AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 893635 10.1155/2012/893635 893635 Research Article Some Results on an Infinite Family of Nonexpansive Mappings and an Inverse-Strongly Monotone Mapping in Hilbert Spaces Cheng Peng 1 Zhang Anshen 2 Qin Xiaolong 1 School of Mathematics and Information Science North China University of Water Resources and Electric Power Zhengzhou 450011 China ncwu.edu.cn 2 No. 13 Zhongxue Feng-Feng Kuangqu Hangdan 056000 China 2012 10 11 2012 2012 08 09 2012 10 10 2012 2012 Copyright © 2012 Peng Cheng and Anshen Zhang. 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 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:CH be a mapping. Let PC be the metric projection from H onto the subset C. The classical variational inequality is to find uC such that (1.1)Au,v-u0,vC. In this paper, we use VI(C,A) to denote the solution set of the variational inequality. For a given point zH, uC satisfies the inequality (1.2)u-z,v-u0,vC, 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-PCyPCx-PCy2,x,yH. One can see that the variational inequality (1.1) is equivalent to a fixed point problem. The point uC is a solution of the variational inequality (1.1) if and only if uC 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-y0,x,yC.

(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-yαx-y2,x,yC.

(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-yαAx-Ay2,x,yC.

(d) A mapping S:CC is said to be nonexpansive if and only if (1.7)Sx-Syx-y,x,yC.

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

(e) A mapping f:CC 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)κx-y,x,yC.

(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,xγ-x2,xH.

(g) A set-valued mapping T:H2H is called monotone if and only if for all x,yH, fTx, and gTy imply x-y,f-g0. A monotone mapping T:H2H 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-g0 for every (y,g)G(T) implies fTx. Let A be a monotone map of C into H and let NCv be the normal cone to C at vC, that is, NCv={wH:v-u,w0, for all uC} and define (1.10)Tv={Av+NCv,vC,,vC.

Then T is maximal monotone and 0Tv if and only if vVI(C,A); see  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  introduced the following iterative process: (1.11)x0C,xn+1=αnxn+(1-αn)SPC(xn-λnAxn),n0, 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 zF(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  considered the problem by the following iterative process: (1.12)x0C,xn+1=αnx+(1-αn)SPC(xn-λnAxn),n0, 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 zF(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,  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)minxF(S)12Bx,x-x,b, where B is a linear bounded self-adjoint operator, and b is a given point in H. In , it is proved that the sequence {xn} defined by the iterative method below, with the initial guess x0H chosen arbitrarily, (1.14)xn+1=(I-αnB)Sxn+αnb,n0, strongly converges to the unique solution of the minimization problem (1.13) provided that the sequence {αn} satisfies certain conditions.

Recently, Marino and Xu  considered the problem by viscosity approximation method. They study the following iterative process: (1.15)x0C,xn+1=(I-αnB)Sxn+αnγf(xn),n0, 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,xC, which is the optimality condition for the minimization problem minxF(S)(1/2)Bx,x-h(x), where h is a potential function for δf (i.e., h'(x)=δf(x) for xH).

Concerning a family of nonlinear mappings has been considered by many authors; see, for example,  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.  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)x0C,xn+1=λnγf(xn)+βnxn+((1-βn)I-λnA)Wnxn,n0, 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 0γ1, 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=1F(Tn) is nonempty, and let γ1,γ2, be real numbers such that 0<γnη<1 for any n1. Then, for every xC and kN, the limit limnUn,kx exists.

Using Lemma 1.1, one can define the mapping W of C into itself as follows. Wx=limnWnx=limnUn,1x, for every xC. Such a W is called the W-mapping generated by T1,T2, and γ1,γ2,. Throughout this paper, we will assume that 0<γnη<1 for all n1.

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=1F(Tn) is nonempty, and let γ1,γ2, be real numbers such that 0<γnη<1 for any n1. Then, F(W)=n=1F(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<ρB-1. Then I-ρB1-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=1γn=;

limsupnδn/γn0 or n=1|δn|<.

Then limnαn=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<liminfnβnlimsupnβn<1. Suppose xn+1=(1-βn)yn+βnxn for all integers n0 and (1.20)limsupn(yn+1-yn-xn+1-xn)0. Then limnyn-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:CC} be a family of infinitely nonexpansive mappings with i=1F(Ti), {γn} be a real sequence such that 0<γnb<1 for each n1. If C is any bounded subset of K, then limnsupxCWx-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 xtH of the contraction xtγf(x)+(I-tB)Tx. Then {xt} converges strongly as t0 to a fixed point x- of T, which solves the variational inequality (1.21)(A-γf)x-,z-x-0,zF(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:CH be an α-inverse-strongly monotone mapping and f:CC a κ-contraction. Let {Ti}i=1 be an infinite family of nonexpansive mappings from C into itself such that F:=i=1F(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)x1C,yn=βnγf(xn)+(I-βnB)WnPC(I-rnA)xn,xn+1=αnxn+(1-αn)PCyn,n1, 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:

limnβn=0, n=1βn=;

0<liminfnαnlimsupnαn<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 qF, where q=PF(γf+(I-B))(q), which solves the variation inequality (2.2)γf(q)-Bq,p-q0,pF.

Proof.

First, we show that the mapping I-rnA is nonexpansive. Notice that (2.3)(I-rnA)x-(I-rnA)y2=x-y-rn(Ax-Ay)2=x-y2-2rnx-y,Ax-Ay+rn2Ax-Ay2x-y2+rn(rn-2α)Ax-Ay2x-y2,x,yC, 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<ρB-1, then I-ρB1-ργ-. Letting pF, we have (2.4)yn-p=βn(γf(xn)-Bp)+(I-βnB)(WnPC(I-rnA)xn-p)βnγf(xn)-Bp+(1-βnγ-)WnPC(I-rnA)xn-pβnγf(xn)-f(p)+βnγf(p)-Bp+(1-βnγ-)xn-p=[1-βn(γ--κγ)]xn-p+βnγf(p)-Bp. On the other hand, we have (2.5)xn+1-p=αn(xn-p)+(1-αn)(PCyn-p)αnxn-p+(1-αn)yn-pαnxn-p+(1-αn)[(1-βn(γ--γκ))xn-p+βnγf(p)-Bp]. By simple induction, we have (2.6)xn-pmax{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+1xn-xn+1+|rn+1-rn|M1, where M1 is an appropriate constant such that M1supn1{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+1κxn+1-xn, where M2 is an appropriate constant such that (2.9)M2max{supn1{BWnρn},γsupn1{f(xn)}}. Since Ti and Un,i are nonexpansive, we have from (1.18) that (2.10)Wn+1ρn-Wnρn=γ1T1Un+1,2ρn-γ1T1Un,2ρnγ1Un+1,2ρ-Un,2ρn=γ1γ2T2Uu+1,3ρn-γ2T2Un,3ρnγ1γ2Uu+1,3ρn-Un,3ρnγ1γ2γnUn+1,n+1ρn-Un,n+1ρnM3i=1nγi, where M30 is an appropriate constant such that Un+1,n+1ρn-Un,n+1ρnM3, for all n0. 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 M4max{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)limnyn-xn=0. On the other hand, we have (2.14)xn+1-xn=(1-αn)xn-PCynxn-yn. This implies from (2.13) that (2.15)limnxn+1-xn=0.

Next, we show limnWρn-ρn=0. Observing that (2.16)yn-Wnρn=βn(γf(xn)-BWnρn) and the condition (i), we can easily get (2.17)limnWnρn-yn=0. Notice that (2.18)ρn-p2=PC(I-rnA)xn-PC(I-rnA)p2(xn-p)-rn(Axn-Ap)2=xn-p2-2rnxn-p,Axn-Ap+rn2Axn-Ap2xn-p2-2rnαAxn-Ap2+rn2Axn-Ap2=xn-p2-rn(2α-rn)Axn-Ap2. On the other hand, we have (2.19)yn-p2=βn(γf(xn)-Bp)+(I-βnB)(Wnρn-p)2(βnγf(xn)-Bp+(1-βnγ-)ρn-p)2βnγf(xn)-Bp2+ρn-p2+2βnγf(xn)-Bpρn-p, from which it follows that (2.20)xn+1-p2=αn(xn-p)+(1-αn)(PCyn-p)2αnxn-p2+(1-αn)yn-p2αnxn-p2+(1-αn)×[βnγf(xn)-Bp2+  ρn-p2+2βnγf(xn)-Bpρn-p]. Substituting (2.18) into (2.20), we arrive at (2.21)xn+1-p2xn-p2+βn  γf(xn)-Bp2  -(1-αn)rn(2α-rn)Axn-Ap2+2βnγf(xn)-Bpρn-p. It follows that (2.22)(1-αn)rn(2α-rn)Axn-Ap2βnγf(xn)-Bp2+xn-p2-xn+1-p2+2βnγf(xn)-Bpρn-pβnγf(xn)-p2+(xn-p+xn+1-p)xn-xn+1+2βnγf(xn)-Bpρn-p. In view of the restrictions (i), and (iv), we find from (2.15) that (2.23)limnAxn-Ap=0. Observe that (2.24)ρn-p2=PC(I-rnA)xn-PC(I-rnA)p2(I-rnA)xn-(I-rnA)p,ρn-p=12{(I-rnA)xn-(I-rnA)p2+ρn-p2  -(I-rnA)xn-(I-rnA)p-(ρn-p)2}12{xn-p2+ρn-p2-(xn-ρn)-rn(Axn-Ap)2}=12{xn-p2+ρn-p2-xn-ρn2-rn2Axn-Ap2  +2rnxn-ρn,Axn-Ap{xn-p2}}, which yields that (2.25)ρn-p2xn-p2-ρn-xn2+2rnρn-xnAxn-Ap. Substituting (2.25) into (2.20), we have (2.26)xn+1-p2xn-p2+βnγf(xn)-Bp2+2rnρn-xnAxn-Ap-(1-αn)ρn-xn2+2βnγf(xn)-Bpρn-p. This implies that (2.27)(1-αn)ρn-xn2xn-p2-xn+1-p2+βnγf(xn)-Bp2+2rnρn-xnAxn-Ap+2βnγf(xn)-Bpρn-p(xn-p+xn+1-p)xn-xn+1+βnγf(xn)-Bp2+2rnρn-xnAxn-Ap+2βnγf(xn)-Bpρn-p. In view of the restrictions (i) and (ii), we find from (2.15) and (2.23) that (2.28)limnρn-xn=0. On the other hand, we have (2.29)ρn-Wnρnxn-ρn+xn-yn+yn-Wnρn. It follows from (2.13), (2.17) and (2.28) that limnWnρn-ρn=0. From Lemma 1.6, we find that Wρn-Wnρn0 as n. Notice that (2.30)Wρn-ρnWnρn-ρn+Wnρn-Wρn, from which it follows that (2.31)limnWρn-ρn=0.

Next, we show limsupnγf(q)-Bq,xn-q0, where q=PF(γf+(I-B))(q). To show it, we choose a subsequence {xni} of {xn} such that (2.32)limsupnγf(q)-Bq,xn-q=limiγf(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 xnip. Hence we have pF. Indeed, let us first show that pVI(C,A). Put (2.33)Tw1={Aw1+NCw1,w1C,,w1C. Since A is inverse-strongly monotone, we see that T is maximal monotone. Let (w1,w2)G(T). Since w2-Aw1NCw1 and ρnC, we have (2.34)w1-ρn,w2-Aw10. On the other hand, from ρn=PC(I-rnA)xn, we have (2.35)w1-ρn,ρn-(I-rnA)xn0 and hence (2.36)w1-ρn,ρn-xnrn+Axn0. It follows that (2.37)w1-ρni,w2w1-ρni,Aw1w1-ρni,Aw1-w1-ρni,ρni-xnirni+Axniw1-ρni,Aw1-ρni-xnirni-Axni=w1-ρni,Aw1-Aρni+w1-ρni,Aρni-Axni-w1-ρni,ρni-xnirniw1-ρni,Aρni-Axni-w1-ρni,ρni-xnirni, which implies from (2.28) that w1-p,w20. We have pT-10 and hence pVI(C,A). Next, let us show pi=1F(Ti). Since Hilbert spaces are Opial’s spaces, from (2.31), we have (2.38)liminfiρni-p<liminfiρni-Wp=liminfiρni-Wρni+Wnρni-WpliminfiWρni-Wpliminfiρni-p, which derives a contradiction. Thus, we have from Lemma 1.2 that pF(W)=i=1F(Ti). On the other hand, we have (2.39)limsupnγf(q)-Bq,xn-q=limnγf(q)-Bq,xni-q=γf(q)-Bq,p-q0.

Finally, we show xnq strongly as n. Notice that (2.40)yn-q2=βn(γf(xn)-Bq)+(I-βnB)(Wnρn-q)2(1-βnγ-)2Wnρn-q2+2βnγf(xn)-Bq,yn-q(1-βnγ-)2xn-q2+κγβn(xn-q2+yn-q2)+2βnγf(q)-Bq,yn-q. Therefore, we have (2.41)yn-q2(1-βnγ-)2+βnγκ1-βnγκxn-q2+2βn1-αnγκγf(q)-Bq,yn-q=(1-2βnγ-+βnκγ)1-βnγκxn-q2+βn2γ-21-βnγκxn-q2+2βn1-βnγκγf(q)-Bq,yn-q[1-2βn(γ--κγ)1-βnγκ]xn-q2+2βn(γ--κγ)1-βnγκ[1γ--κγγf(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-p2=αn(xn-p)+(1-αn)(PCyn-p)2αnxn-p2+(1-αn)PCyn-p2αnxn-p2+(1-αn)yn-p2. Substitute (2.41) into (2.42) yields that (2.43)  xn+1-p2[1-(1-αn)2βn(γ--αγ)1-βnγα]xn-q2+(1-αn)2βn(γ--αγ)1-βnγα[1γ--αγγf(q)-Bq,yn-q+βnγ-22(γ--αγ)M5]. Put ln=(1-αn)(2βn(γ--αnγ)/(1-βnαγ)) and (2.44)tn=1γ--αγγf(q)-Bq,yn-q+βnγ-22(γ--αγ)M5. That is, (2.45)xn+1-q2(1-ln)xn-q2+lntn. Notice that (2.46)γf(q)-Bq,yn-q=γf(q)-Bq,yn-xn+γf(q)-Bq,xn-qγf(q)-Bqyn-xn+γf(q)-Bq,xn-q. From (2.13) and (2.39) that (2.47)limsupnγf(q)-Aq,yn-q0. It follows from the condition (i) and (2.47) that (2.48)limnln=0,n=1ln=,limsupntn0. Apply Lemma 1.4 to (2.45) to conclude xnq 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:CH be an α-inverse-strongly monotone mapping and f:CC 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)x1C,yn=βnγf(xn)+(I-βnB)TPC(I-rnA)xn,xn+1=αnxn+(1-αn)PCyn,n1, where {αn} and {βn} are real number sequences in (0,1). Assume that the control sequence {αn}, {βn} and {rn} satisfy the following restrictions:

limnβn=0, n=1βn=;

0<liminfnαnlimsupnαn<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 qF, where q=PF(γf+(I-B))(q), which solves the variation inequality (2.50)γf(q)-Bq,p-q0,pF.

Corollary 2.3.

Let H be a real Hilbert space and C a nonempty closed convex subset of H. Let f:CC 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)x1C,yn=βnγf(xn)+(I-βnB)Txn,xn+1=αnxn+(1-αn)PCyn,n1, where {αn} and {βn} are real number sequences in (0,1). Assume that the control sequence {αn}, and {βn} satisfy the following restrictions:

limnβn=0,n=1βn=;

0<liminfnαnlimsupnαn<1.

Assume that 0<γ<γ-/κ. Then {xn} strongly converges to some point q, where qF(T), where q=PF(γf+(I-B))(q), which solves the variation inequality (2.52)γf(q)-Bq,p-q0,pF(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:CH be an α-inverse-strongly monotone mapping and f:CC a κ-contraction. Let {Ti}i=1 be an infinite family of nonexpansive mappings from C into itself such that F:=i=1F(Ti)VI(C,A). Let {xn} be a sequence generated in (2.53)x1C,yn=βnf(xn)+(1-βn)WnPC(I-rnA)xn,xn+1=αnxn+(1-αn)yn,  n1, 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:

limnβn=0, n=1βn=;

0<liminfnαnlimsupnαn<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 qF, where q=PFf(q), which solves the variation inequality (2.54)f(q)-q,p-q0,pF.

Rockafellar R. T. On the maximality of sums of nonlinear monotone operators Transactions of the American Mathematical Society 1970 149 75 88 0282272 10.1090/S0002-9947-1970-0282272-5 ZBL0222.47017 Takahashi W. Toyoda M. Weak convergence theorems for nonexpansive mappings and monotone mappings Journal of Optimization Theory and Applications 2003 118 2 417 428 10.1023/A:1025407607560 2006529 ZBL1055.47052 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 2123081 ZBL1093.47058 Xu H. K. An iterative approach to quadratic optimization Journal of Optimization Theory and Applications 2003 116 3 659 678 10.1023/A:1023073621589 1977756 ZBL1043.90063 Marino G. Xu H.-K. A general iterative method for nonexpansive mappings in Hilbert spaces Journal of Mathematical Analysis and Applications 2006 318 1 43 52 10.1016/j.jmaa.2005.05.028 2210870 ZBL1095.47038 Qin X. Cho S. Y. Kang S. M. On hybrid projection methods for asymptotically quasi-φ-nonexpansive mappings Applied Mathematics and Computation 2010 215 11 3874 3883 10.1016/j.amc.2009.11.031 2578853 ZBL1225.47105 Deutsch F. Yamada I. Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings Numerical Functional Analysis and Optimization 1998 19 1-2 33 56 10.1080/01630569808816813 1606921 ZBL0913.47048 Cho S. Y. Kang S. M. Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process Applied Mathematics Letters 2011 24 2 224 228 10.1016/j.aml.2010.09.008 2735146 Cho S. Y. Kang S. M. Approximation of common solutions of variational inequalities via strict pseudocontractions Acta Mathematica Scientia Series B 2012 32 4 1607 1618 10.1016/S0252-9602(12)60127-1 2927448 Cho S. Y. Qin X. Kang S. M. Hybrid projection algorithms for treating common fixed points of a family of demicontinuous pseudocontractions Applied Mathematics Letters 2012 25 5 854 857 10.1016/j.aml.2011.10.031 2888085 ZBL1242.65112 Qin X. Chang S.-s. Cho Y. J. Iterative methods for generalized equilibrium problems and fixed point problems with applications Nonlinear Analysis. Real World Applications 2010 11 4 2963 2972 10.1016/j.nonrwa.2009.10.017 2661959 ZBL1192.58010 Chouhan A. P. Ranadive A. S. Absorbing maps and common fixed point theorem in Menger space Advances in Fixed Point Theory 2012 2 108 119 Ye J. Huang J. Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces Journal of Mathematical and Computational Science 2011 1 1 1 18 2913374 Kadelburg Z. Radenovic S. Coupled fixed point results under tvs-cone metric and w-cone-distance Advances in Fixed Point Theory 2012 2 29 46 Cho Y. J. Qin X. Kang S. M. Some results for equilibrium problems and fixed point problems in Hilbert spaces Journal of Computational Analysis and Applications 2009 11 2 287 294 2508917 ZBL1223.47075 Deutsch F. Hundal H. The rate of convergence of Dykstra's cyclic projections algorithm: the polyhedral case Numerical Functional Analysis and Optimization 1994 15 5-6 537 565 10.1080/01630569408816580 1281561 ZBL0807.41019 Bauschke H. H. The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space Journal of Mathematical Analysis and Applications 1996 202 1 150 159 10.1006/jmaa.1996.0308 1402593 ZBL0956.47024 Qin X. Shang M. Su Y. Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems Mathematical and Computer Modelling 2008 48 7-8 1033 1046 10.1016/j.mcm.2007.12.008 2458216 ZBL1187.65058 Shimoji K. Takahashi W. Strong convergence to common fixed points of infinite nonexpansive mappings and applications Taiwanese Journal of Mathematics 2001 5 2 387 404 1832176 ZBL0993.47037 Bhardwaj R. Some fixed point theorems in polish space using new type of contractive conditions Advances in Fixed Point Theory 2012 2 313 325 Qin X. Cho Y. J. Kang J. I. Kang S. M. Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces Journal of Computational and Applied Mathematics 2009 230 1 121 127 10.1016/j.cam.2008.10.058 2532296 ZBL1170.47043 Xu H. K. An iterative approach to quadratic optimization Journal of Optimization Theory and Applications 2003 116 3 659 678 10.1023/A:1023073621589 1977756 ZBL1043.90063 Suzuki T. Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals Journal of Mathematical Analysis and Applications 2005 305 1 227 239 10.1016/j.jmaa.2004.11.017 2128124 ZBL1068.47085