# Viscosity Approximation Methods for a Family of Nonexpansive Mappings in CAT(0) Spaces

Authors: Jinfang Tang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The purpose of this paper is using the viscosity approximation method to study the strong convergence problem for a family of nonexpansive mappings in CAT(0) spaces. Under suitable conditions, some strong convergence theorems for the proposed implicit and explicit iterative schemes to converge to a common fixed point of the family of nonexpansive mappings are proved which is also a unique solution of some kind of variational inequalities. The results presented in this paper extend and improve the corresponding results of some others.


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## Body

## 1. Introduction

Throughout this paper, we assume thatX is a CAT(0) space, ℕ is the set of positive integers, ℝ is the set of real numbers, ℝ+ is the set of nonnegative real numbers, and C is a nonempty closed and convex subset of a complete CAT(0) space X.A mappingT:C→C is called anonexpansive mapping, if
(1)d(T(x),T(y))≤d(x,y),∀x,y∈C.It is well-known that one classical way to study nonexpansive mappings is to use the contractions to approximate nonexpansive mappings. More precisely, taket∈(0,1) and define a contraction Tt:C→C by
(2)Tt=tu+(1-t)Tx,∀x∈C,
where u∈C is an arbitrary fixed element. In the case of T having a fixed point, Browder [1] proved that xt converged strongly to a fixed point of T that is nearest to u in the framework of Hilbert spaces. Reich [2] extended Browder’s result to the setting of a uniformly smooth Banach space and proved that xt converged strongly to a fixed point of T.Halpern [3] introduced the following explicit iterative scheme (3) for a nonexpansive mapping T on a subset C of a Hilbert space:
(3)xn+1=αnu+(1-αn)Txn.
He proved that the sequence {xn} converged to a fixed point of T.Fixed-point theory in CAT(0) spaces was first studied by Kirk (see [4, 5]). 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. In 2012, using Moudafi’s viscosity approximation methods, Shi and Chen [6] studied the convergence theorems of the following Moudafi’s viscosity iterations for a nonexpansive mapping T:
(4)xt=tf(xt)⊕(1-t)Txt,(5)xn+1=αnf(xn)⊕(1-αn)Txn.
They proved that {xt} defined by (4) and {xn} defined by (5) converged strongly to a fixed point of T in the framework of CAT(0) space which satisfies the property 𝒫.Motivated and inspired by the researches going on in this direction, especially inspired by Shi and Chen [6], the purpose of this paper is to study the strong convergence theorems of Moudafi’s viscosity approximation methods for a family of nonexpansive mappings in CAT(0) spaces. We prove that the implicit and explicit iteration algorithms both converge strongly to the same point x~ such that x~=Pℱf(x~), which is the unique solution to the variational inequality (35), where ℱ is the set of common fixed points of the family of nonexpansive mappings.


## 2. Preliminaries and Lemmas

In this paper, we write(1-t)x⊕ty for the unique point z in the geodesic segment joining from x to y such that
(6)d(x,z)=td(x,y),d(y,z)=(1-t)d(x,y).Lemma 1 (see [7]).
A geodesic spaceX is a CAT(0) space if and only if the following inequality
(7)d2((1-t)x⊕ty,z)≤(1-t)d2(x,z)+td2(y,z)-t(1-t)d2(x,y)
is satisfied for all x,y,z∈X and t∈[0,1]. In particular, if x,y,z are points in a CAT(0) space and t∈[0,1], then
(8)d((1-t)x⊕ty,z)≤(1-t)d(x,z)+td(y,z).Lemma 2 (see [8]).
LetX be a CAT(0) space, p,q,r,s∈X, and λ∈[0,1]. Then
(9)d(λp⊕(1-λ)q,λr⊕(1-λ)s)≤λd(p,r)+(1-λ)d(q,s).
By induction, one writes
(10)⨁m=1nλmxm:=(1-λn)(λ11-λnx1⊕λ21-λnx2iiiiiiiiiiiiiiii⊕⋯⊕λn-11-λnxn-1)⊕λnxn.Lemma 3.
LetX be a CAT(0) space; then, for any sequence {λm}m=1n in [0,1] satisfying ∑m=1nλm=1 and for any {xm}m=1n⊂X, the following conclusions hold:
(11)d(⨁m=1nλmxm,x)≤∑m=1nλmd(xm,x),x∈X;(12)d2(⨁m=1nλmxm,x)≤∑m=1nλmd2(xm,x)-λ1λ2d2(x1,x2),x∈X.Proof.
It is obvious that (11) holds for n=2. Suppose that (11) holds for some k≥2. Next we prove that (11) is also true for k+1. From (8) and (10) we have
(13)d(⨁m=1k+1λmxm,x)=d((1-λk+1)(λ11-λk+1x1⊕λ21-λk+1x2iiiiiiiiiiiiiiiiiiiiiiiiiiiii⊕⋯⊕λk1-λk+1xk)ii⊕λk+1xk+1,xλ11-λn+1(λ11-λk+1x1⊕λ21-λk+1x2)≤(1-λk+1)d(λ11-λk+1x1⊕λ21-λk+1x2iiiiiiiiiiiiiiiiiiiiiiiiii⊕⋯⊕λk1-λk+1xk,x)+λk+1d(xk+1,x)≤λ1d(x1,x)+λ2d(x2,x)+⋯+λkd(xk,x)+λk+1d(xk+1,x)=∑m=1k+1λmd(xm,x).
This implies that (11) holds.
Next, we prove that (12) holds.
Indeed, it is obvious that (12) holds for n=2. Suppose that (12) holds for some k≥2. Next we prove that (12) is also true for k+1.
In fact, we have(14)d2(⨁m=1k+1λmxm,x)=d2(⨁m=1kλmxm⊕λk+1xk+1,x).
From (7) and (10) and the assumption of induction, we have
(15)d2(⨁m=1k+1λmxm,x)=d2(⨁m=1kλmxm⊕λk+1xk+1,x)=d2((1-λk+1)⨁m=1kλm1-λk+1xm⊕λk+1xk+1,x)≤(1-λk+1)d2(⨁m=1kλm1-λk+1xm,x)+λk+1d2(xk+1,x)≤(1-λk+1)∑m=1kλm1-λk+1d2(xm,x)-λ1λ2d2(x1,x2)+λk+1d2(xk+1,x)=∑m=1k+1λmd2(xm,x)-λ1λ2d2(x1,x2).
This completes the proof of (12).Clearly, every CAT(0) spaceX is strictly convex: if, in X, d(u,y0)=d(v,y0) and x=αu⊕βv∈[u,v], then u=x=v whenever d(x,y0)=d(v,y0). Dhompongsa et al. [9] showed the following conclusion which is calledCondition (A):(A)
ify0 and vn belong to X and d(vn,y0)=d(x,y0) for all n, where x=⨁n=1∞λnvn, then vn=x for all n.The concept ofΔ-convergence introduced by Lim [10] in 1976 was shown by Kirk and Panyanak [11] in CAT(0) spaces to be very similar to the weak convergence in Banach space setting. Now, we give the concept of Δ-convergence.Let{xn} be a bounded sequence in a CAT(0) space X. For x∈X, we set
(16)r(x,{xn})=limsup⁡n→∞⁡d(x,xn).
The asymptotic radius r({xn}) of {xn} is given by
(17)r({xn})=inf⁡x∈X{r(x,{xn})},
and the asymptotic center A({xn}) of {xn} is the set
(18)A({xn})={x∈X:r(x,{xn})=r({xn})}.
It is known from Proposition 7 of [12] that, in a complete CAT(0) space, A({xn}) consists of exactly one point. A sequence {xn}⊂X is said to Δ-converge to x∈X if A({xnk})={x} for every subsequence {xnk} of {xn}.The uniqueness of an asymptotic center implies that a CAT(0) spaceX satisfiesOpial’s property; that is, for given {xn}⊂X such that {xn}  Δ-converges to x and given y∈X with y≠x,(19)limsup⁡n→∞d(xn,x)<limsup⁡n→∞⁡d(xn,y).Lemma 4 (see [11]).
Every bounded sequence in a complete CAT(0) space always has aΔ-convergent subsequence.
Berg and Nikolaev [13] introduced the concept of quasilinearization as follows. Let one denote a pair (a,b)∈X×X by ab→ and call it a vector. Then quasilinearization is defined as a map 〈·,·〉:(X×X)×(X×X)→ℝ defined by
(20)〈ab→,cd→〉=12(d2(a,d)+d2(b,c)-d2(a,c)-d2(b,d))iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii(a,b,c,d∈X).
It is easily seen that 〈ab→,cd→〉=〈cd→,ab→〉,〈ab→,cd→〉=-〈ba→,cd→〉 and 〈ax→,cd→〉+〈xb→,cd→〉=〈ab→,cd→〉 for all a,b,c,d∈X. One says that X satisfies the Cauchy-Schwarz inequality if
(21)〈ab→,cd→〉≤d(a,b)d(c,d),
for all a,b,c,d∈X.
LetC be a nonempty closed convex subset of CAT(0) space X. The metric projection PC:X→C is defined by
(22)u=PC(x)⟺d(u,x)=inf⁡{d(y,x):y∈C},∀x∈X.
Recently, Dehghan and Rooin [14] presented a characterization of metric projection in CAT(0) spaces as follows.Lemma 5.
LetC be a nonempty convex subset of a complete CAT(0) space X,  x∈X and u∈C. Then u=PC(x) if and only if
(23)〈yu→,ux→〉≤0,∀y∈C.Lemma 6 (see [15]).
LetX be a complete CAT(0) space, let {xn} be a sequence in X, and x∈X. Then {xn}  Δ-converges to x if and only if limsup⁡n→∞〈xxn→,xy→〉≤0 for all y∈X.Lemma 7 (see [16]).
LetX be a complete CAT(0) space. Then, for all u,x,y∈X, the following inequality holds:
(24)d2(x,u)≤d2(y,u)+2〈xy→,xu→〉.Lemma 8 (see [16]).
LetX be a complete CAT(0) space. For any t∈[0,1] and u,v∈X, let ut=tu⊕(1-t)v. Then, for any x,y∈X, the following inequality holds:
(25)〈utx→,uty→〉≤t〈ux→,uty→〉+(1-t)〈vx→,uty→〉.Lemma 9 (see [17]).
Let{an} be a sequence of nonnegative real numbers satisfying the property an+1≤(1-αn)an+αnβn,n≥0, where {αn}⊂(0,1) and {βn}⊂ℝ such that(i)
∑n=0∞αn=∞,(ii)
limsup⁡n→∞βn≤0 or ∑n=0∞|αnβn|<∞.
Then{an} converges to zero as n→∞.


## 3. Viscosity Approximation Iteration Algorithms

In this section, we present the strong convergence theorems of Moudafi’s viscosity approximation implicit and explicit iteration algorithms for a family of nonexpansive mappings{Tn:C→C}n=1∞ in CAT(0) spaces.Lemma 10.
LetC be a nonempty closed convex subset of a complete CAT(0) space X and let {λn} be a given sequence in (0,1) such that ∑n=1∞λn=1 and w1=T1; one defines a sequence {wn:C→C} as follows:
(26)wn=λ1∑i=1nλiT1⊕λ2∑i=1nλiT2⊕⋯⊕λn∑i=1nλiTn,∀n≥2.
Then the following holds:(i)
wn=(∑i=1n-1λi/∑i=1nλi)wn-1⊕(λn/∑i=1nλi)Tn;(ii)
wn is nonexpansive;(iii)
for anyx∈B, the sequence {wn(x)} converges uniformly to an element T(x)∈C, writing T(x)=⨁n=1∞λnTn(x), where B is a bounded subset of C.Proof.
(i) For eachn we introduce
(27)αin=λi∑j=1nλj,(i=1,2,…,n);
thus
(28)wn=α1nT1⊕α2nT2⊕⋯⊕αnnTn=(1-αnn)(α1n1-αnnT1⊕α2n1-αnnT2⊕⋯⊕αn-1n1-αnnTn-1)⊕αnnTn=(1-αnn)×(λ1∑i=1n-1λiT1⊕λ2∑i=1n-1λiT2⊕⋯⊕λn-1∑i=1n-1λiTn-1)⊕αnnTn=∑i=1n-1λi∑i=1nλiwn-1⊕λn∑i=1nλiTn.
(ii) We will show by induction thatwn is nonexpansive for all n∈ℕ. Since w1=T1, w1 is nonexpansive. Suppose wn is nonexpansive. We consider
(29)d(wn+1(x),wn+1(y))=d(∑i=1nλi∑i=1n+1λiwn(x)⊕λn+1∑i=1n+1λiTn+1(x),ii∑i=1nλi∑i=1n+1λiwn(y)⊕λn+1∑i=1n+1λiTn+1(y))≤∑i=1nλi∑i=1n+1λid(wn(x),wn(y))+λn+1∑i=1n+1λid(Tn+1(x),Tn+1(y))≤d(x,y).
Thus wn+1 is nonexpansive.
(iii) In view of thatlim⁡n→∞λn=0, for any x∈B, we have
(30)d(wn+1(x),wn(x))=d(∑i=1nλi∑i=1n+1λiwn(x)⊕λn+1∑i=1n+1λiTn+1(x),wn(x))≤λn+1∑i=1n+1λid(Tn+1(x),wn(x))≤λn+1λ1d(Tn+1(x),wn(x))⟶0(n⟶∞).
This implies that the sequence {wn(x)} converges uniformly to an element T(x)=⨁n=1∞λnTn(x)∈X. Since C is closed, T(x)∈C.Lemma 11.
LetC be a nonempty closed convex subset of a complete CAT(0) space X, and let {Tn:C→C}n=1∞ be a family of nonexpansive mappings satisfying ℱ:=∩n=1∞F(Tn)≠∅. Define T:C→C by T(x)=⨁n=1∞λnTn(x) for all x∈C, where {λn}⊂(0,1) with ∑n=1∞λn=1. Then T is nonexpansive and F(T)=∩n=1∞F(Tn).Proof.
For anyx,y∈C, we have
(31)d(T(x),T(y))≤d(T(x),wn(x))+d(wn(x),wn(y))+d(wn(y),T(y))≤d(T(x),wn(x))+d(x,y)+d(wn(y),T(y))⟶d(x,y)(n⟶∞).
This implies that T is nonexpansive.
It is easy to see that∩n=1∞F(Tn)⊂F(T). We only show that F(T)⊂∩n=1∞F(Tn). Let q∈F(T). For given p∈∩n=1∞F(Tn), from Lemma 10(iii) we have
(32)d(q,p)=d(T(q),p)=lim⁡n→∞⁡d(wn(q),p)≤lim⁡n→∞⁡(∑n=1∞λnd(Tn(q),p)λ1d(T1(q),p)+λ2d(T2(q),p)iiiiiii+⋯+λnd(Tn(q),p)∑n=1∞λnd(Tn(q),p))=∑n=1∞λnd(Tn(q),p)≤d(q,p).
In view of that
(33)d(Tn(q),p)=d(Tn(q),Tn(p))≤d(q,p),∀n∈ℕ,
we obtain that d(Tn(q),p)=d(q,p) for all n∈ℕ. By condition (A), Tn(q)=q for all n∈ℕ. Thus we complete the proof of Lemma 10.Now we are in a position to state and prove our main results.Theorem 12.
LetC be a closed convex subset of a complete CAT(0) space X, and let {Tn:C→C}n=1∞ be a family of nonexpansive mappings satisfying ℱ:=∩n=1∞F(Tn)≠∅. Let f be a contraction on C with coefficient α∈(0,1), {wn} and let {λn} be as in Lemma 10. Suppose the sequence {xn} is given by
(34)xn=αnf(xn)⊕(1-αn)wn(xn),
for all n≥0, where {αn}⊂(0,1) satisfies lim⁡n→∞αn=0. Then {xn} converges strongly to x~ such that x~=Pℱf(x~), which is equivalent to the following variational inequality:
(35)〈x~f(x~)→,xx~→〉≥0,∀x∈ℱ.Proof.
We will divide the proof of Theorem12 into five steps.
Step
1. The sequence {xn} defined by (34) is well defined for all n≥0.
In fact, let us define the mappingG:C→C by
(36)Gn(x):=αnf(x)⊕(1-αn)wn(x),x∈C.
For any x,y∈C, from Lemma 2, we have
(37)d(Gn(x),Gn(y))=d(∑n=1∞λnd(Tn(q),p)αnf(x)⊕(1-αn)wn(x),iiiiiiiiiiiiiiiαnf(y)⊕(1-αn)wn(y)∑n=1∞λnd(Tn(q),p))≤αnd(f(x),f(y))+(1-αn)d(wn(x),wn(y))≤αnαd(x,y)+(1-αn)d(x,y)=(1-αn(1-α))d(x,y).
This implies that Gn is a contraction mapping. Hence, the sequence {xn} is well defined for all n≥0.
Step
2. The sequence {xn} is bounded.
For anyp∈ℱ, from Lemma 3, we have that
(38)d(xn,p)=d(αnf(xn)⊕(1-αn)wn(xn),p)≤αnd(f(xn),p)+(1-αn)d(wn(xn),p)≤αnd(f(xn),p)+(1-αn)d(xn,p).
Then
(39)d(xn,p)≤d(f(xn),p)≤d(f(xn),f(p))+d(f(p),p)≤αd(xn,p)+d(f(p),p).
This implies that
(40)d(xn,p)≤11-αd(f(p),p).
Hence {xn} is bounded.
Step
3. lim⁡n→∞d(xn,T(xn))=0, where T=⨁n=1∞λnTn.
From (34) and lim⁡n→∞αn=0, we have
(41)d(xn,wn(xn))=d(αnf(xn)⊕(1-αn)wn(xn),wn(xn))≤αnd(f(xn),wn(xn))⟶0(n⟶∞).
From Lemma 10, we get
(42)d(xn,T(xn))≤d(xn,wn(xn))+d(wn(xn),T(xn))⟶0(n⟶∞).
Step
4. The sequence {xn} contains a subsequence converging strongly to x~ such that x~=Pℱf(x~), which is equivalent to (35).
Since{xn} is bounded, by Lemma 4, there exists a subsequence {xnj} of {xn} (without loss of generality we denote it by {xj}) which Δ-converges to a point x~.
First we claim thatx~∈ℱ. Since every CAT(0) space has Opial’s property, if T(x~)≠x~, we have
(43)limsup⁡j→∞d(xj,T(x~))≤limsup⁡j→∞(d(xj,T(xj))+d(T(xj),T(x~)))≤limsup⁡j→∞(d(xj,T(xj))+d(xj,x~))=limsup⁡j→∞d(xj,x~)<limsup⁡j→∞d(xj,T(x~)).
This is a contraction, and hence x~∈ℱ.
Next we prove that{xj} converges strongly to x~. Indeed, it follows from Lemma 8 that
(44)d2(xj,x~)=〈xjx~→,xjx~→〉≤αj〈f(xj)x~→,xjx~→〉+(1-αj)〈wj(xj)x~→,xjx~→〉≤αj〈f(xj)x~→,xjx~→〉+(1-αj)d(wj(xj),x~)d(xj,x~)≤αj〈f(xj)x~→,xjx~→〉+(1-αj)d2(xj,x~).
It follows that
(45)d2(xj,x~)≤〈f(xj)x~→,xjx~→〉=〈f(xj)f(x~)→,xjx~→〉+〈f(x~)x~→,xjx~→〉≤d(f(xj),f(x~))d(xj,x~)+〈f(x~)x~→,xjx~→〉≤αd2(xj,x~)+〈f(x~)x~→,xjx~→〉,
and thus
(46)d2(xj,x~)≤11-α〈f(x~)x~→,xjx~→〉.
Since {xj}  Δ-converges to x~, by Lemma 6 we have
(47)limsup⁡n→∞〈f(x~)x~→,xjx~→〉≤0.
It follows from (46) that {xj} converges strongly to x~.
Next we show thatx~ solves the variational inequality (35). Applying Lemma 1, for any q∈ℱ, we have
(48)d2(xj,q)=d2(αjf(xj)⊕(1-αj)wj(xj),q)≤αjd2(f(xj),q)+(1-αj)d2(wj(xj),q)-αj(1-αj)d2(f(xj),wj(xj)).
This together with Lemma 10(ii) implies that
(49)d2(xj,q)≤d2(f(xj),q)-(1-αj)(d(f(xj),xj)+d(xj,wj(xj)))2.
Taking the limit through j→∞, we can obtain
(50)d2(x~,q)≤d2(f(x~),q)-d2(f(x~),x~).
On the other hand, from (20) we have
(51)〈x~f(x~)→,qx~→〉=12[12d2(x~,x~)+d2(f(x~),q)cccccc-d2(x~,q)-d2(f(x~),x~)12].
From (50) and (51) we have
(52)〈x~f(x~)→,qx~→〉≥0,∀q∈ℱ.
That is, x~ solves the inequality (35).
Step
5. The sequence {xn} converges strongly to x~.
Assume thatxni→x^ as n→∞. By the same argument, we get that x^∈ℱ which solves the variational inequality (35); that is,
(53)〈x~f(x~)→,x~x^→〉≤0,(54)〈x^f(x^)→,x^x~→〉≤0.
Adding up (53) and (54), we get that
(55)0≥〈x~f(x~)→,x~x^→〉-〈x^f(x^)→,x~x^→〉=〈x~f(x^)→,x~x^→〉+〈f(x^)f(x~)→,x~x^→〉-〈x^x~→,x~x^→〉-〈x~f(x^)→,x~x^→〉=〈x~x^→,x~x^→〉-〈f(x^)f(x~)→,x^x~→〉≥〈x~x^→,x~x^→〉-d(f(x^),f(x~))d(x^,x~)≥d2(x~,x^)-αd2(x^,x~)=(1-α)d2(x~,x^).
Since 0<α<1, we have that d(x~,x^)=0, and so x~=x^. Hence the sequence {xn} converges strongly to x~, which is the unique solution to the variational inequality (35).
This completes the proof.Theorem 13.
LetC be a closed convex subset of a complete CAT(0) space X, and let {Tn:C→C}n=1∞ be a family of nonexpansive mappings satisfying ℱ:=∩n=1∞F(Tn)≠∅. Let f be a contraction on C with coefficient α∈(0,1) and let {wn} be as in Lemma 10. Suppose x0∈C and the sequence {xn} is given by
(56)xn+1=αnf(xn)⊕(1-αn)wn(xn),
such that d(wn(xn),wn+1(xn+1))≤d(xn,xn+1)+εn   for all n∈ℕ, where ∑n=1∞‍εn<∞ and {αn}⊂(0,1) satisfies(i)
lim⁡n→∞αn=0;(ii)
∑n=1∞αn=∞;(iii)
∑n=1∞|αn+1-αn|<∞ or lim⁡n→∞(αn+1/αn)=1.Then {xn} converges strongly to x~ such that x~=Pℱf(x~), which is equivalent to the variational inequality (35).Proof.
We first show that the sequence{xn} is bounded. For any p∈ℱ, we have that
(57)d(xn+1,p)=d(αnf(xn)⊕(1-αn)wn(xn),p)≤αnd(f(xn),p)+(1-αn)d(wn(xn),p)≤αn(d(f(xn),f(p))+d(f(p),p))+(1-αn)d(wn(xn),p)≤αnαd(xn,p)+αnd(f(p),p)+(1-αn)d(xn,p)=(1-αn(1-α))d(xn,p)+αn(1-α)·11-αd(f(p),p)≤max⁡{d(xn,p),11-αd(f(p),p)}.
By induction, we have
(58)d(xn,p)≤max⁡{d(x0,p),11-αd(f(p),p)},
for all n≥0. Hence {xn} is bounded, so are {wn(xn)} and {f(xn)}.
From (56), we have
(59)d(xn+1,xn)=d(∑n=1∞λnd(Tn(q),p)αnf(xn)⊕(1-αn)wn(xn),cccccccccccccαn-1f(xn-1)⊕(1-αn-1)wn-1(xn-1)∑n=1∞λnd(Tn(q),p))≤d(∑n=1∞λnd(Tn(q),p)αnf(xn)⊕(1-αn)wn(xn),cccccccccccccαnf(xn-1)⊕(1-αn)wn-1(xn-1)∑n=1∞λnd(Tn(q),p))+d(∑n=1∞λnd(Tn(q),p)αnf(xn-1)⊕(1-αn)wn-1(xn-1),ccccccccccccccαn-1f(xn-1)⊕(1-αn-1)wn-1(xn-1)∑n=1∞λnd(Tn(q),p))≤αnd(f(xn),f(xn-1))+(1-αn)d(wn(xn),wn-1(xn-1))+|αn-αn-1|d(f(xn-1),wn-1(xn-1))≤αnαd(xn,xn-1)+(1-αn)d(xn,xn-1)+|αn-αn-1|d(f(xn-1),wn-1(xn-1))+εn≤(1-(1-α)αn)d(xn,xn-1)+|αn-αn-1|d(f(xn-1),wn-1(xn-1))+εn.
From Lemma 9 and conditions (ii) and (iii) we obtain
(60)lim⁡n→∞⁡d(xn+1,xn)=0.
From condition (i), we have
(61)d(xn,wn(xn))=d(xn,xn+1)+d(xn+1,wn(xn))=d(xn,xn+1)+d(αnf(xn)⊕(1-αn)wn(xn),wn(xn))≤d(xn,xn+1)+αnd(f(xn),wn(xn))⟶0(n⟶∞).
From Lemma 10(iii) we can obtain
(62)d(wm(xn+1),xn+1)≤d(wm(xn+1),wn+1(xn+1))+d(wn+1(xn+1),xn+1)⟶0(m⟶∞,n⟶∞).
Without loss of generality, we can choose the sequence {αm} such that
(63)d(wm(xn+1),xn+1)=o(αm)(m⟶∞,n⟶∞).
Let{zm} be a sequence in C such that
(64)zm=αmf(zm)⊕(1-αm)wm(zm).
It follows from Theorem 12 that {zm} converges strongly to a fixed point x~∈ℱ, which solves the variational inequality (35).
Now we claim that(65)limsup⁡n→∞〈f(x~)x~→,xn+1x~→〉≤0.
Indeed, it follows from Lemma 8 that
(66)d2(zm,xn+1)=〈zmxn+1→,zmxn+1→〉≤αm〈f(zm)xn+1→,zmxn+1→〉+(1-αm)〈wm(zm)xn+1→,zmxn+1→〉=αm〈f(zm)f(x~)→,zmxn+1→〉+αm〈f(x~)x~→,zmxn+1→〉+αm〈x~zm→,zmxn+1→〉+αm〈zmxn+1→,zmxn+1→〉+(1-αm)〈wm(zm)wm(xn+1)→,zmxn+1→〉+(1-αm)〈wm(xn+1)xn+1→,zmxn+1→〉≤αmαd(zm,x~)d(zm,xn+1)+αm〈f(x~)x~→,zmxn+1→〉+αmd(x~,zm)d(zm,xn+1)+αmd2(zm,xn+1)+(1-αm)d2(zm,xn+1)+(1-αm)d(wm(xn+1),xn+1)d(zm,xn+1)≤αmαd(zm,x~)M+αm〈f(x~)x~→,zmxn+1→〉+αmd(x~,zm)M+d2(zm,xn+1)+(1-αm)d(wm(xn+1),xn+1)M,
where
(67)M≥sup⁡m,n≥1{d(zm,xn)}.
This implies that
(68)〈f(x~)x~→,xn+1zm→〉≤(1+α)Md(zm,x~)+d(wm(xn+1),xn+1)αmM.
Taking the upper limit as m→∞ and n→∞, from (63) we get
(69)limsup⁡m,n→∞〈f(x~)x~→,xn+1zm→〉≤0.
Furthermore, we have
(70)〈f(x~)x~→,xn+1x~→〉=〈f(x~)x~→,xn+1zm→〉+〈f(x~)x~→,zmx~→〉≤〈f(x~)x~→,xn+1zm→〉+d(f(x~),x~)d(zm,x~).
Thus, by taking the upper limit as n→∞ first and then m→∞, it follows from zm→x~ and (69) that
(71)limsup⁡n→∞〈f(x~)x~→,xn+1x~→〉≤0.
Finally, we prove thatxn→x~ as n→∞. In fact, for any n≥0, let
(72)yn=αnx~⊕(1-αn)wn(xn).
From Lemmas7 and 8 we have that
(73)d2(xn+1,x~)≤d2(yn,x~)+2〈xn+1yn→,xn+1x~→〉≤(1-αn)2d2(wn(xn),x~)+2[αn〈f(xn)yn→,xn+1x~→〉ccccc+(1-αn)〈wn(xn)yn→,xn+1x~→〉]≤(1-αn)2d2(xn,x~)+2[(1-αn)2αn2〈f(xn)x~→,xn+1x~→〉cccccccccccc+αn(1-αn)〈f(xn)wn(xn)→,xn+1x~→〉cccccccccccc+αn(1-αn)〈wn(xn)x~→,xn+1x~→〉cccccccccccc+(1-αn)2〈wn(xn)wn(xn)→,xn+1x~→〉]≤(1-αn)2d2(xn,x~)+2[αn2〈f(xn)x~→,xn+1x~→〉cccccccccccc+αn(1-αn)〈f(xn)x~→,xn+1x~→〉αn2]=(1-αn)2d2(xn,x~)+2αn〈f(xn)x~→,xn+1x~→〉=(1-αn)2d2(xn,x~)+2αn〈f(xn)f(x~)→,xn+1x~→〉+2αn〈f(x~)x~→,xn+1x~→〉≤(1-αn)2d2(xn,x~)+2αnαd(xn,x~)d(xn+1,x~)+2αn〈f(x~)x~→,xn+1x~→〉≤(1-αn)2d2(xn,x~)+αnα(d2(xn,x~)+d2(xn+1,x~))+2αn〈f(x~)x~→,xn+1x~→〉.
This implies that
(74)d2(xn+1,x~)≤1-(2-α)αn+αn21-ααnd2(xn,x~)+2αn1-ααn〈f(x~)x~→,xn+1x~→〉=(1-αn(2-2α-αn)1-ααn)d2(xn,x~)+2αn1-ααn〈f(x~)x~→,xn+1x~→〉.
Then it follows that
(75)d2(xn+1,x~)≤(1-αn′)d2(xn,x~)+αn′βn′,
where
(76)αn′=αn(2-2α-αn)1-ααn,βn′=22-2α-αn〈f(x~)x~→,xn+1x~→〉.
Applying Lemma 9, we can conclude that xn→x~ as n→∞. This completes the proof.



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*Source: 389804-2014-05-12.xml*