The purpose of this paper is to introduce a new modified relaxed extragradient method and
study for finding some common solutions for a general system of variational inequalities
with inversestrongly monotone mappings and nonexpansive mappings in the framework of
real Banach spaces. By using the demiclosedness principle, it is proved that the iterative
sequence defined by the relaxed extragradient method converges strongly to a common
solution for the system of variational inequalities and nonexpansive mappings under quite
mild conditions.

1. Introduction

Let H be a real Hilbert space with inner product 〈·,·〉 and norm ||·||, and C be a nonempty closed convex subset of H. Let PC be the projection of H onto C, it is known that projection operator PC is nonexpansive and satisfies the following:
(1.1)〈x-y,PCx-PCy〉≥‖PCx-PCy‖2,∀x,y∈H.
Moreover, PCx is characterized by the properties PCx∈C and 〈x-PCx,PCx-y〉≥0 for all y∈C.

Let A:C→H be a mapping. Recall that the classical variational inequality, denoted by VI(C,A), is to find u∈V such that
(1.2)〈Au,v-u〉≥0,∀v∈C.

One can see that the variational inequality (1.2) is equivalent to a fixed point problem. A element x*∈C is a solution of the variational inequality (1.2) if and only if x*∈C is a fixed point of the mapping PC(I-λA), where I is the identity mapping and λ>0 is a constant.

Variational inequalities introduced by Stampacchia [1] in the early sixties have had a great impact and influence in the development of almost all branches of pure and applied sciences and have witnessed an explosive growth in theoretical advances, algorithmic development, and so forth; see, for example, [1–18] and the references therein.

For a monotone mapping A:C→H, Noor [2] studied the following problem of finding (x*;y*)∈C×C such that:
(1.3)〈λAy*+x*-y*,x-x*〉≥0,∀x∈C,〈μAx*+y*-x*,x-y*〉≥0,∀x∈C,
where λ,μ>0 are constants. If we add up the requirement that x*=y*, then the problem (1.3) is reduced to the classical variational inequality (1.2). The problem of finding solutions of (1.3) by using iterative methods has been studied by many authors; see [2–9] and the references therein.

Recently, some authors also studied the problem of finding a common element of the fixed point set of nonexpansive mappings and the solution set of variational inequalities for α-inversestrongly monotone mappings in the framework of real Hilbert spaces [10] and Banach spaces [11].

On the other hands, Ceng et al. [12] introduce the following general system of variational inequalities involving two different operators. For two given operators, consider the problem finding x*,y*∈C such that
(1.4)〈λAy*+x*-y*,x-x*〉≥0,∀x∈C,〈μBx*+y*-x*,x-y*〉≥0,∀x∈C,
where λ,μ>0 are constants. To illustrate the applications of this system, we can refer to an example of related nonlinear optimization problem put forward by Zhu and Marcotte [13]. Very recently, Yao et al. [14] extend the system of variational inequality problems (1.4) to Banach spaces.

In the present paper, motivated and inspired by the methods of Ceng et al. [12], Iiduka and Takahashi [10], Qin et al. [11], and Yao et al. [14], we consider the following general system of variational inequalities in Banach spaces.

Let C be a nonempty closed convex subset of a real smooth Banach space. Let A,B:C→E be α-inversestrongly accretive mapping and β-inverse-strongly accretive mapping. Find (x*,y*)∈C×C such that
(1.5)〈λAy*+x*-y*,j(x-x*)〉≥0,∀x∈C,〈μBx*+y*-x*,j(x-y*)〉≥0,∀x∈C,
where λ,μ>0 are constants, and j is the normalized duality mapping. For more details of j, one may see Li [15]. In a real Hilbert space, j=I is the identity mapping, and the system (1.5) is reduced to (1.4). If we add up the requirement that A=B, then the problem (1.4) is reduced to the generalized variational inequality (1.3), in particular.

And we consider the problem of finding a common element of the fixed point set of nonexpansive mappings and the solution set of the general system of variational inequalities for α-inversestrongly monotone mappings in the framework of real Banach spaces. By using the demiclosedness principle, we prove that under quite mild conditions the iterative sequence defined by the relaxed extragradient method converges strongly to a common solution of this system of variational inequalities and nonexpansive mappings. Our results improve and extend the corresponding results announced by other authors, such as [2–4, 6, 8, 10–12, 14].

2. Preliminaries

Let C be a nonempty closed convex subset of a Banach space of E. Let E* be the dual space of E, and let 〈·,·〉 denote the pairing between E and E*. For q>1, the generalized duality mapping Jq:E→2E* is defined by
(2.1)Jq(x)={f∈E*:〈x,f〉=‖x‖q,‖f‖=‖x‖q},
for all x∈E. In particular, J=J2 is called the normalized duality mapping. It is known that Jq(x)=||x||q-2J(x) for all x∈E. If E is a Hilbert space, then J=I is the identity mapping. Further, we have the following properties of the generalized duality mapping Jq:

Jq(x)=||x||q-2J2(x) for all x∈E with x≠0,

Jq(tx)=tq-1Jq(x) for all x∈E and t∈[0,∞),

Jq(-x)=-Jq(x) for all x∈E.

Let U={x∈E:||x||=1}. E is said to be uniformly convex if, for any ϵ∈(0,2], there exists δ>0 such that for any x,y∈U. ||x-y||≥ϵ implies ||(x+y)/2||≤(1-δ).

It is known that a uniformly convex Banach space is reflexive and strictly convex, E is said to be Ga^teaux differentiable if the limit
(2.2)Limt→0‖x+ty‖-‖x‖t
exists for each x,y∈U. In this case, E is said to be smooth. The norm of E is said to be uniformly Ga^teaux differentiable if, for each y∈U, the limit (2.2) is attained uniformly for x∈U. The norm of E is said to be Fre^chet differentiable, if, for each x∈U, the limit (2.2) is attained uniformly for y∈U. The norm of E is said to be uniformly Fre^chet differentiable if the limit (2.2) is attained uniformly for x,y∈U. It is well-known that (uniform) Fre^chet differentiable of the norm of E implies (uniform) Ga^teaux differentiability of the norm of E.

The modulus of smoothness of E is defined by
(2.3)ρ(τ)=sup{12(||x+y||+||x-y||)-1:x,y∈E,||x||=1,||y||≤t},
where ρ:[0,∞)→[0,∞) is a function. It is known that a Banach space E is uniformly smooth if and only if lim(n→∞)(ρ(t)/t)=0. Let q be a fixed real number with 1<q≤2. A Banach space E is said to be q uniformly smooth if there exists a fixed constant c>0 such that ρ(t)≤ctq, for all t>0.

Next, we always assume that E is a smooth Banach space. Let C be a nonempty closed convex subsets of E. Recall that an operator A of C into E is said to be accretive if
(2.4)〈Ax-Ay,j(x-y)〉≥0,∀x,y∈C.A is said to be α-inversestrongly accretive if there exists a constant α>0 such that
(2.5)〈Ax-Ay,j(x-y)〉≥α‖Ax-Ay‖2,∀x,y∈C.

Let D be a subset of C and Q be a mapping of C into D. Then Q is said to be sunny if
(2.6)Q(Qx+t(x-Qx))=Qx,
whenever Qx+t(x-Qx)∈C for x∈C and t≥0. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D.

In order to prove the main result, we also need the following lemmas. The following Lemma 2.2 describes characterization of sunny nonexpansive retraction on a smooth Banach space.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B16">16</xref>]).

Let E be a real 2 uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:
(2.7)‖x+y‖2≤‖x‖2+2〈y,jx〉+2‖Ky‖2,∀x,y∈E.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B17">17</xref>]).

Let C be a closed convex subset of a smooth Banach space E, let D be a nonempty subset of C, and let Q be a retraction form C onto D. Then Q is sunny and nonexpansive if and only if
(2.8)〈u-Qu,j(y-Qu)〉≤0,
for all u∈C and y∈D.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B18">18</xref>]).

Let {xn} and {yn} be bounded sequences in a Banach space E and a sequence {βn} in [0,1] with
(2.9)0<liminfn→∞βn≤limsupn→∞βn<1.
Suppose that xn+1=(1-βn)yn+βnxn for all integers n≥0 and
(2.10)limsupn→∞(||yn+1-yn||-||xn+1-xn||)≤0.
Then, lim(n→∞)||yn-xn||=0.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B19">19</xref>]).

Let C be a nonempty closed convex subset of a real uniformly smooth Banach space. Let S1 and S2 be two nonexpansive mappings from C into itself with a common fixed point. Define a mapping S:C→C by
(2.11)Sx=δS1x+(1-δ)S2x,∀x∈C,
where δ is a constant in (0,1). Then S is nonexpansive and F(S)=F(S1)⋂F(S2).

Lemma 2.5 (see [<xref ref-type="bibr" rid="B20">20</xref>]).

Assume that {αn} is a sequence of nonnegative real numbers such that
(2.12)αn+1≤(1-γn)αn+δn,
where {γn} is a sequence in (0,1) and {δn} is a sequence such that

∑n=1∞γn=∞,

lim(n→∞)(δn/γn)≤0,or∑n=1∞|δn|<∞.

Then lim(n→∞)αn=0.
Lemma 2.6.

Let C be a nonempty closed convex subset of a real 2 uniformly smooth Banach space E with the best smooth constant K. Let the mappings A,B:C→E be α-inverse strongly accretive and β-inverse strongly accretive, respectively, and then one has I-λA and I-μB are nonexpansive, where λ∈(0,α/K2), μ∈(0,β/K2).

Proof.

Indeed, for all x,y∈C, from Lemma 2.1, we have
(2.13)‖(I-λA)x-(I-λA)y‖2=‖(x-y)-λ(Ax-Ay)‖2≤‖x-y‖2-2λ〈Ax-Ay,j(x-y)〉+2K2λ2‖Ax-Ay‖2≤‖x-y‖2-2λα‖Ax-Ay‖2+2K2λ2‖Ax-Ay‖2≤‖x-y‖2+2λ(K2λ-α)‖Ax-Ay‖2≤‖x-y‖2.
This shows that I-λA is nonexpansive mapping, so is I-μB.

Lemma 2.7.

Let C be a nonempty closed convex subset of a real 2 uniformly smooth Banach space E. Let QC be the sunny nonexpansive retraction from E onto C. Let A,B:C→E be α-inverse strongly accretive mapping and β-inverse strongly accretive mapping, respectively. Let G:C→C be a mapping defined by
(2.14)G(x)=QC[QC(x-μBx)-λAQC(x-μBx)],∀x∈C.
If λ∈(0,α/K2), μ∈(0,β/K2), then G:C→C is nonexpansive.

Proof.

For all x,y∈C, from Lemma 2.6, we have
(2.15)||G(x)-G(y)||=‖QC[QC(x-μBx)-λAQC(x-μBx)]-QC[QC(y-μBy)-λAQC(y-μBy)]‖≤‖QC(x-μBx)-λAQC(x-μBx)-[QC(y-μBy)-λAQC(y-μBy)]‖=‖(I-λA)QC(I-μB)x-(I-λA)QC(I-μB)y‖≤‖QC(I-μB)x-QC(I-μB)y‖≤‖(I-μB)x-(I-μB)y‖≤‖x-y‖.
Therefore, from (2.15), we obtain immediately that the mapping G is nonexpansive.

Lemma 2.8.

Let C be a nonempty closed convex subset of a real smooth Banach space E. Let QC be the sunny nonexpansive retraction from E onto C. Let A,B:C→E be two possibly nonlinear mappings. For given x*,y*∈C, (x*,y*) is a solution of problem (1.5) if and only if x*=QC(y*-λAy*), where y*=QC(x*-μBx*).

Proof.

We note that we can rewrite (1.5) as
(2.16)〈x*-(y*-λAy*),j(x-x*)〉≥0,∀x∈C,〈y*-(x*-λBx*),j(x-y*)〉≥0,∀x∈C.
From Lemma 2.2, we can deduce that (2.16) is equivalent to
(2.17)x*=QC(y*-λAy*),y*=QC(x*-μBx*).
This completes the proof.

Remark 2.9.

From Lemma 2.8, we note that x*=QC[QC(x*-μBx*)-λAQC(x*-μBx*)], which implies that x* is a fixed point of the mappings G.

3. Main Result

To solve the general system of variation inequality problem (1.5), now we are in a position to state and prove the main result in this paper.

Theorem 3.1.

Let E be a uniformly convex and 2 uniformly smooth Banach space with the best smooth constant K, C a nonempty closed convex subset of E, and QC be the sunny nonexpansive retraction from E onto C. Let A,B:C→E be α-inverse strongly accretive mapping and β-inverse strongly accretive mapping, respectively, and S:C→C a nonexpansive mapping with a fixed point. Assume that F=F(S)⋂F(G)≠∅, where G is defined as Lemma 2.7. Let {xn} be a sequence generated in the following manner:
(3.1)x1=u∈C,yn=QC(xn-μBxn),xn+1=αnu+βnxn+γn[δnSxn+(1-δn)QC(yn-λAyn)],n≥1,
where δn⊂[0,1], λ∈(0,α/K2), μ∈(0,β/K2), and {αn},{βn}, and {γn} are three sequences in (0,1), and the following conditions are satisfied

αn+βn+γn=1,foralln≥1,

lim(n→∞)αn=0,∑n=0∞αn=∞,

0<liminf(n→∞)βn≤limsup(n→∞)βn<1,

lim(n→∞)δn=δ∈(0,1).

Then the sequence {xn} defined by (3.1) converges strongly to x-=QFu, and (x-,y-) is a solution of the problem (1.5), where y-=QC(x--μBx-), QF is a sunny nonexpansive retraction of C onto F.

Proof.

We divide the proof of Theorem 3.1 into six steps.

Step 1. First, we prove that F is closed and convex. We know that F(S) is closed and convex. Next, we show that F(G) is closed and convex. From Lemma 3.1 and 3.2, we can see that I-λA,I-μB,G are nonexpansive. This shows that F=F(S)⋂F(G) is closed and convex.

Step 2. Now we prove that the sequences {xn},{yn},{tn},{Ayn}, and {Bxn} are bounded. Let x*∈F(S)⋂F(G), and from Remark 2.9, we obtain that
(3.2)x*=QC[QC(x*-μBx*)-λAQC(x*-μBx*)].
Putting y*=QC(x*-μBx*), we see that
(3.3)x*=QC(y*-λAy*).
Putting tn=δnSxn+(1-δn)QC(yn-λAyn) for each n≥1, we arrive at
(3.4)‖tn-x*‖=‖δnSxn+(1-δn)QC(yn-λAyn)-x*‖≤δn‖Sxn-x*‖+(1-δn)‖QC(yn-λAyn)-x*‖≤δn‖xn-x*‖+(1-δn)‖QC(yn-λAyn)-QC(y*-λAy*)‖≤δn‖xn-x*‖+(1-δn)‖yn-y*‖=δn‖xn-x*‖+(1-δn)‖QC(xn-μBxn)-QC(x*-μBx*)‖≤δn‖xn-x*‖+(1-δn)‖xn-x*‖=‖xn-x*‖.
Hence, it follows that
(3.5)‖xn+1-x*‖=‖αnu+βnxn+γntn-x*‖≤αn‖u-x*‖+βn‖xn-x*‖+γn‖tn-x*‖≤αn‖u-x*‖+βn‖xn-x*‖+γn‖xn-x*‖≤αn‖u-x*‖+(1-αn)‖xn-x*‖≤max{‖x1-x*‖,‖u-x*‖}=‖u-x*‖.
Therefore, {xn} is bounded. Hence {xn},{yn},{tn},{Ayn}, and {Bxn} are bounded.

On the other hand, we have
(3.6)‖tn+1-tn‖=‖δn+1Sxn+1+(1-δn+1)QC(yn+1-λAyn+1)-(δnSxn)+(1-δn)QC(yn-λAyn)‖=‖δn+1(Sxn+1-Sxn)+(1-δn+1)(QC(yn+1-λAyn+1)-QC(yn-λAyn))+(δn+1-δn)(Sxn-QC(yn-λAyn))‖≤δn+1‖(Sxn+1-Sxn)‖+(1-δn+1)‖QC(yn+1-λAyn+1)-QC(yn-λAyn)‖+|δn+1-δn|⋅‖Sxn-QC(yn-λAyn)‖≤δn+1‖xn+1-xn‖+(1-δn+1)‖yn+1-yn‖+|δn+1-δn|⋅‖Sxn-QC(yn-λAyn)‖=δn+1‖xn+1-xn‖+(1-δn+1)+‖QC(xn+1-λBxn+1)-QC(xn-λBxn)‖+|δn+1-δn|⋅‖Sxn-QC(yn-λAyn)‖≤δn+1‖xn+1-xn‖+(1-δn+1)‖xn+1-xn‖+|δn+1-δn|⋅‖Sxn-QC(yn-λnAyn)‖≤‖xn+1-xn‖+|δn+1-δn|M,
where M is an appropriate constant such that
(3.7)M≥supn≥1‖Sxn-QC(yn-λnAyn)‖.

Step 3. We prove that lim(n→∞)||xn+1-xn||=0.

Setting wn=(x(n+1)-βnxn)/(1-βn) for each n≥1, we see that
(3.8)xn+1=(1-βn)wn+βnxn,∀n≥1.
Now, we compute ||wn+1-wn|| from
(3.9)‖wn+1-wn‖=‖αn+1u+γn+1tn+11-βn+1-αnu+γntn1-βn‖=‖αn+11-βn+1u+1-βn+1-αn+11-βn+1tn+1-αn1-βnu-1-βn-αn1-βntn‖=‖αn+11-βn+1(u-tn+1)+αn1-βn(tn-u)+tn+1-tn‖≤αn+11-βn+1‖u-tn+1‖+αn1-βn‖tn-u‖+‖tn+1-tn‖.
Combining (3.6) and (3.9), we arrive at
(3.10)‖wn+1-wn‖-‖xn+1-xn‖≤αn+11-βn+1‖u-tn+1‖+αn1-βn‖tn-u‖+|δn+1-δn|M1.
It follows from the conditions (1.3), (1.4), and (1.5) that
(3.11)limsupn→∞(‖wn+1-wn‖-‖xn+1-xn‖)≤0.
Hence, by Lemma 2.3, we obtain that lim(n→∞)||wn-xn||=0. Consequently,
(3.12)limn→∞‖xn+1-xn‖=limn→∞(1-βn)‖wn-xn‖=0.
On the other hand, it follows from the algorithm (3.1) that
(3.13)xn+1-xn=αn(u-xn)+γn(tn-xn).
From the condition (1.3) and formula (3.12), we see that
(3.14)limn→∞‖tn-xn‖=0.

Step 4. We prove that lim(n→∞)||xn-Vxn||=0. Define a mapping V:C→C by
(3.15)Vx=δSx+(1-δ)QC(QC(I-μB)-λAQC(I-μB))x,∀x∈C,
where lim(n→∞)δn=δ∈(0,1). From Lemma 2.4, we see that V is a nonexpansive mapping with
(3.16)F(V)=F(S)⋂F(QC(QC(I-μB)-λAQC(I-μB)))=F(S)⋂F(QC(I-λA)QC(I-μB))=F(S)⋂F(G)=F.
On the other hand, we have
(3.17)‖xn-Vxn‖≤‖xn-xn+1‖+‖xn+1-Vxn‖≤‖xn-xn+1‖+αn‖un-Vxn‖+βn‖xn-Vxn‖+γn‖tn-Vxn‖=‖xn-xn+1‖+αn‖un-Vxn‖+βn‖xn-Vxn‖+γn‖δnSxn+(1-δn)QC(yn-λAyn)-δSxn-(1-δ)QC(yn-λAyn)‖≤‖xn-xn+1‖+αn‖un-Vxn‖+βn‖xn-Vxn‖+γn|δn-δ|⋅‖Sxn-QC(yn-λnyn)‖≤‖xn-xn+1‖+αn‖un-Vxn‖+βn‖xn-Vxn‖+γn|δn-δ|M.
This implies that (3.18)(1-βn)‖xn-Vxn‖≤‖xn-xn+1‖+αn‖un-Vxn‖+γn|δn-δ|M.
It follows from the conditions (1.3), (1.4), (1.5), and (3.18) that
(3.19)limn→∞‖xn-Vxn‖=0.

Step 5. Next, we show that limsup(n→∞)〈u-x-,j(xn-x-)〉≤0.

Let zt be the fixed point of the contraction z↦tu+(1-t)Vzt, where t∈(0,1). That is, zt=tu+(1-t)Vzt. It follows that
(3.20)‖zt-xn‖=‖(1-t)(Vzt-xn)+t(u-xn)‖.
On the other hand, for any t∈(0,1), we see that
(3.21)‖zt-xn‖2=〈(1-t)(Vzt-xn)+t(u-xn),j(zt-xn)〉=(1-t)〈Vzt-xn,j(zt-xn)〉+t〈u-xn,j(zt-xn)〉=(1-t)〈Vzt-Vxn,j(zt-xn)〉+(1-t)〈Vxn-xn,j(zt-xn)〉+t〈u-zt,j(zt-xn)〉+t〈zt-xn,j(zt-xn)〉≤(1-t)‖zt-xn‖2+(1-t)‖Vxn-xn‖⋅‖zt-xn‖+t〈u-zt,j(zt-xn)〉+t‖zt-xn‖2≤‖zt-xn‖2+‖Vxn-xn‖⋅‖zt-xn‖+t〈u-zt,j(zt-xn)〉.
It follows that
(3.22)〈zt-u,j(zt-xn)〉≤1t‖Vxn-xn‖⋅‖zt-xn‖,∀t∈(0,1).
In view of (3.19), we see that
(3.23)limsupn→∞〈zt-u,j(zt-xn)〉≤0.
On the other hand, we see that QF(V)u=lim(t→0)zt and F(V)=F. It follows that zt→x-=QFu as t→0. Owing to the fact that j is strong to weak* uniformly continuous on bounded subsets of E, we see that
(3.24)limt→0|〈u-x-,j(xn-x-)〉-〈zt-u,j(zt-xn)〉|≤|〈u-x-,j(xn-x-)〉-〈u-x-,j(xn-zt)〉|+|〈u-x-,j(xn-zt)〉-〈zt-x-,j(zt-xn)〉|≤|〈u-x-,j(xn-x-)-j(xn-zt)〉|+|〈zt-x-,j(xn-zt)〉|≤‖u-x-‖⋅‖j(xn-x-)-j(xn-zt)‖+‖zt-x-‖⋅‖xn-zt‖=0.
Hence, for any ε>0, there exists δ>0 such that forallt∈(0,δ), the following inequality holds:
(3.25)〈u-x-,j(xn-x-)〉≤〈zt-u,j(zt-xn)〉+ε.
Since ε is arbitrary and (3.23), we see that
(3.26)limsupn→∞〈u-x-,j(xn-x-)〉≤0.

Step 6. Finally, we show that xn→x- as n→∞. Observe that
(3.27)‖xn+1-x-‖2=〈αnu+βnxn+γntn-x-,j(xn+1-x-)〉=αn〈u-x-,j(xn+1-x-)〉+βn〈xn-x-,j(xn+1-x-)〉+γn〈tn-x-,j(xn+1-x-)〉≤αn〈u-x-,j(xn+1-x-)〉+βn‖xn-x-‖⋅‖xn+1-x-‖+γn‖tn-x-‖⋅‖(xn+1-x-)‖≤αn〈u-x-,j(xn+1-x-)〉+βn‖xn-x-‖⋅‖xn+1-x-‖+γn‖xn-x-‖⋅‖(xn+1-x-)‖=αn〈u-x-,j(xn+1-x-)〉+(1-αn)‖xn-x-‖⋅‖xn+1-x-‖≤αn〈u-x-,j(xn+1-x-)〉+1-αn2(‖xn-x-‖2+‖xn+1-x-‖2)
which implies that
(3.28)‖xn+1-xn‖2≤(1-αn)‖xn-x-‖2+2αn〈u-x-,j(xn+1-x-)〉.
From the conditions (1.3) and (3.26) and applying Lemma 2.5 to (3.28), we obtain that
(3.29)limn→∞‖xn-x-‖=0.
This completes the proof.

Remark 3.2.

Since Lp for all p≥2 is uniformly convex and 2 uniformly smooth, we see that Theorem 3.1 is applicable to Lp for all p≥2. There are a number of sequences satisfying the restrictions (C1)–(C3), for example, αn=1/(n+1),βn=n/(2n+1),γn=n2/(2n2+3n+1) for each n≥1.

Corollary 3.3.

Let H be a real Hilbert space and C a nonempty closed convex subset of H. Let A,B:C→H be α-inverse strongly monotone mapping and β-inverse strongly monotone mapping, respectively, and S:C→C nonexpansive mappings with a fixed point. Assume that F=F(S)⋂F(G′)≠∅, where G′=PC[PC(x-μBx)-λAPC(x-μBx)]. Suppose that {xn} is generated by
(3.30)x1=u∈C,yn=PC(xn-μBxn),xn+1=αnu+βnxn+γn[δnSxn+(1-δn)PC(yn-λAyn)],n≥1,
where δn⊂[0,1], λ∈(0,2α), μ∈(0,2β) and {αn},{βn}, and {γn} are three sequences in (0,1), and the following conditions are satisfied:

αn+βn+γn=1,foralln≥1,

lim(n→∞)αn=0,∑n=0∞αn=∞,

0<liminf(n→∞)βn≤limsup(n→∞)βn<1,

lim(n→∞)δn=δ∈(0,1).

Then the sequence {xn} defined by (3.30) converges strongly to x-=PFu, and (x-,y-) is a solution of the problem (1.4), where y-=PC(x--μBx-), PC is the projection of H onto C, and PF is the projection of C onto F.

Remark 3.4.

Theorem 3.1 and Corollary 3.3 improve and extend the corresponding results announced by other authors, such as [2–4, 6, 8, 10–12, 14].

Acknowledgments

The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the paper. This work was partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696) and NSFC (11071169).

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