# Fixed Points of MultivaluedG-Nonexpansive Mappings in Hadamard Spaces Endowed with Graphs

Authors: Bancha Panyanak
Journal: Journal of Function Spaces
 (2020)

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## Abstract

A fixed-point theorem for multivaluedG-nonexpansive mappings in Hadamard spaces is proved. The strong convergence theorems of Browder’s and Halpern’s iterations are also established. Our results generalize and improve the results of Tiammee et al. (2015), Alfuraidan and Khamsi (2015), Anakkamatee and Tongnoi (2019), and many others.


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## Body

## 1. Introduction

Fixed-point theory is an important tool for finding solutions of problems in the form of equations or inequalities. One of the fundamental and celebrated results in metric fixed-point theory is the so-called Banach contraction principle which states that every contraction mapping on a complete metric space always has a unique fixed point (see [1]).In 1967, Browder [2] applied the Banach contraction principle to prove the strong convergence of an implicit iteration for nonexpansive mappings in Hilbert spaces. Later on, Halpern [3] used Browder’s result to prove the strong convergence of an explicit iteration for nonexpansive mappings in Hilbert spaces. Notice that both Browder’s and Halpern’s iterations can be applied to convex optimization problems, variational inequality problems, and image recovery problems (see, e.g., [4]).In 2008, by combining the concepts of fixed-point theory and graph theory, Jachymski [5] defined a single-valued G-contraction mapping on a metric space and proved a generalization of the Banach contraction principle in a complete metric space endowed with a graph. In 2015, Tiammee et al. [6] gave the analogous results of Browder and Halpern in Hilbert spaces endowed with graphs. Recently, Anakkamatee and Tongnoi [7] extended Tiammee et al.’s results from Hilbert spaces to the general setting of Hadamard spaces.Fixed-point theory for multivalued mappings has many useful applications in applied sciences, for instance, in game theory and optimization theory. It is natural to study the extensions of the known fixed-point results for single-valued mappings to the setting of multivalued mappings. In 2010, Beg et al. [8] defined a multivalued G-contraction mapping on a metric space and proved the existence of fixed points for such kind of mappings in metric spaces endowed with graphs. In 2015, Alfuraidan and Khamsi [9] introduced the concept of multivalued G-nonexpansive mappings as well as the concept of G-compact sets and obtained a fixed-point theorem for a multivalued G-nonexpansive mapping defined on a G-compact convex subset of a complete hyperbolic metric space endowed with a graph.In this paper, by using the concept of propertyG introduced by Tiammee et al. [6], we can prove the existence of fixed points for multivalued G-nonexpansive mappings in Hadamard spaces endowed with graphs. We also show that, under some appropriate conditions, the sequence of Halpern’s iteration converges strongly to a fixed point of a certain multivalued G-nonexpansive mapping. Our results generalize and improve many results in the literature.


## 2. Preliminaries

LetG be a directed graph with a set of vertices VG and a set of edges EG which contain all loops; that is, x,x∈EG for every x∈VG. We also assume that G has no parallel edges. Let x,y∈VG. We say that x dominates y if x,y∈EG. Let A and B be nonempty subsets of VG. We say that A dominates B if a,b∈EG for all a∈A and b∈B.LetM,d be a metric space and D be a nonempty subset of M. We denote by CBD the family of nonempty closed bounded subsets of D and by KD the family of nonempty compact subsets of D. Let H⋅,⋅ be the Hausdorff distance on CBD, i.e.,
(1)HA,B=maxsupa∈Adista,B,supb∈Bdistb,A,A,B∈CBD,where dista,B≔infda,b:b∈B is the distance from the point a to the set B.A multivalued mappingS:D⟶CBD is called a contraction if there exists k∈0,1 such that
(2)HSx,Sy≤kdx,yforallx,y∈D.Moreover, if (2) is valid when k=1, then S is called nonexpansive. It is clear that every contraction mapping is nonexpansive and, in general, the converse is not true.LetG=VG,EG be a directed graph. Throughout this paper, we assume that VG is a nonempty subset of D. A multivalued mapping S:D⟶CBD is said to be monotone increasing G-contraction [9] if there exists λ∈0,1 such that for any x,y∈D with x,y∈EG and any x′∈Sx, there exists y′∈Sy such that
(3)x′,y′∈EG,dx′,y′≤λdx,y.If (3) is valid when λ=1, then S is called monotone increasing G-nonexpansive. Obviously, every monotone increasing G-contraction mapping is monotone increasing G-nonexpansive.Proposition 1.
LetG=VG,EG be a directed graph such that VG=D and EG=D×D. Let S:D⟶KD be a multivalued mapping. Then, the following statements hold:
(i)
S is contraction if and only if S is monotone increasing G-contraction(ii)
S is nonexpansive if and only if S is monotone increasing G-nonexpansiveProof.
This proof is patterned after the proof of Proposition 1 in [10].
(i)
Suppose thatS is a contraction mapping with a constant k∈0,1. Let x,y∈D with x,y∈EG and let x′∈Sx. Since Sy is compact, there exists y′∈Sy such that(4)dx′,y′=distx′,Sy≤HSx,Sy≤kdx,y.
SinceEG=D×D, x′,y′∈EG. This shows that S is monotone increasing G-contraction. Conversely, suppose that S is a monotone increasing G-contraction with a constant λ∈0,1. Let x,y∈D. Since EG=D×D, x,y∈EG. For each x′∈Sx, there exists y′∈Sy such that x′,y′∈EG and dx′,y′≤λdx,y. This implies that
(5)supx′∈Txdistx′,Sy≤λdx,y.
Hence,HSx,Sy≤λdx,y.
The proof of (ii) is similar to (i).A pointx∈D is said to be a fixed point of S if and only if x∈Sx. We shall denote by FS the set of all fixed points of S. We say that S satisfies the endpoint condition [11] if FS≠∅ and Sx=x for all x∈FS. In [9], the authors obtain a fixed-point theorem for a monotone increasing G-contraction mapping as the following result.Theorem 2.
LetD be a nonempty closed subset of a complete metric space M,d and G=VG,EG a directed graph. Suppose that the following property holds:
∗ for any sequence xn in D, if xn⟶x and xn,xn+1∈EG for all n∈ℕ, then there exists a subsequence xnk of xn such that xnk,x∈EG for all k∈ℕ.
LetS:D⟶CBD be a monotone increasing G-contraction mapping and DS≔x∈D:x,y∈EGforsomey∈Sx. Then, FS≠∅ if and only if DS≠∅.LetM,d be a metric space and x,y∈M. If ϕ is a mapping from a closed interval 0,l to M such that ϕ0=x, ϕl=y, and dϕs,ϕt=s−t for all s,t∈0,l, then ϕ is called a geodesic joining x to y. Furthermore, the image of ϕ is called a geodesic segment joining x and y which is denoted by x,y whenever it is unique. If every two points of M are joined by a geodesic, then M is called a geodesic space. Moreover, if every two points of M are joined by exactly one geodesic, then M is called a uniquely geodesic space. For a subset D of M, if every two points in D can be joined by a geodesic in M and the image of every such geodesic is contained in D, then we call D a convex subset of M. A geodesic space M is said to be a Hadamard space if M is a complete metric space and for each x,y,z∈M, we have
(6)d2x,m≤12d2x,y+12d2x,z−14d2y,z,where m is the midpoint of y,z. We refer the reader to [12–14] for elementary properties of Hadamard spaces. It is known that every Hadamard space is uniquely geodesic. Notice also that R-trees and Hilbert spaces are important examples of Hadamard spaces (see [12, 15]).LetD be a nonempty closed convex subset of a Hadamard space M,d. By Proposition 2.4 in [12], for each x in M, there exists a unique point x0 in D such that
(7)dx,x0=infdx,y:y∈D.The metric projection ofM onto D is the mapping PD:M⟶D defined by PDx≔x0, where x0 is the unique point in D satisfying (7). From [16], for each x,y∈M and t∈0,1, there exists a unique point z in x,y such that
(8)dx,z=tdx,yanddy,z=1−tdx,y.The unique pointz satisfying (8) is denoted by 1−tx⊕ty.LetD be a nonempty convex subset of M and G=VG,EG be a directed graph. We say that G is convex if for any x,y,z,w∈D and t∈0,1 such that x,z and y,w are in EG, we have 1−tx⊕ty,1−tz⊕tw∈EG.Lemma 3 (see [16]).
LetM,d be a Hadamard space, x,y,z∈M and t∈0,1. Then,
(i)
d1−tx⊕ty,z≤1−tdx,z+tdy,z(ii)
d1−tx⊕ty,1−tx⊕tz≤tdy,z(iii)
d21−tx⊕ty,z≤1−td2x,z+td2y,z−t1−td2x,yFor a bounded sequencexn in M, we define
(9)rxn≔inflimsupn→∞dxn,x:x∈M,Zxn≔x∈M:limsupn→∞dxn,x=rxn.The real numberrxn and the set Zxn are called the asymptotic radius and the asymptotic center of xn, respectively. It is known that Zxn consists of exactly one point (see, e.g., [17]). Let xn be a sequence in M and x∈M. Then, xn is Δ-convergent to x if Zxnk=x for every subsequence xnk of xn. In this case, we write Δ−limn→∞xn=x.Lemma 4.
LetM,d be a Hadamard space and D a nonempty closed convex subset of M. Then,
(i)
every bounded sequence inM always has a Δ-convergent subsequence [18](ii)
the asymptotic center of any bounded sequence inD belongs to D [19]LetM,d be a metric space. We use the notation ab→ for an ordered pair a,b in M×M. The quasilinearization is a mapping ⋅,⋅:M×M×M×M⟶R which is defined by
(10)ab→,ce→≔12d2a,e+d2b,c−d2a,c−d2b,eforalla,b,c,e∈M.We say thatM,d satisfies the Cauchy-Schwarz inequality if
(11)ab→,ce→≤da,bdc,eforalla,b,c,e∈M.It is known that every Hadamard space satisfies the Cauchy-Schwarz inequality (see [20]).Lemma 5.
LetM,d be a Hadamard space. Then, the following statements hold:
(i)
LetD be a nonempty closed convex subset of M,x∈M and z∈D [21]. Then,(12)z=PDxifandonlyifzx→,wz→≥0forallw∈D(ii)
Letu,v∈M. For each t∈0,1, we set xt=tu⊕1−tv [22]. Then, for z∈M, we have(13)xtz→,xtz→≤tuz→,xtz→+1−tvz→,xtz→(iii)
Letxn be a sequence in M and z∈M [23]. Then, xn is Δ-convergent to z if and only if limsupn→∞wz→,xnz→≤0 for all w∈M
We denote byℓ∞ the Banach space of bounded real sequences. Let μ be a continuous linear functional on ℓ∞ and let a=a1,a2,⋯∈ℓ∞. We shall denote by μnan the value μa. A continuous linear functional μ on ℓ∞ is called a Banach limit if μ=μ1,1,⋯=1 and μnan=μnan+1 for all an∈ℓ∞.Lemma 6 (see [24]).
Leta1,a2,⋯∈ℓ∞ be such that μnan≤0 for each Banach limit μ and suppose that limsupn→∞an+1−an≤0. Then, limsupn→∞an≤0.Lemma 7 (see [25]).
Letcn be a sequence of nonnegative real numbers satisfying
(14)cn+1≤1−γncn+γnηnforalln∈ℕwhere γn⊂0,1 and ηn⊂R such that
(i)
∑n=1∞γn=∞(ii)
∑n=1∞γnηn<∞ or limsupn→∞ηn≤0
Then,limn→∞cn=0.


## 3. Main Results

This section begins by proving a crucial lemma which is an extension of Lemma 2.1 in [26].Lemma 8.
LetM,d be a Hadamard space, D a nonempty closed convex subset of M, and G=VG,EG a convex directed graph. Let S:D⟶CBD be a monotone increasing G-nonexpansive mapping such that u∈DS for some u∈D. Suppose that ∗ holds. Then, for each t∈0,1, the mapping St:D⟶CBD defined by
(15)Stx≔tu⊕1−tSxhas a fixed point xt in D; that is,
(16)xt∈Stxt=tu⊕1−tSxt.Proof.
Letx,y∈D be such that x,y∈EG. Since S is monotone increasing G-nonexpansive, for each x′∈Sx, there exists y′∈Sy such that
(17)x′,y′∈EG,dx′,y′≤dx,y.By the convexity ofG, we have
(18)tu⊕1−tx′,tu⊕1−ty′∈EG.By (17) and Lemma 3 (ii), we have
(19)dtu⊕1−tx′,tu⊕1−ty′≤1−tdx′,y′≤1−tdx,y.This implies thatSt is monotone increasing G-contraction. Since u∈DS, there exists v∈Su such that u,v∈EG. By the convexity of G, we have u,tu⊕1−tv∈EG. This implies that u∈DSt and hence DSt≠∅. By Theorem 2, St has a fixed point xt in D.In [6], the authors introduce a property that is stronger than the condition ∗ in Lemma 8.Definition 9.
LetM,d be a Hadamard space, D a nonempty subset of M, and G=VG,EG a directed graph. Then, D is said to have property G if for any sequence xn in D such that Δ−limn→∞xn=x∈D, there exists a subsequence xnk of xn such that xnk,x∈EG for all k∈ℕ.The following result is an extension of Lemma 3.2 in [27].Lemma 10.
LetM,d be a Hadamard space and D a nonempty closed convex subset of M. Let G=VG,EG be a directed graph and S:D→KM a monotone increasing G-nonexpansive mapping and xn a bounded sequence in D. Suppose that D has property G. If limn→∞distxn,Sxn=0 and Δ−limn→∞xn=x, then x∈Sx.Proof.
Forn∈ℕ, let yn∈Sxn be such that dxn,yn=distxn,Sxn. By Lemma 4 (ii), x∈D. Since D has property G, there exists a subsequence xnk of xn such that xnk,x∈EG for all k∈ℕ. Since S is monotone increasing G−nonexpansive, for each k∈ℕ there exists znk∈Sx such that
(20)ynk,znk∈EG,dynk,znk≤dxnk,x.SinceSx is compact, without loss of generality, we may assume that znk⟶z∈Sx. Thus,
(21)dxnk,z≤dxnk,ynk+dynk,znk+dznk,z≤distxnk,Sxnk+dxnk,x+dznk,z.This implies thatlimsupk→∞dxnk,z≤limsupk→∞dxnk,x. Thus, z∈Zxnk=x and hence, x=z∈Sx.By applying Lemmas8 and 10, we can obtain a fixed-point theorem for monotone increasing G-nonexpansive mappings in Hadamard spaces.Theorem 11.
LetM,d be a Hadamard space and D a nonempty bounded closed convex subset of X. Let G=VG,EG be a convex directed graph and suppose that D has property G. Let S:D⟶KD be a monotone increasing G-nonexpansive mapping such that u∈DS for some u∈D. Then, S has a fixed point in D.Proof.
For eachn∈ℕ, we define a mapping Sn:D⟶KD by
(22)Snx≔1nu⊕1−1nSxforallx∈D.
Then, by Lemma8, there exists xn in D such that xn∈Snxn=1/nu⊕1−1/nSxn. For each n∈ℕ, there exists yn∈Sxn such that
(23)xn=1nu⊕1−1nyn.
This implies that(24)distxn,Sxn≤dxn,yn=1ndu,yn⟶0asn⟶∞.
Sincexn is bounded, by Lemma 4, there exists a subsequence xnk of xn and a point x in D such that Δ−limk→∞xnk=x. By Lemma 10, we have x∈Sx and hence, the proof is complete.As a consequence of Theorem11, by putting VG=D and EG=D×D, we obtain a result of Dhompongsa et al. [28].Corollary 12.
LetM,d be a Hadamard space and D a nonempty bounded closed convex subset of M. Let S:D⟶KD be a nonexpansive mapping. Then, S has a fixed point.The following result gives conditions that guarantee the closedness and the convexity of the fixed-point set of a monotone increasingG-nonexpansive mapping on a closed convex subset of a Hadamard space. Notice also that it is an extension of Theorem 3.2 in [6].Lemma 13.
LetM,d be a Hadamard space and D a nonempty closed convex subset of M. Let G=VG,EG be a convex directed graph and suppose that D has property G. If S:D⟶CBD is a monotone increasing G-nonexpansive mapping satisfying the endpoint condition and FS×FS⊆EG, then FS is closed and convex.Proof.
Letxn be a sequence in FS such that limn→∞xn=x. Since D has property G, there is a subsequence xnk of xnsuch that xnk,x∈EG for all k∈ℕ. Since S is monotone increasing G-nonexpansive, there exists znk∈Sx such that
(25)xnk,znk∈EG,dxnk,znk≤dxnk,x.Thus(26)distx,Sx≤dx,znk≤dx,xnk+dxnk,znk≤2dxnk,x⟶0ask⟶∞.This implies thatx∈Sx and hence, FS is closed. Let x,y∈FS and z=αx⊕1−αy for some α∈0,1. Since FS×FS⊆EG, both x,x and y,x belong to EG. By the convexity of G, we have z,x∈EG. Take any point v∈Sz. Since S is monotone increasing G-nonexpansive, there exists w∈Sx=x such that v,w∈EG and dv,w≤dz,x. This implies that dv,x≤dz,x. Similarly, we can show that dv,y≤dz,y. It follows from Lemma 3 (iii) that
(27)d2v,z≤αd2v,x+1−αd2v,y−α1−αd2x,y≤αd2z,x+1−αd2z,y−α1−αd2x,y≤α1−α2d2x,y+1−αα2d2x,y−α1−αd2x,y=0,which yields z=v∈Sz. Therefore, z∈FS and hence, FS is convex.Now, we prove the strong convergence of Browder’s iteration for monotone increasingG-nonexpansive mappings in Hadamard spaces.Theorem 14.
LetM,d be a Hadamard space and D a nonempty closed convex subset of M. Let G=VG,EG be a convex directed graph and suppose that D has property G. Let S:D⟶KD be a monotone increasing G-nonexpansive mapping satisfying the endpoint condition and having u∈DS for some u∈D. Suppose that FS×FS⊆EG. Then, the following statements hold:
(i)
If the netxt defined by (16) dominates FS, then xt converges strongly to PFSu as t⟶0(ii)
Ifun is a bounded sequence in D such that limn→∞distun,Sun=0, then d2u,PFSu≤μnd2u,un for each Banach limit μProof.
(i) We first show thatxt is bounded. Let p∈FS. Since xt dominates FS, xt,p∈EG. From (16), there exists yt∈Sxt such that xt=tu⊕1−tyt. It follows from the G-nonexpansiveness of S and the endpoint condition that yt,p∈EG and dyt,p≤dxt,p. This yields
(28)dxt,p≤tdu,p+1−tdyt,p≤tdu,p+1−tdxt,p,which implies that dxt,p≤du,p and hence, xt is bounded. Notice also that yt is bounded and distxt,Sxt≤dxt,yt≤tdu,yt⟶0ast⟶0.
Next, we show thatxt converges strongly to PFSu as t⟶0. Let a sequence tn in 0,1 converging to 0 and put xn≔xtn. It suffices to show that there exists a subsequence of xn which converges to PFSu. By Lemmas 4 and 10, there exists a subsequence xnk of xn and a point z in FS such that Δ−limk→∞xnk=z. By Lemma 5 (ii), we have
(29)d2xnk,z=xnkz→,xnkz→≤tnkuz→,xnkz→+1−tnkynkz→,xnkz→≤tnkuz→,xnkz→+1−tnkdynk,zdxnk,z≤tnkuz→,xnkz→+1−tnkd2xnk,z.
Thus,d2xnk,z≤uz→,xnkz→. Since Δ−limk→∞xnk=z, by Lemma 5 (iii), we have limsupk→∞uz→,xnkz→≤0 and this in turn implies that limk→∞xnk=z. Next, we show that z=PFSu. Since xnk dominates FS, for any q∈FS, we have
(30)d2xnk,q≤tnkd2u,q+1−tnkd2ynk,q−tnk1−tnkd2u,ynk≤tnkd2u,q+1−tnkd2xnk,q−tnk1−tnkd2u,ynk.
Thus,(31)d2xnk,q≤d2u,q−1−tnkd2u,ynk.
Letk⟶∞, then
(32)d2z,q≤d2u,q−d2u,z.
Hence,(33)0≤12d2z,z+d2u,q−d2z,q−d2u,z=zu→,qz→forallq∈FS.
By Lemma5 (i), z=PFSu. In summary, for any given sequence tn in 0,1 converging to 0, the sequence xn≔xtn has a subsequence converging to PFSu. This shows that xn⟶PFSu and in turn implies that xt⟶PFSu as t⟶0.
(ii) The proof is similar to the proof of Theorem 3.4 in [27], so we omit it.Now, we give an example which supports Theorem14.Example 1.
LetM=R,⋅, D=0,1 and let G=VG,EG be such that VG=0,1/2 and EG=x,y:x,y∈0,1/2. It is easy to see that G is convex. Let S:D⟶KDbe defined by
(34)Sx≔x2ifx≠1,0ifx=1.Then,S is a monotone increasing G-nonexpansive mapping satisfying the endpoint condition. Notice also that FS=0. Thus, FS×FS⊆EG. Fix u∈DS=0,1/2. For each t∈0,1, let xt be defined by (16). Obviously, xt dominates FS and
(35)xt=tu+1−txt2.Thus,xt/2=tu−xt/2⟶0 as t⟶0. This implies that xt⟶0=PFSu. However, S is not nonexpansive since
(36)HS1,S34=0−38=38>14=1−34.As a consequence of Theorem14, by putting VG=D and EG=D×D, we obtain a result of Dhompongsa et al. [27].Corollary 15.
LetM,d be a Hadamard space, D a nonempty closed convex subset of M, and S:D→KD a nonexpansive mapping satisfying the endpoint condition. Fix u∈D; for each t∈0,1, let xt be a fixed point of St:D⟶KD defined by
(37)Stx≔tu⊕1−tSx,forallx∈D.
Then, the following statements hold:(i)
xt converges strongly to PFSu as t⟶0(ii)
Ifun is a bounded sequence in D such that limn→∞distun,Sun=0, then d2u,PFSu≤μnd2u,un for each Banach limit μNext, we will extend the strong convergence of Halpern’s iteration from single-valuedG-nonexpansive mappings [7] to the general setting of multivalued G-nonexpansive mappings. Notice that our approach does not require the transitivity of the graph G.LetM,d be a Hadamard space and D a nonempty closed convex subset of M. Let G=VG,EG be a convex directed graph and S:D⟶KD a monotone increasing G-nonexpansive mapping for which DS≠∅. Let αn be a sequence in 0,1. Fix u∈DS, and let x1=u. Then, there exists y1∈Sx1 such that x1,y1∈EG. Let
(38)x2=α1u⊕1−α1y1.Sincex1,x1∈EG and G is convex, x1,x2∈EG. Since S is monotone increasing G-nonexpansive, there exists y2∈Sx2 such that y1,y2∈EG and dy1,y2≤dx1,x2. Let
(39)x3=α2u⊕1−α2y2.Sinceu,u∈EG and y1,y2∈EG, by the convexity of G, we have x2,x3∈EG. Inductively, we can construct sequences xn and yn in D such that
(40)xn+1=αnu⊕1−αnyn,yn∈Sxn,(41)yn,yn+1∈EG,xn,xn+1∈EG,(42)dyn,yn+1≤dxn,xn+1foralln∈ℕ.Theorem 16.
LetM,d be a Hadamard space and D a nonempty closed convex subset of M. Let G=VG,EG be a convex directed graph and suppose that D has property G. Let S:D⟶KD be a monotone increasing G-nonexpansive mapping satisfying the endpoint condition and having u∈DS for some u∈D. Suppose that FS×FS⊆EG, and let αn be a sequence in 0,1 satisfying
(C1)
limn→∞αn=0(C2)
∑n=1∞αn=∞,(C3)
∑n=1∞αn−αn+1<∞ or limn→∞αn/αn+1=1
For eachn∈ℕ, let xn be defined by (40) and suppose that xn dominates PFSu. Then, xn converges strongly to PFSu.Proof.
We divide the proof into three steps.Step 1.
We show thatxn and yn are bounded sequences. Let z≔PFSu. Since xn dominates z,xn,z∈EG. It follows from the G-nonexpansiveness of S and the endpoint condition that yn,z∈EG and dyn,z≤dxn,z. Thus,
(43)dxn+1,z≤αndu,z+1−αndyn,z≤αndu,z+1−αndxn,z≤maxdu,z,dxn,zfor all n∈ℕ. This implies that xn is bounded and so is yn.Step 2.
We show thatlimn→∞distxn,Sxn=0. It follows from (40)–(42) that
(44)dxn,xn+1=dαn−1u⊕1−αn−1yn−1,αnu⊕1−αnyn≤dαn−1u⊕1−αn−1yn−1,αnu⊕1−αnyn−1+dαnu⊕1−αnyn−1,αnu⊕1−αnyn≤αn−1−αndu,yn−1+1−αndyn−1,yn≤1−αndxn−1,xn+αn−1−αndu,yn−1.
Putting, in Lemma7, cn=dxn−1,xn,γn=αn and ηn=1−αn−1/αndu,yn−1, we get by (C2) and (C3) that limn→∞dxn,xn+1=0. Consequently, by (C1), we have
(45)distxn,Sxn≤dxn,yn≤dxn,xn+1+dxn+1,yn=dxn,xn+1+dαnu⊕1−αnyn,yn≤dxn,xn+1+αndu,yn⟶0asn⟶∞.Step 3.
We show thatxn converges strongly to z. By Lemma 3 (iii), we get that
(46)d2xn+1,z≤αnd2u,z+1−αnd2yn,z−αn1−αnd2u,yn≤1−αnd2xn,z+αnd2u,z−1−αnd2u,yn.
By Theorem14 (ii), we have μnd2u,z−d2u,xn≤0 for each Banach limit μ. Moreover, since limn→∞dxn,xn+1=0,
(47)limsupn→∞d2u,z−d2u,xn+1−d2u,z−d2u,xn=0.
It follows from Lemma6 that
(48)limsupn→∞d2u,z−1−αnd2u,yn=limsupn→∞d2u,z−d2u,xn≤0.
Applying Lemma7 to the inequality (46), we can conclude that xn⟶z as n⟶∞, and hence, the proof is complete.


## 4. Concluding Remarks and Open Questions

In this paper, we show that ifD is a nonempty subset of a metric space and G=VG,EG is a directed graph such that VG=D and EG=D×D, then the concept of monotone increasing G-nonexpansive mappings introduced by Alfuraidan and Khamsi [9] coincides with the concept of multivalued nonexpansive mappings. Moreover, by using the concept of property G introduced by Tiammee et al. [6], we prove the analogue results of Theorem 3.4 in [27], Theorem 3.2 in [6], and Theorem 3.6 in [7] in the setting of monotone increasing G-nonexpansive mappings in Hadamard spaces endowed with graphs. However, in view of Example 1, we still have the strong convergence result of Browder’s iteration for a monotone increasing G-nonexpansive mapping T on a set D which has no property G. Thus, it is unclear whether property G is necessary for Theorem 14. Therefore, the following question should be of interest.Question 1.
Can we omit propertyG from Theorem 14?



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*Source: 5849262-2020-05-15.xml*