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*〉=∥x∥q,∥x*∥=∥x∥q-1},
where 〈·,·〉 denotes the duality pairing. In particular, J=J2 is called the normalized duality map. It is well known (see e.g., [1]) that Jq is single valued if E is smooth and that
(2)Jq(x)=∥x∥q-2J(x),x≠0.
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)〉≥α∥x-y∥q.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)〉≥η∥Ax-Ay∥q,for everyx,y∈D(A).
Let C be a nonempty, closed, and convex subset of E and, let A:C→E be a nonlinear mapping. The variational inequality problem is to
(5)findu∈Csuchthat〈Au,j(v-u)〉≥0,∀v∈C,
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)findu∈Csuch that〈Au,v-u〉≥0,∀v∈C,
which was introduced and studied by Stampacchia [2].

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, [3–12]).

In 1976, Korpelevič [4] introduced the following well-known extragradient method:
(7)yn=PC(xn-γAxn),xn+1=PC(xn-γAyn),n≥0,
where PC is the metric projection from ℝn onto its subset C, for some γ>0, and A:C→ℝn is an accretive operator. He proved that the sequence {xn} converges to a solution of the variational inequality (6).

Furthermore, Noor [6] proved that the iterative scheme, given by
(8)yn=PC(xn-γAxn),xn+1=PC(yn-γAyn),n≥0,
where A:C⊆ℝn→ℝn 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. [13] 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],n≥0,
where QC is a sunny nonexpansive retraction from E onto a closed and convex C, A:C→E 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 z∈V(C,A).

Recently, Yao et al. [8] 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),n≥0,
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:C→E is called λ-strictly pseudocontractive of Browder-Petryshyn type [14] if for all x,y∈D(T) there exist λ>0 and jq(x-y)∈Jq(x-y) such that
(11)〈Tx-Ty,jq(x-y)〉≤∥x-y∥q-λ∥x-y-(Tx-Ty)∥q.T is called Lipschitz if there exists L≥0 such that
(12)∥Tx-Ty∥≤L∥x-y∥∀x,y∈D(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-Ty∥2≤∥x-y∥2+k∥x-y-(Tx-Ty)∥2,k=(1-2λ),
and we can assume also that k≥0, so that k∈[0,1). A point x∈C 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)={x∈C:Tx=x}.

In 2001, Yamada [7] 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),n≥0,
converges strongly to the unique solution of the variational inequality
(15)Findx*∈F(T)such that〈Ax*,x-x*〉≥0,∀x∈F(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. [8] and Yamada [7], 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. [8], Aoyama et al. [13], 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 [15], the following geometric inequality holds:
(16)∥x+y∥q≤∥x∥q+q〈y,jq(x)〉+cq∥y∥q,
for all x,y∈E and some real constant cq>0.

It is well known (see e.g., [16]) that
(17)Lp(lp)orWmpis{p-uniformly smoothif 1<p<2,2-uniformly smoothif p≥2.
The Banach space E is said to be uniformly convex if, given ε>0, there exists δ>0, such that, for all x,y∈E with ∥x∥≤1, ∥y∥≤1 and ∥x-y∥≥ε, ∥(1/2)(x+y)∥≤1-δ. It is well known that Lp, ℓp, and Sobolev spaces Wmp, (1<p<∞) are uniformly convex.

Let C⊆E 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 x∈E and t≥0. 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 z∈R(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.

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

Qis sunny nonexpansive,

〈x-Q(x),J(y-Q(x))〉≤0 for all x∈E and y∈K.

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,n≥n0,
where {αn}⊂(0,1) and {δn}⊂ℝ satisfying the following conditions: limn→∞αn=0,∑n=1∞αn=∞, and limsupn→∞δn≤0. Then, limn→∞an=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,∀x∈C}.

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 xn→x weakly and xn-Txn→0 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,y∈E, the following inequality holds:
(20)∥x+y∥2≤∥x∥2+2〈y,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:K→E,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:C→E, 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:C→E 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:C→E 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-Aγy∥q=∥(x-y)-γ(Ax-Ay)∥q≤∥x-y∥q-qγ〈Ax-Ay,jq(x-y)〉+γqcq∥Ax-Ay∥q≤∥x-y∥q-qγη∥Ax-Ay∥2+γqcq∥Ax-Ay∥2≤∥x-y∥q-γ(qη-γq-1cq)∥Ax-Ay∥q,≤∥x-y∥q.
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+⋯+αkxk∥2≤∑i=0kαi∥xi∥2-αsαtg(∥xs-xt∥),
for xi∈BR(0):={x∈E:∥x∥≤R}, 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∈ℕ. Then, there exists a nondecreasing sequence {mk}⊂ℕ such that mk→∞, and the following properties are satisfied by all (sufficiently large) numbers k∈ℕ:
(23)amk≤amk+1,ak≤amk+1.
In fact, mk=max{j≤k: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:C→C, for i=1,…,N, be a λi-strictly pseudocontractive mappings, and let A:C→E be an η-inverse strongly accretive mapping. Then, in what follows, we will study the variational inequality
(24)Findx*∈∩i=1NF(Ti)such that〈Ax*,J(x-x*)〉≥0,∀x∈∩i=1NF(Ti),
and the following iteration process:
(25)x0∈C,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) limn→∞αn=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:C→C, for i=1,…,N, be λi-strictly pseudocontractive mappings, and let A:C→E be η-inverse strongly accretive mapping. Let {xn} be a sequence defined by (25). Assume that ℱ:=F∩VI(C,A)≠∅, where F=∩i=1NF(Ti)=F(S). Then, {xn} converges strongly to Qℱ(0), where Qℱ is a sunny nonexpansive retraction of E onto ℱ, 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∈ℱ and, let yn:=βnxn+(1-βn)QC[I-γA]zn. Then from (25), Lemmas 8 and 9 we have that
(26)∥zn-p∥≤cn∥xn-p∥+(1-cn)∥Sxn-p∥≤βn∥xn-p∥+(1-βn)∥xn-p∥≤∥xn-p∥,(27)∥yn-p∥≤βn∥xn-p∥+(1-βn)∥QC[I-γA]zn-QC[p-γAp]∥≤βn∥xn-p∥+(1-βn)∥zn-p∥≤βn∥xn-p∥+(1-βn)∥xn-p∥≤∥xn-p∥.

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

Therefore, by induction,
(29)∥xn+1-p∥≤max{∥x0-p∥,∥p∥},∀n≥0,
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∥+cn∥xn+1-xn∥+|cn+1-cn|∥Sxn+1∥+(1-cn)∥xn+1-xn∥≤∥xn+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∥+βn∥xn+1-xn∥+|βn+1-βn|∥QC[zn+1-γAzn+1]∥+(1-βn)×∥zn+1-zn∥≤βn∥xn+1-xn∥+|βn+1-βn|[∥xn+1∥+∥QC[zn+1-γAzn+1]∥]+(1-βn)∥zn+1-zn∥≤βn∥xn+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-xn∥→0, as n→∞, which implies from (31) that ∥yn+1-yn∥→0, as n→∞. Again from (25), we have that ∥xn+1-yn∥=∥QC[(1-αn)yn]-QCyn∥=αn∥yn∥→0, as n→∞. Consequently,
(33)∥xn-yn∥⟶0,asn⟶∞.

Now, we prove that {xn} converges strongly to the point x*=Qℱ(0). Let tn=QC[I-γA]zn, and let dn=(1-αn)yn. Then, since αn→0, we have that
(34)∥dn-yn∥=αn∥yn∥⟶0,asn⟶∞.
Furthermore, from (25), Lemma 5, and Lemma 10, we get that
(35)∥xn+1-x*∥2=∥QC[(1-αn)yn]-QCx*∥2≤∥αn(-x*)+(1-αn)(yn-x*)∥2≤(1-αn)∥yn-x*∥2-2αn〈x*,j(dn-x*)〉≤(1-αn)[βn∥xn-x*∥2+(1-βn)∥tn-x*∥2-βn(1-βn)g(∥tn-xn∥)βn∥xn-x*∥2]-2αn〈x*,j(dn-x*)〉≤(1-αn)βn∥xn-x*∥2+(1-αn)(1-βn)∥zn-x*∥2-βn(1-βn)(1-αn)g(∥tn-xn∥)-2αn〈x*,j(dn-x*)〉≤(1-αn)βn∥xn-x*∥2+(1-αn)(1-βn)×[cn∥xn-p∥2+(1-cn)∥Sxn-p∥2j-cn(1-cn)g(∥Sxn-xn∥)∥xn-p∥2]-βn(1-βn)(1-αn)g(∥tn-xn∥)-2αn〈x*,j(dn-x*)〉≤(1-αn)βn∥xn-x*∥2+(1-αn)(1-βn)×[cn∥xn-p∥2+(1-cn)∥Sxn-p∥2d-cn(1-cn)g(∥Sxn-xn∥)∥xn-p∥2]-βn(1-βn)(1-αn)g(∥tn-xn∥)-2αn〈x*,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∥)-2αn〈x*,j(dn-x*)〉(37)≤(1-αn)∥xn-x*∥2-2αn〈x*,j(dn-x*)〉.

Now, following the method of proof of Lemma 3.2 of Maingé [5], we consider two cases.

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

In addition, since {dn} is bounded subset of a reflexive space E, we can choose a subsequence {dni} of {dn} such that dni⇀z and limsupn→∞〈x*,j(dn-x*)〉=limi→∞〈x*,j(dni-x*)〉. This implies from (34) and (33) thatxni⇀z. Then, from (39) and Lemma 4, we have that z∈F(S)=∩i=1NF(Ti). Moreover, from (39) and Lemma 4, we have that z∈F(QC[I-γA]), and by Lemma 3, we get z∈VI(C,A), and hence z∈ℱ. Therefore, using the fact that E has a weakly sequentially continuous duality mapping and Lemma 1, we immediately obtain that
(40)limsupn→∞〈x*,j(dn-x*)〉=limi→∞〈x*,j(dni-x*)〉=〈x*,j(z-x*)〉≥0.
Then, it follows from (37), (40), and Lemma 2 that ∥xn-x*∥→0, as n→∞. Consequently, xn→x*=Qℱ0.

Case 2. Suppose that there exists a subsequence {ni} of {n} such that
(41)∥xni-x*∥<∥xni+1-x*∥,
for all i∈ℕ. Then, by Lemma 11, there exists a nondecreasing sequence {mk}⊂ℕ such that mk→∞, and
(42)∥xmk-x*∥≤∥xmk+1-x*∥,∥xk-x*∥≤∥xmk+1-x*∥,
for all k∈ℕ. Now, from (36) and the fact that αn→0, we get that xmk-Sxmk→0 and ∥QC[I-γA]xmk-xmk∥→0, as k→∞. Thus, like in Case 1, we obtain that
(43)limsupk→∞〈x*,j(dmk-x*)〉≥0.
Moreover, from (37), we have that
(44)∥xmk+1-x*∥2≤(1-αmk)∥xmk-x*∥2-2αmk〈x*,j(dmk-x*)〉,
which implies from (42) and (44) that
(45)αmk∥xmk-x*∥2≤∥xmk-x*∥2-∥xmk+1-x*∥2-2αmk〈x*,j(dmk-x*)〉≤-2αmk〈x*,j(dmk-x*)〉.
Now, since αmk>0, we obtain that
(46)∥xmk-x*∥2≤-2〈x*,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∈ℕ; thus, we obtain that xk→x*. Therefore, from both cases, we can conclude that {xn} converges strongly to x*=Pℱ(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:C→C be a λ-strictly pseudocontractive mapping, and let A:C→E 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)suchthat〈Ax*,J(x-x*)〉≥0,∀x∈F(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:C→C, for i=1,2,…,N, be nonexpansive mappings, and let A:C→E be an η-inverse strongly accretive mapping. For x0∈C, 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 ℱ:=F∩VI(C,A)≠∅, where F:=∩i=1NF(Ti)=F(T). Then, {xn} converges strongly to Qℱ(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:C→C be a nonexpansive mapping, and let A:C→E be an η-inverse strongly accretive mapping. For x0∈C, 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)suchthat〈Ax*,J(x-x*)〉≥0,∀x∈F(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:C→E be an η-inverse strongly accretive mapping. For x0∈C, 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:C→C, for i=1,…,N, be λi-strictly pseudocontractive mappings, and Let A:C→E be an α-strongly accretive and L-Lipschitzian continuous mapping. Let {xn} be a sequence defined by (25) for η=α/L2. Assume that ℱ:=F∩VI(C,A)≠∅, where F:=∩i=1NF(Ti)=F(S). Then, {xn} converges strongly to Qℱ(0), which is a solution of the variational inequality problem
(52)Findx*∈∩i=1NF(Ti)suchthat〈Ax*,J(x-x*)〉≥0,∀x∈∩i=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)〉≥α∥x-y∥2≥αL2∥Ax-Ay∥2,∀x,y∈C,
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:C→C, for i=1,…,N, be λi-strictly pseudocontractive mappings, and let A:C→E be an η-inverse strongly accretive mapping. For x0∈C, 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 ℱ:=F∩VI(C,A)≠∅, where F=∩i=1NF(Ti)=F(S). Then, {xn} converges strongly to Pℱ(0), which is a solution of the variational inequality
(55)Findx*∈∩i=1NF(Ti)suchthat〈Ax*,x-x*〉≥0,∀x∈∩i=1NF(Ti).

Remark 19.

Theorem 12 complements Theorem 3.2 of Yao et al. [8] in more general Banach spaces for η-inverse strongly accretive mappings. Moreover, Theorem 12 improves Theorem 3.1 of Aoyama et al. [13] and Theorem 3.7 of Saejung et al. [23] in the sense that our convergence is strong in the set of common fixed points of finite family of strictly pseudocontractive mappings.

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