We study the problem of approximating a common element in the common fixed point set of an infinite family of nonexpansive mappings and in the solution set of a variational inequality involving an inverse-strongly monotone mapping based on a viscosity approximation iterative method. Strong convergence theorems of common elements are established in the framework of Hilbert spaces.

1. Introduction and Preliminaries

Let H be a real Hilbert space, whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. Let C be a nonempty, closed, and convex subset of H. Let A:C→H be a mapping. Let PC be the metric projection from H onto the subset C. The classical variational inequality is to find u∈C such that
(1.1)〈Au,v-u〉≥0,∀v∈C.
In this paper, we use VI(C,A) to denote the solution set of the variational inequality. For a given point z∈H, u∈C satisfies the inequality
(1.2)〈u-z,v-u〉≥0,∀v∈C,
if and only if u=PCz. It is known that projection operator PC is nonexpansive. It is also know that PC satisfies
(1.3)〈x-y,PCx-PCy〉≥∥PCx-PCy∥2,∀x,y∈H.
One can see that the variational inequality (1.1) is equivalent to a fixed point problem. The point u∈C is a solution of the variational inequality (1.1) if and only if u∈C satisfies the relation u=PC(u-λAu), where λ>0 is a constant.

Recall the following definitions.

(a) A is said to be monotone if and only if
(1.4)〈Ax-Ay,x-y〉≥0,x,y∈C.

(b) A is said to be α-strongly monotone if and only if there exists a positive real number α such that
(1.5)〈Ax-Ay,x-y〉≥α∥x-y∥2,x,y∈C.

(c) A is said to be α-inverse-strongly monotone if and only if there exists a positive real number α such that
(1.6)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C.

(d) A mapping S:C→C is said to be nonexpansive if and only if
(1.7)∥Sx-Sy∥≤∥x-y∥,∀x,y∈C.

In this paper, we use F(S) to denote the fixed point set of S.

(e) A mapping f:C→C is said to be a κ-contraction if and only if there exists a positive real number κ∈(0,1) such that
(1.8)∥f(x)-f(y)∥≤κ∥x-y∥,∀x,y∈C.

(f) A linear bounded operator B on H is strongly positive if and only if there exists a positive real number γ- such that
(1.9)〈Bx,x〉≥γ-∥x∥2,∀x∈H.

(g) A set-valued mapping T:H→2H is called monotone if and only if for all x,y∈H, f∈Tx, and g∈Ty imply 〈x-y,f-g〉≥0. A monotone mapping T:H→2H is maximal if the graph of G(T) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x,f)∈H×H, 〈x-y,f-g〉≥0 for every (y,g)∈G(T) implies f∈Tx. Let A be a monotone map of C into H and let NCv be the normal cone to C at v∈C, that is, NCv={w∈H:〈v-u,w〉≥0, for all u∈C} and define
(1.10)Tv={Av+NCv,v∈C,∅,v∉C.

Then T is maximal monotone and 0∈Tv if and only if v∈VI(C,A); see [1] and the reference therein.

For finding a common element in the fixed point set of nonexpansive mappings and in the solution set of the variational inequality involving inverse-strongly mappings, Takahashi and Toyoda [2] introduced the following iterative process:
(1.11)x0∈C,xn+1=αnxn+(1-αn)SPC(xn-λnAxn),∀n≥0,
where A is an α-inverse-strongly monotone mapping, {αn} is a real number sequence in (0,1), and {λn} is a real number sequence in (0,2α). They showed that the sequence {xn} generated in (1.11) weakly converges to some point z∈F(S)∩VI(C,A) provided that F(S)∩VI(C,A) is nonempty.

In order to obtain a strong convergence theorem of common elements, Iiduka and Takahashi [3] considered the problem by the following iterative process:
(1.12)x0∈C,xn+1=αnx+(1-αn)SPC(xn-λnAxn),∀n≥0,
where x is a fixed element in C, A is an α-inverse-strongly monotone mapping, {αn} is a real number sequence in (0,1), and {λn} is a real number sequence in (0,2α). They showed that the sequence {xn} generated in (1.12) strongly converges to some point z∈F(S)∩VI(C,A) provided that F(S)∩VI(C,A) is nonempty.

Iterative methods for nonexpansive mappings have been applied to solve convex minimization problems; see, for example, [4–8] and the references therein. A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping S on a real Hilbert space H:
(1.13)minx∈F(S)12〈Bx,x〉-〈x,b〉,
where B is a linear bounded self-adjoint operator, and b is a given point in H. In [4], it is proved that the sequence {xn} defined by the iterative method below, with the initial guess x0∈H chosen arbitrarily,
(1.14)xn+1=(I-αnB)Sxn+αnb,n≥0,
strongly converges to the unique solution of the minimization problem (1.13) provided that the sequence {αn} satisfies certain conditions.

Recently, Marino and Xu [5] considered the problem by viscosity approximation method. They study the following iterative process:
(1.15)x0∈C,xn+1=(I-αnB)Sxn+αnγf(xn),n≥0,
where f is a contraction. They proved that the sequence {xn} generated by the above iterative scheme strongly converges to the unique solution of the variational inequality
(1.16)〈(B-γf)x*,x-x*〉≥0,x∈C,
which is the optimality condition for the minimization problem minx∈F(S)(1/2)〈Bx,x〉-h(x), where h is a potential function for δf (i.e., h'(x)=δf(x) for x∈H).

Concerning a family of nonlinear mappings has been considered by many authors; see, for example, [9–21] and the references therein. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings. The problem of finding an optimal point that minimizes a given cost function over common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance; see, for example, [16, 17].

Recently, Qin et al. [18] considered a general iterative algorithm for an infinite family of nonexpansive mapping in the framework of Hilbert spaces. To be more precise, they introduced the following general iterative algorithm:
(1.17)x0∈C,xn+1=λnγf(xn)+βnxn+((1-βn)I-λnA)Wnxn,n≥0,
where f is a contraction on H, A is a strongly positive bounded linear operator, Wn are nonexpansive mappings which are generated by a finite family of nonexpansive mapping T1,T2,… as follows:
(1.18)Un,n+1=I,Un,n=γnTnUn,n+1+(1-γn)I,Un,n-1=γn-1Tn-1Un,n+(1-γn-1)I,⋮Un,k=γkTkUn,k+1+(1-γk)I,un,k-1=γk-1Tk-1Un,k+(1-γk-1)I,⋮Un,2=γ2T2Uu,3+(1-γ2)I,Wn=Un,1=γ1T1Un,2+(1-γ1)I,
where {γ1},{γ2},… are real numbers such that 0≤γ≤1, T1,T2,… become an infinite family of mappings of C into itself. Nonexpansivity of each Ti ensures the nonexpansivity of Wn.

Concerning Wn we have the following lemmas which are important to prove our main results.

Lemma 1.1 (see [<xref ref-type="bibr" rid="B19">19</xref>]).

Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let T1,T2,… be nonexpansive mappings of C into itself such that ⋂n=1∞F(Tn) is nonempty, and let γ1,γ2,… be real numbers such that 0<γn≤η<1 for any n≥1. Then, for every x∈C and k∈N, the limit limn→∞Un,kx exists.

Using Lemma 1.1, one can define the mapping W of C into itself as follows. Wx=limn→∞Wnx=limn→∞Un,1x, for every x∈C. Such a W is called the W-mapping generated by T1,T2,… and γ1,γ2,…. Throughout this paper, we will assume that 0<γn≤η<1 for all n≥1.

Lemma 1.2 (see [<xref ref-type="bibr" rid="B19">19</xref>]).

Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let T1,T2,… be nonexpansive mappings of C into itself such that ⋂n=1∞F(Tn) is nonempty, and let γ1,γ2,… be real numbers such that 0<γn≤η<1 for any n≥1. Then, F(W)=⋂n=1∞F(Tn).

Motivated by the above results, in this paper, we study the problem of approximating a common element in the common fixed point set of an infinite family of nonexpansive mappings, and in the solution set of a variational inequality involving an inverse-strongly monotone mapping based on a viscosity approximation iterative method. Strong convergence theorems of common elements are established in the framework of Hilbert spaces.

In order to prove our main results, we need the following lemmas.

Lemma 1.3 (see [<xref ref-type="bibr" rid="B5">5</xref>]).

Assume B is a strongly positive linear bounded operator on a Hilbert space H with coefficient γ->0 and 0<ρ≤∥B∥-1. Then ∥I-ρB∥≤1-ργ-.

Lemma 1.4 (see [<xref ref-type="bibr" rid="B22">22</xref>]).

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

∑n=1∞γn=∞;

limsupn→∞δn/γn≤0 or ∑n=1∞|δn|<∞.

Then limn→∞αn=0.
Lemma 1.5 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

Let {xn} and {yn} be bounded sequences in a Banach space X and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose xn+1=(1-βn)yn+βnxn for all integers n≥0 and
(1.20)limsupn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0.
Then limn→∞∥yn-xn∥=0.

Lemma 1.6 (see [<xref ref-type="bibr" rid="B14">14</xref>, <xref ref-type="bibr" rid="B15">15</xref>]).

Let K be a nonempty closed convex subset of a Hilbert space H, {Ti:C→C} be a family of infinitely nonexpansive mappings with ⋂i=1∞F(Ti), {γn} be a real sequence such that 0<γn≤b<1 for each n≥1. If C is any bounded subset of K, then limn→∞supx∈C∥Wx-Wnx∥=0.

Lemma 1.7 (see [<xref ref-type="bibr" rid="B5">5</xref>]).

Let H be a Hilbert space. Let B be a strongly positive linear bounded self-adjoint operator with the constant γ->0 and f a contraction with the constant κ. Assume that 0<γ<γ-/κ. Let T be a nonexpansive mapping with a fixed point xt∈H of the contraction x↦tγf(x)+(I-tB)Tx. Then {xt} converges strongly as t→0 to a fixed point x- of T, which solves the variational inequality
(1.21)〈(A-γf)x-,z-x-〉≤0,∀z∈F(T).
Equivalently, we have PF(T)(I-A+γf)x-=x-.

2. Main Results Theorem 2.1.

Let H be a real Hilbert space and C a nonempty closed convex subset of H. Let A:C→H be an α-inverse-strongly monotone mapping and f:C→C a κ-contraction. Let {Ti}i=1∞ be an infinite family of nonexpansive mappings from C into itself such that F:=⋂i=1∞F(Ti)∩VI(C,A)≠∅. Let B be a strongly positive linear bounded self-adjoint operator of C into itself with the constant γ->0. Let {xn} be a sequence generated in
(2.1)x1∈C,yn=βnγf(xn)+(I-βnB)WnPC(I-rnA)xn,xn+1=αnxn+(1-αn)PCyn,n≥1,
where Wn is generated in (1.18), {αn} and {βn} are real number sequences in (0,1). Assume that the control sequence {αn}, {βn}, and {rn} satisfy the following restrictions:

limn→∞βn=0, ∑n=1∞βn=∞;

0<liminfn→∞αn≤limsupn→∞αn<1;

limn→∞|rn+1-rn|=0;

{rn}⊂[a,b] for some a,b with 0<a<b<2α.

Assume that 0<γ<γ-/κ. Then {xn} strongly converges to some point q, where q∈F, where q=PF(γf+(I-B))(q), which solves the variation inequality
(2.2)〈γf(q)-Bq,p-q〉≤0,∀p∈F.Proof.

First, we show that the mapping I-rnA is nonexpansive. Notice that
(2.3)∥(I-rnA)x-(I-rnA)y∥2=∥x-y-rn(Ax-Ay)∥2=∥x-y∥2-2rn〈x-y,Ax-Ay〉+rn2∥Ax-Ay∥2≤∥x-y∥2+rn(rn-2α)∥Ax-Ay∥2≤∥x-y∥2,∀x,y∈C,
which implies that the mapping I-rnA is nonexpansive. Since the condition (i), we may assume, with no loss of generality, that βn<∥B∥-1 for all n. From Lemma 1.3, we know that if 0<ρ≤∥B∥-1, then ∥I-ρB∥≤1-ργ-. Letting p∈F, we have
(2.4)∥yn-p∥=∥βn(γf(xn)-Bp)+(I-βnB)(WnPC(I-rnA)xn-p)∥≤βn∥γf(xn)-Bp∥+(1-βnγ-)∥WnPC(I-rnA)xn-p∥≤βnγ∥f(xn)-f(p)∥+βn∥γf(p)-Bp∥+(1-βnγ-)∥xn-p∥=[1-βn(γ--κγ)]∥xn-p∥+βn∥γf(p)-Bp∥.
On the other hand, we have
(2.5)∥xn+1-p∥=∥αn(xn-p)+(1-αn)(PCyn-p)∥≤αn∥xn-p∥+(1-αn)∥yn-p∥≤αn∥xn-p∥+(1-αn)[(1-βn(γ--γκ))∥xn-p∥+βn∥γf(p)-Bp∥].
By simple induction, we have
(2.6)∥xn-p∥≤max{∥x0-p∥,∥Bp-γf(p)∥γ--γκ},
which gives that the sequence {xn} is bounded, so is {yn}.

Next, we prove limn→∞|xn+1-xn∥=0. Put ρn=PC(I-rnA)xn. Next, we compute
(2.7)∥ρn-ρn+1∥=∥PC(I-rnA)xn-PC(I-rn+1A)xn+1∥≤∥(I-rnA)xn-(I-rn+1A)xn+1∥=∥(xn-rnAxn)-(xn+1-rnAxn+1)+(rn+1-rn)Axn+1∥≤∥xn-xn+1∥+|rn+1-rn|M1,
where M1 is an appropriate constant such that M1≥supn≥1{∥Axn∥}. It follows that
(2.8)∥yn-yn+1∥=∥(I-βn+1B)(Wn+1ρn+1-Wnρn)-(βn+1-βn)BWnρn+γ[βn+1(f(xn+1)-f(xn))+f(xn)(βn+1-βn)]∥≤(1-βn+1γ-)(∥ρn+1-ρn∥+∥Wn+1ρn-Wnρn∥)+|βn+1-βn|M2+γβn+1κ∥xn+1-xn∥,
where M2 is an appropriate constant such that
(2.9)M2≥max{supn≥1{∥BWnρn∥},γsupn≥1{∥f(xn)∥}}.
Since Ti and Un,i are nonexpansive, we have from (1.18) that
(2.10)∥Wn+1ρn-Wnρn∥=∥γ1T1Un+1,2ρn-γ1T1Un,2ρn∥≤γ1∥Un+1,2ρ-Un,2ρn∥=γ1∥γ2T2Uu+1,3ρn-γ2T2Un,3ρn∥≤γ1γ2∥Uu+1,3ρn-Un,3ρn∥≤⋯≤γ1γ2⋯γn∥Un+1,n+1ρn-Un,n+1ρn∥≤M3∏i=1nγi,
where M3≥0 is an appropriate constant such that ∥Un+1,n+1ρn-Un,n+1ρn∥≤M3, for all n≥0. Substitute (2.7) and (2.10) into (2.8) yields that
(2.11)∥yn-yn+1∥≤[1-βn+1(γ--κγ)]∥xn+1-xn∥+M4(|rn+1-rn|+|βn+1-βn|+∏i=1nγi),
where M4 is an appropriate appropriate constant such that M4≥max{M1,M2,M3}. From the conditions (i) and (iii), we have
(2.12)limsupn→∞{∥yn+1-yn∥-∥xn+1-xn∥}≤0.
By virtue of Lemma 1.5, we obtain that
(2.13)limn→∞∥yn-xn∥=0.
On the other hand, we have
(2.14)∥xn+1-xn∥=(1-αn)∥xn-PCyn∥≤∥xn-yn∥.
This implies from (2.13) that
(2.15)limn→∞∥xn+1-xn∥=0.

Next, we show limn→∞∥Wρn-ρn∥=0. Observing that
(2.16)yn-Wnρn=βn(γf(xn)-BWnρn)
and the condition (i), we can easily get
(2.17)limn→∞∥Wnρn-yn∥=0.
Notice that
(2.18)∥ρn-p∥2=∥PC(I-rnA)xn-PC(I-rnA)p∥2≤∥(xn-p)-rn(Axn-Ap)∥2=∥xn-p∥2-2rn〈xn-p,Axn-Ap〉+rn2∥Axn-Ap∥2≤∥xn-p∥2-2rnα∥Axn-Ap∥2+rn2∥Axn-Ap∥2=∥xn-p∥2-rn(2α-rn)∥Axn-Ap∥2.
On the other hand, we have
(2.19)∥yn-p∥2=∥βn(γf(xn)-Bp)+(I-βnB)(Wnρn-p)∥2≤(βn∥γf(xn)-Bp∥+(1-βnγ-)∥ρn-p∥)2≤βn∥γf(xn)-Bp∥2+∥ρn-p∥2+2βn∥γf(xn)-Bp∥∥ρn-p∥,
from which it follows that
(2.20)∥xn+1-p∥2=∥αn(xn-p)+(1-αn)(PCyn-p)∥2≤αn∥xn-p∥2+(1-αn)∥yn-p∥2≤αn∥xn-p∥2+(1-αn)×[βn∥γf(xn)-Bp∥2+∥ρn-p∥2+2βn∥γf(xn)-Bp∥∥ρn-p∥].
Substituting (2.18) into (2.20), we arrive at
(2.21)∥xn+1-p∥2≤∥xn-p∥2+βn∥γf(xn)-Bp∥2-(1-αn)rn(2α-rn)∥Axn-Ap∥2+2βn∥γf(xn)-Bp∥∥ρn-p∥.
It follows that
(2.22)(1-αn)rn(2α-rn)∥Axn-Ap∥2≤βn∥γf(xn)-Bp∥2+∥xn-p∥2-∥xn+1-p∥2+2βn∥γf(xn)-Bp∥∥ρn-p∥≤βn∥γf(xn)-p∥2+(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+2βn∥γf(xn)-Bp∥∥ρn-p∥.
In view of the restrictions (i), and (iv), we find from (2.15) that
(2.23)limn→∞∥Axn-Ap∥=0.
Observe that
(2.24)∥ρn-p∥2=∥PC(I-rnA)xn-PC(I-rnA)p∥2≤〈(I-rnA)xn-(I-rnA)p,ρn-p〉=12{∥(I-rnA)xn-(I-rnA)p∥2+∥ρn-p∥2-∥(I-rnA)xn-(I-rnA)p-(ρn-p)∥2}≤12{∥xn-p∥2+∥ρn-p∥2-∥(xn-ρn)-rn(Axn-Ap)∥2}=12{∥xn-p∥2+∥ρn-p∥2-∥xn-ρn∥2-rn2∥Axn-Ap∥2+2rn〈xn-ρn,Axn-Ap〉{∥xn-p∥2}},
which yields that
(2.25)∥ρn-p∥2≤∥xn-p∥2-∥ρn-xn∥2+2rn∥ρn-xn∥∥Axn-Ap∥.
Substituting (2.25) into (2.20), we have
(2.26)∥xn+1-p∥2≤∥xn-p∥2+βn∥γf(xn)-Bp∥2+2rn∥ρn-xn∥∥Axn-Ap∥-(1-αn)∥ρn-xn∥2+2βn∥γf(xn)-Bp∥∥ρn-p∥.
This implies that
(2.27)(1-αn)∥ρn-xn∥2≤∥xn-p∥2-∥xn+1-p∥2+βn∥γf(xn)-Bp∥2+2rn∥ρn-xn∥∥Axn-Ap∥+2βn∥γf(xn)-Bp∥∥ρn-p∥≤(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+βn∥γf(xn)-Bp∥2+2rn∥ρn-xn∥∥Axn-Ap∥+2βn∥γf(xn)-Bp∥∥ρn-p∥.
In view of the restrictions (i) and (ii), we find from (2.15) and (2.23) that
(2.28)limn→∞∥ρn-xn∥=0.
On the other hand, we have
(2.29)∥ρn-Wnρn∥≤∥xn-ρn∥+∥xn-yn∥+∥yn-Wnρn∥.
It follows from (2.13), (2.17) and (2.28) that limn→∞∥Wnρn-ρn∥=0. From Lemma 1.6, we find that ∥Wρn-Wnρn∥→0 as n→∞. Notice that
(2.30)∥Wρn-ρn∥≤∥Wnρn-ρn∥+∥Wnρn-Wρn∥,
from which it follows that
(2.31)limn→∞∥Wρn-ρn∥=0.

Next, we show limsupn→∞〈γf(q)-Bq,xn-q〉≤0, where q=PF(γf+(I-B))(q). To show it, we choose a subsequence {xni} of {xn} such that
(2.32)limsupn→∞〈γf(q)-Bq,xn-q〉=limi→∞〈γf(q)-Bq,xni-q〉.
As {xni} is bounded, we have that there is a subsequence {xnij} of {xni} converges weakly to p. We may assume, without loss of generality, that xni⇀p. Hence we have p∈F. Indeed, let us first show that p∈VI(C,A). Put
(2.33)Tw1={Aw1+NCw1,w1∈C,∅,w1∉C.
Since A is inverse-strongly monotone, we see that T is maximal monotone. Let (w1,w2)∈G(T). Since w2-Aw1∈NCw1 and ρn∈C, we have
(2.34)〈w1-ρn,w2-Aw1〉≥0.
On the other hand, from ρn=PC(I-rnA)xn, we have
(2.35)〈w1-ρn,ρn-(I-rnA)xn〉≥0
and hence
(2.36)〈w1-ρn,ρn-xnrn+Axn〉≥0.
It follows that
(2.37)〈w1-ρni,w2〉≥〈w1-ρni,Aw1〉≥〈w1-ρni,Aw1〉-〈w1-ρni,ρni-xnirni+Axni〉≥〈w1-ρni,Aw1-ρni-xnirni-Axni〉=〈w1-ρni,Aw1-Aρni〉+〈w1-ρni,Aρni-Axni〉-〈w1-ρni,ρni-xnirni〉≥〈w1-ρni,Aρni-Axni〉-〈w1-ρni,ρni-xnirni〉,
which implies from (2.28) that 〈w1-p,w2〉≥0. We have p∈T-10 and hence p∈VI(C,A). Next, let us show p∈⋂i=1∞F(Ti). Since Hilbert spaces are Opial’s spaces, from (2.31), we have
(2.38)liminfi→∞∥ρni-p∥<liminfi→∞∥ρni-Wp∥=liminfi→∞∥ρni-Wρni+Wnρni-Wp∥≤liminfi→∞∥Wρni-Wp∥≤liminfi→∞∥ρni-p∥,
which derives a contradiction. Thus, we have from Lemma 1.2 that p∈F(W)=⋂i=1∞F(Ti). On the other hand, we have
(2.39)limsupn→∞〈γf(q)-Bq,xn-q〉=limn→∞〈γf(q)-Bq,xni-q〉=〈γf(q)-Bq,p-q〉≤0.

Finally, we show xn→q strongly as n→∞. Notice that
(2.40)∥yn-q∥2=∥βn(γf(xn)-Bq)+(I-βnB)(Wnρn-q)∥2≤(1-βnγ-)2∥Wnρn-q∥2+2βn〈γf(xn)-Bq,yn-q〉≤(1-βnγ-)2∥xn-q∥2+κγβn(∥xn-q∥2+∥yn-q∥2)+2βn〈γf(q)-Bq,yn-q〉.
Therefore, we have
(2.41)∥yn-q∥2≤(1-βnγ-)2+βnγκ1-βnγκ∥xn-q∥2+2βn1-αnγκ〈γf(q)-Bq,yn-q〉=(1-2βnγ-+βnκγ)1-βnγκ∥xn-q∥2+βn2γ-21-βnγκ∥xn-q∥2+2βn1-βnγκ〈γf(q)-Bq,yn-q〉≤[1-2βn(γ--κγ)1-βnγκ]∥xn-q∥2+2βn(γ--κγ)1-βnγκ[1γ--κγ〈γf(q)-Bq,yn-q〉+αnγ-22(γ--κγ)M5],
where M5 is an appropriate constant. On the other hand, we have
(2.42)∥xn+1-p∥2=∥αn(xn-p)+(1-αn)(PCyn-p)∥2≤αn∥xn-p∥2+(1-αn)∥PCyn-p∥2≤αn∥xn-p∥2+(1-αn)∥yn-p∥2.
Substitute (2.41) into (2.42) yields that
(2.43)∥xn+1-p∥2≤[1-(1-αn)2βn(γ--αγ)1-βnγα]∥xn-q∥2+(1-αn)2βn(γ--αγ)1-βnγα[1γ--αγ〈γf(q)-Bq,yn-q〉+βnγ-22(γ--αγ)M5].
Put ln=(1-αn)(2βn(γ--αnγ)/(1-βnαγ)) and
(2.44)tn=1γ--αγ〈γf(q)-Bq,yn-q〉+βnγ-22(γ--αγ)M5.
That is,
(2.45)∥xn+1-q∥2≤(1-ln)∥xn-q∥2+lntn.
Notice that
(2.46)〈γf(q)-Bq,yn-q〉=〈γf(q)-Bq,yn-xn〉+〈γf(q)-Bq,xn-q〉≤∥γf(q)-Bq∥∥yn-xn∥+〈γf(q)-Bq,xn-q〉.
From (2.13) and (2.39) that
(2.47)limsupn→∞〈γf(q)-Aq,yn-q〉≤0.
It follows from the condition (i) and (2.47) that
(2.48)limn→∞ln=0,∑n=1∞ln=∞,limsupn→∞tn≤0.
Apply Lemma 1.4 to (2.45) to conclude xn→q as n→∞. This completes the proof.

For a single nonexpansive mapping, we have from Theorem 2.1 the following.

Corollary 2.2.

Let H be a real Hilbert space and C a nonempty closed convex subset of H. Let A:C→H be an α-inverse-strongly monotone mapping and f:C→C a κ-contraction. Let T be a nonexpansive mapping from C into itself such that F:=F(T)∩VI(C,A)≠∅. Let B be a strongly positive linear bounded self-adjoint operator of C into itself with the constant γ->0. Let {xn} be a sequence generated in
(2.49)x1∈C,yn=βnγf(xn)+(I-βnB)TPC(I-rnA)xn,xn+1=αnxn+(1-αn)PCyn,n≥1,
where {αn} and {βn} are real number sequences in (0,1). Assume that the control sequence {αn}, {βn} and {rn} satisfy the following restrictions:

limn→∞βn=0, ∑n=1∞βn=∞;

0<liminfn→∞αn≤limsupn→∞αn<1;

limn→∞|rn+1-rn|=0;

{rn}⊂[a,b] for some a, b with 0<a<b<2α.

Assume that 0<γ<γ-/κ. Then {xn} strongly converges to some point q, where q∈F, where q=PF(γf+(I-B))(q), which solves the variation inequality
(2.50)〈γf(q)-Bq,p-q〉≤0,∀p∈F.Corollary 2.3.

Let H be a real Hilbert space and C a nonempty closed convex subset of H. Let f:C→C be a κ-contraction. Let T be a nonexpansive mapping from C into itself such that F(T)≠∅. Let B be a strongly positive linear bounded self-adjoint operator of C into itself with the constant γ->0. Let {xn} be a sequence generated in
(2.51)x1∈C,yn=βnγf(xn)+(I-βnB)Txn,xn+1=αnxn+(1-αn)PCyn,n≥1,
where {αn} and {βn} are real number sequences in (0,1). Assume that the control sequence {αn}, and {βn} satisfy the following restrictions:

limn→∞βn=0,∑n=1∞βn=∞;

0<liminfn→∞αn≤limsupn→∞αn<1.

Assume that 0<γ<γ-/κ. Then {xn} strongly converges to some point q, where q∈F(T), where q=PF(γf+(I-B))(q), which solves the variation inequality
(2.52)〈γf(q)-Bq,p-q〉≤0,∀p∈F(T).

If B is the identity mapping, then Theorem 2.1 is reduced to the following.

Corollary 2.4.

Let H be a real Hilbert space and C a nonempty closed convex subset of H. Let A:C→H be an α-inverse-strongly monotone mapping and f:C→C a κ-contraction. Let {Ti}i=1∞ be an infinite family of nonexpansive mappings from C into itself such that F:=⋂i=1∞F(Ti)∩VI(C,A)≠∅. Let {xn} be a sequence generated in
(2.53)x1∈C,yn=βnf(xn)+(1-βn)WnPC(I-rnA)xn,xn+1=αnxn+(1-αn)yn,n≥1,
where Wn is generated in (1.18), {αn} and {βn} are real number sequences in (0,1). Assume that the control sequence {αn}, {βn}, and {rn} satisfy the following restrictions:

limn→∞βn=0, ∑n=1∞βn=∞;

0<liminfn→∞αn≤limsupn→∞αn<1;

limn→∞|rn+1-rn|=0;

{rn}⊂[a,b] for some a, b with 0<a<b<2α.

Then {xn} strongly converges to some point q, where q∈F, where q=PFf(q), which solves the variation inequality
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