AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 832548 10.1155/2013/832548 832548 Research Article Extragradient Method for Solutions of Variational Inequality Problems in Banach Spaces Zegeye H. 1 Shahzad N. 2 Chen Ru Dong 1 Department of Mathematics University of Botswana Private Bag 00704, Gaborone Botswana ub.bw 2 Department of Mathematics King Abdulaziz University P.O. Box 80203 Jeddah 21589 Saudi Arabia kau.edu.sa 2013 11 7 2013 2013 27 03 2013 11 05 2013 2013 Copyright © 2013 H. Zegeye and N. Shahzad. 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.

We introduce an iterative process which converges strongly to solutions of a certain variational inequity problem for η-inverse strongly accretive mappings in the set of common fixed points of finite family of strictly pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

1. Introduction

Let E be a real normed linear space with dual E*. For 1<q<, we denote by Jq the generalized duality mapping from E to 2E* defined by (1)Jq(x):={x*E*:x,x*=xq,x*=xq-1}, where ·,· denotes the duality pairing. In particular, J=J2 is called the normalized duality map. It is well known (see e.g., ) that Jq is single valued if E is smooth and that (2)Jq(x)=xq-2J(x),x0. In the sequel, we will denote the single-valued generalized map by jq.

A mapping A with domain D(A)E and range R(A) in E is called α-strongly accretive if there exist α(0,1) and jq(x-y)Jq(x-y) such that (3)Ax-Ay,jq(v-u)αx-yq.A is calledη-inverse strongly accretive if there exist η(0,1) and jq(x-y)Jq(x-y) such that (4)Ax-Ay,jq(x-y)ηAx-Ayq,for every  x,yD(A). Let C be a nonempty, closed, and convex subset of E and, let A:CE be a nonlinear mapping. The variational inequality problem is to (5)finduCsuch  thatAu,j(v-u)0,vC, for some j(v-u)J(v-u). The set of solutions of variational inequality problem is denoted by VI(C,A). If E:=H, a real Hilbert space, the variational inequality problem reduces to (6)finduCsuch thatAu,v-u0,vC, which was introduced and studied by Stampacchia .

Variational inequality theory has emerged as an important tool in studying a wide class of related problems arising in mathematical, physical, regional, engineering, and nonlinear optimization sciences (see, for instance, ).

In 1976, Korpelevič  introduced the following well-known extragradient method: (7)yn=PC(xn-γAxn),xn+1=PC(xn-γAyn),n0, where PC is the metric projection from n onto its subset C, for some γ>0, and A:Cn is an accretive operator. He proved that the sequence {xn} converges to a solution of the variational inequality (6).

Furthermore, Noor  proved that the iterative scheme, given by (8)yn=PC(xn-γAxn),xn+1=PC(yn-γAyn),n0, where A:Cnn is an accretive operator, converges to a solution of the variational inequality (6).

We note that the above algorithms give strong convergence to a solution of the variational inequality (6). However, both algorithms fail, in general, to converge strongly in the setting of infinite-dimensional Hilbert spaces.

In 2006, Aoyama et al.  introduced and studied the following iterative algorithm in a uniformly convex and 2-uniformly smooth Banach spaces possessing weakly sequentially continuous duality mapping: (9)xn+1=αnxn+(1-αn)QC[xn-λnAxn],n0, where QC is a sunny nonexpansive retraction from E onto a closed and convex C, A:CE is an η-inverse strongly accretive mapping and {αn} and {λn} subsets of real numbers, satisfy certain conditions. They proved that the sequence in (9) converges weakly to a point zV(C,A).

Recently, Yao et al.  introduced and considered the following iterative method for η-strongly accretive mappings in a uniformly convex and 2-uniformly smooth Banach space possessing weakly sequentially continuous duality mapping: (10)yn=QC(xn-λnAxn),xn+1=αnu+βnxn+γnQC(yn-λnAyn),n0, where QC is a sunny nonexpansive retraction from E onto C. They proved that the sequence {xn} defined by (10) converges strongly to QVI(C,A)u provided that real sequences {αn}, {βn}, {γn}, and {λn} satisfy certain conditions.

Let C be a nonempty subset of a real Banach space E. A mapping T:CE is called λ-strictly pseudocontractive of Browder-Petryshyn type  if for all x,yD(T) there exist λ>0 and jq(x-y)Jq(x-y) such that (11)Tx-Ty,jq(x-y)x-yq-λx-y-(Tx-Ty)q.T is called Lipschitz if there exists L0 such that (12)Tx-TyLx-yx,yD(T). If L<1 in (12), then T is called contraction, while T is said to be nonexpansive if L=1.

If E=H, a real Hilbert space, then (11) is equivalent to the inequality (13)Tx-Ty2x-y2+kx-y-(Tx-Ty)2,k=(1-2λ), and we can assume also that k0, so that k[0,1). A point xC is a fixed point of T if Tx=x, and we denote by F(T) the set of fixed points of T; that is, F(T)={xC:Tx=x}.

In 2001, Yamada  introduced a hybrid steepest descent method which relates solutions of variational inequality problems with fixed point of mappings in Hilbert spaces. He proved that if T is nonexpansive self-map on C and A is an η-strongly accretive mapping from C into E satisfying certain conditions, then the sequence defined by (14)xn+1=Txn-μλnA(Txn),n0, converges strongly to the unique solution of the variational inequality (15)Findx*F(T)such thatAx*,x-x*0,xF(T). The above results naturally bring us to the following question.

Question. Could we produce an iterative scheme which approximates a solution of variational inequality (5) for η-inverse strongly accretive mappings in Banach spaces?

In this paper, motivated by Yao et al.  and Yamada , it is our purpose to introduce an iterative scheme which converges strongly to a solution of the variational inequality (5) for η-inverse strongly accretive mapping in the set of common fixed points of finite family of strictly pseudocontractive mappings in a uniformly convex and q-uniformly smooth Banach space E possessing weakly sequentially continuous duality mapping. Our results complement or improve the results of Yao et al. , Aoyama et al. , and some authors.

2. Preliminaries

Let E be a real Banach space. The modulus of smoothness of E is the function ρE:[0,)[0,) defined by ρE(τ):=sup{(1/2)(x+y+x-y)-1:x=1,y=τ}. If ρE(τ)>0 for all τ>0, then E is said to be smooth. If there exists a constant c>0 and a real number 1<q<, such that ρE(τ)cτq, then E is said to be q-uniformly smooth.

If E is a real q-uniformly smooth Banach space, then by , the following geometric inequality holds: (16)x+yqxq+qy,jq(x)+cqyq, for all x,yE and some real constant cq>0.

It is well known (see e.g., ) that (17)Lp(lp)orWmpis{p-uniformly  smoothif  1<p<2,2-uniformly smoothif  p2. The Banach space E is said to be uniformly convex if, given ε>0, there exists δ>0, such that, for all x,yE with x1, y1 and x-yε, (1/2)(x+y)1-δ. It is well known that Lp, p, and Sobolev spaces Wmp, (1<p<) are uniformly convex.

Let CE be closed convex and Q a mapping of E onto C. Then, Q is said to be sunny if Q(Q(x)+t(x-Q(x)))=Q(x) for all xE and t0. A mapping Q of E into C is said to be a retraction if Q2=Q. If a mapping Q is a retraction, then Q(z)=(z) for every zR(Q), range of Q. A subset C of E is said to be a sunny nonexpansive retract of E if there exists a sunny nonexpansive retraction of E onto C, and it is said to be a nonexpansive retract of E if there exists a nonexpansive retraction of E onto C. If E=H, the metric projection PC is a sunny nonexpansive retraction from H to any closed convex subset of H. Moreover, if C is a nonempty closed convex subset of a uniformly convex and uniformly smooth real Banach space E and T is a nonexpansive mapping of C into itself with F(T), then the set F(T) is a sunny nonexpansive retract of C.

In what follows, we will make use of the following lemmas.

Lemma 1 (see, e.g., [<xref ref-type="bibr" rid="B4">17</xref>]).

Let E be a smooth Banach space, and let K be a nonempty subset of E. Let Q:EK be a retraction, and let J be the normalized duality map on E. Then, the following are equivalent:

Q  is sunny nonexpansive,

x-Q(x),J(y-Q(x))0 for all xE and yK.

Lemma 2 (see [<xref ref-type="bibr" rid="B14">18</xref>]).

Let {an} be a sequence of nonnegative real numbers satisfying the following relation: (18)an+1(1-αn)an+αnδn,nn0, where {αn}(0,1) and {δn} satisfying the following conditions: limnαn=0,n=1αn=, and limsupnδn0. Then, limnan=0.

Lemma 3 (see [<xref ref-type="bibr" rid="B3">13</xref>]).

Let C be a nonempty closed convex subset of a smooth Banach space E. Let QC be a sunny nonexpansive retraction from E onto C, and let A be an accretive operator of C into E. Then, for all λ>0, (19)VI(C,A)=F(QC(I-λA)), where VI(C,A)={x*C:Ax*,J(x-x*)0,xC}.

Lemma 4 (see [<xref ref-type="bibr" rid="B4">17</xref>]).

Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space E, and let T be nonexpansive mapping of C into itself. If {xn} is a sequence of C such that xnx weakly and xn-Txn0 strongly, then x is a fixed point of T.

Lemma 5 (see [<xref ref-type="bibr" rid="B10">19</xref>]).

Let E be a real Banach space. Then, for any given x,yE, the following inequality holds: (20)x+y2x2+2y,j(x+y),j(x+y)J(x+y).

Lemma 6 (see [<xref ref-type="bibr" rid="B6">20</xref>]).

Let K be a nonempty closed convex subset of a strictly convex Banach space E. Let Ti:KE,i=1,2,,r, be a family of nonexpansive mappings such that i=1rF(Ti). Let α0,α1,α2,,αr be real numbers in (0,1) such that i=0rαi=1, and let T:=α0I+α1T1++αrTr. Then, T is nonexpansive, and F(T)=i=1rF(Ti).

Lemma 7 (see [<xref ref-type="bibr" rid="B24">21</xref>]).

Let C be a nonempty, closed and convex subset of a real uniformly convex and smooth Banach space E. Let Ti:CE, i=1,,N, be λi-strictly pseudocontractive mappings such that i=1NF(Ti). Let T:=θ1T1+θ2T2++θNTN with θ1+θ2++θN=1. Then T is λ-strictly pseudocontractive with λ:=min{λi:i=1,,N} and F(T)=i=1NF(Ti).

Lemma 8 (see [<xref ref-type="bibr" rid="B23">22</xref>]).

Let C be a nonempty closed and convex subset of a real q-uniformly smooth Banach space E for 1<q<. Let T:CE be a λ-strictly pseudocontractive mapping. Then, for 0<μ<μ0=min{1,(qλ/cq)1/(q-1)}, where L is the Lipchitz constant of T and cq is a constant in (16), the mapping Tμx:=(1-μ)x+μTx is nonexpansive, and F(Tμ)=F(T).

Lemma 9.

Let C be a nonempty closed and convex subset of a a real q-uniformly smooth Banach space E for 1<q<. Let A:CE be an η-inverse strongly accretive mapping. Then, for 0<γ<(qη/cq)1/(q-1), the mapping Aμx:=(x-γAx) is nonexpansive.

Proof.

Now, using inequality (16), we get that (21)Aγx-Aγyq=(x-y)-γ(Ax-Ay)qx-yq-qγAx-Ay,jq(x-y)+γqcqAx-Ayqx-yq-qγηAx-Ay2+γqcqAx-Ay2x-yq-γ(qη-γq-1cq)Ax-Ayq,x-yq. The proof is complete.

Lemma 10 (see [<xref ref-type="bibr" rid="B19">10</xref>]).

Let E be a uniformly convex Banach space, and let BR(0) be a closed ball of E. Then, there exists a continuous strictly increasing convex function g:[0,)[0,) with g(0)=0 such that (22)α0x0+α1x1+α2x2++αkxk2i=0kαixi2-αsαtg(xs-xt), for xiBR(0):={xE:xR}, i=0,1,2,,k with i=0kαi=1.

Lemma 11 (see [<xref ref-type="bibr" rid="B9">5</xref>]).

Let {an} be sequences of real numbers such that there exists a subsequence {ni} of {n} such that ani<ani+1, for all i. Then, there exists a nondecreasing sequence {mk} such that mk, and the following properties are satisfied by all (sufficiently large) numbers k: (23)amkamk+1,akamk+1. In fact, mk=max{jk:aj<aj+1}.

3. Main Results

Let C be a nonempty closed convex subset of a real uniformly convex and q-uniformly smooth Banach space E(1<q<). Let Ti:CC, for i=1,,N, be a λi-strictly pseudocontractive mappings, and let A:CE be an η-inverse strongly accretive mapping. Then, in what follows, we will study the variational inequality (24)Findx*i=1NF(Ti)such thatAx*,J(x-x*)0,xi=1NF(Ti), and the following iteration process: (25)x0C,zn=cnxn+(1-cn)Sxn,xn+1=QC[(1-αn)(βnxn+(1-βn)QC[I-γA]zn)], where S:=[(1-μ)I+μT], for T:=θ1T1+θ2T2++θnTN, such that θ1+θ2++θN=1, 0<μ<μ0=min{1,(qλ/cq)1/(q-1)}, for λ:=min{λi:i=1,2,,N}, and 0<γ<(qη/cq)1/(q-1), and cq is the real number in (16). In addition, we assume {αn}(0,c)(0,1) and {βn},{cn}[a,b](0,1) as real sequences satisfying the following control conditions: (i) limnαn=0, (ii) αn=, limn(|βn-βn-1|/αn-1)=0,limn(|αn-αn-1|/αn-1)=0, and  limn(|cn-cn-1|/αn-1)=0.

We now prove our main theorem.

Theorem 12.

Let C be a nonempty closed convex subset of a real uniformly convex and q-uniformly smooth Banach space E possessing weakly sequentially continuous duality mapping. Let Ti:CC, for i=1,,N, be λi-strictly pseudocontractive mappings, and let A:CE be η-inverse strongly accretive mapping. Let {xn} be a sequence defined by (25). Assume that :=FVI(C,A), where F=i=1NF(Ti)=F(S). Then, {xn} converges strongly to Q(0), where Q is a sunny nonexpansive retraction of E onto , which is a solution of the variational inequality (24).

Proof.

By Lemmas 7 and 8 we have that S is nonexpansive. In addition, by Lemma 9 we get that QC[I-γA] is nonexpansive. Let p and, let yn:=βnxn+(1-βn)QC[I-γA]zn. Then from (25), Lemmas 8 and 9 we have that (26)zn-pcnxn-p+(1-cn)Sxn-pβnxn-p+(1-βn)xn-pxn-p,(27)yn-pβnxn-p+(1-βn)QC[I-γA]zn-QC[p-γAp]βnxn-p+(1-βn)zn-pβnxn-p+(1-βn)xn-pxn-p.

Thus, from (25) and (27), we get that (28)xn+1-p=QC[(1-αn)yn]-QCp(1-αn)yn-p+αnp(1-αn)xn-p+αnp.

Therefore, by induction, (29)xn+1-pmax{x0-p,p},n0, which implies that {xn} and hence {yn}, {zn}, and {Axn} are bounded. Furthermore, from (25), we obtain that (30)zn+1-zn=cn+1xn+1+(1-cn+1)Sxn+1-(cnxn+(1-cn)Sxn)=(cn+1xn+1-cnxn)+(1-cn+1)Sxn+1-(1-cn)Sxn|cn+1-cn|xn+1+cnxn+1-xn+|cn+1-cn|Sxn+1+(1-cn)xn+1-xnxn+1-xn+|cn+1-cn|[xn+1+Sxn+1],(31)yn+1-yn=βn+1xn+1+(1-βn+1)QC[zn+1-γAzn+1]j-(βnxn+(1-βn)QC[zn-γAzn])=(βn+1xn+1-βnxn)+(1-βn+1)j×QC[zn+1-γAzn+1]-(1-βn)QC[zn-γAzn]|βn+1-βn|xn+1+βnxn+1-xn+|βn+1-βn|QC[zn+1-γAzn+1]+(1-βn)×zn+1-znβnxn+1-xn+|βn+1-βn|[xn+1+QC[zn+1-γAzn+1]]+(1-βn)zn+1-znβnxn+1-xn+|βn+1-βn|[xn+1+QC[zn+1-γAzn+1]]+(1-βn)xn+1-xn+|cn+1-cn|[xn+1+Sxn+1]xn+1-xn+|βn+1-βn|×[xn+1+QC[zn+1-γAzn+1]]+|cn+1-cn|[xn+1+Sxn+1]. And, hence, from (25) and (31), we have that (32)xn+1-xn=QC[(1-αn)yn]-QC[(1-αn-1)yn-1](1-αn)yn-(1-αn-1)yn-1(1-αn)yn-(1-αn-1)yn+(1-αn-1)yn-(1-αn-1)yn-1|αn-αn-1|yn+(1-αn-1)yn-1-yn(1-αn-1)xn-xn-1+|αn-αn-1|M+|βn-βn-1|M+|cn-cn-1|M, for some M>0. Thus, using the properties of {αn}, {βn}, {cn}, (32), and Lemma 2, we obtain that xn+1-xn0, as n, which implies from (31) that yn+1-yn0, as n. Again from (25), we have that xn+1-yn=QC[(1-αn)yn]-QCyn=αnyn0, as n. Consequently, (33)xn-yn0,asn.

Now, we prove that {xn} converges strongly to the point x*=Q(0). Let tn=QC[I-γA]zn, and let dn=(1-αn)yn. Then, since αn0, we have that (34)dn-yn=αnyn0,asn    . Furthermore, from (25), Lemma 5, and Lemma 10, we get that (35)xn+1-x*2=QC[(1-αn)yn]-QCx*2αn(-x*)+(1-αn)(yn-x*)2(1-αn)yn-x*2-2αnx*,j(dn-x*)(1-αn)[βnxn-x*2+(1-βn)tn-x*2-βn(1-βn)g(tn-xn)βnxn-x*2]-2αnx*,j(dn-x*)(1-αn)βnxn-x*2+(1-αn)(1-βn)zn-x*2-βn(1-βn)(1-αn)g(tn-xn)-2αnx*,j(dn-x*)(1-αn)βnxn-x*2+(1-αn)(1-βn)×[cnxn-p2+(1-cn)Sxn-p2j-cn(1-cn)g(Sxn-xn)xn-p2]-βn(1-βn)(1-αn)g(tn-xn)-2αnx*,j(dn-x*)(1-αn)βnxn-x*2+(1-αn)(1-βn)×[cnxn-p2+(1-cn)Sxn-p2d-cn(1-cn)g(Sxn-xn)xn-p2]-βn(1-βn)(1-αn)g(tn-xn)-2αnx*,j(dn-x*), which implies that (36)xn+1-x*2(1-αn)xn-x*2-cn(1-cn)(1-αn)(1-βn)×g(Sxn-xn)-βn(1-βn)(1-αn)g(tn-xn)-2αnx*,j(dn-x*)(37)(1-αn)xn-x*2-2αnx*,j(dn-x*).

Now, following the method of proof of Lemma 3.2 of Maingé , we consider two cases.

Case 1. Suppose that there exists n0 such that {xn-x*} is decreasing for all nn0. Then, we get that {xn-x*} is convergent. Thus, from (36) and the fact that αn0, as n, we have that (38)g(Sxn-xn)0,g(tn-xn)0,asn, which implies that (39)Sxn-xn0,tn-xn=QC[I-γA]xn-xn0,asn.

In addition, since {dn} is bounded subset of a reflexive space E, we can choose a subsequence {dni} of {dn} such that dniz and limsupnx*,j(dn-x*)=limix*,j(dni-x*). This implies from (34) and (33) thatxniz. Then, from (39) and Lemma 4, we have that zF(S)=i=1NF(Ti). Moreover, from (39) and Lemma 4, we have that zF(QC[I-γA]), and by Lemma 3, we get zVI(C,A), and hence z. Therefore, using the fact that E has a weakly sequentially continuous duality mapping and Lemma 1, we immediately obtain that (40)limsupnx*,j(dn-x*)=limix*,j(dni-x*)=x*,j(z-x*)0. Then, it follows from (37), (40), and Lemma 2 that xn-x*0, as n. Consequently, xnx*=Q0.

Case 2. Suppose that there exists a subsequence {ni} of {n} such that (41)xni-x*<xni+1-x*, for all i. Then, by Lemma 11, there exists a nondecreasing sequence {mk} such that mk, and (42)xmk-x*xmk+1-x*,xk-x*xmk+1-x*, for all k. Now, from (36) and the fact that αn0, we get that xmk-Sxmk0 and QC[I-γA]xmk-xmk0, as k. Thus, like in Case 1, we obtain that (43)limsupkx*,j(dmk-x*)0. Moreover, from (37), we have that (44)xmk+1-x*2(1-αmk)xmk-x*2-2αmkx*,j(dmk-x*), which implies from (42) and (44) that (45)αmkxmk-x*2xmk-x*2-xmk+1-x*2-2αmkx*,j(dmk-x*)-2αmkx*,j(dmk-x*). Now, since αmk>0, we obtain that (46)xmk-x*2-2x*,j(dmk-x*), and using (43), we get that xmk-x*0. This together with (44) implies that xmk+1-x*0, as k. But xk-x*xmk+1-x*, for all k; thus, we obtain that xkx*. Therefore, from both cases, we can conclude that {xn} converges strongly to x*=P(0), which is a solution of the variational inequality (24), and the proof is complete.

If in Theorem 12, we consider that N=1, we get the following corollary.

Corollary 13.

Let C be a nonempty closed convex subset of a real uniformly convex and q-uniformly smooth Banach space E possessing weakly sequentially continuous duality mapping. Let T:CC be a λ-strictly pseudocontractive mapping, and let A:CE be an η-inverse strongly accretive mapping. Let {xn} be a sequence defined by (25), where S:=[(1-μ)I+μT]. Assume that F:=F(T)VI(C,A). Then, {xn} converges strongly to QF(0) which is a solution of the variational inequality (47)Findx*F(T)suchthatAx*,J(x-x*)  0,xF(T).

If in Theorem 12, we assume that Ti, for i=1,2,,N, are nonexpansive, we get the following corollary.

Corollary 14.

Let C be a nonempty closed convex subset of a real uniformly convex and q-uniformly smooth Banach space E possessing weakly sequentially continuous duality mapping. Let Ti:CC, for i=1,2,,N, be nonexpansive mappings, and let A:CE be an η-inverse strongly accretive mapping. For x0C, let the sequence {xn} be generated iteratively by (48)zn=cnxn+(1-cn)Txn,xn+1=QC[(1-αn)(βnxn+(1-βn)QC[I-γA]zn)], where T:=θ0I+θ1T1++θNTN for {θi}i=1N, {αn}, {βn}, γ are as in (24). Assume that :=FVI(C,A), where F:=i=1NF(Ti)=F(T). Then, {xn} converges strongly to Q(0), which is a solution of the variational inequality problem (24).

Proof.

Lemma 6 and the method of proof of Theorem 12 provide the required assertion.

If in Corollary 14, we consider that N=1, we get the following corollary.

Corollary 15.

Let C be a nonempty closed convex subset of a real uniformly convex and q-uniformly smooth Banach space E possessing weakly sequentially continuous duality mapping. Let T:CC be a nonexpansive mapping, and let A:CE be an η-inverse strongly accretive mapping. For x0C, let the sequence {xn} be generated iteratively by (49)zn=cnxn+(1-cn)Txn,xn+1=QC[(1-αn)(βnxn+(1-βn)QC[I-γA]zn)]. Assume that F:=F(T)VI(C,A). Then, {xn} converges strongly to QF(0), which is a solution of the variational inequality problem (50)Findx*F(T)suchthatAx*,J(x-x*)  0,xF(T).

If in Corollary 14, we assume that T=T1=T2==TN=I, we obtain the following corollary.

Corollary 16.

Let C be a nonempty closed convex subset of a real uniformly convex and q-uniformly smooth Banach space E possessing weakly sequentially continuous duality mapping. Let A:CE be an η-inverse strongly accretive mapping. For x0C, let the sequence {xn} be generated iteratively by (51)xn+1=QC[(1-αn)(βnxn+(1-βn)QC[I-γA]xn)]. Assume that VI(C,A). Then, {xn} converges strongly to QVI(C,A)(0), where QVI(C,A) is a sunny nonexpansive retraction of E onto VI(C,A).

If in Theorem 12, we assume that A is an α-strongly accretive and L-Lipschitzian continuous mapping, we obtain the following corollary.

Corollary 17.

Let C be a nonempty closed convex subset of a real uniformly convex and q-uniformly smooth Banach space E possessing weakly sequentially continuous duality mapping. Let Ti:CC, for i=1,,N, be λi-strictly pseudocontractive mappings, and Let A:CE be an α-strongly accretive and L-Lipschitzian continuous mapping. Let {xn} be a sequence defined by (25) for η=α/L2. Assume that :=FVI(C,A), where F:=i=1NF(Ti)=F(S). Then, {xn} converges strongly to Q(0), which is a solution of the variational inequality problem (52)Findx*i=1NF(Ti)suchthatAx*,J(x-x*)0,xi=1NF(Ti).

Proof.

We note that if A is an α-strongly accretive and L-Lipschitzian continuous mapping of C into E, then we have that (53)Ax-Ay,j(x-y)αx-y2αL2Ax-Ay2,x,yC, and hence, A is an η-inverse strongly accretive mapping with η=α/L2. Thus, the conclusion follows from Theorem 12.

If E=H, a real Hilbert space, then E is a uniformly convex and q-uniformly smooth Banach space E for 1<q< possessing weakly sequentially continuous duality mapping. In this case, we have that QC=PC, projection mapping from H onto C. Thus, we have the following corollary.

Corollary 18.

Let C be a nonempty closed convex subset of a real Hilbert space H. Let Ti:CC, for i=1,,N, be λi-strictly pseudocontractive mappings, and let A:CE be an η-inverse strongly accretive mapping. For x0C, let the sequence {xn} be generated iteratively by (54)zn=cnxn+(1-cn)Sxn,xn+1=PC[(1-αn)(βnxn+(1-βn)PC[I-γA]zn)], where S:=[(1-μ)I+μT], for T:=θ1T1+θ2T2++θnTN, such that θ1+θ2++θN=1, 0<μ<min{1,2λ}, for λ:=min{λi:i=1,2,,N}, and 0<γ<2η. Assume that :=FVI(C,A), where F=i=1NF(Ti)=F(S). Then, {xn} converges strongly to P(0), which is a solution of the variational inequality (55)Findx*i=1NF(Ti)suchthatAx*,x-x*0,xi=1NF(Ti).

Remark 19.

Theorem 12 complements Theorem 3.2 of Yao et al.  in more general Banach spaces for η-inverse strongly accretive mappings. Moreover, Theorem 12 improves Theorem 3.1 of Aoyama et al.  and Theorem 3.7 of Saejung et al.  in the sense that our convergence is strong in the set of common fixed points of finite family of strictly pseudocontractive mappings.

Agarwal R. P. ORegan D. Sahu D. R. Fixed Point Theory for Lipschitzian-Type Mappings with Applications 2000 New York, NY, USA Springer Stampacchia G. Formes bilinéaires coercitives sur les ensembles convexes Comptes Rendus de l'Académie des Sciences 1964 258 4413 4416 MR0166591 ZBL0124.06401 Iiduka H. Takahashi W. Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings Nonlinear Analysis. Theory, Methods & Applications 2005 61 3 341 350 10.1016/j.na.2003.07.023 MR2123081 ZBL1093.47058 Korpelevič G. M. An extragradient method for finding saddle points and for other problems Èkonomika i Matematicheskie Metody 1976 12 4 747 756 MR0451121 Maingé P.-E. Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization Set-Valued Analysis 2008 16 7-8 899 912 10.1007/s11228-008-0102-z MR2466027 ZBL1156.90426 Noor M. A. A class of new iterative methods for general mixed variational inequalities Mathematical and Computer Modelling 2000 31 13 11 19 10.1016/S0895-7177(00)00108-4 MR1773134 ZBL0953.49016 Yamada I. The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings Inherently parallel algorithms in feasibility and optimization and their applications 2001 8 Amsterdam, The Netherlands North-Holland 473 504 10.1016/S1570-579X(01)80028-8 MR1853237 ZBL1013.49005 Yao Y. Liou Y.-C. Li C.-L. Lin H.-T. Extended extragradient methods for generalized variational inequalities Journal of Applied Mathematics 2012 2012 14 237083 10.1155/2012/237083 MR2852852 ZBL1235.49029 Yao Y. Xu H.-K. Iterative methods for finding minimum-norm fixed points of nonexpansive mappings with applications Optimization 2011 60 6 645 658 10.1080/02331930903582140 MR2826133 Zegeye H. Ofoedu E. U. Shahzad N. Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings Applied Mathematics and Computation 2010 216 12 3439 3449 10.1016/j.amc.2010.02.054 MR2661700 ZBL1198.65100 Zegeye H. Shahzad N. A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems Nonlinear Analysis. Theory, Methods & Applications 2011 74 1 263 272 10.1016/j.na.2010.08.040 MR2734995 ZBL1250.47078 Zegeye H. Shahzad N. Strong convergence theorems for a common zero of a countably infinite family of α-inverse strongly accretive mappings Nonlinear Analysis. Theory, Methods & Applications 2009 71 1-2 531 538 10.1016/j.na.2008.10.091 MR2518059 Aoyama K. Iiduka H. Takahashi W. Weak convergence of an iterative sequence for accretive operators in Banach spaces Fixed Point Theory and Applications 2006 2006 35390 10.1155/FPTA/2006/35390 MR2235489 ZBL1128.47056 Browder F. E. Petryshyn W. V. Construction of fixed points of nonlinear mappings in Hilbert space Journal of Mathematical Analysis and Applications 1967 20 197 228 MR0217658 10.1016/0022-247X(67)90085-6 ZBL0153.45701 Xu H. K. Inequalities in Banach spaces with applications Nonlinear Analysis. Theory, Methods & Applications 1991 16 12 1127 1138 10.1016/0362-546X(91)90200-K MR1111623 ZBL0757.46033 Alber Y. I. Iusem A. N. Solodov M. V. Minimization of nonsmooth convex functionals in Banach spaces Journal of Convex Analysis 1997 4 2 235 254 MR1613463 ZBL0895.90150 Browder F. E. Nonlinear Operators and nonlinear equations of evolution in Banach spaces Nonlinear Functional Analysis 1976 Rhode Island, New England American Mathematical scociety 1 308 MR0405188 ZBL0327.47022 Xu H.-K. Iterative algorithms for nonlinear operators Journal of the London Mathematical Society 2002 66 1 240 256 10.1112/S0024610702003332 MR1911872 ZBL1013.47032 Morales C. H. Jung J. S. Convergence of paths for pseudocontractive mappings in Banach spaces Proceedings of the American Mathematical Society 2000 128 11 3411 3419 10.1090/S0002-9939-00-05573-8 MR1707528 Chidume C. E. Zegeye H. Shahzad N. Convergence theorems for a common fixed point of a finite family of nonself nonexpansive mappings Fixed Point Theory and Applications 2005 2 233 241 MR2199943 ZBL1106.47054 Zhang Y. Guo Y. Weak convergence theorems of three iterative methods for strictly pseudocontractive mappings of Browder-Petryshyn type Fixed Point Theory and Applications 2008 2008 13 672301 10.1155/2008/672301 Zhang H. Su Y. Strong convergence theorems for strict pseudo-contractions in q-uniformly smooth Banach spaces Nonlinear Analysis. Theory, Methods & Applications 2009 70 9 3236 3242 10.1016/j.na.2008.04.030 MR2503069 Saejung S. Wongchan K. Yotkaew P. Another weak convergence theorems for accretive mappings in Banach spaces Fixed Point Theory and Applications 2011 article 26 2011, http://www.fixedpointtheoryandapplications.com/content/2011/1/26 MR2831138