ACCRETIVE OPERATORS IN BANACH SPACES
KOJI AOYAMA, HIDEAKI IIDUKA, AND WATARU TAKAHASHI Received 21 November 2005; Accepted 6 December 2005
LetCbe a nonempty closed convex subset of a smooth Banach spaceEand letAbe an accretive operator ofCintoE. We first introduce the problem of finding a pointu∈C such thatAu,J(v−u) ≥0 for allv∈C, whereJ is the duality mapping ofE. Next we study a weak convergence theorem for accretive operators in Banach spaces. This theorem extends the result by Gol’shte˘ın and Tret’yakov in the Euclidean space to a Banach space.
And using our theorem, we consider the problem of finding a fixed point of a strictly pseudocontractive mapping in a Banach space and so on.
Copyright © 2006 Koji Aoyama et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
LetHbe a real Hilbert space with norm · and inner product (·,·), letCbe a nonempty closed convex subset ofHand letAbe a monotone operator ofCintoH. The variational inequality problem is formulated as finding a pointu∈Csuch that
(v−u,Au)≥0 (1.1)
for allv∈C. Such a pointu∈Cis called a solution of the problem. Variational inequali- ties were initially studied by Stampacchia [13,17] and ever since have been widely studied.
The set of solutions of the variational inequality problem is denoted by VI(C,A). In the case whenC=H, VI(H,A)=A−10 holds, whereA−10= {u∈H:Au=0}. An element ofA−10 is called a zero point ofA. An operatorAofCintoHis said to be inverse strongly monotone if there exists a positive real numberαsuch that
(x−y,Ax−Ay)≥αAx−Ay2 (1.2)
for allx,y∈C; see Browder and Petryshyn [5], Liu and Nashed [18], and Iiduka et al.
[11]. For such a case,Ais said to beα-inverse strongly monotone. LetTbe a nonexpansive mapping ofCinto itself. It is known that if A=I−T, thenA is 1/2-inverse strongly
Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 35390, Pages1–13 DOI10.1155/FPTA/2006/35390
monotone andF(T)=VI(C,A), whereIis the identity mapping ofHandF(T) is the set of fixed points ofT; see [11]. In the case ofC=H=RN, for finding a zero point of an inverse strongly monotone operator, Gol’shte˘ın and Tret’yakov [8] proved the following theorem.
Theorem 1.1 (see Gol’shte˘ın and Tret’yakov [8]). LetRNbe theN-dimensional Euclidean space and letAbe anα-inverse strongly monotone operator ofRNinto itself withA−10= ∅. Let{xn}be a sequence defined as follows:x1=x∈RNand
xn+1=xn−λnAxn (1.3)
for everyn=1, 2,..., where{λn}is a sequence in [0, 2α]. If{λn}is chosen so thatλn∈[a,b]
for somea,bwith 0< a < b <2α, then{xn}converges to some element ofA−10.
For finding a solution of the variational inequality for an inverse strongly monotone operator, Iiduka et al. [11] proved the following weak convergence theorem.
Theorem 1.2 (see Iiduka et al. [11]). LetCbe a nonempty closed convex subset of a real Hilbert spaceH and let Abe anα-inverse strongly monotone operator ofC intoH with VI(C,A)= ∅. Let{xn}be a sequence defined as follows:x1=x∈Cand
xn+1=PC
αnxn+1−αn
PC
xn−λnAxn
(1.4)
for everyn=1, 2,..., wherePCis the metric projection fromHontoC,{αn}is a sequence in [−1, 1], and{λn}is a sequence in [0, 2α]. If{αn}and{λn}are chosen so thatαn∈[a,b] for somea,bwith−1< a < b <1 andλn∈[c,d] for somec,dwith 0< c < d <2(1 +a)α, then {xn}converges weakly to some element of VI(C,A).
A mappingTofCinto itself is said to be strictly pseudocontractive [5] if there existsk with 0≤k <1 such that
Tx−T y2≤ x−y2+k(I−T)x−(I−T)y2 (1.5) for allx,y∈C. For such a case,Tis said to bek-strictly pseudocontractive. For finding a fixed point of ak-strictly pseudocontractive mapping, Browder and Petryshyn [5] proved the following weak convergence theorem.
Theorem 1.3 (Browder and Petryshyn [5]). LetKbe a nonempty bounded closed convex subset of a real Hilbert spaceHand letTbe ak-strictly pseudocontractive mapping ofKinto itself. Let{xn}be a sequence defined as follows:x1=x∈Kand
xn+1=αxn+ (1−α)Txn (1.6)
for everyn=1, 2,..., whereα∈(k, 1). Then{xn}converges weakly to some element ofF(T).
In this paper, motivated by the above three theorems, we first consider the following generalized variational inequality problem in a Banach space.
Problem 1.4. LetEbe a smooth Banach space with norm · , letE∗denote the dual of E, and letx,fdenote the value of f ∈E∗atx∈E. LetCbe a nonempty closed convex
subset ofEand letAbe an accretive operator ofCintoE. Find a pointu∈Csuch that
Au,J(v−u)≥0, ∀v∈C, (1.7)
whereJis the duality mapping ofEintoE∗.
This problem is connected with the fixed point problem for nonlinear mappings, the problem of finding a zero point of an accretive operator and so on. For the problem of finding a zero point of an accretive operator by the proximal point algorithm, see Kamimura and Takahashi [12]. Second, in order to find a solution ofProblem 1.4, we introduce the following iterative scheme for an accretive operatorAin a Banach spaceE:
x1=x∈Cand
xn+1=αnxn+1−αnQCxn−λnAxn (1.8) for everyn=1, 2,..., whereQCis a sunny nonexpansive retraction fromEontoC,{αn} is a sequence in [0, 1], and{λn}is a sequence of real numbers. Then we prove a weak con- vergence (Theorem 3.1) in a Banach space which is generalized simultaneously Gol’shte˘ın and Tret’yakov’s theorem (Theorem 1.1) and Browder and Petryshyn’s theorem (Theorem 1.3).
2. Preliminaries
LetEbe a real Banach space with norm · and letE∗denote the dual ofE. We denote the value of f ∈E∗atx∈Ebyx,f. When{xn}is a sequence inE, we denote strong convergence of{xn}tox∈Ebyxn→xand weak convergence byxnx.
LetU= {x∈E:x =1}. A Banach spaceEis said to be uniformly convex if for each ε∈(0, 2], there existsδ >0 such that for anyx,y∈U,
x−y ≥εimpliesx+y 2
≤1−δ. (2.1)
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Ba- nach spaceEis said to be smooth if the limit
limt→0
x+ty − x
t (2.2)
exists for allx,y∈U. It is also said to be uniformly smooth if the limit (2.2) is attained uniformly forx,y∈U. The norm ofEis said to be Fre´chet differentiable if for eachx∈U, the limit (2.2) is attained uniformly for y∈U. And we define a functionρ: [0,∞)→ [0,∞) called the modulus of smoothness ofEas follows:
ρ(τ)=sup 1
2
x+y+x−y
−1 :x,y∈E,x =1,y =τ
. (2.3) It is known thatEis uniformly smooth if and only if limτ→0ρ(τ)/τ=0. Letqbe a fixed real number with 1< q≤2. Then a Banach spaceEis said to beq-uniformly smooth if there exists a constantc >0 such thatρ(τ)≤cτqfor allτ >0. For example, see [1,23] for more details. We know the following lemma [1,2].
Lemma 2.1 [1,2]. Letqbe a real number with 1< q≤2 and letEbe a Banach space. Then Eisq-uniformly smooth if and only if there exists a constantK≥1 such that
1 2
x+yq+x−yq
≤ xq+K yq (2.4)
for allx,y∈E.
The best constantK inLemma 2.1is called theq-uniformly smoothness constant ofE;
see [1]. Let qbe a given real number with q >1. The (generalized) duality mappingJq
fromEinto 2E∗ is defined by
Jq(x)= x∗∈E∗:x,x∗= xq,x∗= xq−1
(2.5) for allx∈E. In particular,J=J2 is called the normalized duality mapping. It is known that
Jq(x)= xq−2J(x) (2.6)
for allx∈E. IfEis a Hilbert space, thenJ=I. The normalized duality mappingJhas the following properties:
(1) ifEis smooth, thenJis single-valued;
(2) ifEis strictly convex, thenJis one-to-one andx−y,x∗−y∗>0 holds for all (x,x∗), (y,y∗)∈Jwithx=y;
(3) ifEis reflexive, thenJis surjective;
(4) ifEis uniformly smooth, thenJis uniformly norm-to-norm continuous on each bounded subset ofE.
See [22] for more details. It is also known that
qy−x,jx≤ yq− xq (2.7)
for allx,y∈Eand jx∈Jq(x). Further we know the following result [25]. For the sake of completeness, we give the proof; see also [1,2].
Lemma 2.2 [25]. Letqbe a given real number with 1< q≤2 and letEbe aq-uniformly smooth Banach space. Then
x+yq≤ xq+qy,Jq(x)+ 2K yq (2.8) for allx,y∈E, whereJq is the generalized duality mapping ofEandK is theq-uniformly smoothness constant ofE.
Proof. Letx,y∈Ebe given arbitrarily. From (2.7), we haveqy,Jq(x) ≥ xq− x− yq. Thus, it follows fromLemma 2.1that
qy,Jq(x)≥ xq− x−yq
≥ xq−
2xq+ 2K yq− x+yq
= −xq−2K yq+x+yq.
(2.9)
Hence we havex+yq≤ xq+qy,Jq(x)+ 2K yq.
LetEbe a Banach space and letCbe a subset ofE. Then a mappingTofCinto itself is said to be nonexpansive if
Tx−T y ≤ x−y (2.10)
for allx,y∈C. We denote byF(T) the set of fixed points ofT. A closed convex subset Cof a Banach spaceEis said to have normal structure if for each bounded closed convex subsetDofCwhich contains at least two points, there exists an element ofDwhich is not a diametral point ofD. It is well known that a closed convex subset of a uniformly convex Banach space has normal structure and a compact convex subset of a Banach space has normal structure. We know the following theorem [14] related to the existence of fixed points of a nonexpansive mapping.
Theorem 2.3 (Kirk [14]). Let E be a reflexive Banach space and letD be a nonempty bounded closed convex subset ofEwhich has normal structure. LetT be a nonexpansive mapping ofDinto itself. Then the setF(T) is nonempty.
To prove our main result, we also need the following theorem [4].
Theorem 2.4 (see Browder [4]). LetDbe a nonempty bounded closed convex subset of a uniformly convex Banach spaceEand letT be a nonexpansive mapping ofDinto itself. If {uj}is a sequence ofDsuch thatuju0and limj→∞uj−Tuj =0, thenu0 is a fixed point ofT.
LetDbe a subset ofCand letQbe a mapping ofCintoD. ThenQis said to be sunny if
QQx+t(x−Qx)=Qx (2.11)
wheneverQx+t(x−Qx)∈Cforx∈Candt≥0. A mappingQofCinto itself is called a retraction ifQ2=Q. If a mappingQofCinto itself is a retraction, thenQz=zfor every z∈R(Q), whereR(Q) is the range ofQ. A subsetDofCis called a sunny nonexpansive retract ofCif there exists a sunny nonexpansive retraction fromContoD. We know the following two lemmas [15,20] concerning sunny nonexpansive retractions.
Lemma 2.5 [15]. Let Cbe a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach spaceEand letTbe a nonexpansive mapping ofCinto itself with F(T)= ∅. Then the setF(T) is a sunny nonexpansive retract ofC.
Lemma 2.6 (see [20]; see also [6]). LetCbe a nonempty closed convex subset of a smooth Banach spaceEand letQCbe a retraction fromEontoC. Then the following are equivalent:
(i)QCis both sunny and nonexpansive;
(ii)x−QCx,J(y−QCx) ≤0 for allx∈Eandy∈C.
It is well known that if Eis a Hilbert space, then a sunny nonexpansive retraction QCis coincident with the metric projection fromEontoC. LetCbe a nonempty closed convex subset of a smooth Banach spaceE, letx∈Eand letx0∈C. Then we have from Lemma 2.6thatx0=QCxif and only ifx−x0,J(y−x0) ≤0 for ally∈C, whereQCis a sunny nonexpansive retraction fromEontoC.
LetEbe a Banach space and letCbe a nonempty closed convex subset ofE. An oper- atorAofCintoEis said to be accretive if there exists j(x−y)∈J(x−y) such that
Ax−Ay,j(x−y)≥0 (2.12)
for allx,y∈C. We can characterize the set of solutions ofProblem 1.4by using sunny nonexpansive retractions.
Lemma 2.7. LetCbe a nonempty closed convex subset of a smooth Banach spaceE. LetQC
be a sunny nonexpansive retraction fromEontoCand letAbe an accretive operator ofC intoE. Then for allλ >0,
S(C,A)=FQC(I−λA), (2.13) whereS(C,A)= {u∈C:Au,J(v−u) ≥0,∀v∈C}.
Proof. We have fromLemma 2.6thatu∈F(QC(I−λA)) if and only if
(u−λAu)−u,J(y−u)≤0 (2.14) for ally∈Candλ >0. This inequality is equivalent to the inequality−λAu,J(y−u) ≤ 0. Sinceλ >0, we haveu∈S(C,A). This completes the proof.
Now, we define an extension of the inverse strongly monotone operator (1.2) in Ba- nach spaces. LetCbe a subset of a smooth Banach spaceE. Forα >0, an operatorAofC intoEis said to beα-inverse strongly accretive if
Ax−Ay,J(x−y)≥αAx−Ay2 (2.15) for allx,y∈C. Evidently, the definition of the inverse strongly accretive operator is based on that of the inverse strongly monotone operator. It is obvious from (2.15) that
Ax−Ay ≤1
αx−y (2.16)
for allx,y∈C. Letqbe a given real number withq≥2. We also have from (2.6), (2.15), and (2.16) that
Ax−Ay,Jq(x−y)= x−yq−2
Ax−Ay,J(x−y)
≥ x−yq−2αAx−Ay2
≥
αAx−Ayq−2
αAx−Ay2
=αq−1Ax−Ayq
(2.17)
for allx,y∈C. One should note that no Banach space isq-uniformly smooth forq >2;
see [23] for more details. So, in this paper, we study a weak convergence theorem for inverse strongly accretive operators in uniformly convex and 2-uniformly smooth Ba- nach spaces. It is well known that Hilbert spaces and the LebesgueLp(p≥2) spaces are
uniformly convex and 2-uniformly smooth. LetX be a Banach space and letLp(X)= Lp(Ω,Σ,μ;X), 1≤p≤ ∞, be the Lebesgue-Bochner space on an arbitrary measure space (Ω,Σ,μ). Let 1< q≤2 and letq≤p <∞. ThenLp(X) isq-uniformly smooth if and only ifXisq-uniformly smooth; see [23]. For convergence theorems in the Lebesgue spaces Lp(1< p≤2), see Iiduka and Takahashi [9,10].
We can know the following property for inverse strongly accretive operators in a 2- uniformly smooth Banach space.
Lemma 2.8. LetCbe a nonempty closed convex subset of a 2-uniformly smooth Banach space E. Letα >0 and letAbe anα-inverse strongly accretive operator ofCintoE. If 0< λ≤α/K2, thenI−λAis a nonexpansive mapping ofCintoE, whereKis the 2-uniformly smoothness constant ofE.
Proof. We have fromLemma 2.2that for allx,y∈C, (I−λA)x−(I−λA)y2=(x−y)−λ(Ax−Ay)2
≤ x−y2−2λAx−Ay,J(x−y)+ 2K2λ2Ax−Ay2
≤ x−y2−2λαAx−Ay2+ 2K2λ2Ax−Ay2
≤ x−y2+ 2λ(K2λ−α)Ax−Ay2.
(2.18) So, if 0< λ≤α/K2, thenI−λAis a nonexpansive mapping ofCintoE.
Remark 2.9. Ifq≥2, we have from (2.17) that forx,y∈C,
(I−λA)x−(I−λA)yq≤ x−yq+λ2Kqλq−1−qαq−1Ax−Ayq. (2.19) Since, forq >2, there exists no Banach space which isq-uniformly smooth, we consider only 2-uniformly smooth Banach spaces. For 1< q <2, the inequalities (2.17) and (2.19) do not hold.
ApplyingTheorem 2.3, Lemmas2.7and2.8, we have that ifDis a nonempty bounded closed convex subset of a uniformly convex and 2-uniformly smooth Banach spaceE,D is a sunny nonexpansive retract ofEandA is an inverse strongly accretive operator of DintoE, then the setS(D,A) is nonempty. We know also the following theorem which was proved by Reich [21]; see also Lau and Takahashi [16], Takahashi and Kim [24], and Bruck [7].
Theorem 2.10 (see Reich [21]). LetCbe a nonempty closed convex subset of a uniformly convex Banach space with a Fre´chet differentiable norm. Let{T1,T2,...}be a sequence of nonexpansive mappings ofCinto itself with∞n=1F(Tn)=∅. Letx∈CandSn=TnTn−1···T1
for alln≥1. Then the set ∞ n=1
co Smx:m≥n∩ ∞ n=1
FTn
(2.20) consists of at most one point, wherecoDis the closure of the convex hull ofD.
3. Weak convergence theorem
In this section, we obtain the following weak convergence theorem for finding a solution ofProblem 1.4for an inverse strongly accretive operator in a uniformly convex and 2- uniformly smooth Banach space.
Theorem 3.1. LetEbe a uniformly convex and 2-uniformly smooth Banach space and let Cbe a nonempty closed convex subset ofE. LetQCbe a sunny nonexpansive retraction from EontoC, letα >0 and letAbe anα-inverse strongly accretive operator ofCintoEwith S(C,A)= ∅. Supposex1=x∈Cand{xn}is given by
xn+1=αnxn+ (1−αn)QC
xn−λnAxn
(3.1)
for everyn=1, 2,..., where{λn}is a sequence of positive real numbers and{αn}is a sequence in [0, 1]. If{λn}and{αn}are chosen so thatλn∈[a,α/K2] for somea >0 andαn∈[b,c]
for someb,cwith 0< b < c <1, then{xn}converges weakly to some elementz ofS(C,A), whereKis the 2-uniformly smoothness constant ofE.
Proof. Putyn=QC(xn−λnAxn) for everyn=1, 2,....Letu∈S(C,A). We first prove that {xn}and{yn}are bounded and limn→∞xn−yn =0. We have from Lemmas2.7and2.8 that
yn−u=QC
xn−λnAxn
−QC
u−λnAu
≤xn−λnAxn−
u−λnAu≤xn−u (3.2) for everyn=1, 2,....It follows from (3.2) that
xn+1−u=αn
xn−u+1−αn
yn−u
≤αnxn−u+1−αnyn−u
≤αnxn−u+1−αnxn−u=xn−u
(3.3)
for every n=1, 2,.... Therefore, {xn−u} is nonincreasing and hence there exists limn→∞xn−u. So,{xn}is bounded. We also have from (3.2) and (2.16) that{yn}and {Axn}are bounded.
Next we will show limn→∞xn−yn =0. Suppose that limn→∞xn−yn =0. Then there areε >0 and a subsequence{xni−yni}of{xn−yn}such thatxni−yni ≥εfor eachi=1, 2,....SinceEis uniformly convex, the function · 2is uniformly convex on bounded convex setB(0,x1−u), whereB(0,x1−u)= {x∈E:x ≤ x1−u}. So, forε, there isδ >0 such that
x−y ≥εimpliesλx+ (1−λ)y2≤λx2+ (1−λ)y2−λ(1−λ)δ (3.4) wheneverx,y∈B(0,x1−u) andλ∈(0, 1). Thus, for eachi=1, 2,...,
xni+1−u2=αni
xni−u+1−αni
yni−u2
≤αnixni−u2+1−αniyni−u2−αni
1−αni
δ. (3.5)
Therefore, for eachi=1, 2,...,
0< b(1−c)δ≤αni1−αniδ≤xni−u2−xni+1−u2. (3.6) Since the right-hand side of the inequality above converges to 0, we have a contradiction.
Hence we conclude that
nlim→∞xn−yn=0. (3.7)
Since{xn}is bounded, we have that a subsequence{xni}of{xn}converges weakly toz.
And sinceλniis in [a,α/K2] for somea >0, it holds that{λni}is bounded. So, there exists a subsequence{λni j}of{λni}which converges toλ0∈[a,α/K2]. We may assume without loss of generality thatλni→λ0. We next provez∈S(C,A). SinceQCis nonexpansive, it holds fromyni=QC(xni−λniAxni) that
QC
xni−λ0Axni
−xni≤QC
xni−λ0Axni
−yni+yni−xni
≤xni−λ0Axni
−
xni−λniAxni+yni−xni
≤Mλni−λ0+yni−xni,
(3.8)
whereM=sup{Axn:n=1, 2,...}. We obtain from the convergence of{λni}, (3.7), and (3.8) that
limi→∞QCI−λ0Axni−xni=0. (3.9) On the other hand, fromLemma 2.8, we have thatQC(I−λ0A) is nonexpansive. So, by (3.9),Lemma 2.7, andTheorem 2.4, we obtainz∈F(QC(I−λ0A))=S(C,A).
Finally, we prove that{xn}converges weakly to some element ofS(C,A). We put Tn=αnI+1−αn
QC
I−λnA (3.10)
for everyn=1, 2,....Then we havexn+1=TnTn−1···T1xandz∈∞
n=1co{xm:m≥n}. We have fromLemma 2.8thatTn is a nonexpansive mapping ofCinto itself for every n=1, 2,....And we also have fromLemma 2.7that∞n=1F(Tn)=∞
n=1F(QC(I−λnA))= S(C,A). ApplyingTheorem 2.10, we obtain
∞ n=1
co xm:m≥n∩S(C,A)= {z}. (3.11) Therefore, the sequence{xn}converges weakly to some element ofS(C,A). This com-
pletes the proof.
4. Applications
In this section, we prove some weak convergence theorems in a uniformly convex and 2-uniformly smooth Banach space by usingTheorem 3.1. We first study the problem of finding a zero point of an inverse strongly accretive operator. The following theorem is a generalization of Gol’shte˘ın and Tret’yakov’s theorem (Theorem 1.1).
Theorem 4.1. LetEbe a uniformly convex and 2-uniformly smooth Banach space. Letα >0 and letAbe anα-inverse strongly accretive operator ofEinto itself withA−10= ∅, where A−10= {u∈E:Au=0}. Supposex1=x∈Eand{xn}is given by
xn+1=xn−rnAxn (4.1)
for everyn=1, 2,..., where{rn}is a sequence of positive real numbers. If{rn} is chosen so thatrn∈[s,t] for somes,twith 0< s < t < α/K2, then{xn}converges weakly to some elementzofA−10, whereKis the 2-uniformly smoothness constant ofE.
Proof. By assumption, we note that 1−tK2/α∈(0, 1). We define sequences {αn} and {λn}by
αn=1−tK2
α , λn= rn
1−αn (4.2)
for everyn=1, 2,..., respectively. Then it is easy to check thatλn∈(0,α/K2) andS(E, A)=A−10. It follows from the definition of{xn}that
xn+1=xn−rnAxn=αnxn+1−αn
xn− rn 1−αnAxn
=αnxn+1−αnIxn−λnAxn,
(4.3) whereIis the identity mapping ofE. Obviously, the identity mappingIis a sunny non- expansive retraction fromEonto itself. Therefore, by usingTheorem 3.1,{xn}converges
weakly to some elementzofA−10.
We next study the problem of finding a fixed point of a strictly pseudocontractive mapping. Let 0≤k <1. LetEbe a Banach space and letCbe a subset ofE. Then a map- pingT of Cinto itself is said to be k-strictly pseudocontractive [5,19] if there exists
j(x−y)∈J(x−y) such that
Tx−T y,j(x−y)≤ x−y2−1−k
2 (I−T)x−(I−T)y2 (4.4) for allx,y∈C. Then the inequality (4.4) can be written in the form
(I−T)x−(I−T)y,j(x−y)≥1−k
2 (I−T)x−(I−T)y2. (4.5) IfEis a Hilbert space, then the inequality (4.4) (and hence (4.5)) is equivalent to the inequality (1.5). The following theorem is a generalization of Browder and Petryshyn’s theorem (Theorem 1.3).
Theorem 4.2. LetEbe a uniformly convex and 2-uniformly smooth Banach space and let Cbe a nonempty closed convex subset and a sunny nonexpansive retract ofE. LetT be a k-strictly pseudocontractive mapping ofCinto itself withF(T)= ∅. Supposex1=x∈C and{xn}is given by
xn+1=
1−βnxn+βnTxn (4.6)
for everyn=1, 2,..., where{βn}is a sequence in (0, 1). If{βn}is chosen so thatβn∈[β,γ]
for someβ,γwith 0< β < γ <(1−k)/(2K2), then{xn}converges weakly to some elementz ofF(T), whereKis the 2-uniformly smoothness constant ofE.
Proof. By assumption, note that 1−2γK2/(1−k)∈(0, 1). We define sequences{αn}and {λn}by
αn=1−γ 2K2
1−k, λn= βn
1−αn (4.7)
for everyn=1, 2,..., respectively. Then we can readily verify that 0< λn≤1−k
2K2 ≤ 1
2 <1 (4.8)
for every n=1, 2,.... Put A=I−T. We have from (4.5) that A is (1−k)/2-inverse strongly accretive. It is easy to show that
S(C,A)=S(C,I−T)=F(T)= ∅. (4.9) SinceCis a sunny nonexpansive retract ofEandλn∈(0, 1), there exists a sunny nonex- pansive retractionQCsuch that (1−λn)xn+λnTxn=QC((1−λn)xn+λnTxn) for every n=1, 2,....It follows from the definition of{xn}that
xn+1= 1−βn
xn+βnTxn
= 1−λn
1−αn
xn+λn
1−αn
Txn
=αnxn+1−αn
1−λn
xn+λnTxn
=αnxn+1−αnQC1−λnxn+λnTxn
=αnxn+1−αn)QCxn−λn(I−T)xn
=αnxn+1−αnQCxn−λnAxn.
(4.10)
Therefore, by usingTheorem 3.1,{xn}converges weakly to some elementzofF(T).
LetCbe a subset of a smooth Banach spaceE. Letα >0. An operatorAofCintoEis said to beα-strongly accretive if
Ax−Ay,J(x−y)≥αx−y2 (4.11) for allx,y∈C. Letβ >0. An operatorAofCintoEis said to beβ-Lipschitz continuous if
Ax−Ay ≤βx−y (4.12)
for allx,y∈C. LetCbe a nonempty closed convex subset of a Hilbert spaceH. One method of finding a pointu∈VI(C,A) is the projection algorithm which starts with any x1=x∈Cand updates iterativelyxn+1according to the formula
xn+1=PCxn−λAxn (4.13)
for everyn=1, 2,..., wherePC is the metric projection fromH ontoC,A is a mono- tone (accretive) operator ofCintoH, andλis a positive real number. It is well known that ifAis anα-strongly accretive andβ-Lipschitz continuous operator ofCintoHand λ∈(0, 2α/β2), then the operatorPC(I−λA) is a contraction ofCinto itself. Hence, the Banach contraction principle guarantees that the sequence generated by (4.13) converges strongly to the unique solution of VI(C,A); see [3]. Motivated by this result, we prove the following weak convergence theorem for strongly accretive and Lipschitz continuous operators.
Theorem 4.3. LetEbe a uniformly convex and 2-uniformly smooth Banach space and let Cbe a nonempty closed convex subset ofE. LetQCbe a sunny nonexpansive retraction from EontoC, letα >0, letβ >0, and letAbe anα-strongly accretive andβ-Lipschitz continuous operator ofCintoEwithS(C,A)= ∅. Supposex1=x∈Cand{xn}is given by
xn+1=αnxn+1−αnQCxn−λnAxn (4.14) for everyn=1, 2,..., where{λn}is a sequence of positive real numbers and{αn}is a sequence in [0, 1]. If{λn}and{αn}are chosen so thatλn∈[a,α/(K2β2)] for somea >0 andαn∈ [b,c] for someb,cwith 0< b < c <1, then{xn}converges weakly to a unique elementzof S(C,A), whereKis the 2-uniformly smoothness constant ofE.
Proof. SinceAis anα-strongly accretive andβ-Lipschitz continuous operator ofCinto E, we have
Ax−Ay,J(x−y)≥αx−y2≥ α
β2Ax−Ay2 (4.15) for allx,y∈C. Therefore,Aisα/β2-inverse strongly accretive. SinceAis strongly accre- tive andS(C,A)= ∅, the setS(C,A) consists of one point z. UsingTheorem 3.1,{xn}
converges weakly to a unique elementzofS(C,A).
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