# Strong Convergence of New Two-Step Viscosity Iterative Approximation Methods for Set-Valued Nonexpansive Mappings inCAT(0) Spaces

Authors: Ting-jian Xiong; Heng-you Lan
Journal: Journal of Function Spaces
 (2018)

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## Abstract

This paper is for the purpose of introducing and studying a class of new two-step viscosity iteration approximation methods for finding fixed points of set-valued nonexpansive mappings inCAT(0) spaces. By means of some properties and characteristic to CAT(0) space and using Cauchy-Schwarz inequality and Xu’s inequality, strong convergence theorems of the new two-step viscosity iterative process for set-valued nonexpansive and contraction operators in complete CAT(0) spaces are provided. The results of this paper improve and extend the corresponding main theorems in the literature.


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## Body

## 1. Introduction

In [1], the fixed point theory in CAT(0) spaces was first introduced and studied by Kirk. Further, Kirk [1] presented that each nonexpansive (single-valued) mapping on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. On the other hand, fixed point theory for set-valued mappings has been applied to applied sciences, game theory, and optimization theory. This promotes the rapid development of fixed point theory for single-valued (set-valued) operators in CAT(0) spaces, and it is natural and particularly meaningful to extensively study fixed point theory of set-valued operators. Particularly, some old relative works on Ishikawa iterations for multivalued mappings can be found in [2–4]. For more detail, we refer to [5–14] and the references therein.Definition 1.
Letg:X→X be a nonlinear operator on a metric space (X,d) and G:E→BC(X) be a set-valued operator, where E⊂X is a nonempty subset and BC(X) is the family of nonempty bounded closed subsets of X. Then
(i)g is said to be acontraction, if there exists a constant κ∈[0,1) such that (1)dgx,gy≤κdx,y∀x,y∈X.Here, g is callednonexpansive when κ=1 in (1).
(ii)G is said to be a nonexpansive, if (2)HGx,Gy≤dx,y,where H(·,·) is Hausdorff distance on BC(X), i.e., (3)HA,B=max⁡supa∈Ainfb∈Bda,b,supb∈Binfa∈Ada,b,∀A,B∈BCX.Recently, Shi and Chen [5] first considered the following Moudafi’s viscosity iteration for a nonexpansive mapping g:E→E with ∅≠Fix(g)={x∣x=g(x)} and a contraction mapping f:E→E in CAT(0) space X: (4)xα=αfxα⊕1-αgxα,and (5)xn+1=αnfxn⊕1-αngxn,n≥1,where α,αn∈(0,1) and x1 is an any given element in a nonempty closed convex subset E⊆X. xα∈E is called unique fixed point of contraction x↦αf(x)⊕(1-α)g(x). Shi and Chen [5] showed that {xα} defined by (4) converges strongly to x~∈Fix(g) as α→0+, where x~=PFix(g)f(x~) in CAT(0) space (X,d) satisfies the following property P: for all x,u,y1,y2∈X, (6)dx,m1dx,y1≤dx,m2dx,y2+dx,udy1,y2,that is, an extra condition on the geometry of CAT(0) spaces is requested, where mi=P[x,yi]u for i=1,2. Further, the authors also found that the sequence {xn} generated by (5) converges strongly to x~∈Fix(g) under some suitable conditions about {αn}. Afterwards, based on the concept of quasilinearization introduced by Berg and Nikolaev [15], Wangkeeree and Preechasilp [6] explored strong convergence results of (4) and (5) in CAT(0) spaces without the property P and presented that the iterative processes (4) and (5) converges strongly to x~∈Fix(g) such that x~=PFix(g)f(x~) is the unique solution of the following variational inequality: (7)x~fx~→,yx~→≥0,∀y∈Fixg.In [16], Panyanak and Suantai extended (4) and (5) to T being a set-valued nonexpansive mapping from E to BC(X). That is, for each α∈(0,1), let a set-valued contraction Gα on E be defined by (8)Gαx=αfx⊕1-αTx,∀x∈E.By Nadler’s [17] theorem, it is easy to know that there exists xα∈E such that xα is a fixed point of Gα, which does not have to be unique, and (9)xα∈αfxα⊕1-αTxα,i.e., for each xα, there exists yα∈Txα such that (10)xα=αfxα⊕1-αyα.Further, when the contraction constant coefficient of f is k∈[0,1/2) and {αn}⊂(0,1/(2-k)) satisfying some suitable conditions, Panyanak and Suantai [16] proved strong convergence of one-step viscosity approximation iteration defined by (10) or the following iterative process in CAT(0) spaces: (11)xn+1=αnfxn⊕1-αnyn,yn∈Mxn,and d(yn,yn+1)≤d(xn,xn+1) for all n∈N, where M is a set-valued nonexpansive operator from E to C(E), the family of nonempty compact subsets of E, f:E→E is a contraction, and {αn}⊆(0,1). Moreover, Chang et al. [7] affirmatively answered the open question proposed by Panyanak and Suantai [16, Question 3.6]: “If k∈[0,1) and {αn}⊂(0,1) satisfying the same conditions, does {xn} converge to x~=PF(M)f(x~)”, where F(M) denotes the set of all fixed points of M.On the other hand, Piatek [18] introduced and studied the following two-step viscosity iteration in complete CAT(0) spaces with the nice projection property N: (12)yn=αnfxn⊕1-αngxn,xn+1=βnxn⊕1-βnyn,∀n≥1,where x1∈E is an given element and {αn},{βn}⊆(0,1) satisfying some suitable conditions and the contraction coefficient of f is k∈[0,1/2).Based on the ideas of Wangkeeree and Preechasilp [6] and Piatek [18] intensively, Kaewkhao et al. [19] omit the nice projection property N. We note that the two-step viscosity iteration (12) is also considered and studied by Chang et al. [8] when the property N is not satisfied and k∈[0,1), which is due to the open questions in [19], where the property N depends on whether its metric projection onto a geodesic segment preserves points on each geodesic segment, that is, for every geodesic segment χ⊂X and x,y∈X, m∈[x,y] implies Pχm∈[Pχx,Pχy], where Pχ denotes the metric projection from X onto χ. For more works on the convergence analysis of (viscosity) iteration approximation method for (split) fixed point problems, one can refer to [20–27].Motivated and inspired mainly by Panyanak and Suantai [16] and Piatek [18] and so on, we consider the following two-step viscosity iteration for set-valued nonexpansive operator T:E→C(E): (13)xn+1=βnxn⊕1-βnyn,yn=αnfxn⊕1-αnzn,∀n≥1,where E is a nonempty closed convex subset of complete CAT(0) space (X,d), x1∈E is an given element and {αn},{βn}⊆(0,1), f:E→E is a contraction mapping, and zn∈T(xn) satisfying d(zn,zn+1)≤d(xn,xn+1) for any n∈N.By using the method due to Chang et al. [7, 8], the purpose of this paper is to prove some strong convergence theorems of the viscosity iteration procedure (13) in complete CAT(0) spaces. Hence, the results of Chang et al. [7, 8] and many others in the literature can be special cases of main results in this paper.


## 2. Preliminaries

Throughout of this paper, let(X,d) be a metric space. A geodesic path joining x∈X to y∈X (or, more briefly, a geodesic from x to y) is a map ξ:R⊇[0,l]→X such that ξ(0)=x,ξ(l)=y, and d(ξ(s),ξ(t))=s-t for each s,t∈[0,l]. In particular, ξ is a isometry and d(x,y)=l. The image of ξ is called a geodesic segment (or metric) joining x and y if unique is bespoke by [x,y]. The space (X,d) is called a geodesic space when every two points in X are joined by a geodesic, and X is called uniquely geodesic if there is exactly one geodesic joining x and y for any x,y∈X. A subset E of X is said to be convex if E includes every geodesic segment joining any two of its points. A geodesic triangle △(p,q,r) in a geodesic space (X,d) consists of three points p,q,r in X (the vertices of △) and a choice of three geodesic segments [p,q],[q,r],[r,p] (the edge of △) joining them. A comparison triangle for geodesic triangle △(p,q,r) in X is a triangle △¯(p-,q-,r-) in the Euclidean plane R2 such that (14)dR2p-,q-=dp,q,dR2q-,r-=dq,r,dR2r-,p-=dr,p.A point u-∈[p-,q-] is said to be a comparison point for u∈[p,q] if d(p,u)=dR2(p-,u-). Similarly, we can give the definitions of comparison points on [q-,r-] and [r-,p-].Definition 2.
Suppose that△ is a geodesic triangle in (X,d) and △¯ is a comparison triangle for △. A geodesic space is said to be a CAT(0) space, if all geodesic triangles of appropriate size satisfy the following comparison axiom (i.e., CAT(0) inequality): (15)du,v≤dR2u-,v-,∀u,v∈△,u-,v-∈△¯.Complete CAT(0) spaces are often called Hadamard spaces (see [28]). For other equivalent definitions and basic properties of CAT(0) spaces, one can refer to [29]. It is well known that every CAT(0) space is uniquely geodesic and any complete, simply connected Riemannina manifold having nonpositive sectional curvature is a CAT(0) space. Other examples for CAT(0) spaces include pre-Hilbert spaces [29], R-trees [9], Euclidean buildings [30], and complex Hilbert ball with a hyperbolic metric [31] as special case.
LetE be a nonempty closed convex subset of a complete CAT(0) space (X,d). By Proposition 2.4 of [29], it follows that, for all x∈X, there exists a unique point x0∈E such that (16)dx,x0=inf⁡dx,y:y∈E.Here, x0 is said to beunique nearest point of x in E.
Assume that(X,d) is a CAT(0) space. For all x,y∈X and t∈[0,1], by Lemma 2.1 of Phompongsa and Panyanak [10], there exists a unique point z∈[x,y] such that (17)dx,z=1-tdx,yanddy,z=tdx,y.We shall denote by tx⊕(1-t)y the unique point z satisfying (17). Now, we give some results about CAT(0) spaces for the proof of our main results.Lemma 3 ([1, 10]).
Let(X,d) be a CAT(0) space. Then for each x,y,z∈X and α∈[0,1],(i)d(αx⊕(1-α)y,z)≤αd(x,z)+(1-α)d(y,z).(ii)d2(αx⊕(1-α)y,z)≤αd2(x,z)+(1-α)d2(y,z)-α(1-α)d2(x,y).(iii)d(αx⊕(1-α)z,αy⊕(1-α)z)≤αd(x,y).Lemma 4 ([11]).
Suppose that(X,d) is a CAT(0) space. Then for all x,y∈X and α,β∈[0,1], (18)dαx⊕1-αy,βx⊕1-βy≤α-βdx,y.Lemma 5 ([12]).
Assume that{xn} and {yn} are two bounded sequences in a CAT(0) space (X,d) and {βn} is a sequence in [0,1] with 0<liminfnβn≤limsupnβn<1. If (19)xn+1=βnxn⊕1-βnyn,∀n∈N,limsupn→∞dyn+1,yn-dxn+1,xn≤0,then limn→∞d(xn,yn)=0.Lemma 6 ([32]).
Suppose that nonnegative real numbers sequence{un} is defined by (20)un+1≤1-αnun+αnβn,∀n≥1,where {αn}⊂[0,1] and {βn}⊂R are two sequences satisfying
(i)∑n=1∞αn=∞; (ii) limsupn→∞βn≤0 or ∑n=1∞αnβn<∞.
Thenlimn→∞{un}=0.Lemma 7 ([13]).
Assume thatE is a closed convex subset of a complete CAT(0) space (X,d). If a set-valued nonexpansive operator T:E→BC(X) satisfies endpoint condition C, i.e., F(T)≠∅ and T(x)={x} for any x∈F(T) (see [33]), then F(T) is closed and convex.In [15], Berg and Nikolaev introduced the concept of quasilinearization. Now we denote a pair (a,b)∈X×X by ab→, which is a vector. Define the quasilinearization by a map 〈·,·〉: (X×X)×(X×X)→R as follows: (21)ab→,cd→=12d2a,d+d2b,c-d2a,c-d2b,d,∀a,b,c,d∈X.One can easily know that (22)ab→,cd→=cd→,ab→,ab→,cd→=-ba→,cd→,ab→,cd→+ad→,bc→=ac→,bd→,ax→,cd→+xb→,cd→=ab→,cd→for every a,b,c,d,x∈X. We say that a geodesic metric space (X,d) satisfies the Cauchy-Schwarz inequality if (23)ab→,cd→≤da,bdc,d,∀a,b,c,d∈X.From [15, Corollary 3], it is known that a geodesic space (X,d) is a CAT(0) space if and only if X satisfies the Cauchy-Schwarz inequality. Further, we give the following other properties of quasilinearization.Lemma 8 ([14]).
Assume thatE is a nonempty closed convex subset of a complete CAT(0) space (X,d). Then for u∈X and x∈E, (24)x=PEuif  and  only  ifxu→,yx→≥0,∀y∈E.Lemma 9 ([6]).
For two pointsu and v in a CAT(0) space (X,d) and any α∈[0,1], letting uα=αu⊕(1-α)v, then for all x,y∈X, the following results hold:(i)〈uαx→,uαy→〉≤α〈ux→,uαy→〉+(1-α)〈vx→,uαy→〉;(ii)〈uαx→,uy→〉≤α〈ux→,uy→〉+(1-α)〈vx→,uy→〉 and 〈uαx→,vy→〉≤α〈ux→,vy→〉+(1-α)〈vx→,vy→〉.Definition 10.
A continuous linear functionalμ is said to be Banach limit on l∞ if (25)μ=μ1,1,⋯=1andμnun=μnun+1,∀un∈l∞.Lemma 11 ([34]).
Suppose that, for real numberα and all Banach limits μ, (u1,u2,⋯)∈l∞ satisfies (26)μnun≤α,limsupnun+1-un≤0.Then limsupnun≤α.Lemma 12 ([16]).
Assume that(X,d) is a complete CAT(0) space, E⊂X is a nonempty closed convex subset, T:E→C(E) is a set-valued nonexpansive operator satisfying endpoint condition C, and f:E→E is a contraction with k∈[0,1). Then we have following results:(i){xα} generated by (10) converges strongly to x~ as α→0+, where x~=PF(T)f(x~).(ii) In addition, if{xn}⊂E is a bounded sequence such that limn→∞dist(xn,T(xn))=0, where dist(a,B) is the distance from a point a∈X to the set B∈C(X), then for all Banach limits μ, (27)d2fx~,x~≤μnd2fx~,xn.


## 3. Main Results

Employing the preliminaries in the previous section, now we will study the strong convergence of the new two-step viscosity iteration (13) for set-valued nonexpansive operators in complete CAT(0) spaces.Theorem 13.
Assume that(X,d) is a complete CAT(0) space, E⊂X is a nonempty closed convex subset, T:E→C(E) is a set-valued nonexpansive operator satisfying endpoint condition C, and f:E→E is contraction with k∈0,1. If sequences {αn},{βn}∈(0,1) satisfy(L1) limn→∞αn=0, (L2) ∑n=1∞αn=∞, and (L3) 0<liminfn→∞βn≤limsupn→∞βn<1, then the sequence {xn} generated by (13) converges strongly to x~, where (28)x~=PFTfx~,x~fx~→,xx~→≥0,∀x∈FT.Proof.
The proof shall be divided into the following four steps.
Step 1. We first prove that sequences {xn}, {f(xn)}, {yn}, and {zn} are bounded. In fact, setting p∈F(T), then from Lemma 3, we know (29)dyn,p≤αndfxn,p+1-αndistzn,Tp≤αndfxn,p+1-αnHTxn,Tp≤αndfxn,p+1-αndxn,p≤αndfxn,fp+αndfp,p+1-αndxn,p≤1-αn1-kdxn,p+αndfp,p,and (30)dxn+1,p≤βndxn,p+1-βndyn,p≤1-αn1-k1-βndxn,p+αn1-k1-βndfp,p1-k≤max⁡dxn,p,dfp,p1-k.Thus, we obtain (31)dxn,p≤max⁡dx1,p,dfp,p1-k.Hence, {xn} is bounded, so is {f(xn)}. By (29), it is easy to know that {yn} is bounded. Since d(zn,p)≤H(T(xn,T(p))≤d(xn,p), one can easily know that the sequence {zn} is also bounded.
Step 2. We present that limn→∞d(xn,yn)=0, limn→∞dist(xn,T(xn))=0, limn→∞d(xn,zn)=0, limn→∞d(xn,xn+1)=0, limn→∞d(zn,zn+1)=0, and limn→∞dist(zn,T(zn))=0. Indeed, by applying Lemmas 3 and 4, we have (32)dyn,yn+1≤dαnfxn⊕1-αnzn,αn+1fxn+1⊕1-αn+1zn+1≤dαnfxn⊕1-αnzn,αnfxn⊕1-αnzn+1+dαnfxn⊕1-αnzn+1,αnfxn+1⊕1-αnzn+1+dαnfxn+1⊕1-αnzn+1,αn+1fxn+1⊕1-αn+1zn+1≤1-αndzn,zn+1+αndfxn,fxn+1+αn-αn+1dfxn+1,zn+1≤1-αn1-kdxn,xn+1+αn-αn+1dfxn+1,zn+1,and so (33)dyn,yn+1-dxn,xn+1≤αn-αn+1dfxn+1,zn+1-1-kαndxn,xn+1.From limn→∞αn=0 and the boundedness of {xn}, {f(xn)}, and {zn}, we know (34)limsupn→∞dyn+1,yn-dxn+1,xn≤0.It follows from Lemma 5 that (35)limn→∞dxn,yn=0.Thus, (36)distxn,Txn≤dxn,zn≤dxn,yn+αndfxn,zn⟶0asn⟶∞.By (36), now we know that (37)limn→∞dzn,xn=0.
Moreover,(38)dxn,xn+1=1-βndxn,yn⟶0and (39)dzn,zn+1≤dxn,xn+1⟶0as n→∞. By (36) and (37), we get (40)distzn,Tzn≤dzn,xn+distxn,Txn+HTxn,Tzn≤2dxn,zn+distxn,Txn⟶0asn⟶∞.
Step 3. Now, we show that (41)limsupn→∞d2fx~,x~-d2fx~,zn≤0,with x~=PF(T)f(x~) satisfying (42)x~fx~→,xx~→≥0,∀x∈FT.Above all, since T(x) is compact for any x∈E, then T(x)∈BC(X). It follows from Lemma 7 that F(T) is closed and convex, which implies that PF(T)u is well defined for any u∈X. By Lemma 12 (i), we know that {xα} generated by (10) converges strongly to x~=PF(T)f(x~) as α→0+. Then by Lemma 8, we know that x~ is the unique solution of the following variational inequality: (43)x~fx~→,xx~→≥0,∀x∈FT.
Next, since{zn} is bounded and limn→∞dist(zn,T(zn))=0, it follows from Lemma 12 (ii) that for all Banach limits μ, (44)d2fx~,x~≤μnd2fx~,zn,and so (45)μnd2fx~,x~-d2fx~,zn≤0.Further, limn→∞d(zn,zn+1)=0 implies that (46)limsupn→∞d2fx~,x~-d2fx~,zn+1-d2fx~,x~-d2fx~,zn=0.By Lemma 11, we have (47)limsupn→∞d2fx~,x~-d2fx~,zn≤0.
Step 4. limn→∞xn=x~ will be verified. In fact, by Lemma 3 and (13), now we know (48)d2xn+1,x~≤βnd2xn,x~+1-βnd2yn,x~-βn1-βnd2xn,yn≤βnd2xn,x~+1-βnd2yn,x~,and (49)d2yn,x~≤αnd2fxn,x~+1-αnd2zn,x~-αn1-αnd2fxn,zn≤1-αnH2Txn,Tx~+αn2d2fxn,zn+αnd2fxn,x~-d2fxn,zn≤1-αnd2xn,x~+αn2d2fxn,zn+αnd2fxn,x~-d2fxn,zn.It follows from (21), Cauchy-Schwarz inequality, and Lemma 9 that (50)αnd2fxn,x~-d2fxn,zn≤2αndfxn,fx~dzn,x~+fx~x~→,znx~→-d2zn,x~≤2αnkdxn,x~dzn,x~+fx~x~→,znx~→-d2zn,x~≤αnkd2xn,x~+d2zn,x~+2αnfx~x~→,znx~→-2αnd2zn,x~≤αnkd2xn,x~+αnd2fx~,x~-d2fx~,zn.From (50) and (49), we know (51)d2yn,x~≤1-αn1-kd2xn,x~+αnd2fx~,x~-d2fx~,zn+αn2d2fxn,zn.Combining (51) and (48), we get (52)d2xn+1,x~≤βnd2xn,x~+1-βn1-αn1-kd2xn,x~+αnd2fx~,x~-d2fx~,zn+αn2d2fxn,zn≤1-1-kαn1-βnd2xn,x~+αn1-βnd2fx~,x~-d2fx~,zn+1-βnαn2d2fxn,zn≤1-1-kαn1-βnd2xn,x~+αn1-βnd2fx~,x~-d2fx~,zn+αn2d2fxn,zn,i.e., (53)un+1≤1-αn′un+αn′βn′,∀n≥1,where un=d2(xn,x~), αn′=(1-k)αn(1-βn), and (54)βn′=1-βnd2fx~,x~-d2fx~,zn+αnd2fxn,zn1-k1-βn.Thus, from the conditions (L1)-(L3) and the inequality (41), it follows that αn′∈(0,1), and (55)∑n=1∞αn′=∞,limsupn→∞⁡βn′≤0.Hence, it follows from Lemma 6 that un→0. This implies that the proof is completed.IfT≡g is a nonexpansive single-valued operator with Fix(g)≠∅, then from Theorem 13, one can easy to obtain the following result.Corollary 14.
Suppose thatf, E, and (X,d) are the same as in Theorem 13, and the conditions (L1)-(L3) in Theorem 13 are satisfied. If g:E→E is a nonexpansive single-valued operator with Fix(g)≠∅, then the sequence {xn} generated by (12) converges strongly to x~=PFix(g)f(x~) with (56)x~fx~→,yx~→≥0,∀y∈Fixg.Remark 15.
Corollary14 is the corresponding results of Theorem 3.1 in [8].
Iff≡I, the identity operator, then by Theorem 13, now we directly have the following theorem.Theorem 16.
Assume thatT, E, and (X,d) are the same as in Theorem 13, and the conditions (L1)-(L3) in Theorem 13 hold. Then for any given u,x1∈E, sequence {xn} generated by (57)yn=αnu⊕1-αnzn,dzn,zn+1≤dxn,xn+1,zn∈Txn,xn+1=βnxn⊕1-βnyn,∀n≥1,converges strongly to the unique nearest point x~ of u in F(T), i.e., x~=PF(T)u, where x~ also satisfies (58)x~u→,vx~→≥0,∀v∈FT.Remark 17.
Theorems13 and 16 also extend and improve the corresponding results of Chang et al. [7], Piatek [18], Kaewkhao et al. [19], Panyanak and Suantai [16], and many others in the literature.


## 4. Concluding Remarks

The purpose of this paper is to introduce and study the following new two-step viscosity iterative approximation for finding fixed points of a set-valued nonlinear mappingG:D→C(D) and a contraction mapping g:D→D: (59)un+1=βnun⊕1-βnvn,vn=αngun⊕1-αnwn,∀n≥1,where D is a nonempty closed convex subset of a metric space E, u1∈D is an any given element and {αn},{βn}⊆(0,1), and wn∈G(un) satisfying d(wn,wn+1)≤d(un,un+1) for any n∈N.By using the method due to Chang et al. [7, 8], Cauchy-Schwarz inequality, and Xu’s inequality, we exposed strong convergence theorems of the new two-step viscosity iteration approximation (59) in complete CAT(0) spaces. The main theorems of this paper extend and improve the corresponding results of Chang et al. [7, 8], Piatek [18], Kaewkhao et al. [19], Panyanak and Suantai [16], and many others in the literature.However, wheng is a set-value contraction operator or is also nonexpansive in (59), whether can our main results be obtained? Furthermore, can our results be obtained when the iterations (13) (i.e., (10)), (12), and (57) become three-step iterations as in [35] or operator T is total asymptotically nonexpansive single-valued (set-valued) operator? These are stillopen questions to be worth further studying.



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