference therein). Motivated by the work of Suzuki [10], Aoyama and Kohsaka [11], Dehaish and Khamsi [9], and Song et al. [8], Pant and Shukla obtained existence results in ordered Banach space for a wider class of nonexpansive mappings [12, 13]. There are many mathematicians who worked on weak and strong convergence of nonexpansive mappings and its generalizations by using one step, two step, and multistep iteration process ([8, 14, 15]). We obtain existence results in partial ordered hyperbolic space for monotone generalized α-nonexpansive and monotone generalized β-nonexpansive map. Particularly, in Section 3, some auxiliary results and existence theorems for monotone α-nonexpansive mappings in ordered hyperbolic spaces are presented. In Section 4, we presented numerical examples and graphical representation. In Section 5, we obtained some existence results for monotone generalized β-nonexpansive mappings in ordered hyperbolic spaces.


## 2. Preliminaries

In1976, the concept of Δ-convergence was given by Lim [14].Lim [14] initiated the idea that in a metric space, Δ-convergence is possible. This concept is adapted for CAT(0) spaces by Kirk and Panyanak [16], and they have indicated that in numerous Banach space, outcomes comprising weak convergence were having exactly accurate analogs in this manner.Definition 1.
A self mapT on W is known as Lipschitz, if there exists k≥0 such that
(1)σTξ,Tη≤kσξ,η,forallξ,η∈W.
T is called to be contractive if k∈0,1, and T is called nonexpansive mapping, if k=1, that is,
(2)σTξ,Tη≤σξ,η,forallξ,η∈W.
Suzuki [10] introduced an interesting generalized nonexpansive mapping as follows.Definition 2.
A self mapT on W is said to satisfy condition (3), if for all ξ,η∈W,
(3)12σξ,Tξ≤σξ,η⇒σTξ,Tη≤σξ,η.Suzuki type generalized nonexpansive mapping is another name of self mapT holding condition (3).Many generalizations of nonexpansive mapping have been introduced in the literature (see [17–19]). Aoyama and Kohsaka [11] defined a new type of nonexpansive mapping that satisfies the condition (3) known as α-nonexpansive mapping as follows.Definition 3.
LetW be a nonempty subset of a Banach space H. A self map T on W can be referred α-nonexpansive mapping, if for all ξ,η∈W and α<1,
(4)Tξ−Tη2≤αTξ−η2+αξ−Tη2+1−2αξ−η2.The concept of monotone nonexpansive mappings was introduced in 2015 by Bachar and Khamsi [20], and they studied common approximate fixed point of monotone nonexpansive semigroup. To determine some order fixed points, Dehaish and Khamsi [9] proposed weak convergence theorems of the Mann iteration process for monotone nonexpansive mappings in uniformly convex ordered Banach spaces.Definition 4.
LetT be a self map on W; then, T is said to be
(1)
monotone [20] if Tξ⪯Tη for all ξ,η∈W with ξ⪯η;(2)
monotone nonexpansive [20] if T is monotone and(5)σTξ,Tη≤σξ,η,
for allξ,η∈W with ξ⪯η;
(3)
monotone quasi-nonexpansive [8] if T is monotone and(6)σTξ,η≤σξ,η,where ξ∈W and η∈FT, and FT is the set of fixed points.Definition 5 [21].
H,σ,S is called a hyperbolic space, if H,σ is a metric space and S:H×H×0,1⟶H is a function holding
(i)
σζ,Sξ,η,λ≤1−λσζ,ξ+λσζ,η;(ii)
σSξ,η,λ1,Sξ,η,λ2≤λ1−λ2σξ,η;(iii)
Sξ,η,λ=Sη,ξ,1−λ;(iv)
σSξ,ζ,λ,Sη,ν,λ≤1−λσξ,η+λσζ,ν,
for allξ,η,ζ,ν∈H and λ,λ1,λ2∈0,1.Here, (i) defines the convexity in metric spaceH,σ that was first considered by Takahashi [22], (ii) provides a unique geodesic between any two elements ξ and η of H by convexity map S that is the space of hyperbolic type in the sense of Goebel and Kirk [23], (iii) provides symmetry along the direction of geodesic, and (iv) defines negative curvature or hyperbolicity of metric space was first considered by Itoh [24].Theorem 6 [11].
LetW be a nonempty closed convex subset of a uniformly convex Banach space H and T:W⟶W be an α-nonexpansive mapping. Then, FT is nonempty iff there exists ξ∈W such that Tnξ is bounded.Since the hyperbolic spaces contain all normed linear spaces and their convex subsets, so uniformly convex Banach space is contained in hyperbolic metric space so it is natural to generalize the above result to hyperbolic metric space.Definition 7 [16].
A bounded sequenceξn in H is known as Δ-converge to an element ξ∈H, if ξ is a unique asymptotic centre of each subsequence ξnk of ξn.In this section, following definitions and lemma are stated in [9].Definition 8.
LetW be a nonempty set of a hyperbolic metric space H,σ. A map τ:W⟶0,∞ is said to be a type function, if there exists a bounded sequence un in H such that
(7)τu=limsupn⟶∞σun,u,forallu∈W.It is known that each bounded sequence generates a unique type function.Lemma 9.
LetH,σ,⪯ be a uniformly convex hyperbolic metric space and W a nonempty closed convex subset of H. Let τ:W⟶0,∞ be a type function. Then, τ is continuous. Furthermore, there exists a unique minimum point ξ∈W such that
(8)τξ=infτu:u∈W.Definition 10.
A hyperbolic spaceH,σ known as uniformly convex, if for every r>0 and ε>0,
(9)δr,ε=inf1−1rσ12ξ⊕12η,a;σξ,a≤r,ση,a≤r,σξ,η≥rε>0,for any a∈H.Definition 11.
LetW be a nonempty subset of a hyperbolic space H and ξυ be a bounded sequence in H. Then, for every ξ∈H, define
(i)
Asymptotic radius ofξυ at ξ by(10)rξ,ξυ=limsupυ⟶∞σξυ,ξ.(ii)
Asymptotic radius of the sequenceξυ relative to the above supposed set W by(11)rξ,ξυ=infrξ,ξυ:ξ∈W.(iii)
Asymptotic centre of the sequenceξυ relative to the above supposed set W by(12)AW,ξυ=rξ,ξυ=rW,ξυ:ξ∈W.Note thatAW,ξυ≠∅. Further, AW,ξυ has exactly one point if H is uniformly convex.From now to onward, we will suppose that the ordered intervals are convex and closed, and they are also contained in ordered hyperbolic spaceH,⪯,F; these are described as follows:
(13)a,⟶≔ξ∈H:a⪯ξand⟵,b≔ξ∈H:ξ⪯bforanya,b∈H.


## 3. Monotoneα-Nonexpansive Mappings

In this section, we will use the following iteration introduced by Kalsoom et al. [25].
(14)ξ∈W,ξυ+1=T1−αυTξυ+αυTηυ,ηυ=1−βυzυ+βυTzυ,zυ=1−γυξυ+γυTξυ,υ∈ℕ,where αυ,βυ and γυ are in 0,1.Now we define monotoneα-nonexpansive mappings in partially ordered hyperbolic metric space H,σ,⪯ as follows.Definition 12.
LetW be a nonempty closed convex subset of an ordered hyperbolic metric space H. A self map T on W is monotone α-nonexpansive mapping, if T is monotone and for some α<1,
(15)σ2Tξ,Tη≤ασ2Tξ,η+ασ2ξ,Tη+1−2ασ2ξ,η,for all ξ,η∈W with ξ⪯η.Lemma 13.
LetW be a nonempty closed convex subset of an ordered hyperbolic metric space H. A self map T on W is monotone α-nonexpansive mapping; then,
(i)
T is monotone quasi.(ii)
For allξ,η∈W with ξ⪯η(16)σ2Tξ,Tη≤σ2ξ,η+2α1−ασ2Tξ,ξ+2α1−ασTξ,ξσξ,η+σTξ,Tη.Proof.
To prove (i), it is followed by definition of monotoneα-nonexpansive mappings that
(17)σ2Tξ,η≤ασ2Tξ,η+ασ2ξ,η+1−2ασ2ξ,η,
impliesσTξ,η≤σξ,η.
Hence,T is monotone quasi for ξ∈W and η∈FT.
Now we will prove (ii), and if0<α<1 then we have
(18)σ2Tξ,Tη≤ασ2Tξ,η+ασ2ξ,Tη+1−2ασ2ξ,η≤ασ2Tξ,ξ+2ασTξ,ξσξ,η+ασ2ξ,η+ασ2Tξ,Tη+2ασTξ,TησTξ,ξ+ασ2Tξ,ξ+1−2ασ2ξ,η,σ2Tξ,Tη≤σ2ξ,η+2α1−ασ2Tξ,ξ+2α1−ασTξ,ξσξ,η+σTξ,Tη.
This completes the proof.☐☐Definition 14.
An ordered hyperbolic metric spaceH,σ,⪯ is said to be uniformly convex, if for an arbitrary h∈H,r>0 and ε>0,
(19)δr,ε=inf1−12σ12ξ⊕12η,h;σξ,h≤r;ση,h≤r;σξ,η≥rε>0.Now, we utilize iteration processes for monotoneα-nonexpansive mappings.Lemma 15.
LetH,σ,⪯ be a uniformly convex partially ordered hyperbolic metric space (in short, UCPOHMS) and W a nonempty closed convex subset of H. Let T:W⟶W be a monotone mapping and ξ1∈W be such that ξ1⪯Tξ1orTξ1⪯ξ1. Then, for sequence ξυ defined by (14), we have
(a)
ξυ⪯Tξυ⪯ξυ+1 (or ξυ+1⪯Tξυ⪯ξυ);(b)
ξυ⪯porp⪯ξυ provided ξυΔ-converges to a point p∈W,∀υ∈ℕ.Theorem 16.
LetW be a nonempty closed convex subset of a UCPOHMS H,σ,⪯ and T:W⟶W be a monotone α-nonexpansive mapping. Assume that there exists ξ1∈W such that ξ1⪯Tξ1 and ξυ defined in (14) is a bounded sequence with ξυ≤η for all η∈W such that
(20)liminfυ⟶∞σξυ,Tξυ=0.
Then,FT≠∅.Proof.
Letξυ defined by (14) be a bounded sequence such that
(21)liminfυ⟶∞σξυ,Tξυ=0.
Then, there exists a subsequenceξυq such that
(22)limk⟶∞σξυq,Tξυq=0.
By Lemma15, we have
(23)ξ1⪯ξυq⪯ξυq+1.
Define(24)Wq=u∈W:ξ1⪯u,for all q∈ℕ. Clearly, for every q∈ℕ,Wq is closed convex. As u∈Wq, it shows that Wq≠∅. Define
(25)W∞=⋂q=1∞Wq≠∅.
Then,W∞ is a closed convex subset of W. Let u∈W∞. Then,
(26)ξυq⪯u,∀q∈ℕ.
As we know,T is a mapping which is monotone; then, for q∈ℕ,
(27)ξυq⪯Tξυq⪯Tu,which implies that TW∞⊂W∞. Let a type function τ:W∞⟶0,∞ generated by ξυq such that
(28)τu=limsupq⟶∞σξυq,u.
Then, there exists a unique pointz∈W∞ such that
(29)τz=infτu:u∈W∞.
By definition of type function,(30)τTz=limsupq⟶∞σξυq,Tz.
By using Lemma13, we get
(31)σ2Tξυq,Tz≤σ2ξυq,z+2α1−ασ2Tξυq,ξυq+2α1−ασTξυq,ξυqσξυq,z+σTξυq,Tz.
From the boundedness of the sequenceξυq and limq⟶∞σξυq,Tξυq=0, we have
(32)limsupq⟶∞σ2Tξυq,Tz≤limsupq⟶∞σ2ξυq,z,implies τTz=τz. It shows that Tz=z, and hence, FT≠∅.☐☐Theorem 17.
LetW be a nonempty closed convex subset of a UCPOHMS H,σ,⪯ and T:W⟶W a monotone α-nonexpansive mapping. Assume that there exists ξ1∈W such that Tξ1⪯ξ1 and ξυ defined by (14) is a bounded sequence with ξυ≤η for all η∈W such that
(33)liminfυ⟶∞σξυ,Tξυ=0.
Then,FT≠∅.Theorem 18.
LetH,σ,⪯ be a UCPOHMS, C a closed convex cone, and T:C⟶C a monotone α-nonexpansive mapping. Assume that ξ1=0 and ξυ defined by (14) is a bounded sequence with ξυ≤η for all η∈W such that
(34)liminfυ⟶∞σξυ,Tξυ=0.
Then,FT≠∅.Proof.
With the help of definition of partial order⪯, we know that ξ1=0⪯T0=Tξ1; then, the proof is directly from Theorem 16.☐☐We now prove some convergence results for monotoneα-nonexpansive mappings.Lemma 19 [9].
SupposeH,σ be a UC hyperbolic space and also monotone modulus of uniform convexity δ, and suppose z∈H and a sequence αυ such that 0<a≤αυ≤b<1. If the sequences uυ and vυ are in H, in such a way that limsupυ⟶∞σuυ,z≤r, limsupυ⟶∞σvυ,z≤r, and limυ⟶∞σαυuυ⊕1−αυvυ,z=r, then limυ⟶∞σuυ,vυ=0.Theorem 20.
LetW be a nonempty closed convex subset of a UCPOHMS H,σ,⪯ and T:W⟶W a monotone α-nonexpansive mapping. Suppose there exists a sequence ξυ defined by (14) with ξ1⪯Tξ1orTξ1⪯ξ1 and FT≠∅. Then,
(1)
ξυ is bounded(2)
σξυ+1,z≤σξυ,z and limυ⟶∞σξυ,z exists for all z∈FT(3)
limυ⟶∞σTξυ,ξυ=0.Proof.
Suppose thatξ1⪯z and z∈FT, and as we know that T is monotone, then Tξ1⪯Tz, and so
(35)ξ1⪯Tξ1⪯z.
It follows from Lemma15.
(36)ξυ⪯ξυ+1⪯Tξυ⪯z,which gives that
(37)ξυ+1⪯z,∀υ∈ℕ.
It shows thatξυ is bounded. On the other hand, by using Lemma 13, we get
(38)σξυ+1,z≤σξυ,z,and so
(39)σξυ+1,z≤βυσξυ,z+1−βυσTξυ,z≤βυσξυ,z+1−βυσξυ,z=σξυ,z⋮≤σξ1,z.
Then, the sequenceσξυ,z is nonincreasing and bounded sequence; hence, (i) and (ii) proved. So limυ⟶∞σξυ,z exists for all z∈FT and υ∈ℕ. Suppose that
(40)limυ⟶∞σξυ,z≤s.
AsT is monotone quasi,
(41)limsupυ⟶∞σTξυ,z≤s,and hence,
(42)limsupυ⟶∞σξυ+1,z≤s,s=limsupυ⟶∞βυσξυ,z+1−βυσTξυ,z=limsupυ⟶∞σξυ,z.
It concludes from Lemma19 that
(43)limsupυ⟶∞σξυ,z=0.
☐☐Theorem 21.
LetW be a nonempty closed convex subset of a UCPOHMS H,σ,⪯ and T:W⟶W a monotone α-nonexpansive mapping. Suppose there exists a sequence ξυ defined by (14) with ξ1⪯Tξ1orTξ1⪯ξ1 and FT≠∅. Then, ξυΔ-converges to a fixed point of T.Proof.
By the Theorem18, we have ξυ is bounded. So, there exists a subsequence ξυq of ξυΔ-converges to some p∈W such that
(44)ξ1⪯ξυq⪯p,∀q∈ℕ.
In the next step, we prove there exists a uniqueΔ-limit in FT corresponding to each Δ-convergent subsequence of ξυ. Consider ξυ has two subsequences ξυq and ξυr which are Δ-convergent to l and m, respectively. Then, ξυ is bounded and
(45)limq⟶∞σTξυq,ξυq=0,which concludes that l∈FT. Let τ:W⟶0,∞ is a type function which is generated by ξυq. Then,
(46)τl=limsupυ⟶∞σξυq,l,τTl=limsupυ⟶∞σξυq,Tl.
By Lemma13, we infer
(47)σ2Tξυq,Tl≤σ2ξυq,l+2α1−ασ2Tξυq,ξυq+2α1−ασTξυq,ξυqσξυq,l+σTξυq,Tl,as
(48)limq⟶∞σTξυq,ξυq=0,limsupq⟶∞σTξυq,Tl≤limsupq⟶∞σξυq,l.
By uniqueness of elementl and definition of Δ-convergence, we conclude that
(49)Tl=l.
Similarly, one can easily show that(50)Tm=m.
By continuity ofτ and definition of Δ-convergence, we get
(51)limsupυ⟶∞σξυ,l≤limsupq⟶∞σξυq,l≤limsupq⟶∞σξυq,m=limsupυ⟶∞σξυ,m=limsupr⟶∞σξυr,m≤limsupr⟶∞σξυr,l,which shows that l=m.☐☐Theorem 22.
LetW be a nonempty closed convex subset of a UCPOHMS H,σ,⪯ and T:W⟶W a monotone α-nonexpansive mapping. Suppose there exists a sequence ξυ defined by (14) with ξ1⪯Tξ1orTξ1⪯ξ1 and
(52)limsupυ⟶∞βυ1−βυ>0.Then,ξυ converges strongly to a fixed point of T.Proof.
By Theorem18, there exists a subsequence ξυq of ξυ which converges strongly to a point η∈W. From Lemma 15, we get
(53)ξ1⪯ξυq⪯η,∀q∈ℕ.
By Theorem17, FT≠∅ and ξυ is bounded, and
(54)liminfυ⟶∞σTξυ,ξυ=0.
Without loss of generality, we get(55)limq⟶∞σTξυq,ξυq=0.
On the other hand, by Lemma13, we derive
(56)σ2Tξυq,Tη≤σ2ξυq,η+2α1−ασ2Tξυq,ξυq+2α1−ασTξυq,ξυqσξυq,η+σTξυq,Tη,as
(57)limq⟶∞σTξυq,ξυq=0,limsupq⟶∞σTξυq,Tη≤limsupq⟶∞σξυq,η.
By boundedness ofξυq, we have
(58)limsupq⟶∞σξυq,η=0,limsupq⟶∞σTξυq,Tη=0,and hence,
(59)limq⟶∞σTξυq,Tη=0.
Therefore,(60)limq⟶∞σξυq,Tη=limq⟶∞σξυq,Tξυq+limq⟶∞σTξυq,Tη=0,which shows that η∈FT. By Theorem 18, limυ⟶∞σξυ,η exists, so
(61)limυ⟶∞σξυ,η=0.
This completes the proof.☐☐Example 1.
LetW=1,4 be a nonempty closed convex subset of a UCPOHMS ℝ,σ,≤ and T:W⟶W defined by Tx=2x+41/3 be a monotone α-nonexpansive mapping. Then, for the sequences αn=3n/2n+14, βn=2n+1/7n+32, and γn=5n+2/3n+23, there exists a sequence ξυ for which all the conditions of Theorem 22 are satisfied by T, and hence, x=2 is the required fixed point.


## 4. Comparison of Iteration Processes

In this section, we are presenting some iterations [25–31] which we will be used in the numerical example.Mann iteration processIn 1953, Mann proposed an iteration, namely,Mann iteration, for calculation of a fixed point for a nonexpansive mapping T, defined as
(62)ξυ+1=βυξυ+1−βυTξυ,for each υ≥1 and βυ⊂0,1.Ishikawa iteration processIn 1974, Ishikawa proposed the two-step iteration process as follows:(63)ξ1=ξ∈W,ξυ+1=1−αυξυ+αυTηυ,ηυ=1−βυξυ+βυTξυ,υ∈ℕ,where αυ and βυ are in 0,1.Noor iteration processIn 2000, Noor proposed the three-step iteration as follows:(64)ξ1=ξ∈W,ξυ+1=1−αυξυ+αυTηυ,ηυ=1−βυξυ+βυTzυ,zυ=1−γυξυ+γυTξυ,υ∈ℕ,where αυ, βυ, and γυ are in 0,1.Agarwal iteration processIn 2007, Agarwal et al. introduced the three-step iteration as follows:(65)ξ1=ξ∈W,ξυ+1=1−αυTξυ+αυTηυ,ηυ=1−βυξυ+βυTξυ,υ∈ℕ,where αυ and βυ are in 0,1.Abbas and Nazir iteration processIn 2014, Abbas and Nazir introduced the three-step iteration as follows:(66)ξ1=ξ∈W,ξυ+1=1−αυTzυ+αυTηυ,ηυ=1−βυTξυ+βυTzυ,zυ=1−γυξυ+γυTξυ,υ∈ℕ,where αυ, βυ, and γυ are in 0,1.Thakur iteration processIn 2016, Thakur et al. proposed the three-step iteration as follows:(67)ξ1=ξ∈W,ξυ+1=Tηυ,ηυ=T1−αυξυ+αυzυ,zυ=1−βυξυ+βυTξυ,υ∈ℕ,where αυ and βυ are in 0,1.Two qualities fastness and stability play a vital role in iteration process to be performed. In [32], Rhoades mentioned that for the increasing functions, Ishikawa [27] iteration process is faster than Mann iteration process [26] but in the case of decreasing function, condition is reverse. In [29], Agarwal et al. proved that their iteration process was more stable than the previous ones. In [31], Thakur iteration process was considered faster convergent than all the abovementioned iteration processes.Recently, Kalsoom et al. [25] introduced a new iteration process and proved it to be the fastest convergent than all. The following example is given to support this claim.Example 2.
Define(68)Tξ=sinξ−1+cosξ,for ξ∈0,2. Then, T is monotone nonexpansive mapping as well as T is monotone α-nonexpansive mapping.Now we will compare abovementioned iteration processes to check the convergence of mappingT. By using MATLAB, we present graphs and table.In Table1, we discussed the convergence behavior of some iteration processes. It is clear that all iterations approach to 0 which is the fixed point of mapping T. In this case, Figures 1–3 show that Kalsoom et al. iteration process converges faster to the fixed point as compared the other iterations.Table 1
Convergence behavior of Mann, Ishikawa, Noor, Agarwal, Abbas, Thakur, and Kalsoom et al. iterations towards fixed point.
Initial points0.50.71.31.8Mann3978398039813981Ishikawa2656265726592600Noor2279228022822283Agarwal1590159015901576Abbas1000100010001001Thakur883883883882Kalsoom795795795786Figure 1
Convergence behavior of Mann, Ishikawa, and Kalsoom et al. iterations towards fixed point.Figure 2
Convergence behavior of Noor, Agarwal, and Kalsoom et al. iterations towards fixed point.Figure 3
Convergence behavior of Abbas, Thakur, and Kalsoom et al. iterations towards fixed point.


## 5. Monotone Generalizedβ-Nonexpansive Mappings

In this section, we define monotone generalizedβ-nonexpansive mapping which generalizes the results of Pandey and Shukla [13] in hyperbolic spaces.Now we will define monotone generalizedβ-nonexpansive mappings in hyperbolic space with nontrivial example.Definition 23.
LetW be a nonempty subset of an ordered hyperbolic metric space H,σ,⪯. A mapping T:W⟶W is said to be monotone generalized β-nonexpansive mapping, if T is monotone and there exists β∈0,1 such that
(69)12σξ,Tξ≤σξ,η⇒σTξ,Tη≤σξ,η+βσξ,η−σTξ,Tη,for all ξ,η∈W with ξ⪯η.Proposition 24.
Every monotone nonexpansive mapping is monotone generalizedβ-nonexpansive mapping, but converse is not true.Proof.
By puttingβ=0 in 69, we have
(70)12σξ,Tξ≤σξ,η⇒σTξ,Tη≤σξ,η,which shows that monotone generalized β-nonexpansive mapping reduces to monotone nonexpansive mapping satisfying the condition (3). The following example will prove the converse statement does not hold.☐☐Example 3.
LetW=0,4 be a subset of H endowed with usual order. Define T:W⟶W by
(71)Tξ=0,ξ≠4,2,ξ=4.
Then, forξ∈2,8/3 and η=4,
(72)12σξ,Tξ≤σξ,η,implies
(73)σTξ,Tη=2>σξ,η,and so T does not hold condition (3). Again, ξ∈2,3 and η=4,(74)12ση,Tη≤σξ,η⇒σTξ,Tη>σξ,η,
andT does not satisfy condition (3). Nevertheless, T is generalized β-nonexpansive mapping with β≥1/3.Proposition 25.
LetW be a nonempty subset of an ordered hyperbolic metric space H,σ,⪯ and T:W⟶W a monotone generalized β-nonexpansive mapping which has a fixed point η∈W with ξ⪯η. Then, T can be referred as a monotone quasi-nonexpansive mapping.Proof.
Letη∈FT and ξ∈W; then, by 69,
(75)12σξ,Tξ≤σξ,η⇒σTξ,Tη≤σξ,η+βσξ,η−σTξ,Tη,σTξ,η≤σξ,η+βσξ,η−βσTξ,η,1+βσTξ,η≤1+βσξ,η,
where1+β>0 as β∈0,1 which shows that T is monotone quasi-nonexpansive mapping.☐☐Proposition 26.
LetW be a nonempty subset of an ordered hyperbolic metric space H,σ,⪯. If T:W⟶W is monotone generalized β-nonexpansive mapping, then FT is closed. Furthermore, if H is strictly convex, then W is convex and FTis also convex.Proof.
Letzυ be a sequence in FT which converges to z∈W. Since
(76)12σzυ,Tzυ≤σzυ,z,with the help of continuity of metric, we have
(77)limυ⟶∞σTzυ,Tz≤limυ⟶∞σzυ,Tz,limυ⟶∞σzυ,Tz≤limυ⟶∞σzυ,z+βσzυ,z−σTzυ,Tz,limυ⟶∞σzυ,Tz≤limυ⟶∞σzυ,z+βlimυ⟶∞σzυ,z−βlimυ⟶∞σzυ,Tz,1+βlimυ⟶∞σzυ,Tz≤1+βlimυ⟶∞σzυ,z,1+β>0asβ∈0,1,limυ⟶∞σzυ,Tz≤limυ⟶∞σzυ,z,Tz=z.
Hence,FT is closed. Now, we assume that H is strictly convex and W is convex. Let λ∈0,1,ξ,η∈FT with ξ≠η; then, put z=λξ⊕1−λη∈W.
(78)12σξ,Tξ=0<σξ,z,σTξ,Tz≤σξ,z+βσξ,z−σTξ,Tz,1+βσTξ,Tz≤1+βσξ,z,1+β>0asβ∈0,1,⇒σTξ,Tz≤σξ,z.
By similar argument, we get(79)12ση,Tη=0<ση,z,σTη,Tz≤ση,z+βση,z−σTη,Tz,1+βσTη,Tz≤1+βση,z,1+β>0asβ∈0,1,⇒σTη,Tz≤ση,z.
Therefore,(80)σξ,η≤σξ,Tz+σTz,η=σTξ,Tz+σTz,Tη≤σξ,z+σz,η≤σξ,η.
From strict convexity ofH, there exists μ∈0,1 such that
(81)Tz=μξ⊕1−μ,σξ,Tξ≤σξ,z,σξ,μξ⊕1−μη≤σξ,z,1−μσξ,η≤σξ,zη.
By using value ofξ, it gives
(82)1−μσξ,η≤1−λσξ,η,and similarly,
(83)ση,Tz≤ση,z.
By putting values ofz and Tz, we infer
(84)μσξ,η≤λσξ,η.
From the above two inequalities, it is concluded that(85)1−μ≤1−λ,μ≤λ,μ=λ,z=Tz⇒z∈FT.
Hence,FT is convex.☐☐Lemma 27.
LetW be a nonempty subset of an ordered hyperbolic metric space H,σ,⪯ and T:W⟶W a monotone generalized β-nonexpansive mapping. Then, for each ξ,η∈W with ξ⪯η,
(a)
σTξ,T2ξ≤σξ,Tξ.(b)
Either1/2σξ,Tξ≤σξ,η or 1/2σTξ,T2ξ≤σTξ,η.(c)
EitherσTξ,Tη≤σξ,η+βσξ,ησTξ,Tη or σT2ξ,T2η≤σTξ,η+β−σTξ,η−σT2ξ,T2η.Proof.
Since(86)12σξ,Tξ≤σξ,Tξ,implies
(87)σTξ,T2ξ≤σξ,Tξ+βσξ,Tξ−σTξ,T2ξ,1+βσTξ,T2ξ≤1+βσξ,Tξ,1+β>0,β∈0,1,σTξ,T2ξ≤σξ,Tξ.
Hence, part (a) is satisfied. Now, we will prove part (b); we argue with contradiction, and suppose(88)12σξ,Tξ>σξ,η,12σTξ,T2ξ>σTξ,η.
By (a) and triangular inequality,(89)σξ,Tξ≤σξ,η+σTξ,η<12σξ,Tξ+12σTξ,T2ξ≤12σξ,Tξ+12σξ,Tξ≤σξ,Tξ,which is contradiction to our supposition, hence proved. The proof of (c) is in a similar way, so we omit that.☐☐Lemma 28.
LetW be a nonempty subset of an ordered hyperbolic metric space H,σ,⪯ and T:W⟶W a monotone generalized β-nonexpansive mapping. Then, for each ξ,η∈W with ξ⪯η,
(90)σξ,Tη≤2+β1−βσξ,Tξ+1+β1−βσξ,η.Proof.
By the help of Lemma27, we infer either
(91)σTξ,Tη≤σξ,η+βσξ,η−σTξ,Tη,
or(92)σT2ξ,T2η≤σTξ,η+βσTξ,η−σT2ξ,T2η.
In the first case, we have(93)σξ,Tη≤σξ,Tξ+σξ,η+βσξ,η−σTξ,Tη≤σξ,Tξ+σξ,η+βσξ,Tξ+σTξ,η−σTξ,Tη≤σξ,Tξ+σξ,η+βσξ,Tξ+σξ,η−σξ,Tη≤σξ,Tξ+σξ,η+βσξ,Tξ+ση,Tη≤σξ,Tξ+σξ,η+βσξ,Tξ+βσξ,η+βσξ,Tη,and so
(94)σξ,Tη≤1+β1−βσξ,Tξ+1+β1−βσξ,η.
In second case,(95)σξ,Tη≤σξ,Tξ+σTξ,T2ξ+σT2ξ,Tη≤σξ,Tξ+σξ,Tξ+σT2ξ,Tη≤2σξ,Tξ+σTξ,η+βσTξ,η+σT2ξ,Tη≤2σξ,Tξ+σTξ,η+βσξ,Tξ+ση,Tη≤2σξ,Tξ+σξ,η+βσξ,Tξ+βσξ,η+βσξ,Tη,and hence,
(96)σξ,Tη≤2+β1−βσξ,Tξ+1+β1−βσξ,η.
Hence, we get the desired result.☐☐Theorem 29.
LetW be a nonempty convex and closed subset of an ordered hyperbolic metric space H,σ,⪯ and T:W⟶W a monotone generalized β-nonexpansive mapping. Then, FT≠∅iffTυξ is a sequence which is also bounded for some ξ∈W provides that Tυξ⪯z for some z∈W and ξ⪯Tξ.Proof.
LetTυξ be a bounded sequence for some ξ∈W. As we know that T is monotone and ξ⪯Tξ, so we get
(97)Tξ⪯T2ξ.
In the same manner, we get(98)T2ξ⪯T3ξ⪯T4ξ⪯⋯⪯Tυξ⪯Tυ+1ξ⪯⋯.
Defineξυ=Tξυ−1=Tυξ, ∀υ∈ℕ. Then, the asymptotic centre of ξυ w.r.t W is AW,ξυ=z where z is unique and ξυ⪯z for all υ∈ℕ. Now, we claim that σξυ+1,ξυ+2 is a nonincreasing sequence, that is,
(99)σξυ+1,ξυ+2≤σξυ,ξυ+1.
Since(100)12σξυ,Tξυ≤σξυ,ξυ+1,which gives that
(101)σξυ+1,ξυ+2=σTξυ,Tξυ+1≤σξυ,ξυ+1+βσξυ,ξυ+1−σTξυ,Tξυ+1=σξυ,ξυ+1+βσξυ,ξυ+1−σξυ+1,ξυ+2⟹1+βσξυ+1,ξυ+2≤1+βσξυ,ξυ+1⟹σξυ+1,ξυ+2≤σξυ,ξυ+1.
Now, we claim that(102)σξυ,ξυ+1≤2σξυ,zorσξυ+1,ξυ+2≤2σξυ+1,z.
To prove this, we consider the contradiction(103)2σξυ,z<σξυ,ξυ+1and2σξυ+1,z<σξυ+1,ξυ+2.
By using triangular inequality,(104)σξυ,ξυ+1≤σξυ,z+σξυ+1,z<12σξυ,ξυ+1+12σξυ+1,ξυ+2<σξυ,ξυ+1,which is not possible, so (102) is satisfied.
In the first case of (102),
(105)12σξυ,ξυ+1≤σξυ,z12σξυ,Tξυ≤σξυ,zσTξυ,Tz≤σξυ,z+βσξυ,z−σTξυ,Tz1+βσTξυ,Tz≤1+βσξυ,zσTξυ,Tz≤σξυ,z.
Puttinglimsup on both sides,
(106)limsupυ⟶∞σTξυ,Tz≤limsupυ⟶∞σξυ,z,Tz=z.
Similarly, in the second case,(107)12σξυ+1,ξυ+2≤σξυ+1,z12σξυ+1,Tξυ+1≤σξυ+1,zσTξυ+1,Tz≤σξυ+1,z+βσξυ+1,z−σTξυ+1,Tz1+βσTξυ+1,Tz≤1+βσξυ+1,zσTξυ+1,Tz≤σξυ+1,z.
Puttinglimsup on both sides,
(108)limsupυ⟶∞σTξυ+1,Tz≤limsupυ⟶∞σξυ+1,z,Tz=z.
Conversely,FT≠∅; then, there exists some w∈FT and Tυw=w∀υ∈ℕ; then, Tυw is a constant sequence, and hence, it is bounded and this completes the proof.☐☐Example 4.
LetT:W⟶W where W=3,6; then, T is defined as
(109)Tξ=3ξ+4,for any ξ∈W; take αυ=0.7, βυ=0.6, and γυ=0.5. The fixed point of T is 4, and take initial point as ξ0=5. Then, T is monotone generalized β-nonexpansive mapping.In Table2, we discussed the convergence behavior of some iteration processes. It is clear that all iterations approach to 4 which is the fixed point of T. In this case, Figures 4–6 show that Kalsoom et al. iteration process converges faster to the fixed point as compared the other iterations.Table 2
Convergence behavior of Mann, Ishikawa, Noor, Agarwal, Abbas, Thakur and Kalsoom et al. iterations towards fixed point.
StepsMannIshikawaNoorAgarwalAbbasThakurKalsoom14.8076694.4571184.4404064.2647884.1832614.0982554.08014024.6530294.2106124.1951964.0723514.0349414.0101334.00674034.5284654.0974074.0867704.0199424.0067134.0010504.00056944.4279744.0451324.0386234.0055104.0012914.0001084.00004854.3468004.0209284.0172024.0015234.0002484.0000114.00000064.2811604.0097084.0076634.0004214.0000474.0000014.00000074.2280364.0045044.0034144.0001164.0000094.0000004.00000084.1850114.0020904.0015214.0000324.0000014.000000494.1501444.0009704.0006774.0000084.0000004.0000004104.1218754.0004504.0003024.0000024.0000004.0000004114.0989464.0002084.0001344.0000004.0000004.0000004124.0803424.0000964.0000594.0000004.00000044134.0652444.0000444.0000264.000000444144.0529884.0000204.0000114.000000444154.0430384.0000094.0000054.000000444164.0349594.0000044.0000024.000000444174.0283974.0000024.0000014444184.0230694.0000004.0000004444194.0187404.0000004.0000004444204.0152254.0000004.0000004444214.012369444444224.010049444444234.008164444444244.006633444444254.006633444444Figure 4
Convergence behavior of Mann, Ishikawa, and Kalsoom et al. iterations towards fixed point.Figure 5
Convergence behavior of Noor, Agarwal, and Kalsoom et al. iterations towards fixed point.Figure 6
Convergence behavior of Abbas, Thakur, and Kalsoom et al. iterations towards fixed point.


## 6. Conclusion

It concludes that we have approximated fixed point results of monotoneα and generalized β-nonexpansive mappings in hyperbolic spaces. Moreover, we proved some numerical applications and presented the graphical representations by using different iteration processes.



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