# Strong Convergence of Viscosity Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces

Authors: Luo Yi Shi; Ru Dong Chen
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces are studied. Consider a nonexpansive self-mappingT of a closed convex subset C of a CAT(0) space X. Suppose that the set Fix(T) of fixed points of T is nonempty. For a contraction f on C and t∈(0,1), let xt∈C be the unique fixed point of the contraction x↦tf(x)⊕(1-t)Tx. We will show that if X is a CAT(0) space satisfying some property, then {xt} converge strongly to a fixed point of T which solves some variational inequality. Consider also the iteration process {xn}, where x0∈C is arbitrary and xn+1=αnf(xn)⊕(1-αn)Txn for n≥1, where {αn}⊂(0,1). It is shown that under certain appropriate conditions on αn,{xn} converge strongly to a fixed point of T which solves some variational inequality.


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## Body

## 1. CAT(0) Spaces

A metric spaceX is a CAT(0) space if it is geodesically connected and if every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane. The precise definition is given below. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include pre-Hilbert spaces [1], R-trees [2], Euclidean buildings [3], the complex Hilbert ball with a hyperbolic metric [4], and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [1].Fixed-point theory in CAT(0) spaces was first studied by Kirk (see [5, 6]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then, the fixed-point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed, and many papers have appeared [2, 7–17].The purpose of this paper is to study the iterative scheme defined as follows. Consider a nonexpansive self-mappingT of a closed convex subset C of a CAT(0) space X. Suppose that the set Fix(T) of fixed points of T is nonempty. For a contraction f on C and t∈(0,1), let xt∈C be the unique fixed point of the contraction x↦tf(x)⊕(1-t)Tx. Consider the iteration process {xn}, where x0∈C is arbitrary and(1.1)xn+1=αnf(xn)⊕(1-αn)Txn,
for n≥1, where {αn}⊂(0,1). We show that {xn} converge strongly to a fixed point of T under certain appropriate conditions on αn, and the fixed point of T solves some variational inequality.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 c from a closed interval [0,l]⊂R to X such that c(0)=x,c(l)=y, and d(c(t),c(t′))=|t-t′| for all t,t∈[0,l]. In particular, c is an isometry and d(x,y)=l. The image α of c is called a geodesic(ormetric)segment joining x and y. When it is unique, this geodesic segment is denoted by [x,y]. The space (X,d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x,y∈X. A subset Y⊆X is said to be convex if Y includes every geodesic segment joining any two of its points.Ageodesic triangle▵(x1,x2,x3) in a geodesicmetric space (X,d) consists of three points x1,x2, and x3 in X (the vertices of ▵) and a geodesic segment between each pair of vertices (the edges of ▵). A comparison triangle for the geodesic triangle ▵(x1,x2,x3) in (X,d) is a triangle ▵¯(x1,x2,x3):=▵(x1¯,x2¯,x3¯) in the Euclidean plane 𝔼2 such that d𝔼2(xi¯,xj¯)=d(xi,xj) for i,j∈1,2,3.A geodesic space is said to be a CAT(0) space if all geodesic triangles satisfy the following comparison axiom.CAT(0): let▵ be a geodesic triangle in X, and let ▵¯ be a comparison triangle for ▵. Then, ▵ is said to satisfy the CAT(0) inequality if for all x,y∈▵ and all comparison points x¯,y¯∈▵¯,(1.2)d(x,y)≤dE2(x¯,y¯).Letx,y∈X, and by Lemma 2.1(iv) of [18] for each t∈[0,1], there exists a unique point z∈[x,y] such that(1.3)d(x,z)=td(x,y),d(y,z)=(1-t)d(x,y).
From now on, we will use the notation (1-t)x⊕ty for the unique point z satisfying the above equation.We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results.Lemma 1.1.
LetX be a CAT(0) space. Then,(i)
(see [18, Lemma 2.4]) for each x,y,z∈X and t∈[0,1], one has(1.4)d((1-t)x⊕ty,z)≤(1-t)d(x,z)+td(y,z),(ii)
(see [7]) for each x,y,z∈X and t,s∈[0,1], one has(1.5)d((1-t)x⊕ty,(1-s)x⊕sy)≤|t-s|d(x,y),(iii)
(see [6, Lemma 3]) for each x,y,z∈X and t∈[0,1], one has(1.6)d((1-t)z⊕tx,(1-t)z⊕ty)≤td(x,y),(iv)
(see [18, Lemma 2.5]) for each x,y,z∈X and t∈[0,1], one has(1.7)d((1-t)x⊕ty,z)2≤(1-t)d(x,z)2+td(y,z)2-t(1-t)d(x,y)2.LetX be a complete CAT(0) space, let {xn} be a bounded sequence in a complete X, and for x∈X, set(1.8)r(x,{xn})=limsup⁡n→∞d(x,xn).Theasymptotic radiusr({xn}) of {xn} is given by(1.9)r({xn})=inf⁡{r(x,{xn}):x∈X},and theasymptotic centerA({xn}) of {xn} is the set(1.10)A({xn})={x∈X:r(x,{xn})=r({xn})}.It is known (see, e.g., [11, Proposition 7]) that in a CAT(0) space, A({xn}) consists of exactly one point.A sequence{xn} in X is said to ▵-converge to x∈X if x is the unique asymptotic center of {un} for every subsequence {un} of {xn}. In this case, we write ▵-limnxn=x and call x the ▵-limit of {xn}.Lemma 1.2.
Assume thatX is a CAT(0) space. Then,(i)
(see [14]) every bounded sequence in X has a ▵-convergent subsequence,(ii)
(see [14, Proposition 3.7]) if K is a closed convex subset of X, and f:K→X is a nonexpansive mapping, then the conditions {xn}▵-converge to x and d(xn,f(xn))→0 and imply x∈K and f(x)=x.Lemma 1.3 (see [19, Proposition 3.5]).
Assume thatX is a CAT(0) space, C is a closed convex subset of X. Then the metric (nearest point) projection PC:X→C,PC(x):=inf{d(x,y);y∈C} is a nonexpansive mapping. one calls a CAT(0) space X satisfying property 𝒫 if for x,u,y1,y2∈X,
(1.11)d(x,P[x,y1](u))d(x,y1)≤d(x,P[x,y2](u))d(x,y2)+d(x,u)d(y1,y2).Remark 1.4.
The property𝒫 in Hilbert space corresponds to the inequality
(1.12)|〈u-x,y1-x〉|≤|〈u-x,y2-x〉|+‖u-x‖⋅‖y1-y2‖.
Recall that a continuous linear functionalμ on l∞, the Banach space of bounded real sequences, is called a Banach limit if ∥μ∥=μ(1,1,…)=1 and μn(an)=μn(an+1) for all {an}∈l∞.Lemma 1.5 (see [20, Proposition 2]).
Let(a1,a2,…)∈l∞ be such that μn(an)≤0 for all Banach limits μ and limsupn(an+1-an)≤0. Then limsupnan≤0.Lemma 1.6 (see [21, Lemma 2.3]).
Let{sn} be a sequence of nonnegative real numbers, {αn} a sequence of real numbers in [0,1] with ∑n=1∞αn=∞, {un} a sequence of nonnegative real numbers with ∑n=1∞un<∞, and {tn} a sequence of real numbers with limsupntn≤0. Suppose that
(1.13)sn+1=(1-αn)sn+αntn+un,∀n∈N.
Then limn→∞sn=0.


## 2. Viscosity Iteration for a Single Mapping

In this section, we prove the main results of this paper.Lemma 2.1.
LetC be a closed convex subset of a complete CAT(0) space X, and let T:C→C be a nonexpansive mapping. Let f be a contraction on C with coefficient α<1. For each t∈[0,1], the mapping St:C→C defined by
(2.1)Stx=tf(x)⊕(1-t)Tx,forx∈C
has a unique fixed point xt∈C, that is,
(2.2)xt=tf(xt)⊕(1-t)Txt.Proof.
Forx,y∈C, according to Lemma 1.1, we have the following:
(2.3)d(St(x),St(y))=d(tf(x)⊕(1-t)Tx,tf(y)⊕(1-t)Ty)≤d(tf(x)⊕(1-t)Tx,tf(x)⊕(1-t)Ty)+d(tf(y)⊕(1-t)Tx,tf(y)⊕(1-t)Ty)≤td(f(x),f(y))+(1-t)d(Tx,Ty)≤(1-t(1-α))d(x,y).
This implies that St is a contraction mapping, and hence, the conclusion follows.The following result is to prove that the net{xt} converge strongly to a fixed point of T.Theorem 2.2.
LetC be a closed convex subset of a complete CAT(0) space X satisfying the property 𝒫, and let T:C→C be a nonexpansive mapping. Let f be a contraction on C with coefficient α<1. For each t∈[0,1], let {xt} be given by
(2.4)xt=tf(xt)⊕(1-t)Txt.
Then one has limt→0xt=:x̃ and x̃=PFix(T)f(x̃).Proof.
We first show that{xt} is bounded. Indeed choose a p∈Fix(T), and using Lemma 1.1 and the nonexpansive of T, we derive that
(2.5)d(xt,p)=d(tf(xt)⊕(1-t)Txt,p)≤td(f(xt),p)+(1-t)d(Txt,p)≤td(f(xt),p)+(1-t)d(xt,p).
It follows that
(2.6)d(xt,p)≤d(f(xt),p)≤d(f(xt),f(p))+d(f(p),p)≤αd(xt,p)+d(f(p),p).
Hence,
(2.7)d(xt,p)≤11-αd(f(p),p),
and {xt} is bounded, so are {Txt} and {f(xt)}. As a result, we can get that
(2.8)lim⁡t→0d(xt,Txt)=lim⁡t→0d(tf(xt)⊕(1-t)Txt,Txt)=lim⁡t→0td(f(xt),Txt)=0.
Assume that{tn}⊆(0,1) is such that tn→0 as n→∞. Put xn:=xtn. We will show that {xn} contains a subsequence converging strongly to x̃, where x̃∈Fix(T).
Since{xn} is bounded, by Lemma 1.2(i),(ii), we may assume that {xn}▵-converges to a point x̃, and x̃∈Fix(T).
Next we will prove that{xn} converge strongly to x̃.
Indeed, according to Lemma1.1 and the property of T and f, we can get that
(2.9)d2(xn,x̃)=d2(tnf(xn)⊕(1-tn)Txn,x̃)≤tnd2(f(xn),x̃)+(1-tn)d2(Txn,x̃)-tn(1-tn)d2(f(xn),Txn)≤tnd2(f(xn),x̃)+(1-tn)d2(xn,x̃)-tn(1-tn)d2(f(xn),Txn).
It follows that
(2.10)d2(xn,x̃)≤d2(f(xn),x̃)-(1-tn)d2(f(xn),Txn)=d2(f(xn),x̃)-d2(f(xn),Txn)+tnd2(f(xn),Txn).
Sincelimt→0d(xt,Txt)=0, we can get that
(2.11)lim⁡⁡sup⁡n→∞d2(xn,x̃)≤lim⁡⁡sup⁡n→∞d2(f(xn),x̃)-d2(f(xn),xn).
Let Δ(x̃¯,xn¯,f(xn)¯) be a comparison triangle for ▵(x̃,xn,f(xn)) in 𝔼2. Then,
(2.12)d2(f(xn),x̃)-d2(f(xn),xn)=d2(f(xn)¯,x̃¯)-d2(f(xn)¯,xn¯)=〈f(xn)¯-x̃¯,f(xn)¯-x̃¯〉-〈f(xn)¯-x̃n¯,f(xn)¯-x̃n¯〉=2〈f(xn)¯-x̃¯,xn¯-x̃¯〉-〈xn¯-x̃¯,xn¯-x̃¯〉=2〈f(xn)¯-x̃¯,xn¯-x̃¯〉-d2(xn¯,x̃¯)=2〈f(xn)¯-x̃¯,xn¯-x̃¯〉-d2(xn,x̃).
Hence,
(2.13)lim⁡sup⁡n→∞d2(xn,x̃)≤lim⁡sup⁡n→∞〈f(xn)¯-x̃¯,xn¯-x̃¯〉.
Let▵(x̃¯,xn¯,f(x̃)¯) be a comparison triangle for ▵(x̃,xn,f(x̃)) in 𝔼2. For each n, let un¯ be the point of the segment [x̃¯,f(x̃)¯] which is nearest to xn¯, and let un be the point of the segment [x̃,f(x̃)] for which d(un,x̃)=d(un¯,x̃¯).
By passing to subsequences again, we may suppose that{un¯} converges to u¯∈[x̃¯,f(x̃)¯], {un} converges to u∈[x̃,f(x̃)].
Since{xn}▵-converges to a point x̃, we have
(2.14)r({xn})=lim⁡n⁡sup⁡d(x,xn)=lim⁡n⁡sup⁡d(x¯,xn¯)≥lim⁡n⁡sup⁡d(un¯,xn¯)=lim⁡n⁡sup⁡d(u¯,xn¯)≥lim⁡n⁡sup⁡d(u,xn).
Thus, r(u,{xn})≤r({xn}). This implies that u=x by uniqueness of the asymptotic center. Hence, u¯=x¯. That is to say, {un¯} converges to x̃¯, and {un} converges to x̃.
Moreover, sinceX satisfies the property 𝒫, we can get that
(2.15)|〈f(xn)¯-x̃¯,xn¯-x̃¯〉|=d(x̃¯,P[x̃¯,f(xn)¯](xn¯))⋅d(x̃¯,f(xn)¯)=d(x̃,P[x̃,f(xn)](xn))⋅d(x̃,f(xn))≤d(x̃,P[x̃,f(x̃)](xn))⋅d(x̃,f(x̃))+d(x̃,xn)⋅d(f(xn),f(x̃))=d(x̃¯,P[x̃¯,f(x̃)̅](xn¯))⋅d(x̃¯,f(x̃)̅)+d(x̃,xn)⋅d(f(xn),f(x̃))≤d(un¯,x̃¯)d(x̃¯,f(x̃)¯)+αd2(xn,x̃).
It follows that
(2.16)lim⁡⁡sup⁡n→∞d2(xn,x̃)≤11-αlim⁡⁡sup⁡n→∞d(un¯,x̃¯)d(f(x̃)¯,x̃¯).
Since {un¯} converges to x̃¯, we obtain that limsupn→∞d2(xn,x̃)=0, that is, {xn} converge strongly to x̃. Since {tn}⊆(0,1) is such that tn→0 as n→∞ is arbitrarily selected, we can get that limt→0xt=x̃.
Finally, we will prove thatx̃ satisfy the equation x̃=PFix(T)f(x̃).
Indeed, for anyy∈Fix(T),
(2.17)d(xt,y)=d(tf(xt)⊕(1-t)Txt,y)=td(f(xt),y)+(1-t)d(Txt,y)≤td(f(xt),y)+(1-t)d(xt,y).
It follows that
(2.18)d(xt,y)≤d(f(xt),y).
Since limt→0xt=x̃, we can get that
(2.19)d(x̃,y)≤d(f(x̃),y).
Hence,(2.20)d(f(x̃),y)≥|d(x̃,y)-d(x̃,f(x̃))|,d(x̃,f(x̃))≥d(f(x̃),y).
That is to say, x̃=PFix(T)f(x̃).Consider now the iteration process(2.21)x0∈C,xn+1=αnf(xn)⊕(1-αn)Txn,n≥0,
where {αn}⊆(0,1) satisfies(H1)
αn→0,(H2)
∑n=0∞αn=∞,(H3)
either∑n=0∞|αn+1-αn|<∞ or limn→∞(αn+1/αn)=1.Theorem 2.3.
LetX be a CAT(0) space satisfying the property 𝒫, C a closed convex subset of X, T:C→C a nonexpansive mapping with Fix(T)≠∅, and f:C→C a contraction with coefficient α<1. Let x0∈C, {xn} be generated by xn+1=αnf(xn)⊕(1-αn)Txn,n≥0. Then under the hypotheses (H1 )–(H3 ), xn→x̃, where x̃=PFix(T)f(x̃).Proof.
We first show that the sequence{xn} is bounded. Let p∈Fix(T). Then,
(2.22)d(xn+1,p)=d(αnf(xn)⊕(1-αn)Txn,p)≤αnd(f(xn),p)+(1-αn)d(Txn,p)≤αn[d(f(xn),f(p))+d(f(p),p)]+(1-αn)d(xn,p)≤max⁡{d(xn,p),11-αd(f(p),p)}.
By induction, we have
(2.23)d(xn,p)≤max⁡{d(x0,p),11-αd(f(p),p)},
for all n∈ℕ. This implies that {xn} is bounded and so is the sequence {Txn} and {f(xn)}.
We claim thatd(xn+1,xn)→0. Indeed, we have
(2.24)d(xn+1,xn)=d(αnf(xn)⊕(1-αn)Txn,αn-1f(xn-1)⊕(1-αn-1)Txn-1)≤d(αnf(xn)⊕(1-αn)Txn,αnf(xn)⊕(1-αn)Txn-1)+d(αnf(xn)⊕(1-αn)Txn-1,αnf(xn-1)⊕(1-αn)Txn-1)+d(αnf(xn-1)⊕(1-αn)Txn-1,αn-1f(xn-1)⊕(1-αn-1)Txn-1)≤(1-αn)d(Txn,Txn-1)+αnd(f(xn),f(xn-1))+|αn-αn-1|d(f(xn-1),Txn-1)≤(1-αn)d(xn,xn-1)+αnαd(xn,xn-1)+|αn-αn-1|d(f(xn-1),Txn-1).
By the conditions H2 and H3, we have
(2.25)d(xn+1,xn)⟶0.
Consequently, by the condition H1,
(2.26)d(xn,Txn)≤d(xn,xn+1)+d(xn+1,Txn)=d(xn,xn+1)+d(αnf(xn)⊕(1-αn)Txn,Txn)=d(xn,xn+1)+αnd(f(xn),Txn)⟶0.
Since{xn} is bounded, we may assume that {xn}▵-converges to a point x̂. By Lemma 1.2, we have x̂∈Fix(T).
Next we will prove that{xn} converge strongly to x̂ and x̂=x̃. Indeed, according to Lemma 1.1 and the property of T and f, we can get that
(2.27)d2(xn+1,x̂)=d2(tnf(xn)⊕(1-tn)Txn,x̂)≤αnd2(f(xn),x̂)+(1-αn)d2(Txn,x̂)-αn(1-αn)d2(f(xn),Txn)≤(1-αn)d2(xn,x̂)+αn[d2(f(xn),x̂)-(1-αn)d2(f(xn),Tn)].
With a minor modification of the proof of the analogous statement in Theorem 2.2, we can get that
(2.28)d2(f(xn),x̂)-d2(f(xn),xn)=2〈f(xn)¯-x̂¯,xn¯-x̂¯〉-d2(xn,x̂)≤2d(un¯,x̂¯)d(f(x̂)¯,x̂¯)+2αd2(xn,x̂)-d2(xn,x̂),
and d(un¯,x̂¯)→0.
Thus,(2.29)d2(xn+1,x̂)≤(1-αn)d2(xn,x̂)+αn[d2(f(xn),x̂)-(1-αn)d2(f(xn),xn)+(1-αn)d2(f(xn),xn)-(1-αn)d2(Txn,xn)]≤(1-2(1-α)αn)d2(xn,x̂)+2(1-α)αn[1(1-α)d(un¯,x̂¯)d(f(x̂)¯,x̂¯)+1(1-α)βn],
where βn=(1-αn)d2(f(xn),xn)-(1-αn)d2(Txn,xn)]. Since d(un¯,x̂¯)→0 and d(xn,Txn)→0, we obtain that
(2.30)lim⁡nsup⁡1(1-α)d(un¯,x̂¯)d(f(x̂)¯,x̂¯)+1(1-α)βn≤0.
According to Lemma1.6, we can get d2(xn,x̂)→0.
Finally, we prove thatx̂=x̃.
Indeed, for anyz∈Fix(T),
(2.31)d2(xn+1,z)≤αnd2(z,f(xn))+(1-αn)d2(z,Txn)-αn(1-αn)d2(f(xn),Txn)≤αnd2(z,f(xn))+(1-αn)d2(z,xn)-αn(1-αn)d2(f(xn),Txn).
Letμ be a Banach limit. Then,
(2.32)μnd2(xn+1,z)≤μnd2(z,f(xn))-μnd2(f(xn),Txn).
Sincexn→x̂, we obtain that
(2.33)d2(x̂,z)≤d2(z,f(x̂))-d2(f(x̂),x̂).
It follows that
(2.34)d2(f(x̂),x̂)≤d2(z,f(x̂)),
that is to say, x̂=PFix(T)f(x̂). Since PFix(T)f is a contraction and x̃=PFix(T)f(x̃), we know that x̂=x̃.



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