# Extragradient Method for Fixed Points in CAT(0) Spaces

Authors: Yu-Pei Lv; Khurram Shabbir; Sundus Shahzeen; Farman Ali; Jeevan Kafle
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This paper is dedicated to construct a viscosity extragradient algorithm for finding fixed points in a CAT(0) space. The mappings we consider are nonexpansive. Strong convergence of the algorithm is obtained. The results established in this work extend and improve some recent discovers in the literature.


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## Body

## 1. Introduction

LetX be a CAT(0) space and K be any closed convex subset of X. Let ϕ:K×K⟶ℝ be a bifunction with ϕv,v=0 for all v∈K. The main equilibrium problem is to get r∈K satisfying
(1)ϕr,w≥0,forallw∈K.Finding solutions for equilibrium problems aids in solving problems in other areas of science like physics, optimization, and economics. The set of all solutions of the equilibrium problem is denoted byEPϕ, i.e.,
(2)EPϕ=r∈K:ϕr,w≥0,∀w∈K.Numerous iterative algorithms for monotome equilibrium have been investigated previously; for finding the solutions, see [1, 2]. Here, we will find the iterative algorithm for pseudomonotone bifunction. The bifunction ϕ is said to be pseudomonotone if
(3)ϕv,w≥0⟶ϕw,v≤0,forallv,w∈K.An extragradient method, to solve a pseudomonotone equilibrium problem inℝn, was introduced in [3]. The method of extragradient is as follows: given v0∈K, find successively wn and vn+1 by
(4)wn=argminw∈Kλnϕvn,w+12w−vn2,vn+1=argminw∈Kλnϕwn,w+12w−vn2,where λn⊂0,∞ and ϕ is such that the Lipschitz (type) condition holds. In [4], the following algorithm was introduced by Anh for finding a fixed point of a nonexpansive mapping T which is also the solution of the equilibrium problem for pseudomonotone bifunction ϕ in a Hilbert space:
(5)wn=argminw∈Kλnϕvn,w+12w−vn2,vn+1=argminw∈Kλnϕwn,w+12w−vn2,where αn,λn⊂0,1 and ϕ is such that the Lipschitz (type) condition holds. The convergence (strong) of vn with limn⟶∞vn+1−vn=0 and under certain considerations on αn,λn∈0,1.In 2012, an algorithm for hybrid projection was considered by Vuong et al. [5] for
(6)wn=argminw∈Kλnϕvn,w+12w−vn2,vn+1=argminw∈Kλnϕvn,w+12w−vn2,sn=αnv0+1−αnβnz+1−βnSzn,Kn=z∈K:sn−z≤vn−z,Dn=z∈K:vn−z,v0−vn≥0,vn+1=PKn∩Dnv0,where αn,λn∈0,1 and ϕ is such that the Lipschitz (type) condition holds. The convergence proved was strong.The Armijo-type method for pseudomonotone equilibrium problems was formulated in [6] in the setting of Hilbert spaces. After that, the authors in [7] presented the convergence of weak and strong types for the algorithms in order to solve the equilibrium problem. The admirable outcomes are due to Dinh and Kim in which there is no restriction on monotonicity of the bifunction. The current results of equilibrium problems are given for pseudomonotone type. For further references, see [8, 9]. To the current knowledge, the authors modified the “hybrid projection algorithm” in order to get convergence of strong type for iterative algorithms of equilibrium problems of pseudomonotone type [10, 11].The aim of this paper is to construct an extragradient algorithm of viscosity type for finding the same element for the solution set of a pseudomonotone equilibrium problem and fixed point set of a nonexpansive mapping in the framework of a CAT(0) space and derive its strong convergence.


## 2. Definitions and Known Results

In this section, we present basic definitions and known results. The notions that are not defined in this paper can be seen in [12–14].Throughout this paper,X,d denotes the geodesic metric space with a geodesic triangle Δv1,v2,v3 in X,d, where v1, v2, and v3 represent the vertices of Δ in X,d. We will represent X,d as X henceforth. A comparison triangle in X is a triangle Δ¯v1,v2,v3≔Δv1¯,v2¯,v3¯ in the Euclidean plane E2 such that dE2vi,vj=dvi,vj for i,j=1,2,3.A geodesic space is called a CAT(0) space, if all the geodesic triangles satisfy the following comparison axiom(7)dv,w≤dE2v¯,w¯.Letv,w∈X, and by Lemma 2.1(iv) of [15] for each s∈0,1, there exists a unique point z∈v,w such that
(8)dv,z=sdv,w,(9)dw,z=1−sdv,w,for all v,w∈Δv1,v2,v3 with v¯,w¯ the corresponding points of v,w in Δ¯v1,v2,v3.Lemma 1 (see [16]).
LetX be a CAT(0) space. Then the following assertions are true:
(i)
For anyv,w,z∈X and s∈0,1,(10)d1−sv⊕sy,z≤1−sdv,z+sdw,z(ii)
For anyv,w,z∈X and s∈0,1,(11)d21−sv⊕sy,z≤1−s2dv,z+sd2w,z−s1−sd2v,wLetX,d be a uniquely geodesic metric space; that is, for each v,w∈X, there exists a unique isometry c:0,dv,w⟶X such that c0=v and cdv,w=w, and in this case, we write v,w=ct:t∈0,dx,y. For each s∈0,1, we write 1−sv⊕sw for the element z∈v,w such that dz,v=sdv,w and dz,w=1−sdv,w.Hadamard spaces are the complete CAT(0) spaces; for details, see [16]. If v,w1,w2 are points of a CAT(0) space and w0 is the midpoint of the segment w1,w2, which we will denote by w1/2⊕w2/2, then the CAT(0) inequality gives
(12)d2v,w12⊕w22≤12d2v,w1+12d2v,w2−14d2w1,w2.This inequality is called the (CN) inequality of Bruhat and Tits [16]. In fact, a geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality (cf. [16], page 163). Berg and Nikolaev [17] introduced the idea of the quasilinearization as follows: Let us denote the pair α,β∈X×X by αβ⟵ and call it a vector. Then, quasilinearization is defined as a map
(13).,.:X×X×X×X⟶ℝdefined as
(14)αβ⟶,νϑ⟶=12d2α,ϑ+d2β,ν−d2α,ν−d2β,ϑ.It is easy to see thatαβ⟶,νϑ⟶=νϑ⟶,αβ⟶, αβ⟶,νϑ⟶=−βα⟶,νϑ⟶, and αx⟶,νϑ⟶+xβ⟶,νϑ⟶=αβ⟶,νϑ⟶ for all α,β,ν,ϑ∈X. We say that X satisfies the Cauchy-Schwarz inequality if
(15)αβ⟵,νϑ⟵≤dα,βdν,ϑfor all α,β,ν,ϑ∈X. It is well known [17] that a geodesically connected metric space is a CAT(0) space if and only if it satisfies the Cauchy-Schwarz inequality.In 1976, Lim gave the definition ofΔ-convergence in a general metric space case (see [18]). He named this type of convergence as strong Δ-convergence, proving that CAT(0) spaces provide a natural framework for the Lim concept and, in the setting of Δ-convergence, provide many properties of the usual notion of weak convergence in Banach spaces. Let us recall this type of convergence for the case of CAT(0) spaces, as follows.Definition 2.
LetX,d be a CAT(0) space. A sequence vn in X is said to Δ-converge to v∈X if and only if v is the unique asymptotic center of all subsequences of vn. In this case, we write Δ−limn⟶∞vn=v, and v is called the Δ-limit of vn.Note that the asymptotic centerAvn of vn is the set
(16)Avn≔v∈X:rv,vn=rvn,and the asymptotic radius rvn of xn is given by
(17)rvn≔infv∈Xrx,vn,where rv,vn≔limsupn⟶∞dvn,v, for vn is a bounded sequence in X.Ahmadi Kakavandi and Amini introduced in [19] another variant of weak convergence in complete CAT(0) spaces, taking into account the concept of quasilinearization.Definition 3 (see [19]).
LetX,d be a complete CAT(0) space. A sequence vn in X is said to w-converge to an element v∈X if for each w∈X, limn⟶∞vvn⟶,vw⟶=0.Obviously, the convergence in the metric impliesw-convergence, and the w-convergence implies Δ-convergence (see Proposition 2.5 in [19]). But the converse is not true (see [20]). The following result proves an explicit connection between w-convergence and Δ-convergence.Theorem 4 (see [20]).
LetX,d be a complete CAT(0) space. Then a sequence vn in XΔ-converges to v∈X if and only if, for every w∈X, limsupn⟶∞vvn⟶,vw⟶≤0.Further, let us recall some definitions for the case of a CAT(0) space.Definition 5.
LetX be a CAT(0) space and T:X⟶X be a mapping. Then T is called nonexpansive if
(18)dTv,Tw≤dv,w,foreveryv,w∈K.Definition 6.
LetX be a CAT(0) space and T:X⟶X be a mapping. Then T is called a contraction if
(19)dTv,Tw≤θdv,w,foreveryv,w∈K,θ∈0,1.Throughout this paper, we denote byFixT=v∈C∣Tv=v the set of fixed points of T.Concerning the convexity in CAT(0) spaces, we remark that, in this type of spaces, angles exist in a strong sense, the distance function is convex, and one has both uniform convexity and orthogonal projection onto convex subsets. Moreover, CAT(0) spaces turn up to represent a real framework for convexity theory.Remark 7.
Considering the CAT(0) space case, a subsetC⊂X is said to be convex if C includes every geodesic segment joining any two of its points, i.e., 1−tx⊕ty∈C, for every x,y∈C and t∈0,1.Definition 8.
LetX be a CAT(0) space. A function f:X⟶−∞,+∞ is said to be convex if
(20)ftv⊕1−tw≤tfv+1−tfw,for all v,w∈X, and t∈0,1.LetX be a CAT(0) space and K⊂X a convex and closed subset. Further, let a bifunction ϕ:K×K⟶ℝ; then ϕ is called
(1)
ς-strong monotone ς<0 on K if ∀v,w∈K, we have(21)ϕv,w+ϕw,v≤ςd2v,w(2)
monotone onK if for each v,w∈K, one has(22)ϕv,w+ϕw,v≤0(3)
pseudomonotone onK if for each v,w∈K, one has(23)ϕv,w≥0⇒ϕw,v≤0The above bifunctionϕ is Lipschitz-type continuous on K, if there exist two constants c1>0 and c2>0 such that
(24)ϕv,z≤ϕv,w+ϕw,z+c1d2v,w+c2d2w,z,foreveryv,w∈K.In [19], Kakavandi and Amini define the notion of a subdifferential of a function as follows.Definition 9 (see [19]).
Letf:X⟶−∞,+∞ be a proper function with efficient domain Df=v:fv<+∞, then the subdifferential of f is the multifunction ∂f:X⇉X∗ defined by
(25)∂fv∗=v∗:fz−fv≥v∗,vz⟶,forallz∈X,when v∈Df and ∂fv=∅.We mention thatX∗ denote the notion of dual space of the metric space X,d. For more details, see [19].In [21], Georgiou and Papadopoulos gave a strong discussion concerning the convergence types and topologies on function spaces. Then, let us consider X,Y,Z as three topological spaces. A mapping f:X⟶Y into a is called weakly continuous at v∈X if for every open neighbourhood V of fv there exists an open neighbourhood U of v such that fU⊆ClV. ClV denotes the closure of V. The mapping f is weakly continuous on X, if it is weakly continuous at each point of X. In the following, WCY,Z denotes the set of all weakly continuous maps of Y into Z. If τ is a topology on the set WCY,Z, then the corresponding topological space is denoted by WCτY,Z.In this conditions, we can recall the notion ofjointly weakly continuous function.Definition 10.
A topologyτ on WCY,Z is called weakly jointly continuous if for every X, the weak continuity of a map G:X⟶WCτY,Z implies the weak continuity of the map G~:X×Y⟶Z.For more details and results concerning the topology and the convergence types in function spaces, see [21–23].Further, taking into account the previous notions, let us consider the following properties ofϕ:
(1)
ϕv,v=0 for all v∈K, and ϕ is taken pseudomonotone on the subset K(2)
ϕ is taken to be Lipschitz-type continuous on the subset K(3)
For allv∈K, w⟶ϕv,w is subdifferentiable and convex(4)
ϕv,w is taken jointly weakly continuous on K×KWith conditions (6), (7), (8), and (10), the set EPϕ is convex and closed.Lemma 11 (see [24]).
LetX be a CAT(0) space. Assume that EPϕ≠∅ and v∈K. Further, let w,s∈K be solutions of
(26)w=argminz∈Kλϕv,w+12d2v,z,(27)s=argmina∈Kλϕv,w+12d2w,a,strongly convex problems, where λ>0, then
(28)λϕv,z−ϕv,w≥wv⟶,wz⟶,z∈K,d2s,x≤d2v,x−1−2λc1d2v,w−1−2λc2d2s,x,∀x∈EPϕ.Lemma 12 (the demiclosedness principle).
LetK be a nonempty closed convex subset of the CAT(0) space X and T:K⟶K such that
(29)vn⇀v∈K,dvn,Tvn⟶0.
Then,v=Tv. (Here, ⟶ (respectively, ⇀) denotes strong (respectively, weak) convergence.) The notion of weak convergence is the same as defined in [25].Lemma 13 (see [26]).
LetX be a CAT(0) space and K be any closed convex subset of X. For each point v∈X, there exists a unique nearest point of K, denoted by PKv, such that dv,PKv≤dv,w for all w∈K. Such a PK is metric projection onto K from X. Then,
(1)
for allv∈X and z∈K, z=PKv iff(30)vz⟶,zw⟶≥0,forallw∈K(2)
for allv∈X and z∈K, it holds(31)d2PKv,z≤d2v,z−d2PKv,vFor more information, see Section 3 of [27].Lemma 14.
LetX be a complete CAT(0) space. For all v,w∈X and 0=0,1∈X, the following hold:
(1)
d2v⊕w,0≤d2v,0+2w0⟶,v0⟶+2d2w,0(2)
d2sv⊕1−sw,0=sd2v,0+1−sd2w,0−s1−sd2v,w for all s∈0,1Lemma 15 (see [28]).
Assume thatan is a sequence of nonnegative real numbers such that
(32)an+1≤1−βnan+δn,foralln≥0,where βn is a sequence in 0,1 and δn is a sequence with
(1)
∑n=0∞βn=∞(2)
limsupn⟶∞δn/βn≤0 or ∑n=0∞∣δn∣<∞
Then,limn⟶∞an=0.Another interesting result useful in the proof of our main results is the following.Lemma 16 (see [29]).
Assume thatan is a sequence of nonnegative real numbers such that there exists subsequences nj of n such that ani<ani+1 for all i∈ℕ. Then there exists a nondecreasing sequence mk⊂ℕ such that mk⟶∞ as k⟶∞, and the following properties are satisfied all (sufficiently large) number k∈ℕ:
(33)amk<amk+1,ak<amk+1.
In fact,mk=maxj≤k:aj≤aj+1.Example 17.
LetX=Rnn≥2 and K=v1,v2,⋯,vn:vi≥1,i=1,2,⋯,n. Consider a bifunction f:K×K⟶R defined as fv,w=2wn−vndv,w for each v,w∈K. Then we have the inequality
(34)fv,w=2wn−vndv,w≤2wn−rndv,r+2wn−rndw,r+2rn−vndv,r+2rn−vndr,w≤2dw,rdv,r+2dr,vdr,w+fv,r+fr,w≤2d2v,r+2d2r,w+fv,r+fr,w.
Then,f is Lipschitz-type continuous on K with c1=2=c2.


## 3. Strong Convergence of the Proposed Algorithm

Suppose that a nonexpansive mappingT:K⟶K and a bifunction ϕ:K×K⟶R satisfy the conditions (6)–(10) and FixT∩EPϕ≠∅. Further, let p:K⟶K be a ρ-contraction. Because PFixT∩EPϕp is a contraction mapping on K, we have q^∈K, such that q^=PFixT∩EPϕpq^.Before presenting the first result of this section, let us recall a crucial theorem given by Lim in [18], which is used in the proof of our main results.Theorem 18 ([18], Theorem 5.2).
Every bounded sequencexn in a Hadamard space X,d has a Δ-convergent subsequence.The following algorithm is useful in finding a common element of a solution set of pseudomonotone equilibrium problem onϕ and fixed point set of T.Algorithm 19
Considerλn⊂δ1,δ2 with 0<δ1<δ2<min1/2c1,1/2c2,αn⊂0,1/2−ρ,βn⊂0,1, and v0∈K. Consider n=0.Step 1.
(35)wn=argminw∈Kλnϕvn,w+12d2w,vn,vn=argmins∈Kλnϕwn,s+12d2s,vn.Step 2.
Ifwn=vn=Tvn, then vn∈FixT∩EPϕ, stop the process; otherwise, go to Step 3.Step 3.
Generatevn+1=αnpvn⊕1−αnβnvn⊕1−βnTsn. Set n=n+1 and go to Step 1. Obviously, if wn=vn for n∈N, by using (26), it gives ϕvn,z≥0 for all z∈K and vn∈EPϕ. It gives vn∈FixT∩EPϕ: from vn=Tvn. Further, for convergence of the algorithm, let us consider that Step 2 is not true for n∈N.Lemma 20.
Considervn to be bounded sequence. If dvn,wn⟶0, dvn,sn⟶0, dsn,Tsn⟶0 as n⟶∞, then
(36)limsupn⟶∞vnq^⟶,pq^q^⟶≤0.Proof.
We consider the following bounded sequencevn. Using Theorem 18, there exists vnk a subsequence of vn such that vnk weakly converges to v∈K with
(37)limsupn⟶∞vnq^⟶,pq^q^⟶=limk⟶∞vnkq^⟶,pq^q^⟶=vq^⟶,pq^q^⟶.
Sincedvn,wn⟶0, we have wnk⇀v.
By (26) with v=vnk and w=wnk, we have
(38)λϕvnk,z−ϕvnk,wnk≥wnkvnk⟶,yz⟶,forallz∈K.
Takingk⟶∞ on λn, (6), and (10), we achieve
(39)ϕv,z≥0forallz∈K.
This givesv∈EPϕ. Also snk⇀v.
By Lemma12 with dvn,wn⟶0, we obtain v∈FixT. Then v∈FixT∩EPϕ. Using Lemma 13 with (36), the conclusion follows.☐Theorem 21.
Consider the sequencesαn and βn satisfying the conditions:
(40)limn⟶∞αn=0,∑n=0∞αn=∞,0<liminfn⟶∞βn≤limsupn⟶∞βn<1,then the sequence vn strongly converges to q^.Proof.
Considerln=βnvn⊕1−βnTsn∀n∈ℕ. Lemma 11 follows
(41)d2ln,q^≤βnd2vn,q^+1−βnd2Tsn,q^≤βnd2vn,q^+1−βnd2sn,q^≤βnd2vn,q^+1−βnd2vn,q^−1−λnc1d2vn,wn−1−λnc2d2vn,sn≤βnd2vn,q^+1−βn1−λnc1d2vn,wn−1−βn1−λnc2d2wn,sn≤d2vn,q^.
Then(42)dvn+1,q^≤αndpvn,q^+1−αndln,q^≤αndpvn,pq^+αndpq^,q^+1−αndvn,q^≤αnρdvn,q^+αndpq^,q^+1−αndvn,q^=1−αn1−ρvn,q^+αndpq^,q^≤maxpq^,q^1−ρ,dvn,q^⋯≤maxdpq^,q^1−ρ,dv0,q^.
Hence,vn is bounded; then, pvn, wn, and sn are bounded too. On the other hand, by (41), we have
(43)d2vn+1,q^≤αnd2pvn,q^+1−αnd2ln,q^≤αnd2pvn,q^+1−αnd2vn,q^−1−βn1−λnc1d2vn,wn+1−λnc2d2wn,sn.
Let us consider(44)M=supd2pvn,q^−d2vn,q^+1−βn1−λnc1d2vn,wn+1−λnc2d2wn,sn:n∈ℕ.
Combining (43) and (44), we get
(45)1−βn1−λnK1d2vn,wn+1−λnc2d2wn,sn≤d2vn,q^−d2vn+1,q^+αnM.
By Lemma14 and (40), we have
(46)d2vn+1,q^=d2αnpvn−q^⊕1−αnln−q^,0≤1−αn2d2ln,q^+2αnpvnq^⟶vn+1q^⟶≤1−αn2d2vn,q^+2αnpvnpq^⟶,vn+1q^⟶+2αnpq^q^⟶,vn+1q^⟶≤1−αn2d2vn,q^+2αndpvn,pq^dvn+1,q^+2αnpvnq^⟶,vn+1q^⟶≤1−αn2d2vn,q^+2αnρdvn,q^dvn+1,q^+2αnpqnq^⟶,vn+1q^⟶≤1−αn2d2vn,q^+αnρd2vn,q^+d2vn+1,q^+2αnpqnq^⟶,vn+1q^⟶.
Therefore,(47)d2vn+1,q^≤1−αn2+αnρ1−αnρd2vn,q^+2αn1−αnρpq^q^⟶,vn+1q^⟶≤1−2αn+αnρ1−αnρd2vn,q^+αn21−αnρd2vn,q^+2αn1−αnρpq^q^⟶,vn+1q^⟶≤1−21−ραn1−αnρd2vn,q^+21−ραn21−αnM0+2αn1−αnρpq^q^⟶,vn+1q^⟶,where M0=supd2vn,q^:n∈ℕ. Taking γn=21−ραn/1−αnρ for all n∈ℕ. As αn⊂0,1/2−ρ, then γn⊂0,1. By the conditions on αn, we get
(48)limn⟶∞γn=0,∑n=0∞γn=∞.
Then, we have(49)d2vn+1,q^≤1−γnd2vn,q^+αnγnM021−ρ+γn1−ρpq^q^⟶,vn+1q^⟶.
The rest of the proof will be divided into two parts.☐Case 1.
Suppose that there existsn0∈ℕ such that dvn,q^n=n0∞ is nonincreasing. In this situation, dvn,q^ is convergent. This together with the hypothesis on γn, αn, βn, and (45) gives
(50)limn⟶∞dvn,wn=limn⟶∞dvn,sn=0.
On the other hand, by Lemmas11 and 14, we have
(51)d2vn+1,q^=d2αnpvn−q^⊕1−αnβnvn⊕1−βnTsn−q^,0≤αnd2pvn,q^+1−αnd2βnvn⊕1−βnTsn−q^,0≤αnd2pvn,q^+1−αnβnd2vn,q^+1−βnd2Tsn,q^−βn1−βnd2vn,Tsn≤αnd2pvn,q^+1−αnβnd2vn,q^+1−βnd2vn,q^−βn1−βnd2vn,Tsn≤αnd2pvn,q^+1−αnβnd2vn,q^−βn1−βn1−αnd2vn,Tsn.
Hence,(52)βn1−βn1−αnd2vn,Tsn≤d2vn,q^−d2vn+1,q^+αnd2pvn,q^.
Then,dvn,q^ converges. Also, αn⊂0,1/2−ρ, and conditions on αn and βn gives
(53)limn⟶∞dvn,Tsn=0.
Combining (49) and (52), we get
(54)dsn,Tsn≤dsn,vn+dvn,Tsn⟶0,asn⟶∞.
Lemma20 and (49) and (53) give
(55)limsupn⟶∞pqnq^⟶,vn+1q^⟶≤0.
The conclusion follows from Lemma15 and (48), (49), and (54).Case 2.
Letvni be a subsequence of vn with
(56)dvni,q^<dvni,q^∀i∈ℕ.
Then, by Lemma16, there exists a subsequence mk such that mk⟶∞:
(57)dvnk,q^<dvmk+1,q^,dvk,q^<dvnk+1,q^,∀k∈ℕ.
The above expression with (44) concludes
(58)1−βmk1−λmkc1d2vmk,wmk+1−λmkc2d2wmk,smk≤d2vmk,q^−d2vmk+1,q^+αmkM≤αmkM∀k∈ℕ.
By the hypothesis onαn,βn, and γn, it follows that
(59)limk⟶∞d2vmk,wmk=d2wmk,smk=0.
Using (51), we have
(60)βmk1−βmk1−αmkd2vmk,Tsmk≤d2vmk,q^−d2vmk+1,q^+αmkd2pvmk,q^≤αmkd2pvmk,q^⟶0,ask⟶∞.
By the hypothesis onαn and βn, it follows that
(61)limk⟶∞dvmk,Tsmk=0.
In a similar way as Case1, we arrive at
(62)limsupk⟶∞pvmk+1q^⟶,pq^q^⟶≤0.
Note that(63)d2vmk+1,q^≤1−αmkd2vmk,q^+2αmkpq^q^⟶,vmk+1q^⟶.
Since(64)dvmk,q^<dvmk+1,q^forallk∈ℕ,we have
(65)αmkd2vmk,q^≤d2vmk,q^−d2vmk+1,q^+2αmkpq^q^⟶,vmk+1q^⟶+2αmkpq^q^⟶,vmk+1q^⟶.
Asαmk>0, we have
(66)d2vmk,q^≤pq^q^⟶,vmk+1q^⟶.
From (61), it follows that dvmk,q^⟶0 as k⟶∞. By the above arguments, using relations (61) and (62) and conditions of αn, we obtain
(67)dvmk+1,q^⟶0,with k⟶∞.
Asdvk,q^<dvmk,q^ for all k∈ℕ, we get vk⟶q^ when k⟶∞.Corollary 22.
Consider a CAT(0) spaceX and nonempty, convex, and closed subset K of X. Further, consider a bifunction ϕ:K×K⟶R that fulfils conditions (6)–(10) and a nonexpensive mapping T:K⟶K with FixT∩EPϕ≠∅. Then, define a sequence vn as follows: for any u,v0∈K, consider
(68)wn=argminw∈Kλnϕvn,w+12d2w,vn,sn=argminw∈Kλnϕvn,s+12d2s,vn,vn+1=αnu⊕1−αnβnvn⊕1βnTsn,n∈ℕ.
Initially, we chooseλn⊂δ1,δ2 with 0<δ1<δ2<min1/2c1,1/2c2,αn⊂0,1/2−ρ,βn⊂0,1, and take v0∈K. Set n=0 and the sequences αn, βn as in Theorem 21, then vn converges strongly to q^=PFixT∩EPϕu.Corollary 23.
Consider a CAT(0) spaceX and nonempty, convex, and closed subset K of X. Further consider a bifunction ϕ:K×K⟶R that fulfils conditions (6)–(10) such that EPϕ≠ϕ, the sequence vn with u,v0∈K and
(69)wn=argminw∈Kλnϕvn,w+12d2w,vn,sn=argmins∈Kλnϕsn,w+12d2s,vn,vn+1=αnu⊕1−αnβnvn⊕1−βnTsn,n∈ℕ,where λn⊂δ1,δ2 with 0<δ1<δ2<min1/2c1,1/2c2,αn,βn⊂0,1. If the sequences αn and βn are same as in Theorem 21, then the vn converges strongly to q^=PEPϕu.Example 24.
LetH=ℝ3 and K=v1,v2,v3:v1,v2,v3=0. Let ϕv,w=w3−v3v for allv=v1,v2,v3, w=w1,w2,w3∈K. Then ϕ is Lipschitz-type continuous with the constants c1=c2=1.
It is easy to see thatϕ satisfies conditions (6)–(10). Let Tv=v1,v2,arctan0.1v3 and pv=0.5v1,0.4v2,0.2v3 for all v=v1,v2,v3∈K. It follows that T is a nonexpansive mapping and p is a contraction. Take the initial point v1=2,2,2 and put the sequences αn=1/10n, βn=1/4+4n, and γn=1/4+1/8n for all n∈ℕ. It is easy to see that
(70)FixT∩EPϕ=v1,v2,0:v1,v2=0.
We give somevn and vn−wn by Table 1: from Table 1, we see that after 13 iterations v13=1.70912,1.65541,0.00000∈FixT∩EPϕ.Corollary 25.
Consider a CAT(0) spaceX and nonempty, convex, and closed subset K of X. Further consider a bifunction ϕ:K×K⟶R that fulfils conditions (6)–(10) and a nonexpensive mapping T:K⟶K with FixT∩EPϕ≠∅. Then define a sequence vn as follows.
For anyu,v0∈K, consider
(71)wn=argminw∈Kλnϕvn,w+12d2w,vn,sn=argminw∈Kλnϕvn,s+12d2s,vn,vn+1=αnu⊕1−αnβnvn⊕1βnTsn,n∈ℕ.
Initially chooseλn⊂δ1,δ2 with 0<δ1<δ2<min1/2c1,1/2c2,αn⊂0,1/2−ρ,βn⊂0,1 and take v0∈K. Set n=0 and the sequences αn, βn as in Theorem 21.
Thenvn converges strongly to q^=PFixT∩EPϕu.Corollary 26.
Consider a CAT(0) spaceX and nonempty, convex, and closed subset K of X. Further consider a bifunction ϕ:K×K⟶R that fulfils conditions (6)–(10) such that EPϕ≠ϕ, the sequence vn with u,v0∈K and
(72)wn=argminw∈Kλnϕvn,w+12d2w,vn,sn=argmins∈Kλnϕsn,w+12d2s,vn,vn+1=αnu⊕1−αnβnvn⊕1−βnTsn,n∈ℕ,where λn⊂δ1,δ2 with 0<δ1<δ2<min1/2c1,1/2c2,αn,βn⊂0,1. If the sequences αn and βn are the same as in Theorem 21, then the sequence vn converges strongly to q^=PEPϕu.Remark 27.
Our Theorem21 is an analog of Theorem 3.1 of Wang et al. [30] for the Hilbert space case.Remark 28.
Our results can be extended to anyCATκ space with κ≤0, since any CATκ′ space is a CATκ space, for any κ′>κ (see [17]).Table 1
Somevn and vn−wn generated by Algorithm 19.
Iteration (n)v1nv2nv3nvn−wn12.000002.000002.000001.2990321.900001.880000.980710.8897331.852501.823600.367800.3678041.821621.787130.120970.1209651.798861.760320.037460.0374661.780871.739200.011160.0111671.760261.721810.003230.0032481.753411.707040.000920.0009291.742451.694240.000260.00026101.732771.682950.000070.00007111.724111.672850.000010.00002121.716271.663730.000010.00001131.709121.655410.000000.00000


## 4. Conclusions

It is well known that finding solutions for equilibrium problems play an important role in solving problems in other areas of science like physics, optimization, and economics. In this paper, we construct a new viscosity extragradient algorithm in order to find fixed points in a CAT(0) space for the case of nonexpansive mappings. Also, we prove a strong convergence of the proposed algorithm, and we give some examples to support our results.



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*Source: 7808255-2021-08-26.xml*