Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 105928, 28 pages http://dx.doi.org/10.1155/2014/105928

Research Article

Convex Minimization with Constraints of Systems of Variational Inequalities, Mixed Equilibrium, Variational Inequality, and Fixed Point Problems

Lu-Chuan Ceng,1 Cheng-Wen Liao,2 Chin-Tzong Pang,3 and Ching-Feng Wen4

1 Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China

2 Department of Food and Beverage Management, Vanung University, Chung-Li 320061, Taiwan

3 Department of Information Management and Innovation Center for Big Data and Digital Convergence, Yuan Ze University, Chung-Li 32003, Taiwan

4 Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Correspondence should be addressed to Chin-Tzong Pang; imctpang@saturn.yzu.edu.tw Received 17 March 2014; Accepted 26 March 2014; Published 8 May 2014 Academic Editor: Jen-Chih Yao

Copyright © 2014 Lu-Chuan Ceng et al. 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 and analyze one iterative algorithm by hybrid shrinking projection method for finding a solution of the minimization problem for a convex and continuously Fréchet differentiable functional, with constraints of several problems: finitely many generalized mixed equilibrium problems, finitely many variational inequalities, the general system of variational inequalities and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm by hybrid shrinking projection method for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.

1. Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H and let Pc be the metric projection of H onto C. Let S : C ^ H be a nonlinear mapping on C. We denote by Fix(S) the set of fixed points of S and by R the set of all real numbers. A mapping S : C ^ H is called L-Lipschitz continuous if there exists a constant L> 0 such that

\\Sx-Sy\\< L\\x- y\\, Vx,yeC. (1)

In particular, if L = 1 then S is called a nonexpansive mapping; if L e [0,1) then S is called a contraction. A mapping V is called strongly positive on H if there exists a constant f > 0 such that

{Vx,x)>y\\xf-, VxeH. (2)

Let A : C ^ H be a nonlinear mapping on C. We consider the following variational inequality problem (VIP): find a point x e C such that

{Ax, y-x) >0, Vy eC. (3)

The solution set of VIP (3) is denoted by VI(C, A).

Let f : C ^ R be a real-valued function, let A : H ^ H be a nonlinear mapping, and let & : C x C ^ R be a bifunction. Peng and Yao [1] introduced the following generalized mixed equilibrium problem (GMEP) of finding x eC such that

& (x, y) + f(y)-f (x) + {Ax, y -x) >0, Vy eC. (4)

We denote the set of solutions of GMEP (4) by GMEP(&, f, A). The GMEP (4) is very general in the sense that it includes, as special cases, optimization problems,

variational inequalities, minimax problems, and Nash equilibrium problems in noncooperative games. It covers problems considered in [2-5].

It is assumed as in [1] that 0 : CxC ^ R is a bifunction satisfying conditions (A1)-(A4) and (p : C ^ R is a lower semicontinuous and convex function with restriction (B1) or (B2), where

(A1) 0(%, x) = 0 for all xeC;

(A2) 0 is monotone; that is, 0(x, y) + 0(y, x) < 0 for any x,y e C;

(A3) 0 is upper-hemicontinuous; that is, for each x,y,z e C,

lim sup 0 (tz + (1 - t) x, y) < 0 (x, y) ; (5)

(A4) 0(x, ■) is convex and lower semicontinuous for each xeC;

(B1) for each x e H and r > 0, there exists a bounded subset Dx c C and yx e C such that, for any z e C\DX,

0 (z> yx) + 9 (yx) -9(z) + ~ (yx -z,z-x) < 0; (6)

(B2) C is a bounded set.

Given a positive number r > 0, let T: H ^ C be the solution set of the auxiliary mixed equilibrium problem; that is, for each x e H,

t*^ (x) ■= {yeC:®(y,z)+<p (z) - <p (y)

+ 1 {y - x,z - y) >0, Vz ec \. r J

Let Fi,F2 : C ^ H be two mappings. Consider the following general system of variational inequalities (GSVI) [6] of finding (x*, y*) e CxC such that

{v1F1y* + x* -y*,x-x*)>0, VxeC, {v2F2x* + y* - x*, x - y* )> 0, Vx e C,

where v1 > 0 and v2 > 0 are two constants. In 2008, Ceng et al. [6] transformed the GSVI (8) into a fixed point problem in the following way.

Proposition CWY (see [6]). For given x,y e C, (x,~y) is a solution of the GSVI (8) if and only if x is a fixed point of the mapping G : C ^ C defined by

Gx = Pc (I- ViFi)Pc (I- V2F2)x, VxeC, (9) where y = Pc(I - v2F2)x.

In particular, if the mapping Fj : C ^ H is Cj-inverse-strongly monotone for j = 1,2, then the mapping G is nonexpansive provided Vj e (0,2(j] for j = 1,2. We denote by GSVI(G) the fixed point set of the mapping G.

Let Ani,An2,... ,AnN e (0,1], n > 1. Given the nonexpansive self-mappings T1,T2,...,TN on C, for each n > 1, the mappings Uni, Un,2,..., Un,N are defined by

un,i = K,Ji + (1-K,i)i, Un,2 = K,2TnUn,i + (1-*na)I,

un,n-i = An-Jn-iUn,n + (1 - An-i) I,

Un,N-i = An,N-i^N-iUn,N-2 + (1 - An,N-i) 1 Wn := Un,N = An,NTNUn,N-i + (1 - An,N) 1

The Wn is called the W-mapping generated by Ti ,...,TN and Ani, An2,..., AnN. Note that the nonexpansivity of Ti implies the one of Wn. In 2012, combining the hybrid steepest-descent method in [7] and viscosity approximation method, Ceng et al. [8] proposed and analyzed the following hybrid iterative algorithm for finding a common element of the solution set of GMEP (4) and the fixed point set of finitely many nonexpansive mappings [Ti]1N=i.

Theorem CGY (see [8, Theorem 3.1]). Let C be a nonempty closed convex subset of a real Hilbert space H. Let 0 : C x C ^ R be a bifunction satisfying assumptions (A1)-(A4) and let f : C ^ R be a lower semicontinuous and convex function with restriction (B1) or (B2). Let the mapping A : H ^ H be S-inverse-strongly monotone, and let [Ti]1N=i be a finite family of nonexpansive mappings on H such that Q := nN=iFix(Ti) n GMEP(0, f, A) = 0. Let F : H ^ H be a K-Lipschitzian and q-strongly monotone operator with positive constants K,q > 0 and Q : H ^ H an l-Lipschitzian mapping with constant I > 0. Let 0 < ^ < 2q/K and 0 < yl < t, where t = 1 -- - ^k2). Suppose [an] and {/n] are two sequences in

(0,1), {yn] is a sequence in (0,2S], and {Aniis a sequence in [a,b] with 0 < a < b < 1. For every n > 1, let Wn be the W-mapping generated by Ti,...,TN and Ani,An2,..., AnN. Given xi e H arbitrarily, suppose the sequences {xn] and {un] are generated iteratively by

& (un, y) + <p{y)-<p (un) + (Axn, y - un) + -{y-un,un -xn)>0, VyzQ

rn (11)

Xn+1 = anYQXn + Pnxn + {(1-Pn)1- anVF) Wnu^

\ln > 1,

where the sequences {an}, [pn], and {rn} and the finite family of sequences {Ani ^ satisfy the following conditions:

(i) limn = 0 and !Zl an =

(ii) 0 < lminfn^mpn ^ limsupn< 1

(iii) 0 < liminfn^mrn < limsupn^œrn < 28 and

n^^(rn+1 rn) °>

(iv) limn^m(An+u - Ani) = 0fori =l,2,...,N.

Thenboth {xn} and {un} converge strongly to x* = Pn(I-pF + yQ)x*, which is the unique solution in Q to the VIP

{(pF-yQ)x*,x* -x)<0, VxeQ. (12)

Let f : C ^ R be a convex and continuously Frechet differentiable functional. Consider the convex minimization problem (CMP) of minimizing f over the constraint set C

minimize [f (x) : x e C}. (13)

We denote by r the set of minimizers of CMP (13).

Next, recall some concepts. Let C be a nonempty subset of a normed space X. A mapping S : C ^ C is called uniformly Lipschitzian if there exists a constant L > 0 such that

\\Snx-Sny\\< L\\x-y\\, Vn>1,Vx,yeC. (14)

Recently, Kim and Xu [9] introduced the concept of asymptotically fc-strict pseudocontractive mappings in a Hilbert space as below.

Definition 1. Let C be a nonempty subset of a Hilbert space H. A mapping S : C ^ C is said to be an asymptotically k-strict pseudocontractive mapping with sequence {yn} if there exists a constant fc e [0,1) and a sequence {yn} in [0, ot) with limn^TOyn = 0 such that

\\Snx-Sny\\2 <(l+yn)\\x-yf

+ k\\x-Snx-(y-Sn y)\\2, (15)

Vn>1, Vx,yeC.

It is important to note that every asymptotically k-strict pseudocontractive mapping with sequence {yn} is a uniformly L-Lipschitzian mapping with L = sup{(fc +

^1 + (1- k)yn)/(1 + k) : n > 1}. Subsequently, Sahu et al.

[10] considered the concept of asymptotically fc-strict pseu-docontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.

Definition 2. Let C be a nonempty subset of a Hilbert space H. A mapping S : C ^ C is said to be an asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {yn} if there exist a constant k e [0,1) and a sequence {yn} in [0, ot) with limn^TOyn = 0 such that

limsup sup (\\Snx - Sny\\2 - (1 + y^ \\x - y\\2

X'y (16) -k\\x-Snx-(y-Sny)\\2)<0.

Put Cn := max{0, supx>v6C(ySnx - Sny\\2-(1+yn)\\x - yf -

k\\x-Snx-(y-Sny)\\2)}. Then cn > 0 (Vn > 1), cn ^ 0 (n ^ ot), and there holds the relation

\\Snx-Sn y\\2

< (1 + yn) ||* - y\\2 + k\\x - Snx - (y - Sny)\\2 + Cn, (17)

Vn>1, Vx,yeC.

In 2009, Sahu et al. [10] first established one weak convergence theorem for the following Mann-type iterative scheme:

x1 = x e С chosen arbitrary,

Xn+1 = (1 - an) Xn + ^"х» Vn > h

where 0 < S < an <1-k-S, ancn < txi, and ^^ yn < >x>, and then obtained another strong convergence theorem for the following hybrid CQ iterative scheme:

x1 = x e С chosen arbitrary,

Уп = (l-an)xn + ansnxn,

Cn = [zeC:\\yn -zll2 <||*n -42 +дп}, (19)

Qn = {z e С : (Xn -z,x-xn)>0}, Xn+1 = PC„nQ„ ^ Vn > h

where 0 < S < an < 1 - k, dn = cn + ynAn, and An = sup|y*n - z\\2 : z e Fix(S)} < tx>. Subsequently, the above iterative schemes are extended to develop new iterative algorithms for finding a common solution of the VIP and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense; see, for example, [11-13].

Motivated and inspired by the above facts, we first introduce and analyze one iterative algorithm by hybrid shrinking projection method for finding a solution of the CMP (13) with constraints of several problems: finitely many GMEPs, finitely many VIPs, the GSVI (8), and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. The iterative algorithm is based on shrinking projection method, Korpelevich's extragradient method, hybrid steepest-descent method in [7], viscosity approximation method, averaged mapping approach to the GPA in [14], and strongly positive bounded linear operator technique. On the other hand, we also propose another iterative algorithm by hybrid shrinking projection method for finding a fixed point of infinitely many nonexpansive mappings with the same constraints. We derive its strong convergence under mild assumptions. The results obtained in this paper improve and extend the corresponding results announced by many others.

2. Preliminaries

Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by {■, ■) and \\ ■ \\, respectively. Let С be a nonempty closed convex subset of H. We write xn ^ x to indicate that the sequence {xn} converges weakly to x and xn ^ x to indicate that the sequence {xn} converges strongly to x. Moreover, we use

ww(xn) to denote the weak «-limit set of the sequence {xn}; that is,

alternatively, T is firmly nonexpansive if and only if T can be expressed as

ww (xn) := {x e H : xn, ^ x for some subsequence

К} of {xn}\-

Recall that a mapping A: C ^ H is called

(i) monotone if

(Ax - Ay,x- y) > 0, Vx, y eC; (21)

(ii) ^-strongly monotone if there exists a constant q > 0 such that

(Ax - Ay,x - y) > q\\x - y\\2, Vx,yeC; (22)

(iii) a-inverse-strongly monotone if there exists a constant a > 0 such that

(Ax - Ay,x - y) > a\Ax - Ay\\ , Vx,y e C. (23)

It is obvious that if A is a-inverse-strongly monotone, then A is monotone and 1/a-Lipschitz continuous.

The metric (or nearest point) projection from H onto C is the mapping Pc : H ^ C which assigns to each point x e H

the unique point Pcx e С satisfying the property - Pcx\\ = inf - y\\ =: d (x, C).

Some important properties of projections are gathered in the following proposition.

Proposition 3. Forgiven x e H and z e C,

(i) z = Pcx & {x - z, y - z) < 0, Vy e C;

(ii) z = Pcx & ||x - z\\2 < ||x - y\\2 - \\y - z\\2, Vy e C;

(iii) {Pcx - Pcy, x-y)> \\Pcx - pcyt, Vy e H.

Consequently, Pc is nonexpansive and monotone.

If A is an a-inverse-strongly monotone mapping of С into H, then it is obvious that A is 1/a-Lipschitz continuous. We also have that if A < 2a, then I-XA is a nonexpansive mapping from С to H.

Definition 4. A mapping T : H ^ H is said to be

(a) nonexpansive if

ЦТх-ТуЦкЦх-y\\, Vx,yeH; (25)

(b) firmly nonexpansive if 2T - I is nonexpansive or, equivalently, if T is 1-inverse-strongly monotone (1-ism):

(x - y,Tx- Ту) > \\Tx - Ty\\2, Vx, у eH; (26)

T=±(I + S),

where S : H ^ H is nonexpansive; projections are firmly nonexpansive.

It can be easily seen that if T is nonexpansive, then I - T is monotone. It is also easy to see that a projection Pc is 1-ism. Inverse-strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.

Definition 5. A mapping T : H ^ H is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping; that is,

T = (1 - a) I + aS,

where a e (0,1) and S : H ^ H is nonexpansive. More precisely, when the last equality holds, we say that T is a-averaged. Thus firmly nonexpansive mappings (in particular, projections) are 1/2-averaged mappings.

Proposition 6 (see [15]). Let T:H ^ H be a given mapping.

(i) T is nonexpansive if and only if the complement I - T is 1/2-ism.

(ii) If T is v-ism, then, for у > 0, yT is v/y-ism.

(iii) T is averaged if and only if the complement I-T is v-ism for some v > 1/2. Indeed, for a e (0,1), T is a-averaged ifand only if I - T is 1/2a-ism.

Proposition 7 (see [15]). Let S,T,V : H ^ H be given operators.

(i) If T = (1 - a)S + aV for some a e (0,1) and if S is averaged and V is nonexpansive, then T is averaged.

(ii) T is firmly nonexpansive ifand only if the complement I-T is firmly nonexpansive.

(iii) If T = (1 - a)S + aV for some a e (0,1) and if S is firmly nonexpansive and V is nonexpansive, then T is averaged.

(iv) The composite of finitely many averaged mappings is averaged. That is, if each of the mappings [Ti]1=1 is averaged, then so is the composite T1,..., TN. In particular, if T1 is a1-averaged and T2 is a2-averaged, where a1,a2 e (0,1), then the composite T1T2 is a-averaged, where a = a1 + a2 - a1a2.

(v) If the mappings {Tt^ are averaged and have a common fixed point, then

ÖFix(T,) = Fix(T1,...,TN).

The notation Fix(T) denotes the set of all fixed points of the mapping T; that is, Fix(T) = {x e H : Tx = x}.

Proposition 8 (see [3]). Assume that & : C x C ^ R satisfies (A1)-(A4) and let f : C ^ R be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r > 0 and x e H, define a mapping T^v) :H ^ C as follows:

Т{У (x)= {zeC:©(z,y) + f(y)-f(z)

+ - {y - z,z - x) >0, Vy ec},

for all x e H. Then the following hold:

(i) for each x e H, T®'^ (x) = 0;

(ii) T®'^ is single-valued;

(iii) T®'^ is firmly nonexpansive; that is, for any x, y e H, ¡T^x - T^yf < {T^x - T^y, x-y); (31)

(iv) Fix(Tf'v)) = MEP(©, f);

(v) MEP(&, <p) is closed and convex.

We need some facts and tools in a real Hilbert space H which are listed as lemmas below.

Lemma 9. Let X be a real inner product space. Then there holds the following inequality:

||x + y||2 < \\x\\2 + 2(y,x + y), Ух,уеХ. (32)

Lemma 10. Let A : С ^ H be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 3(i)) implies

ue VI (С, A) ^^ u = Pc (u- XAu), \>0.

Lemma 11 (see [16, demiclosedness principle]). Let С be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on С. Then I-T is demiclosed. That is, whenever {xn} is a sequence in С weakly converging to some x e С and the sequence {(I - T)xn} strongly converges to some y, it follows that (I - T)x = y. Here I is the identity operator of H.

Let {Tn}'^1 be an infinite family of nonexpansive mappings on H and let {An}'^=1 be a sequence of nonnegative numbers in [0,1]. For any n> 1, define a mapping Wn on H as follows:

Un,n = AnTnUn,n+1 + (1 - An)I, Un,n-1 = An-1Tn-1Un,n + (1 - An-1 )I,

un,k = AkTkUnM1 + (l-A k

Un,k-1 = Ak-\Tk-lUn,k + (l - Ak-l) I,

Un,2 = A2T2Un,3 + (1-A 2)I,

Wn = uni =AiTiUn2 + (1-ai)i.

Such a mapping Wn is called the W-mapping generated by T^ T„-^ and ^ ...,Ai.

Lemma 12 (see [17, Lemma 3.2]). Let С be a nonempty closed convex subset of a real Hilbert space H. Let {Tn}'^l be a sequence of nonexpansive self-mappings on С such that Pi^=lFix(Tn) = 0 and let {An} be a sequence in (0, b] for some b e (0,1). Then, for every x e С and к > 1, the limit limn^mUnJcx exists, where Unk is defined as in (34).

Lemma 13 (see [17, Lemma 3.3]). Let С be a nonempty closed convex subset of a real Hilbert space H. Let {Tn}'^l be a sequence of nonexpansive self-mappings on С such that Pi^=lFix(Tn) = 0, and let {An} be a sequence in (0, b] for some b e (0, l). Then, Fix(W) = n™=lFix(Tn).

The following lemma can be easily proven, and, therefore, we omit the proof.

Lemma 14. LetV:H ^ H be a у-strongly positive bounded linear operator with constant у > 1. Then, for у - 1 > 0,

{(V -I)x-(V-I)y,x-y)>(y-

Vx, y eH.

That is,V - I is strongly monotone with constant y - 1.

Let С bea nonempty closed convex subset ofa real Hilbert space H. We introduce some notations. Let A be a number in (0,1] and let ^ > 0. Associating with a nonexpansive mapping T : С ^ H,we define the mapping Tx : С ^ H by

T x := Tx - A^F (Tx), Vx eC,

where F : H ^ H is an operator such that, for some positive constants K,q > 0,F is K-Lipschitzian and q-strongly monotone on H; that is, F satisfies the following conditions:

WFx - -

< KWx -

{Fx - Fy, x - y) >■,

for all x, y e H.

Lemma 15 (see [18, Lemma 3.1]). T is a contraction provided 0 < ^ < 2ц!к2 ; that is,

\\TxX - Txy\\ < (1 - Хт) -

4x,yeC, (38)

where r = 1 - - - ^k2) e (0,1].

Lemma 16 ([10, Lemma 2.5]). Let H be a real Hilbert space. Given a nonempty closed convex subset of H and points x,y,zeH and given also a real number a e R, the set

{v eC:\\y- v||2 < \\x - v||2 + (z, v) + a} (39)

is convex (and closed).

Recall that a set-valued mapping T : D(T) cH ^ 2H is called monotone if, for all x,ye D(T), f e Tx and g e Ty imply

(f-g,x-y)>°.

A set-valued mapping T is called maximal monotone if T is monotone and (I+XT)D(T) = H for each X > 0,where7 is the identity mapping of H. We denote by G(T) the graph of T. It is known that a monotone mapping T is maximal if and only if, for (x,f) e HxH, (f-g,x-y) > 0 for every (y,g) e G(T) implies f e Tx. Let A: C ^ H be a monotone, fc-Lipschitz-continuous mapping and let Ncv be the normal cone to C at V eC; that is,

NCV = {w e H : (v -u,w) >0, Vu e C}. (41)

Define

Av + Ncv, if v e C, 0, if v t C.

Then, T is maximal monotone and 0 e Tv if and only if v e VI(C, A).

Lemma 17 ([10, Lemma 2.6]). Let С be a nonempty subset of a Hilbert space H and let S : С — С be an asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {yn}. Then

\\Snx-Sny\\

(k\\x-y\\ + ^(l + (1-k)Yn)

+ (1-к)сП)

for all x,y e С and n> 1.

Lemma 18 ([10, Lemma 2.7]). Let С be a nonempty subset of a Hilbert space H and let S : С — С be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {yn}. Let {xn} be a

sequence in С such that \\xn-xn as h —> ^o. Then \\xn - Sxn

0 and \\x„-Snx„

Lemma 19 (demiclosedness principle [10, Proposition 3.1]). Let С be a nonempty closed convex subset of a Hilbert space H and let S : С — С be a continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {yn}. Then I-S is demiclosed at zero in the sense that if {xn} is a sequence in С such that xn ^ x e С and

limsupm^œlimsupn-

\\Xn -S"

= 0, then (I - S)x = 0.

Lemma 20 ([10, Proposition 3.2]). Let С be a nonempty closed convex subset of a Hilbert space H and let S : С ^ С be a continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence [yn] such that Fix(S) = 0. Then Fix(S) is closed and convex.

Remark 21. Lemmas 19 and 20 give some basic properties of an asymptotically fc-strict pseudocontractive mapping in the intermediate sense with sequence |yn|.

Lemma 22 (see [19]). Let С be a closed convex subset of a real Hilbert space H. Let {xn} be a sequence in H and и e H. Let q = Pcu. If {xn} is such that ww(xn) с С and satisfies the condition

then xn

\\xn - u\ < \u - q\\, Уп, q as n — œ>.

Lemma 23. Let H be a real Hilbert space. Then the following hold:

(a) \\x - yU2 = \\x\\2 - \\y\\2 - 2{x - y, y) for all x,ye H;

(b) \\Xx + py\\2 = X\\x\\2+^\\y\\2-X^\\x - y\\2 for all x, y e

H and X,^e[0,1] with X + ^ = 1; (c) if {xn} is a sequence in H such that xn

limsup\\*n

= limsup\\xn - x\\

+ \\x -

x, it follows

, VyeH. (45)

0 as n

3. Convex Minimization Problems with Constraints

In this section, we will introduce and analyze one iterative algorithm by hybrid shrinking projection method for finding a solution of the CMP (13) with constraints of several problems: finitely many GMEPs, finitely many VIPs, GSVI (8), and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. This iterative algorithm is based on shrinking projection method, Korpelevich's extragradient method, hybrid steepest-descent method in [7], viscosity approximation method, averaged mapping approach to the GPA in [14], and strongly positive bounded linear operator technique.

Theorem 24. Let С be a nonempty closed convex subset of a real Hilbert space H. Let M, N be two integers. Let f : С — R be a convex functional with L-Lipschitz

continuous gradient V/. Let &k be a bifunction from CxC to R satisfying (A1)-(A4) and let фк ■ С ^ R U be a proper lower semicontinuous and convex function, where к e {1,2,...,M}. Let Bk, A{ ■ H ^ H, and Fj ■ С ^ H be ^k-inverse-strongly monotone, qrinverse-strongly monotone, and £j-inverse-strongly monotone, respectively, where к e {1,2,...,M}, i e {1,2,...,N}, and j e {1,2}. Let S ■ С ^ С be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense for some 0 <k < 1 with sequence lyn} с [0, rn) such that limn^TOyn = 0 and {cn} с [0, rn) such that limn^TOcn = 0. Let V be a y-strongly positive bounded linear operator with у > 1. Let F ■ H ^ H be a к-Lipschitzian and ц-strongly monotone operator with positive constants к,ц > 0. Let Q ■ H ^ H be an l-Lipschitzian mapping with constant I > 0. Let 0 < ^ < 2ц1к

and 0 < yl < т, where r = 1 - - ^(2ц - ^к2). Assume that

П ■= nM=1GMEP(&k,<pk,Bk) n n1=1VI(C,Ai) n GSVI (G) П Fix(S) n Г is nonempty and bounded and that either (B1) or (B2) holds. Let 0 < a < an <1, k < Sn < d < 1 for all n> 1, and let lpn}, {an} be sequences in (0,1]. Pick any x0 e H and set C1 = C, x1 = PCi x0. Let {xn} be a sequence generated by the following algorithm:

Un =1rUin (i-rM,n"MJ * rM-hn

x(l- Гм-шBm-1 ) ■ ■ ■ t^1'91) (I - rl,nBl) Xn, Vn = Pc (I- AN,n An) Pc (I - AN-1,n An-1 J ■■■PC x(I-A2,nA2JPc (I-KnA 1)u^

Zn = pn*n + ((l-pn)l-snV)Tn GVn

+ Sn [Tn*n - ^n (TnXnJ - YQXn)] , (46)

К = SnZn + (l-5n )S"Zn, Уп =(l-an )xn + "n^

Cn+1 = {zeCn ■ bn - zf < \\xn - ZU2 + en} ,

Xn+1 = PC„+1 x0,

Vn> 1,

wherePc(I-XnVf) = snI+(l-sn)Tn (here Tn is nonexpansive; sn = (2 - XnL)/4 e (0,1/2) for each Xn e (0,2/L)), вп = (sn + Yn)(l + Yn)^n + cn, and An = suplH^ - pt + (IIU -V)p\\ + II(yQ - Иf)pii)2/(y - 1) : p e Q} < rn. Suppose that the following conditions are satisfied:

(i) sn e (0,1/2)foreachXn e (0,2/L), limn^œsn = 0 limn^œ К = 2/L);

(ii) lrknJ с [ek,fk] с (0,2^k),{X,J с [a,,b,] с (0,2ц) and Vj e (0,2(j), where k e {1,2, ...,M}, i e ll,2,...,N}, and j e {1,2};

(iii) 0 < liminfn^mßn < limsupn ^mßn < 1

Then one has the following:

(I) [xn] converges strongly as Xn ^ 2/L (^ sn ^ 0) to x = Pnxo;

(II) [xn] converges strongly as Xn ^ 2/L (^ sn ^ 0) to x* = Pnx0 provided \\xn - zn\\ = o(sn) and limn^œ<rn = 0, which is the unique solution in Q to the VIP

{(I-V)x*,p-x* )<0, ypeQ. Equivalently, x* = Pn(2I - V)x*.

Proof. Since V/ is L-Lipschitzian, it follows that V/ is l/L-ism. By Proposition 6(ii) we know that, for X > 0, XVf is l/AL-ism. So by Proposition 6(iii) we deduce that I - XVf is AL/2-averaged. Now since the projection Pc is l/2-averaged, it is easy to see from Proposition 7(iv) that the composite Pc(I - XVf) is (2 + AL)/4-averaged for X e (0,2/L). Hence we obtain that, for each n> l, Pc(I - XnVf) is (2 + XnL)/4-averaged for each Xn e (0,2/L). Therefore, we can write

Pc (1-bnVf) =

2 - XnL 2 + XnL -I +

= SnI + - Sn) Pn

where Tn is nonexpansive and sn := sn(Xn) = (2 - XnL)/4 e (0,1/2) for each Xn e (0,2/L). It is clear that

As \imn^msn = 0 and 0 < liminf< limsupn^TO^n < 1, we may assume, without loss of generality, that {pn} c [a, a] c (0,1) and pn + sn||V|| < 1 for all n> 1. Since V is a y-strongly positive bounded linear operator on H, we know that

IIVII = sup {{Vu, u) :ueH, HuH = 1} > y > 1. (50) Taking into account that + sn||V|| < 1 for all n> 1, we have

(((1 -ßn)l- SnV) u,u) = \-ßn -Sn (Vu, U)

>\-ßn -Sn ||V|| > 0;

that is, (1 - ßn)I - snV is positive. It follows that

\\(l-ßn)l-SnV\\

= sup {(((l-ßn)l-snV)u,u) : и e H,IIu\I = 1} = sup {l - ßn - sn (Vu, и) :u e H, ЦиЦ = 1}

<1-ßn - snY.

Akn = TZ'fk) (l-rknBk)T%-^

■ (I-rk-mBk^-li^ (1-r^Xn

for allk e [1,2,...,M] andn > 1 and

A'n = Pc (I- hnBt) Pc (I - A ,-^B,-!) ...pc (I- X 1>nB1)

for all i e {1,2,..., N], A0 = I, and A0 = I, where I is the

15 5 5 J J n J n r

identity mapping on H. Then we have that un = AMxn and V„ = ANn un.

We divide the rest of the proof into several steps.

Step 1. We show that {xn] is well defined. It is obvious that Cn is closed and convex. As the defining inequality in Cn is equivalent to the inequality

(2(Xn -yJ)Z><|M2 -\\yn ||2 + en, (55)

by Lemma 16 we know that Cn is convex for every n > 1.

First of all, let us show that Q c Cn for all n> 1. Suppose that Q c Cn for some n > 1. Take p e Q arbitrarily. From (46) and Proposition 8(iii), we have

IK -p\\\

_\\r(&M-fM) (T-r R ) AM-1 V ~\\1rM,„ (1 rM'n °M)An Xn

-T?z(l-rM,nBm)Am-1 P\\

< \\(7 - rM„Bm) AM-1 xn - (I - rM,nBm) AM-1 p\\

w " (56)

< \\AM-1 -AM-1 P\\

= K -P\1

Similarly, we have \k -p\\

= H^c (I - *N,n An) AN-1 Un -Pc (I- XN,n An) ANn-1 p\\ < HC - *N,nAn) ANn-1 un - (I - Xn,„An) ANn-1 p\\

<\\AN-1 -AN-1 p\\

<\Kun -A0np\\

= k- P\\ .

Combining (56) and (57), we have

\K - p\\ < \\xn - p\\. (58)

Since p = Gp = PC(I - v1F1)Pc(I - v2F2)p, Fj is Cj-inverse-strongly monotone for j = 1,2, and 0 < < for j = 1,2, we deduce that, for any n > 1,

H^» -p\\2

= \\PC (I- V1F1) Pc (I- V2F2) Vn

-Pc (I- V1F1 )PC (I- V2F2)p\\2 <H(7- V1F1)PC (I- V2F2) Vn

-(I- V1F1) Pc (I- V2F2 )p\\2 = \\[PC (I- V2F2) Vn -pc (I- V2F2)p]

- ] [F1PC (I - V2F2) vn - F1PC (I - V2F2) p] \\2

< \\PC (I - V2F2) vn -Pc (I- V2F2) p\\2 (59)

+ ]1 (]1 -2Q

x \\F1PC (I - V2F2) vn - F1PC (I - V2F2) p\\2 <\\PC (I- V2F2) Vn -Pc (I- V2F2)p\\2 <H(J- ^2) Vn -(I- V2F2)PH2

= \\(Vn -P)- ]2 (F2Vn -P2P)H2

< W -P\\2 + V2 (]2 -2(2)^ -F2 p\\2

< Wv„ -p\\2.

Utilizing Lemma 15, from (46), (52), (58), and (59), we obtain that

K - pW

= \\pn (xn -p) + ((1-l3n)l- SnV) (TnGVn - p)

+ Sn [T„Xn - (^P (T„Xn) - yQx„) - p]

+ (I-V)p\\ = \\P„ (xn-p) + ((1-pn)l-snV) (TnGVn - p)

+ Sn WnY (QXn - Qp) + (I- anVF) T„Xn

-(l-an^F)Tnp] + sn [(I-V)p + an (yQ-i4F)p] \\ < Pn H*« - P\\ + \\(1 -Pn)l- s„VH \\TnGVn - p\\

+ Ky HQ*«- qpW

+ HU - On^P) TnXn -(I- On^P) Tnp\\] + sn \\(I-V)p + an (YQ-^F)pH

<k \\xn-p\\ + (1-pn -SnY)\\GVn -p\\ + Sn KYl \\Xn -P||+(l- °nT) \\xn - p\\]

+ s» (||(/-V)p|| + ||(rQ-pF)p||)

-fk(l-£n - sm7) | | vn

+ (||(/-V)p|| + ||(rQ-^F)p||) -p|| + (l-^n - VF^n -f||

+ *» (l-^n fr-^«*» -p||

+ s» (||(/-V)p|| + ||(rQ-pF)p||)

=(1 - vF) k- p|| + s» (1 - ^n- yz)) k - p|| + s» (||(/-V)p|| + ||(rQ-^F)p||)

< (1-Sny)||^n -p||+S»||*n -p|| + *» (||U-V)p|| + ||(rQ-pF)p||)

= (l-5n (y-l))||Xn -p||

+ *» (||(/-V)p|| + ||(rQ-^F)p||)

= (l-5n (y-l))||Xn -p||

II(J-y)p|HI(rQ-^)p||

+ s» (y-l)

which hence yields

»-p||2

<(l-S» (y-l))||x» -p||2 + 5» (y-l)

x(I|(i-v)p|HI(rQ-^)p||)2

x -ö-

(y-l)2

2 (k-^IMI(yQ-^)p||)2

< K - P« + S»

By Lemma 23(b), we deduce from (46) and (61) that k -p||2

= k (z» -p) + (l-a»)(s"z» -p)||2

= 5»||Z» -P||2 + (l-5»)||Snz» -P||2

-5n (l -5n)||zn -s»Znf

< ^n||z» -p||2 + (l-<y

x [(l + y») k - p||2 + fck- s»z»||2 + c»] -5» (l-a»)||z» -S»z»||2

= [l+y» (l-5n)]|K -p||2

+ (l-5n)(fc-5»)||z» -S»z»||2 + (l-a»K

<(l + y»)||z» -pf + (l-5n)(fc-5»)||Z» -S»Z»||2 +c»

<(l+y»)||z» -p||2 +Cn

<(l + y»)( k

(J-vMI + IKyQ-^II)2 y-l

+ c„.

So, from (46) and (62) we get

k -pii2

= ||(l-«J (*n -P)+«n ft, -P)||2

< (l-«J||*n -pf + «n||*n -pf

< (l - an) k -pf + «»

(l+y»)( K -p||2

a-y)p|HI(rQ-^)p||)2

+ C„

< (l + y»)

x (k- pII +s»

2,_ (k-^IHI(yQ-^)p||)2

+ C„

= k - pf + y»||^n - pf + (l + y») Sn

x(Ila-^)p|HI(yQ-^)p||)2 +r x f-l +c»

< k - pii2 + y»(l + y») k- pII2 +s»(l + y»)

x (II(J-^)^H + II(yQ-^^)^H)2 +r X f-l +C»

< II*»- pf + (y» + O (l + yJ II*»- pf

+ (*» + y») (l + y») ■

U-y)p|HI(yQ-^)p||)2 y-l

+ C„

= ||*n -pf + (*n + y»)(l+y»)

x(||*» - p«2 + (Ila-^)pIHKyQ-^)pII)2

+ C„

< \\xn - pf + (sn + Yn) (l + Yn)&n + cn = \\Xn -p\\2+®n>

where dn — (sn + yn)(1 + yn)A n

(im - V)p\\ + ||(yQ - Vp)p\\)2/(y -1):peQ}<Hence p e Cn+1. This implies that Q c Cn for all n > 1. Therefore, {xn} is well defined.

Step 2. We prove that \\xn - kn\\ ^ 0, \\xn - zn|| ^ 0 and \\Snzn - zn\\ ^ 0 as n ^ rn.

Indeed, let x* — Pnx0. From xn — Pc x0 and x* e Q c Cn, we obtain

Xn *0|| - \\X X0

This implies that {xn} is bounded and hence {un}, {vn}, {zn}, {kn}, and {yn} are also bounded. Since xn+1 e Cn+1 c Cn and xn — Pc x0, we have

xn -Xo\\<\\xn+1 -Xo\\, Vn>l.

Therefore limn^m\xn - %0H exists. From xn — Pc x0,xn+1 e Cn+1 c Cn, by Proposition 3(ii), we obtain

\\Xn+1 Xn\ < \\xo Xn+l\\ \\xo xn\ , (66)

which implies

lim \\Xn+l - xn\\ = 0.

It follows from xn+1 e Cn+1 that \\yn -d„ and hence

\\Xn yn\

<2(\\Xn - xn+lt + \\Xn+1 -ynf)

<2{\\Xn - Xn+l\\2 + \\Xn -Xn+lt + \

— 2(2\\xn -xn+1f + 0n). From (67) and limn^mdn — 0, we have

Am, \\x»- y»\\— 0 (69)

Since yn - xn — an(kn - xn) and 0 < a<an < 1,we have

a \\kn - xn\\ < an \\kn - xn\\ — \\yn - xn\\, (70)

which immediately leads to

hm \\kn -xn\\ = °.

Also, utilizing Lemmas 9 and 23(b) we obtain from (46), (58), (59), and (62) that

\k -p\\2

= \\ßnxn + ((l-ßn)l-SnV)TnGvn

+Sn \JnXn °n № (Tnxn) - yQxn)] - p\\

= \\ßn (xn -p) + (l-ßn)(TnGvn -p)

+Sn \TnXn - °n № (TnXn) - yQXn) - VTnGvn]\\

<\\ßn (xn -p) + (l-ßn)(TnGvn -p)\\2

+ 2sn (T„xn - № (Tnxn) - yQxn) - VTnGvn, zn - p) = ßn\\xn -pf + (l-ßn)\\TnGvn -pf -ßn (l-ßn)\\xn -TnGvn\\2

+ 2sn (T„xn - № (Tnxn) - yQxn) - VTnGvn, zn - p)

< ßn\\xn -pf + (l-ßn)\\Gyn -p\\2 -ßn (l-ßn)\\xn -TnGvn\\2

+ 2sn \\T„Xn - °n № (TnXn) - YQXJ - VT„Gvn\\ X \\zn - p\\

< ßn\\Xn -pf + (l-ßn)\K -pf

-ßn (l-ßn)\\xn -TnGvn\\2

+ 2sn \\T„Xn - °n № (TnXn) - YQXJ - VT„Gvn\\ X \\Zn - p\\

< ßn\\Xn -P\\ +(l-ßn)\\xn -Pi

-ßn (l-ßn)\\xn -TnGvn\\2

+ 2sn \\T„Xn - °n № (TnXn) - YQXJ - VT„Gvn\\ X \\zn - p\\

= \\*« -Pf -ßn (l-ßn)\\xn -TnGvn\\2

+ 2sn \\T„Xn - № {T„Xn) - YQXJ - VT„Gvn\\ X \\zn - p\\ '

and hence

bn -Pf

< {1-«n)\\Xn -p\\2 + «J\K -pf

<(1- On) \\Xn - p\\2 + «n [(1 + Yn) \\Zn - p\\2 + Cn]

< {1-«n)\\Xn -p\\2 + «n

Journal of Applied Mathematics x [(1 + Yn)

-p\\2 -Pn (l-pn)\K -TnGVnW2

+ 2sn \\TnXn - °n (VF (TnXn) - YQxJ - VTnGVn X \\zn -p\\)+ Cn] < (1-an)\\Xn -pf + «n (1 + 7n)

X (\xn -p\\2 -Pn (1-Pn)\\xn -TnGVn\\2

+ 2sn \\TnXn - °n (VF (TnXn) - YQxJ - VTnGVn X \\zn -p\\) + Cn <(1- «J \\Xn - p\\2 + «n (1 + Yn) \\Xn - p\\2

-an (1+Yn)Pn (1-Pn)\\xn -Tn GVn\\

+ (l + Yn)2Sn

X \\TnXn - °n (VF (TnXn) - YQXn) - VTnGvn X \\Zn - p\\ + Cn

<(1 + Yn) \\xn - p\\2 -On (1 + Yn) Pn (1 - Pn)

X Pn - TnGvn\\

+ 2sn (1+Yn)

X \\TnXn - °n (VF (TnXn) - YQXn) - VTnGvn\\ X \\Zn - p\\ + Cn-

So, it follows that

a(l + Yn)a(1-a)\\xn -rPnGvj\

<an (1 + Yn)pn (1-Pn)\\xn -TnGVn\\2

< \\xn -pf -bn -p\\2 + Yn\\Xn -p\ + 2Sn (1+Yn)

X \\TnXn - °n (VF (TnXn) - YQXn) - VTnGvn\\ X \\zn - p\\ + Cn

< \\xn - yn\\ (\\Xn - p\\ + \\yn - Pd + Yn\\Xn - pf + 2Sn (1+Yn)

X \\TnXn - °n (VF (TnXn) - YQXn) - VTnGvn\\

Since limn^mSn = 0' limn^m7n = 0 and lim n^rncn = 0,

it follows from (69) and the boundedness of {xn}' {yn}' {zn}, and {vn} that

lim \\xn - T^v^ = 0.

Note that

\\Zn - Xn\\

= \\(1-l3n)(TnCVn -xn)

+Sn (TnXn - °n (VF (TnXn) - YQXn) - VTnGvn] <(1-pn)\\TnCVn -Xn\\

+ Sn \\TnXn - °n (VF (TnXn) - YQXn) - VTnGvn < \\TnGvn - Xn\

+ Sn \\TnXn - °n (VF (TnXn) - YQXn) - VTnGvn

Hence, it follows from (75) and limn^TOsn = 0 that

lim \\xn - z„\\ = 0.

Note that

\\kn - Zn\\ < \\kn - xn\\ + \\xn - Zn\\ - (78)

Thus, we deduce from (71) and (77) that

A™, \\kn - zn\\ = 0- (79)

Since kn - zn = (1 - Sn)(Snzn - zn) and k < Sn < d < 1,we have

(1 - d) \\snzn - zn\\ < (1 - sn) \\snzn - Zn\ = \\kn - Zn\\ '

which, together with (79), yields

lim \\Snz„ - zJ = 0.

Step 3. We prove that \\xn - un\\ ^ 0, \\xn - vn\\ ^ 0, \\vn -gvJ\ ^ 0, \\Vn -Pc(I-(2/L)Vf)Vn\\ ^ 0,and \\Zn -SzJ ^ 0 as n ^ >x>.

Indeed, from (57), (59), y > 1, and yl <r it follows that

\\zn - p\\

X \\Zn - p\\ + Cn -

= \\Pn (Xn -p) + ((1-pn) I- SnV) (TnCVn - p) + Sn [°nY (QXn - Qp) + (I- TnXn

-(i-On^F)TnP]

+n [(I-V)p + an (yQ-^F)p] ||2

* Wßn (*n-p) + ((l-ßn)l-SnV) (TnGVn - p)

+ Sn WnY (Qxn - Qp) + (l- TnXn -(i-WF)TnP] ||2 + 2Sn ((I-V)p + an (yQ-^F)p,Zn - p)

*[ßn\W -p\\ + \\(l-ßn)l-*nV\\ WTnGVn -p\\ + Sn (onY \\QXn -

+ W(l-anVF)TnXn -(l-anVF)TnP\\)]2 + 2Sn ((I-V)p + an (yQ-^F)p,Zn - p)

* [ßn \K -p\\ + (l-ßn -SnY)\\GVn -p\\

+Sn {(JnYl \\Xn - p\\ + (1 - Ont) \\Xn - p\\)]2 + 2Sn ((I-V)p + On (yQ-^F)p,Zn -p)

= [ßn \K -p\\ + (l-ßn -SnY)\\GVn -p\\ +Sn (1-On (r-yl))\\xn -p\\]2 + 2Sn ((I-V)p + On (yQ-^F)p,Zn -p)

* [ßn \K -p\\ + (l-ßn -SnY)\\GVn -p\\

+Sn\\Xn -p\\]2 + 2Sn ((I-V)p + On (yQ-^F)p,Zn -p)

* [(ßn + SnY) \K - p\\ + (l-ßn - SnY) \\GVn - p\\]2 + 2Sn ((I-V)p + On (yQ-^F)p,Zn -p)

* (ßn + SnY) \K - p\\2 + (l-ßn - Snf) \\GVn - pf + 2Sn ((I-V)p + On (yQ-^F)p,Zn -p)

* (ßn + SnY) K - ft + (l-ßn - SnY) \K - pf + 2Sn ((I-V)p + On (yQ-^F)p,Zn -p)

* (ßn + SnY) K - pf + (l-ßn - SnY) \K - p\\2 + 2Sn ((I-V)p + On (yQ-^F)p,Zn -p).

Next let us show that

For p e Q,we find from (46) that

K*. -p||2

= K'^U - ^k )Ak-lxn - Tty* ^ -

= \\(l-rknnBk)Akn1Xn -(l-rknnBk)pf

* ¡A^Xn - pf + rkn (rkn - 2^k) ¡Bk^Xn - Bkpf

* \W - p\\2 + rk,n (rk,n - 2^k) ¡BkA^Xn - Bkpf.

By (56), (82), and (84), we obtain

\\zn -p\\2

* (ßn + SnY) K - p\\2 + (l-ßn - SnY) \K - p\\2 + 2Sn ((I-V)p + On (yQ-^F)p,Zn -p)

*(ßn + SnY)\\Xn -p\\2 + (l-ßn -SnY)

x\K*n -pf

+ 2Sn ((I-V)p + On (yQ-^F)p,Zn -p) *(ßn + SnY)\\xn -p\\2 + (l-ßn -SnY)

* [IK - p\\2 + rkn (rkn - 2^k) ¡BkA^Xn - Bkpf ] + 2Sn ((I-V)p + On (yQ-^F)p,Zn -p)

= \\Xn - p\\2 + (l-ßn - SnY) rk,n (rk,n - 2Vk)

* ¡BkA^Xn - Bkpf

+ 2Sn ((I-V)p + On (yQ-^F)p,Zn -p),

which immediately yields

(1 -a-sny) rk„ (2^k - rkn) ¡BkA^x» - Bkpf

^{1-Pn - sn y) rkn {2Vk - rkn) ¡BkAknlxn - Bkpf

*\\Xn -p\\ -\\Zn -p\\

+ 2Sn ((I-V)p + an (yQ - pF) p, Zn - p) * \K -Zn\\(\Xn -P\\ + Wzn -P\0 + 2Sn (\\(I-V)p\\ + \\(yQ-vF)p\\)\\Zn -p\\.

lim \\x„ -u„\\ = 0.

Since limn^mSn = 0, [rKn\ c [ek, fk] c (0,2^), and {xJ and [zn] are bounded sequences, it follows from (77) that

nKm ¡Bk Akn-1xn - Bkpj = 0, k = 1,2, ...,M. (87)

By Proposition 8(iii) and (46), we have

K*. -^Il2

= IK:^ (! - r^nBk) Akn-1xn - T^ (I - rk,nBk) pf Z ((I - rk,nBk) Aknlxn -(I- rknBk) p, AknXn - p) = 2 (IIU - rk,nBk) Aknlxn -(I- TknBk) pf + ¡AknXn - pf -||(7 - rknBk) A^Xn -(I- rkn,Bk) p - (Aknxn -

^Kk1 xn -p(+k* -p|2

-¡Akn-1Xn - AknXn - rkn,(BkAkn-1Xn - BkP)f] ,

which implies that

¡A".* -p(

z K1*, -p|2

- ¡A^Xn - AknXn - rk,n (BkAknlXn - Bk p)f

|| a k-1 l|2 \ a k-1 a k ¡2

= IIAn Xn -p\ -IAn Xn -Anxnll -rln\BkAkn1xn -BkPlf + 2rk,n (A^Xn - Aknxn, BkAkn-1Xn - Bkp)

^ ¡Uk-1 j12 ¡a k-1 a k ¡2

Z ||An Xn -P\ - ||An Xn -Anxnll

+ 2rk,n\AknlXn - AknXnII \BkAknlXn - Bkph

^ II ^¡2 \a k-1 a k \2

Z Fn -P\I - ||A n Xn -AnXnI

+ 2rKn \\Akn-1xn - Aknxn\\ \^BkAkn-1xn - Bkp\\. From (82) and (89), we have

\\zn -pf

Z (ßn + SnY) \\Xn - p\\2 + (l-ßn - SnY) \K - pf

+ 2Sn ((I-V)p + an (yQ-^F)p,Zn - p)

< (ßn + SnY) \\Xn - p\\2 + (l-ßn - w) \\AknXn - pf + 2Sn ((I-V)p + an (yQ-^F)p,Zn - p)

< (ßn + SnY) \\Xn - p\\2 + (l-ßn - SnY)

xhx. -p\\2 -K-1 Xn -AknXnf

+ 2Sn ((I-V)p + On (yQ-^F)p,Zn -p) z IK - p\2 -(1-Pn - ^nY) ¡Akn-1Xn - Aknxn\ + 2rk,n ¡A^Xn - AknxJ\ ¡BkAkn-1Xn - Bkpl

+ 2$n ((I-V)p + On (yQ-^F)p,Zn -p), which leads to

(l-a-Sn y)lAkn1Xn -AknXnf

z(l-pn -Sn Y)\Akn-1Xn -Aknxnf

z IK -p\\2 -IK -p\2

+ 2rk,n Kk - AknXn\ ¡BkAkn1xn - Bkp\l (91)

+ 2Sn ((I-V)p + an (yQ - pF) p, z. - p)

z IK -Znll(lxn -pll + llzn -pll)

+ 2rk.n\AknlXn - AknXn\ ¡BkAkn-1Xn - Bkpl

+ 23. (II(I-V)pII + II(yQ-vF)pI\)IIZn -p\.

Since lim.^mSn = 0, {rk.} c [e^fk] c (0,2^k), and {x.} and {zn} are bounded sequences, it follows from (77) and (87) that

lim \\Akn 1Xn - AknxJ\ = 0.

n^m \ n n \

Hence we obtain from (92) that

\\ \\ \U 0 aM

\K -Un\\ = \\AnXn -An Xn\\

< \A°nXn - A\Xn\ + \A\Xn - A\Xn\

+ ---\\AM-1 X -AMX \\ + \\An Xn An Xn\

0 as n —> x.

+2rkn \\Akn-1Xn - AknxJ\ \\BkAkn-1Xn - Bkp\ ]

That is, (83) holds.

Next we show that limn^TO\\AiMnun - Atp\\ = 0, i = 1,2,... ,N. As a matter of fact, observe that

\\A"nUn -p(

= \\pc (I - X hnA) A -1Un -pc (I- X hnA) pf

<\\(I-XtnnA,)A-1Un -(I-XhnA,)pf

A,-1Un - pf + A(A- 2n) \\AAn1 un - A,pf

n - p\2 + A i,n in - 2m) ¡AiK^n - Aipf xn - p\\2 + A,,n (x,,n - 2nd \\A,A'n1un - A.pf.

Combining (57), (82), and (94), we have

IK -pf

z (Pn + sny) \\xn - pf + (\-pn - sny) \\vn - p\\2 + 2sn ((I-V)p + an (yQ-^F)p,Zn - p)

z (Pn + Sny) \\Xn - pf + (1-Pn - *nï) KUn - pf

+ 2Sn ((I-V)p + an (yQ-^F)p,Zn - p) z(Pn + Sny)\\xn -p\\2 + (1-Pn -SnY) (95)

* \\K - p\\2 + A,,n (*■,,n - 2nd \\AA'n1un - A,pf] + 2Sn (\\(I-V)p\\ + \\(yQ-vF)p\\)\\Zn -p\\

= \\Xn - p\\2 + (l-Pn - SnY) Ài,n i,n - W *||A ft~nUn -Aipf

+ 2sn (\\(I-V)p\\ + \\(yQ-vF)p\\)\\Zn -p\\, which leads to

(1 -â-Snf) Xm (2m - Xtn) ¡A,A'n1Un - A,pf z(\-pn - SnY) Ao, (2m - Ahn) \\A,A'n1Un - A,pf

z\\xn -p\\2 -\K -pf (96)

+ 2sn (\\(I-V)p\\ + \\(yQ-vF)p\\)\\Zn -P\\

z \\Xn -Zn\\(\Xn -p\\ + \K -p\\)

+ 2Sn (\\(I-V)p\\ + \\(yQ-vF)p\\)\\Zn -p\\.

Since limn^œsn = 0, [Xin} c [ai,bi] c (0,2^i), and [xn] and [zn] are bounded sequences, it follows from (77) that

nlimo W^'n^n -Atp\\ = °, i=l,2,...,N. (97) By Proposition 3(iii) and Lemma 23(a), we obtain

№n"n -pf

= ||pc (I - X hnA) A 'nlUn -pc (I- X hnA) pf z ((I - X hnA) A1-1 Un -(I- X hnA) p, A\un - p) = 2(\\(I- A ,,nA,) A'n1Un -(I- X hnA,) pf + \\A'nun - pf -\\(I-XhnA,)A'-1Un -(I-X,-a,)p

Journal ofApplied Mathematics zi(|A'-1un -pf + \\A<nun -pf

-WK^n - - A , ,n {AA'nUn - Atp)f)

z2,(\K -p\\2+|a"n»n -pf

-WK^n - - A o, {AtA'n1Un - Atp)f ) z2(\\xn -p\\2+|a"n»n -pf

-WK^n - - A , ,n {AA'nUn - Atp)f) ,

which implies HA'nUn -pf

z \\xn - pf - IIA i-1Un - A^Un - XUn {Afi^Un - Atp)f

= \\Xn - pf - \Klun - ^nUnf

-*l\\AtA'-1Un -A,p\\

+ 21 ,,n (K1un - A'nun, A,A'n1Un - A,p)

z \\Xn - pf - \Klun - ^nUnf

+ m \\K1un - Kun\\ WA.K1"- - A,p\\. Combining (57), (82), and (99), we have

\\zn -p\\2

z (Pn + SnY) \\Xn - p\\2 + (1-Pn - SnY) Ik - pf + 2s- ((I-V)p + an (yQ-^F)p,Zn - p)

z (Pn + SnY) \\Xn - p\\2 + (1-Pn - SnY) \\A'n"n - pf

+ 2s- ((I-V)p + an (yQ-^F)p,Zn - p) z (Pn + SnY)WXn -p\\2 + (1-Pn - SnY) * \\\*n -p\\2 -||A i-1Un -^n"nf

№-rf)

+2XUn ¡k1"- - ^nUn\\ lAiK'Un - AiP\\ + 2s- ((I-V)p + an (yQ-^F)p,Zn - p) z \\xn - pf -(1-Pn - SnY) ik1»- - A'nUn||2 + 2Xm \\K1un - A'nun\\ \\A,K1un - A,p\\

+ 2s- ((I-V)p + an (yQ-^F)p,Zn - p) ,

which yields

(1 - fl-s„y)||A,-1M„ - A>„||2

<(l-& -^»y)||A;1«„ -A>„||2

<k -pf -k -pf + 2A* ¡a;1 M„ - A>„|| | | A,A,-1M„ - A,p|| + 2s„ ((7 - V) p + a„ (yQ - pF) p, zn - p) <K -Z„||(||X„ -p|| + ||z„ -p||) + 2A,,„ ¡a;1 m„ - A>„|| | | A,A,-1M„ - A,p|| + 2s„ (||(/-V)p|| + ||(rQ-^F)p||)||Z„ -p||.

Since = 0 and |zn|, and {«„} are bounded,

from (77) and (97) we get

lim ||a,(j1m„ -a,km„|| = 0. h^TO II " " " "II

From (102) we get

N N Na 0 ,N

II"« - VJ = ||A A - A „ "J

< ||aa - A A|| + ||a A - aAn

+ - + IK"1«„ -A?«„!! 0

as n —> x.

Taking into account that ||%„ - vj| < ||%„ -m„|| + ||m„ - vj,we conclude from (83) and (103) that

lim ||%„ - vJ = 0.

K^ to 11 " ""

On the other hand, for simplicity, we write p = Pc(7 -^^ V„ = PcU - and = Gv„ = Fc(i - ]1^1)V„

for all n > 1. Then

p = Gp = Fc (7 - ViFi) p = Fc (7 - ViFi) Fc (7 - v2F2) p.

We now show that limn^TO||Gvn - vj| = 0; that is, - = As a matter of fact, for p 6 ^ it follows from (58), (59), and (82) that

x||GV„ -p||2

+ 2s„ ((7-V)p + a„ (yQ-^F)pZ„ - p)

=(A, + vF)k- pii2 +(1 - A, - s»y) x IK -P||2

+ 2^„ ((7-V)p + a„ (yQ-^F)pZ„ - p)

<(A, + s„y)||*„ -p||2 + (1-A, -^»y)

x[||v„ -p||2 + ] (] -2Ci)||FiV„ -Fip||2] + 2s„ ((7-V)p + a„ (yQ-^F)p,Z„ - p)

< (A, + ^k -pn2 + (1-A, -^»y)

x [||v„ -p||2 + ]2 (]2 - 2C2) ||f2vk -F2PH2 +] (] -2Ci)||FiV„ -Fipf]

+ 2s„ (||(J-V)p|| + ||(rQ-pF)p||)||Z„ -p|| <(A, + s„y)||*„ -pii2 + (1-A, -^»y) x [ ||*„ -PII2 + ]2 (]2 -2C2)||*2V„ -^2p||2

+] (] -2Ci)||FiV„ -Fipf] + 2s„ ( | | (7 - V) p | | + | | (yQ - ^F) p | | ) | | zn -p||

= k -pii2 + (1-A, -s»y)

x[]2 (]2 - 2C2) ||F2V„ -F2PII2

+] (] -2Ci)||FiV„ - Fip||2] + 2s„ ( | | (7 - V) p | | + | | (yQ - ^F) p | | ) | | zn -p||,

which immediately yields

(l-fl-S„y)[V2 (2C2 - V2)||^2V„ -F2PII2

+] (2Ci - Vi)||FiV„ -Fipf] <(l-j3„ -5„y)[V2 (2C2 - ]2)kV„ -F2PII2

+] (2Ci - Vi)||FiV„ -FiPlI2] (107)

< ||*k -p||2 - IIZK -p||2

+ 2S„ (im-y^II + IKyQ-^pIPK -P||

< II*« -Z„||(||*„ -pkk -Pll)

+ 2S„ ( | | (7 - y) p | | + | | (yQ - ^F) p | | ) | | zn -p||.

Since lim„^TOs„ = 0 and {xn} and {zn} are bounded, from (77) we get

„lim WPiVn - fipw = 0 „Hm 11^ - F^W = 0. (108)

Also, in terms of the firm nonexpansivity of Pc and the inverse strong monotonicity of Fj for j = 1,2,we obtain from Vj e (0,2(j), j = 1,2, and (59) that

\K - pf

= \\PC (I- v2f2) Vn -Pc (I- v2f2)p\\2 < ((I - V2P2) Vn -(I- V2P2) P. Vn - P) = \[W(I- V2P2) Vn -(I- V2F2)p\\2 + \\Vn -p\\2

-||(J- V2P2) Vn -(I- V2F2)p-(Vn -p)\\2]

<\[\vn -p\\2 +W -p\\2

-\\(Vn -Vn)- V2 (F2Vn -F2p)-(p-p)\\2]

= \[\Vn -Pf + \\ -ff

-\\(Vn - Vn)-(P

+ 2v2 {(vn - vn) -(p-p) > p2vu - p2p)

-Vl}\p2Vn - F2

'n -Pf

= \\PC (I- v1F1)Wn -Pc (I- v1F1)p\\2 < ((I- ]1F1) vn -(I- ViF1)p,wn -p)

= \[\\(I- V1F1) Vn -(I- ViFi)p\\2 + \\wn -p\\2

-\\(7- V1F1)Vn -(I- V1F1) p - (wn -p)\\2]

<\[Pn -pf + H -pf

-\\\(v}n -Wn) + (P-P)\\2 + 2]1 (F^ - Frf, (vn - wn) + (p- jj))

-Vi\\F1V« -F1l

< 2 [\\Vn -p\\ + WWn -p\\

-\\\(V}n -Wn) + (P-P)\\2

+2]1 (p1Vn - F1p' (\ - Wn) + (P-P))]-

Thus, we have

< \\Vn -p\\ - \\(vn -Vn)-(P-,

+ 2v2 ((v„ - Vn) -(P-P), F2Vn - P2P)

- V^V« -F2P\\2

\\Wn -pf

< \\vn -P\\2 - \\(Vn -wn) + (p-p) \ \ 2 (111)

+ 2V1 \\F1Vn -F1p\\ \\(v„ -wn) + (p-p)\\. Consequently, from (58), (106), and (110) it follows that

< (ßn + s„Y) \\xn - pf + (l-ßn - s„Y)

x [W -p\\2 + V1 (V1 -2i1)\\F1Vn -F1P\\2] + 2sn ((I-V)p + an (yQ-^F)p,zn - p)

< (ßn + sny) \\xn - P\\2 + (l-ßn - sny) \\Vn - p\\2 + 2sn (\\(I-V)p\\ + \\(yQ-^F)p\\)\\zn -p\\

< (ßn + S„y) \xn - p\\2 + (l-ßn - Sny)

x [\vn -P\\2 - \\(V„ -~Vn)-(p-p)\\2 + 2v2 ((v„ - vj -(p-p), f2v„ - F2P)

-V^Vn -F2P\\2] + 2s„ (\\(I-V)pW + \\(yQ-vF)p\\)\\zn -p\\

< (ßn + S„y) \xn - p\\2 + (l-ßn - Sny)

"[Wtn -P\\2 - \\(V„ - \)-(p-p)\\2

+2V2 \\(vn -Vn)-(p-p)\\\F2Vn -F2PW]

+ 2s„ (\\(I-V)pW + \\(yQ-vF)p\\)\\zn -p\\ <\\*„ -pf -(l-ßn -Sny)\\(vn -Vn)-(p-p)W\2 + 2V2 \\(vn - Vn)-(p-p)\\ \\F2Vn -F2PW + 2sn (\\(I-V)p\\ + \\(yQ-vF)p\\)\\zn - p\\,

which hence leads to

(l-a-s„y)||(v„ - vj-(p-p)||2

^ (1-A,- vf)||(v„ -vj-cp-.

— k -p||2 - k -pf

+ 2]2 ||(V„ - V„)-(p-fO|| HVn -^2p|| + 2s, (||(/-V)p|| + ||(rQ-^F)p||)||Z„ -p||

— k -z„||(||*„ -pkk -p||) + 2]2 ||(v„ -V„)-(p-^|| kv, - F2p|| + 2s, (||(/-V)p|| + ||(rQ-^F)p||)||Z„ -p||.

Since lim„^ms„ = 0 and |x„}, |z„}, |v„}, and {vJ are bounded sequences, we conclude from (77) and (108) that

„Km ll(v„ -V„)-(P-JV)|| = 0-

Furthermore, from (58), (106), and (111) it follows that

k -P||2

<(A. + vy)k -pf + (1-A, -^„y)

x ||w.

+ 2^„ ((7-V)p + a„ (yQ-^F)^ - p>

— (A. + vy)k -p||2 + (1-A, -^»y) x[||v, -P||2 -||(V„ -^„) + (p-p)||2

+2Vl ||F!V„ - FjpH ||(v„ -^) + (p-p)||] + 2S, (||(/-V)p|| + ||(rQ-^F)p||)||Z„ -p||

— (A. + vy)k -p||2 + (1-A, -^»y) x [k -P||2 -kn -^Mf-p)!2

+2]i ||FiV„ -F^ ||(v„ -^) + (p-p)||] + 2S, (||(/-V)p|| + ||(rQ-^F)p||)||Z„ -p||

— k -p||2 -(1-A, -^„y)

xkn -^Mf-p)!!2 + 2] ||FiV„ -FJ|| ||(v„ -^) + (p-p)|| + 2S, (||(/-V)p|| + ||(rQ-^F)p||)||Z„ -p||,

which hence yields

(l-«-s„y)||(v„ -wj + (p-p)||2 <(l-j3„ -s„y)||(v„ -^„) + (p-iv)H2

<k -p|2 -k -p|2

+ 2] 11Fjv„ - ^iivll ll(v„ -^„) + (p-p)|| + 2s„ (ll(/-V)pll + ll(rQ-^F)pll)HZ„ -p|| <k -z„||(||*„ -P|| + ||z„ -p||) + 2] ||FiV„ -Fi^|| ||(v„ -^„) + (p-p)|| + 2s„ (||(/-V)p|| + ||(rQ-^F)p||)||Z„ -p||.

Since lim„^ms„ = 0 and {*„}, |z„}, |w„}, and |v„} are bounded sequences, we conclude from (77) and (108) that

A™, ||(v„ -wk) + (p-jp)|| = 0.

Note that

— ||(v„ - vJ - - p)| + ||(v„ - wj + - jp)||.

Hence from (114) and (117) we get

fem k- GvJ| = Amk- w»|| =0 (119)

Observe that

H v - T v H

|| n n n||

— ||Vn *n || + k - rnGVn| + ||rnGVn - TnVn|| (120)

— ||vn - *n|| + k - TnGvn|| + ||Gvn - Vn|| .

Hence, from (75), (104), and (119) we have

lim ||vn - TnVn|| = 0.

n^œ" n n n"

It is clear that

k (J-AnV/) Vn - Vn||

= |KVn + (1-ÜTnVn - Vn|| = (1-Ü||TnVn - Vn|| — ||rnvn - Vn| >

Vn - Wn

where s„ = (2 - AnL)/4 e (0,1/2) for each A„ e (0,2/L). Hence we have

^c V. - V„|

< |Pc ('-fv/) v" -pc (f-A„V/)

+ ||^c (i-A„V/) V„ - V„||

< |(J-!V/) V" -(J-A»V/) v»

+ ||^c (^-A„V/) V» - V»||

— Q-A»)||V/(v„)|| + ||r„v„ - v„||.

From the boundedness of |v»}, s» ^ 0 (^ A» ^ 2/L), and llr»v» - v„y ^ 0 (due to (121)), it follows that

v» -Pc (*-fV/) v»

In addition, from (67) and (77), we have ||Z»+1 - Z»||

— ||Z»+1 - ^n+l|| + ||*»+1 - *»|| + ll^n - Z»||

as n —> œ.

We note that

Ikv - S»+V II

IIO Z» O Z»ll

0 (125)

— IP Zn Z»|| + ||Zn Z»+11

II c»+1 II , ||c»+1 c»+1

+ ||Z»+1 - ^ Zn+1|| + IP Z»+1 - ^ Z»

From (81), (125), and Lemma 17, we obtain

lim |p»z» - Sn+1zJ| = 0.

»^œ N » "II

In the meantime, we note that

F» - Sz»ll — II

+ IIS £» S £» 11

From (81), (127), and the uniform continuity of S, we have

lim ||z» - Sz»|| = 0.

n^œ 11 » »"

Step 4. We prove that xn ^ x* = Pnx0 as n ^ ra.

Indeed, since |xn} is bounded, there exists a subsequence |xn} which converges weakly to some w. From (77), (83), (104),(92),and(102)wehavethatzn. ^ w, wn. ^ w, V» ^ w, Afc„ xn ^ w, and A*" m„ ^ w, where fc e |1,2,..., M} and m e |1,2,..., N}. Since S is uniformly continuous, by (129) we get limn^ J|zn - Smzn|| = 0 for any m > 1. Hence from

Lemma 19, we obtain w e Ff'x(S). In the meantime, utilizing Lemma 11, we deduce from Vn. ^ w, xn, ^ w, (119), and (124) that w e GSVI(G) and w e Fix^U - (2/L)V/)) = VI(C, V/) = r. Next we prove that w e VI(C, Am). Let

-1 m K

Amv + N"cv, v eC, 0, v iC,

where m e |1,2,..., N}. Let (v, m) e G(Tm). Since m - AmV e N"cV and A"wn e C, we have

(v -Am M»,«-A mv)>0.

On the other hand, from A"wn = Fc(/-Am>nAJA" 1wn and V e C,we have

(v - ^"»> ^"» - (A»i 1w» - Am>»^mA'm 1m»)> > °>

and hence

Am^ - aTV

m » »

v - A" "»>

Therefore we have

(V -Amm,"n,>")

><V -A", Mn, ,AmV> ><V -A" «n, ,AmV>

+ AmAm-1M»)>0. (133)

Am m» - Am-1«.

- < v-A»,"»,.

»_»i + A A™-1«

= (v - ^"», > Amv - ^™A™"», >

+ (v -Am, «» ,AmAm, M»i -AmA;-1«», >

a™ u - Am-1M

/ « m A»i "»i A»i "»i

-< v -A„ m»,---

>(v -a; M», ,AmAm, «», -AmAm-1«», >

a; m» - Am-1«.

- <v-A»,"»,.

From (102) and since Am is Lipschitz continuous, we obtain that limn^J|AmAm«n - AmAT^II = 0. From a"m», ^ |Ai>n} c [flj.fcj] c (0,Vi e |1,2,..., N}, and (102), we have

(v - m) > 0.

Since Tm is maximal monotone, we have w e Tm10 and hence w e VI(C,Am), m = 1,2,...,N, which implies w e n^_1 VI(C, Am). Next we prove that

w e n™=1 GMEP(&k,fk,Bk). Since Aknxn = T^(I -rknBk)Akf-1xn, n>1, k e {l,2,...,M},we have

®k iAknXn' y) + fk (y) - fk iAknXn) + {BkAknlxn,y-AknXn)

+ :L(y-Aknxn,Aknxn -Aknlxn)>°.

By (A2), we have

fk (y) - fk {AknXn) + (Bk^Xn, 7 - Anxn)

+ -1(y- AnXn, AnXn - ^n^n) rk,n

>®k (y,AknXn).

Let zt = ty+ (1- t)w for all t e (0,1] and y eC. This implies that zt e C. Then, we have

(zt - AknXn> BkZt)

> fk {AknXn) - fk (zt) + (zt - AknXn' BkZt) - (zt - AknXn'Bk^n1 xn)

/ »k AknXn -Akn1xn\ ~ I ,.k \ -( Zt - AnXn' --- )+®k (Zt,AnXn)

= fk {AknXn) - fk (zt) + (zt - AknXn' BkZt - BkAknXn)

+ (zt -Aknxn,BkAknXn -BkAkn-1Xn)

/ »k AknXn -Akn1xn\ ~ I » k N Zt - AnXn' -;- ) + ®k (Zt,AnXn) ■

By (92), we have \\BkAknxn - Bk Aknlxn\\ ^ 0 as n ^ rn. Furthermore, by the monotonicity of Bk, we obtain (zt -Aknxn, Bkzt - BkAknxn) > 0. Then, by (A4) we obtain

(zt - w, BkZt) > fk (w) - fk (zt) + ®k (zt>w) ■ (139) Utilizing (A1), (A4), and (139), we obtain ° = @k (ZvZt) + fk (Zt)-fk (Zt) < t&k (Zt, y) + (l-t)&k (Zt, w) + tfk (y) + (l-t)fk (w)-fk (zt) <t[&k (zt,y) + fk (y)-fk (zt)] (140)

+ (l-t) (zt -w,BkZt)

= t[®k (zt,y) + fk (y)-fk (zt)]

+ (\-t)t(y-w,BkZt),

and hence

°<&k (zf y) + fk (y) - fk (zt) + (l-t)(y-w, BkZt) ■

Letting t ^ 0, we have, for each y e C,

0<®k (w, y) + fk (y) - fk (w) + (y-w, Bkw) ■ (142)

This implies that w e GMEP(@k,fk,Bk) and hence w e n^ GMEP(&k,fk,Bk). Consequently, w e nf=l GMEP(@k,fk,Bk) n VI(C, A¡) n GSVI(G) n Fix(S) n r =: Q. This shows that ww(xn) c Q. From (64) and Lemma 22 we infer that xn ^ x* = Pnx0 as n ^ >x>.

Finally, assume additionally that \\xn - zn\\ = o(sn) and

limn^œ<rn = 0. It is clear that

((V - I) x - (V - I) y,x - y) > (y - 1)\\x -

Vx, y eH.

So, we know that V - I is (y - l)-strongly monotone with constant y - l > 0. In the meantime, it is easy to see that V-I is (H^ll + 1)-Lipschitzian with constant ||V|| + l > 0. Thus, there exists a unique solution x in Q to the VIP

((I-V)x,p-x) <0, Vpen.

Equivalently, x = Pn(2I - V)x. Furthermore, from (58), (59), and (82)we get

HZn -pf

Z (Pn + *nY)\\Xn -pf + (1-Pn - SnY)

*\\GVn -p\\2

+ 2Sn ((I -V)p + On (yQ - pF) p, Zn - p)

Z (Pn + SnY)\^Xn -p\\2 + (1-Pn - SnY)

x \\V.

+ 2Sn ((I -V)p + On (yQ - pF) p, Zn - p) Z (Pn + SnY)WXn -p\\2 + (1-Pn - SnY)

X Wxn -pf

+ 2Sn ((I -V)p + On (yQ - pF) p, Zn - p)

= \\xn -pf

+ 2Sn((I -V)p + On (yQ - pF) p, Zn -p), which hence yields

((I-V)p + an (yQ-^F)p,p-Zn)

„ \\Xn -p\\2 -\\zn -p\\2

\\Xn -p\\ + \\Zn -p\d ■

Since \\xn-zj = o(sn), limn^man = 0, limn^m\\xn-x*\\ = 0, and {xn}, {zn} are bounded, we infer from (146) that

((I-V)p,p-x*) <0, VpeQ, (147)

which, together with Minty's Lemma [4], implies that

{(I-V)x* ,p-x*)<0, VpeQ..

This shows that x* is a solution in Q to the VIP (144). Utilizing the uniqueness of solutions in Q to the VIP (144), we get x* = x. This completes the proof. □

Corollary 25. Let C be a nonempty closed convex subset of a real Hilbert space H. Let f : C ^ R be a convex functional with L-Lipschitz continuous gradient Vf. Let 0 be a bifunction from C x C to R satisfying (A1)-(A4) and let f : C ^ R U {+ot} be a proper lower semicontinuous and convex function. Let B, A t : H ^ H, and Fj : C ^ H be inverse-strongly monotone, ^-inverse-strongly monotone, and ^-inverse-strongly monotone, respectively, for i = 1,2 and j = 1,2. Let S : C ^ C be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense for some 0 < k < 1 with sequence {yn} c [0, ot) such that \imn^myn = 0 and {cj c [0, ot) such thatlimn^mcn = 0.Let V be a y-strongly positive bounded linear operator with y > 1. Let F : H ^ H be a K-Lipschitzian and q-strongly monotone operator with positive constants > 0. Let Q : H ^ H be an l-Lipschitzian mapping with constant I > 0. Let 0 < ^ <

2^/k2 and 0 < yl < t, where r = 1 - tJi - - ^k2). Assume that Q := GMEP(0, <p, B) n VI(C, A J n VI(C, A2) n GSVI(G)nFix(S) n r is nonempty and bounded and that either (B1) or (B2) holds. Let 0 < a < an < 1,k <Sn < d < 1 for all n>1, and let {pn}, {an} be sequences in (0,1].Pickany x0 eH and set C1 = C,x1 = Pc x0. Let {xn} be a sequence generated by the following algorithm:

0 {"v y) + f{y)-f {u„) + (Bx^ y - u„) + — (un - x„ y-un)> a Vy e C,

vn = Pc {! - KnA2)Pc {I-* 1,nAi )un, zn = finxn + {{1-pn)l-snV)TnGvn

+ sn [Tnxn - on № {Tnxn) - yQxn)], (149)

K = Snzn + {1 - Sn) S^v

yn ={1-an)xn + an

Cn+1 = {Z e Cn : \\yn - zf < \\xn - zf + dn) ,

Xn+1 =PC„+1 ^ Vn>1,

where Pc(I-XnVf) = snI+(1-sn)Tn (here Tn is nonexpansive; sn = (2 - XnL)/4 e (0,1/2) for each Xn e (0,2/L)), 9n = (sn + Yn)(1 + Yn)kn + c„, and An = sup{\\%„ - p\\2 + (\\(7 -V)p\\ + \\(yQ - PF)p\\)2/(y - 1) : p e Q} < ot. Suppose that the following conditions are satisfied:

(i) sn e (0,1/2) for each Xn e (0,2/L), limn^msn = 0 (^ limn^mXn = 2/L);

(ii) {rn} c [e,f] c (0,20, {X,J c [a„b,] c (0,2n,), and Vj e (0,2^j) for i= 1,2 and j = 1,2;

(iii) 0 < liminf< limsupn^m^n < L

Then one has the following:

(I) {xn} converges strongly as Xn ^ (2/L) sn ^ 0) to x = pqx0;

(II) {xn} converges strongly as Xn ^ (2/L) sn ^ 0) to x* = Pnx0 provided \\xn - zn\\ = o(sn) and limn^man = 0, which is the unique solution in Q to the VIP

((I-V)x*,p-x* )<0, VpeQ. (150) Equivalently, x* = Pn(2I - V)x*.

Corollary 26. Let C be a nonempty closed convex subset of a real Hilbert space H. Let f : C ^ R be a convex functional with L-Lipschitz continuous gradient Vf. Let 0 be a bifunction from C x C to R satisfying (A1)-(A4) and let f : C ^ R U {+ot} be a proper lower semicontinuous and convex function. Let B,A:H ^ H, and Fj : C ^ H be 0inverse-strongly monotone, inverse-strongly monotone, and ^-inverse-strongly monotone, respectively, for j = 1,2. Let S : C ^ C be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense for some 0 < k < 1 with sequence {yn} c [0, ot) such that limn^myn = 0 and {cn} c [0, ot) such that limn^mcn = 0. Let V be a y-strongly positive bounded linear operator with y > 1. Let F : H ^ H be a K-Lipschitzian and ^-strongly monotone operator with positive constants K,q > 0. Let Q : H ^ H be an l-Lipschitzian mapping with constant I > 0. Let0 < ^ < 2^/k2

and 0 < yl < t, where r = 1 - - - ^k2). Assume that Q := GMEP(0, <p, B) n VI(C, A) n GSVI(G) n Fix(S) n r is nonempty and bounded and that either (B1) or (B2) holds. Let 0 < a < an < 1,k < Sn < d < 1 for all n > 1, and let {pn}, {an} be sequences in (0,1]. Pick any x0 e H and set C1 = C,x1 = Pc x0. Let {xn} be a sequence generated by the following algorithm:

0 ^ y) + f{y)-f {un) + (Bx„, y - un) + —{un - xn, y-un) > 0, "Vy e C,

Vn =pc (I- PnA) Un> zn = ßnXn +((l-ßn )l-SnV)Tn GVn

+ Sn [TnXn - an № (TnXn) - yQXn)], (151)

K = SnZn + (1 - Sn) S" yn = (l-an)xn + an Cn+1 = {zsCn : \bn - Zf < \\Xn - Zf + ®n} ,

Xn, Vn > 1,

where Pc(I-XnVf) = snI+(l-sn)Tn (here Tn is nonexpansive; sn = (2 - XnL)/4 e (0,1/2) for each Xn e (0,2/L)), 9n = (s„ + Yn)(l + Yn)An + cn, and An = sup{||%„ - p\\2 + (||(7 -V)p\\ + ||(yQ - ^F)pW)2/(y - 1) : p e £1} < <xi. Suppose that the following conditions are satisfied:

(i) sn e (0,1/2)foreachXn e (0,2/L), lim„^msn = 0 lim„^m K = 2/L);

(ii) {rn} c [e,f] c (0,20, {pn} c [a,b] c (0,2$), and vj e (0,2^) for j =1,2;

(iii) 0 < liminf< limsup„< I.

Then one has the following:

(I) {xn} converges strongly as Xn ^ (2/L) sn ^ 0) to x = pqx0;

(II) {xn} converges strongly as Xn ^ (2/L) sn ^ 0) to x* = Pnx0 provided Wxn - zn|| = o(sn) and limn^TOon = 0, which is the unique solution in £ to the VIP

{(I-V)x* ,p-x*)<0, ypeü.

Equivalently, x* = Pn(2I - V)x*

Corollary 27. Let C be a nonempty closed convex subset of a real Hilbert space H. Let f : C ^ R be a convex functional with L-Lipschitz continuous gradient V/. Let 0 be a bifunction from C x C to R satisfying (A1)-(A4) and let f : C ^ R U be a proper lower semicontinuous and

convex function. Let B,A : H ^ H, and Fj : C ^ H be inverse-strongly monotone, inverse-strongly monotone, and ^-inverse-strongly monotone, respectively, for j = 1,2. Let S : C ^ C be a uniformly continuous asymptotically k-strict pseudocontractive mapping for some 0 < k < 1 with sequence {yn} c [0, rn) such that limn^TOyn = 0. Let V be a y-strongly positive bounded linear operator with y > 1. Let F : H ^ H be a K-Lipschitzian and ^-strongly monotone operator with positive constants K,q > 0. Let Q : H ^ H be an l-Lipschitzian mapping with constant I > 0. Let0 < ^ < 2^/k and 0 < yl < t, where r = 1 - - - ^k2). Assume that £ := GMEP(0, <p, B) n VI(C, A) n GSVI(G) n Fix(S) n r is nonempty and bounded and that either (B1) or (B2) holds. Let 0 < a < an < 1,k < Sn < d < 1 for all n > 1, and let {pn}, {an} be sequences in (0,1]. Pick any x0 e H and set C1 = C,x1 = PCix0. Let {xn} be a sequence generated by the following algorithm:

@ (u„ y) + cp{y)-cp (U„) + {Bx„ y - un)

+ — {un -xn,y-un)>0, УyeC,

Vn = pc (l- PnA)

zn = ßnxn + ((1-ßn)l-snV)TnGvn

+ s„ [T„xn - № (Tnxn) - yQxn)], kn = Snzn + (1 - Sn) Snzn,

yn = (l-an)xn + "n^

Cn+1 = {zeCn ■ \\yn - z\\2 < \\x„ - z\\2 + d„

= PC+ X^

yn > 1,

where PC(I - XnVf) = snI+(1-sn)Tn (here Tn is nonexpansive; sn = (2 - XnL)/4 e (0,1/2) foreach Xn e (0,2/L)), = (s„ + Yn)(— + Yn)A„, and An = sup{||x„ - pf + (||(7 -V)P\\+\\(yQ- ^P)pW)2/(y - 1) ■ p e Q.} < rn. Suppose that

the following conditions are satisfied:

(i) sn e (0,1/2) foreach Xn e (0,2/L), limn^^sn = 0 (^ limn^mXn = 2/L);

(ii) {rn} c [e,f] c (0,2£),{pn} c [a,b] c (0,2$), and Vj e (0,2(j) for j = 1,2;

(iii) 0 < liminfn^mßn < limsupn^mßn < h

Then one has the following:

(I) {xn} converges strongly as Xn ^ (2/L) sn to x = pqxq;

(II) {xn} converges strongly as Xn ^ (2/L) sn ^ 0) to x* = Pnx0 provided Wxn - znW = o(sn) and limn^man = 0, which is the unique solution in £ to the VIP

{(I-V)x*,p-x* )<0, Vpe£. (154) Equivalently, x* = Pn(2I - V)x*. 4. Fixed Point Problems with Constraints

In this section, we will introduce and analyze another implicit iterative algorithm for solving the fixed point problem of infinitely many nonexpansive mappings with constraints of several problems: finitely many GMEPs, finitely many VIPs, the GSVI (8), and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under mild assumptions. This iterative algorithm is based on shrinking projection method, Korpelevich's extragradient method, hybrid steepest-descent method in [7], viscosity approximation method, W-mapping approach to fixed points of infinitely many nonexpansive mappings, and strongly positive bounded linear operator technique.

Theorem 28. Let C be a nonempty closed convex subset of a real Hilbert space H. Let M, N be two integers. Let 0k be a bifunction from C x C to R satisfying (A1)-(A4) and let fk : C ^ R U be a proper lower semicontinuous

and convex function, where к e {1,2,..., M}. Let Bk,Ai : H —> H, and Fj : С — H be ^-inverse-strongly monotone, цгinverse-strongly monotone, and Çj-inverse-strongly monotone, respectively, where к e {1,2,..., M},i e {1,2, ...,N}, and j e {1,2}. Let {Tn}4^=1 be a sequence of nonexpansive mappings on H and let {An} be a sequence in (0, b] for some b e (0,1). Let S : С — С be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense for some 0 < к < 1 with sequence {yn} с [0, m) such that limn^œyn = 0 and {cn} с [0, m) such that limn^œcn = 0. Let V be a y-strongly positive bounded linear operator with y > 1. Let F : H — H be a k-Lipschitzian and ц-strongly monotone operator with positive constants к,ц > 0. Let Q : H — H be an l-Lipschitzian mapping with constant I > 0. Let 0 < ^ < 2ц/к and

0 < yl < t, where r = 1 - - ^(2ц - ^к2). Assume that

Q := n™=1Fix(Tn) n n™=1GMEP(@k, cpk, Bk) n n?=1VI(C, A,) n GSVI(G) n Fix(S) is nonempty and bounded and that either (B1) or (B2) holds. Let 0 < a < an <1, k <Sn < d < 1 for all n > 1, and let {pn}, {en}, and {an} be sequences in (0,1]. Pick any x0 e H andset C1 = C,x1 = PCi x0.Let{xn} beasequence generated by the following algorithm:

Un = lrM,n fl TM,n tiM)1rM-1„

xfl- Гм-inBm-1 ) ■ ■ ■ fl - rUnBi) Xn,

Vn = Pc fI- ^N,n An) Pc fI - ^N-1,n AN-1 )--Pc

*fl-hnB2)Pc fl-KnB1)Un,

Zn = fan +ff1-pn)l- €nV) WnGVn

+ en №nXn - on № fWnXn) - yQXn)],

kn = SnZn + f1 - Sn) SnZn

Уп = f1-an)xn + an

Cn+1 = {zeCn : bn - zf < \\Xn - z\\2 + dn] , Xn+1 = PC„+1 X0, Vn > 1

where Wn isthe W-mapping defined by (34), 9n = (en + yn)(1 + yn)An + cn,and An = sup{||*n - pt + (IIU - V)p\\ + ||(yQ -^F)p\\)2/(y - 1) : p e Q} < m. Suppose that the following conditions are satisfied:

(i) {rkJ с [ek,fk] с (0,2^k),{AtJ с [at,bt] с (0,2ц) and ]j e (0,2Çj), where k e {1,2,...,M},i e {1,2,...,N}, and j e {1,2};

(ii) limn^TO en = 0 and 0 < liminf lim supn^mpn < 1.

Then one has the following:

(I) [xn] converges strongly to x* = Pnx0;

n ^ ttln

(II) [xn] converges strongly to x* = Pnx0 provided \\xn -zn\\ = o(en) and limn^œon = 0, which is the unique solution in Q to the VIP

{(I-V)x*,p-x* )<0, VpeQ..

Equivalently, x* = Pn(2I - V)x*.

Proof. First of all, let us show that the sequence [xn] is well defined. As limn^œen = 0 and 0 < liminf< limsupn^œ^n < 1, we may assume, without loss of generality, that [pn] c [a, a] c (0,1) and pn + en\\V\\ < 1 for all n > 1. Utilizing the arguments similar to those in the proof of Theorem 24, we get

{!-ßn)l-enV\\<1-ßn -eny.

^ = (j - rkn

for all k e {1,2,..., M} and n > 1 and

A'n = Pc (I- A in B,) Pc (I - A ,-mB,-!) ■■■Pc (I- KnBi)

for all i e {1,2,..., N}, A0n = I, and A0n = I, where I is the identity mapping on H. Then we have that un = A^xn and

Vn = Kn Un.

We divide the rest of the proof into several steps.

Step 1. We show that {xn} is well defined. It is obvious that Cn is closed and convex. As the defining inequality in Cn is equivalent to the inequality

(2(xn -yn),z)<\\xnf -|Ы|2 + dn, (160)

by Lemma 16 we know that Cn is convex for every n > 1.

First of all, let us show that Q c Cn for all n> 1. Suppose that Q c Cn for some n > 1. Take p e Q arbitrarily. Utilizing the arguments similar to those in the proof of Theorem 24 we obtain that

-p\\<\\AknXn -AknP\\<\\Xn -Pi (161)

\\Vn -p\\<\WnUn -Л'прЦкЦнп -p\\, (162)

\\Vn - p\\ < \\Xn - p\\ ■.

l|gv„ -p||2

<||pc (7- v2f2) v„ -pc (7- ]2f2)p||2

+ ]1 (]1 -2ci)

x ||fifc (7 - ^2) v„ - fipc (7 - ^2) pf

< ||V„ -P||2 + ]2 (]2 - 2C2) ||^2V„ -F2P||

< ¡V« -P||2> k -p||2

< (1 + y«) ||Z« -P||2

+ (i-5„ )(*-a„ )K -s« z«||2 + c«

< (l+y«)||z« -pf + C«

< (1 + y«)

x ( ||x„-,|2+e„ (t^ p||+||(yQ-^F)p||)2

+ C„

So, from (155) and (165) we get k -pf

< (l-a«)||*" -pf + «#« -pf

< (l - a«) k -pf

(1 + y«)

< (l + y«)( k

(7-v)jP|H|(yQ-^)p||)2

= k- pf + y«k - p||2 +(1 + y«)e« . (||a-^)p|H|(yQ-^)p||)2

+ C„

+ c„

x I ||*« -pf +

a-y)Jp|H|(yQ-^)Jp||)2 y-1

< k - p|| + (e « + y«) (1 + y«) A « + C« = k -pf +0«>

where 0» — (e» + y»)(1 + y„)An

(||(7 - V)p|| + ||(yQ - pF)p||)2/(y-1):ieO)<m. Hence p e C„+1. This implies that Q cC„ for all n > 1. Therefore, |%„} is well defined.

Step 2. We prove that ||%„ - fc„|| ^ 0, ||%„ - z„|| ^ 0, and ||S"z„ - z„|| ^ 0 as n ^ rn.

Indeed, let x* — Fnx0. From — Fc x0 and x* e Q c C„, we obtain

xo|| < ||x

This implies that } is bounded and hence {«„}, |v„}, |z„}, |fc„}, and |y„} are also bounded. Utilizing the arguments similar to those of (67), (75), (77), and (81) in the proof of Theorem 24 we obtain that

lim ||%«+i - x«!! = 0,

«^œ11 «+1 «"

lim ||x« - W«Gv«|| = 0,

lim ||x« - zj = 0,

«^œ 11 " ""

Am kz«- z«|| =0

<k -p|| +(e« + y«)(1 + y«)

Step 3. We prove that ||%„ - wj| ^ 0, ||%„ - v»|| ^ 0, ||v„ -GvJ| ^ 0, ||v„-wv„|| ^ 0,and||z„-SzJ| ^ 0 as n ^ rn. Indeed, from (162), (164), y > 1, and yZ < r, it follows that

II2» -pf

— HA, (*„ - p) + ((1 - A,) ' - *„V) (w»gv„ - p) + sB ky (Q*„- Qp) + U - o^

-(7-a„^F)W„p] [(7-V)p + a„ (yQ-FF)p]||2

< [A, ||*„ - P|| + ||(1 - № - e„V|| ||w„gv„ - p|| + eB (^»y ||Q*B - QP||

+ ||(7 - a»FF) - (7 - a»FF) W„p||)]2 + 2e„ ((7-V)p + a„ (yQ-^F)p,z„ - p>

^ [Pn \\xn-p\\ + (l-pn -enY)\\Gvn -p\\

+en {anyl \\xn - p\\ + (1 - on r) \\xn - p\\)]2 + 2en ((I-V)p + an (yQ-^F)p,Zn - p)

= [pn \K -P\\ + (1-Pn -£nY)\\GVn -p\\

+Cn (1 - an (r-yl))\\xn -p\\]2 + 2en ((I-V)p + an (yQ-^F)p,Zn - p)

< [(Pn + £nï) \\Xn - p\\ + (1-Pn - enf) \\GVn - p\\]2 + 2Sn ((I-V)p + an (yQ-^F)p,Zn - p)

< (pn + enf) \\Xn - p\\2 + (1-Pn - en?) \\GVn - pf + 2en ((I-V)p + an (yQ-^F)p,Zn - p)

< (pn + enf) \\Xn - p\\2 + (1-pn - enf) \\Vn - p\\2

+ 2en ((I-V)p + an (yQ-^F)p,Zn - p)

< (Pn + enf) \\Xn - p\\2 + (1-Pn - enf) bn - p\\2 + 2Sn ((I-V)p + an (yQ-^F)p,Zn - p) .

Utilizing the arguments similar to those of (83), (92), (102), (104), (119), (121), and (129) in the proof of Theorem 24 we obtain that

lim \\xn -u„\\ = 0,

lim \\\Akn-1Xn - AknXn\\ = 0, к=1,2,...,М, (173)

lim \\ti-lu - Л'и! = 0, i=l,2,...,N,

lim \\xn - vn\\ = 0,

lim \\vn - Gvn\\ = 0,

lim \\v - W v \\ = 0,

\ n n n \

lim \\zn - Szn\\ = 0.

In addition, note that

\\v - Wv \\ < \\v - W v \\ + \\W v - Wv \\

\ n n \ \ n n n \ \ n n n \

So, from \\vn - Wnvn\\ ^ 0 and [20, Remark 3.2] it follows that

lim \\vn - Wvn\\ = 0.

x = Pnx0 as n ^ œ>.

Step 4. We prove that xn

Indeed, since [xn] is bounded, there exists a subsequence [xn } which converges weakly to some w. From (169), (172),

(175), (173), and (174) we have that zn ^ w, un ^ w, vn ^ w,Akn xn, ^ w, and A™ un, ^ w, where k e {1,2,...,M} and m e {1,2,..., N}. Since S is uniformly continuous, by (178) we get limn^TO||zn - Smzn\\ = 0 for any m > 1. Hence, from Lemma 19, we obtain w e Fix(S). In the meantime, utilizing Lemma 11, we deduce from (176) and (180) that w e GSVI(G) and w e Fix(W) = n™=1Fix(Tn) (due to Lemma 13). Hence we get w e GSVI(G) n n'^=1Fix(Tn). Repeating the same arguments as in the proof of Theorem 24 we conclude that w e n^=1 VI(C,Am) and w e n™=1 GMEP(@k,<pk,Bk). Consequently, w e n™=1 Fix(Tn) n nf=l GMEP(&k,cpk, Bk) n nf=1 VI(C,A,) n GSVI(G) n Fix(S) =: Q. This shows that ww(xn) c Q. From (167) and Lemma 22 we infer that xn ^ x* = Pnx0 as n ^ x.

Finally, assume additionally that \\xn - zn\\ = o(en) and

limn^œan = 0. It is clear that

{(V -I)x-(V-I)y,x-y)>(f-

Ух, y e H.

So, we know that V - I is (f - 1)-strongly monotone with constant f - 1 > 0. In the meantime, it is easy to see that V-I is (11^11 + 1)-Lipschitzian with constant ||V|| + 1 > 0. Thus, there exists a unique solution x in Q to the VIP

{(I-V)x,p-x) <0, УреП.

Equivalently, x = Pn(21 - V)x. Furthermore, from (163), (164), and (171) we get

bn -p\\

< (pn + enf) bn - p\\2 + (1-Pn - enf) \\Gvn - p\\2 + 2en {(I-V)p + an (yQ-^F)p,Zn - p)

< (Pn + enf) bn - p\\2

+ (l-ßn -enf)bn -p\\2

+ 2Sn {(I-V)p + an (yQ-^F)p,Zn - p)

< (Pn + enf) bn - p\\2 + (1-Pn - enf) bn - p\\2 + 2Sn {(I-V)p + an (yQ-^F)p,Zn - p)

= bn -p\\2

+ 2£n((I -V)p + On (yQ - pF) p, Zn - p), which hence yields

{(I-V)p + On (yQ-^F)p,p-Zn)

„ bn -pf -bn -p\\2

— rr

<Lû_à(\\Xn -p\\ + \\Zn -p\\).

otii^H X II = 0,

Since \\xn-zn\\ = o(sn)Mmn^man = 0,limn and [xn], {zn} are bounded, we infer from (94) that

((I-V)p,p-x*) <0, VpeQ, (185) which, together with Minty's Lemma, implies that

((I-V)x* ,p-x*)<0, VpeQ. (186)

This shows that x* is a solution in Q to the VIP (182). Utilizing the uniqueness of solutions in Q to the VIP (182), we get x* = x. This completes the proof. □

Corollary 29. Let C bea nonempty closed convex subset ofa real Hilbert space H. Let 0 be a bifunction from C x C to R satisfying (A1)-(A4) and let f : C — R U {+x} be a proper lower semicontinuous and convex function. Let B,A t : H — H, and Fj : C — H be inverse-strongly monotone, qr inverse-strongly monotone, and (-inverse-strongly monotone, respectively, for i = 1,2 and j = 1,2. Let {Tn}<^^=1 be a sequence of nonexpansive mappings on H and let {Xn} be a sequence in (0,b] for some b e (0,1). Let S : C — C be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense for some 0 < k < 1 with sequence {yn} c [0, x) such that limn^ixyn = 0 and {cn} c [0, x) such that limn^mcn = 0. Let V be a y-strongly positive bounded linear operator with y > 1. Let F : H — H be a k-Lipschitzian and q-strongly monotone operator with positive constants K,q > 0. Let Q : H — H be an l-Lipschitzian mapping with constant I > 0. Let 0 < ^ < 2^/k2 and

0 < yl < t, where t = 1 - ^1- - ^k2). Assume that Q := n™=1Fix(Tn)nGMEP(0,<p,B)nVI(C,A 1)nVI(C,A2)n GSVI(G) n Fix(S) is nonempty and bounded and that either (B1) or (B2) holds. Let 0 < a < an < 1,k <Sn < d < 1 for all n > 1, and let {pn}, {en}, and {an} be sequences in (0,1]. Pick any x0 e H andset C1 = C,x1 = PCi x0.Let{xn} beasequence generated by the following algorithm:

0 ^ y) + f{y)-f {un) + (Bxn, y - u„) + 1(un - x„, y-un)> a Vy e C,

vn = pc (!-KnA2) Pc (l-KnBi)un,

zn = ßnXn + ((l-ßn)l-enV)WnGVn

+ £n [WnXn - On (j4F (WnXn) - yQXn)] . K = SnZn + (1 - Sn) ^^

yn = (l-an)Xn + Cn+1 = {zsCn : \\yn - Zf < \\Xn - Zf + dn\ ■

Mn> 1,

where Wn isthe W-mapping defined by (34), dn = (en + yn)(1 + yn)An + cn,and An = sup{\\*„ - p\ + (\\U - V)p\\ + \\(yQ -^F)p\\)2/(y - 1) : p e Q} < (Xi. Suppose that the following conditions are satisfied:

(i) [rn] c [e,f] c (0,20, [X,J c [at,bt] c (0,2m), and vj e (0,2Çj) for i= 1,2 and j = 1,2;

(ii) limn = 0 and 0 < liminf^

limsupn < 1

Then one has the following:

(I) [xn] converges strongly to x* = Pnx0;

(II) [xn] converges strongly to x* = Pnx0 provided \\xn -zn\\ = o(en) and limn^œon = 0, which is the unique solution in Q to the VIP

{(I-V)x*,p-x* )<0, ypeü.

Equivalently, x* = Pn(2I - V)x*

Corollary 30. Let C be a nonempty closed convex subset of a real Hilbert space H. Let 0 be a bifunction from C x C to R satisfying (A1)-(A4) and let f : C — R U {+x} be a proper lower semicontinuous and convex function. Let B, A : H —> H, and Fj : C — H be inverse-strongly monotone, inverse-strongly monotone, and (j-inverse-strongly monotone, respectively, for j = 1,2. Let {Tn}<^^=1 be a sequence of nonexpansive mappings on H and let {Xn} be a sequence in (0,b] for some b e (0,1). Let S : C — C be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense for some 0 < k < 1 with sequence {yn} c [0, x) such thatlimn^TOyn = 0 and {cn} c [0, x) such that limn^mcn = 0. Let V be a y-strongly positive bounded linear operator with y > 1. Let F : H — H be a k-Lipschitzian and q-strongly monotone operator with positive constants K,q > 0. Let Q : H — H be an l-Lipschitzian mapping with constant I > 0. Let 0 < ^ < 2^/k2 and

0 < yl < t, where t = 1- ^1- - ^k2). Assume that Q := n^=1Fix(Tn) n GMEP(0, <p, B) n VI(C, A) n GSVI(G) n Fix(S) is nonempty and bounded and that either (B1) or (B2) holds. Let 0 < a < an < 1,k < Sn < d < 1 for all n > 1, and let {pn}, {en}, and {an} be sequences in (0,1]. Pick anyx0 e H and set C1 = C,x1 = Pc x0. Let {xn} be a sequence generated by the following algorithm:

0 ^ y) + f{y)-f {un) + {Bx„, y - u„) + 1(un -xn,y-un)>0, VyeC,

Vn = pc {I- PnA) zn = finxn + {{1-pn)l-enV)WnGvn

+ [Wnxn - on {^F {Wnxn) - yQxn)], kn = Snzn + {1 - Sn) Snzn,

yn = {1-an)xn + «n^

Cn+1 = {z e Cn : \\yn - zf < \\xn - zf + dn),

Xn+1 = PC„+1 X0, Vn > 1

where Wn isthe W-mapping defined by (34), dn = (en + yn)(1 + yn)An + cn,and An = sup{||*„ - pt + (||U - V)p\\ + ||(yQ -^F)p\\)2/(y - 1) : p e Q} < x. Suppose that the following conditions are satisfied:

(i) {rn} c [e,f] c (0,20, {p„} c [a,c] c (0,2$), and vj e (0,2^) for j =1,2;

(ii) lim„^men = 0 and 0 < liminf< limsup„ < 1

Then one has the following:

(I) {xn} converges strongly to x* = Pnx0;

(II) {xn} converges strongly to x* = Pnx0 provided Wxn -znW = o(en) and limn^man = 0, which is the unique solution in Q to the VIP

{(I-V)x* ,p-x*)<0, VpeQ..

Equivalently, x* = Pn(2I - V)x*.

Corollary 31. Let C be a nonempty closed convex subset of a real Hilbert space H. Let 0 be a bifunction from C x C to R satisfying (A1)-(A4) and let f : C — R U {+x} be a proper lower semicontinuous and convex function. Let B, A : H —> H, and Fj : C — H be ^-inverse-strongly monotone, $-inverse-strongly monotone, and (j-inverse-strongly monotone, respectively, for j = 1,2. Let {T.„}' be a sequence of nonexpansive mappings on H and let {Xn} be a sequence in (0,b] for some b e (0,1). Let S : C — C be a uniformly continuous asymptotically k-strict pseudocontractive mapping for some 0 < k < 1 with sequence {yn} c [0, x) such that limn^TOyn = 0. Let V be a y-strongly positive bounded linear operator with y > 1. Let F : H — H be a k-Lipschitzian and q-strongly monotone operator with positive constants K,q > 0. Let Q : H — H be an l-Lipschitzian mapping with constant I > 0. Let 0 < ^ < 2^/k2 and

0 < yl < t, where t = 1- ^1- - ^k2). Assume that Q := n"=1Fix(TJnGMEP(0,<p,B)nVI(C, A)n GSVI (G)nFix(S) is nonempty and bounded and that either (B1) or (B2) holds. Let 0 < a < an < 1,k < Sn < d < 1 for all n > 1, and let {pn}, {en}, and {an} be sequences in (0,1]. Pick any x0 e H and set C1 = C,x1 = PCi x0. Let {xn} be a sequence generated by the following algorithm:

@(u„, y) + cp{y)-cp (un) + {Bx„ y - un)

+—{un -xn,y-un)>0, УyeC,

Vn = pc (l- PnA)

zn = ßn xn + ((l-ßn)l-enV)WnGvn

+ en [Wnxn - on (j4F (Wnxn) - yQxn)], kn = Snzn + (l - Sn) Snzn,

yn = (l-an)xn + "n^

Cn+1 = {zeCn : \\yn - z\\2 < \\x„ - z\\2 + d„

Xn+1 = PC„+, X0,

Vn> l,

where Wn is the W-mapping defined by (34), dn = (en + yn)(1 + yn)An, and An = supUK -p\\2 + (IIU - V)p\\ + IKyQ -^F)p\\)2/(y - 1) : p e Q} < x. Suppose that the following conditions are satisfied:

(i) {rn} c [e,f] c (0,20, ipn} c [a,c] c (0,2$), and Vj e (0,2Çj) for j = 1,2;

(ii)lim„= 0 and 0 < liminf< limsup„ < 1

Then one has the following:

(I) {xn} converges strongly to x* = Pnx0;

(II) {xn} converges strongly to x* = Pnx0 provided \\xn -zn\\ = o(en) and limn^œon = 0, which is the unique solution in Q to the VIP

{(I-V)x\p-x* )<0, ypeD.. Equivalently, x* = Pn(2I - V)x*.

Remark 32. Let A:C — H be ^-inverse-strongly monotone and let Ft : C — H be vj-inverse-strongly monotone for j = 1,2. Let Q : C — C be a ^-contraction with p e [0,1), and let S : C — C be a uniformly continuous asymptotically fc-strict pseudocontractive mapping in the intermediate sense for some 0 < k < 1 with sequence {yn} c [0, x) such that limK^TOyK = 0 and {cj c [0, x) such that limK^TOcK = 0. Assume that Q := VI(C, A) n GSVI(G) n Fix(S) is nonempty and bounded. In [11], Guu et al. introduced and analyzed a hybrid viscosity CQ iterative algorithm for finding a point p e Q:

X1 = xeC choosen arbitrarily,

yn = PC (Xn - ■ PnAxn) ,

t« = anQXn + (l-an) Gy„,

Zn = (l-Vn - Vn) Xn + Vntn + ^Vrßntn,

Cn = {zeC: 22 ■ \K - 4 < \\xn- 4 + on

Qn = [zeC: {xn -Z,X-Xn) > ^

n+1 = Pc„nQ„x , Vn>l,

where v} e (0,2<;-) for j = 1,2,0„ = (a„ + y„)A „ + c„;

A „ = sup{||x„ - pt + (1 + y„)/(1 - p)||(J - Q),p||2 :peO}< ш; {#J is a sequence in (0,2£); and {аи|, and {vn} are three sequences in [0,1] such that ^n + vn < 1 for all n > 1. The authors of [11] proved that under suitable conditions {хи| converges strongly to Pnx; see [11, Theorem 3.1] for more details.

Theorem 28 extends, improves, supplements, and develops [11, Theorem 3.1] in the following aspects.

(i) The problem of finding a point p e n^=1Fix(T„) П n^ GMEP(0fc, <pfc, Bfc) n n|!1 VI(C, A;) n GSVI(G) n Fi'x(S) in Theorem 28 is very different from the problem of finding a point p e VI(C, A) n GSVI(G) n Fi'x(S) in [11, Theorem 3.1]. There is no doubt that our problem of finding a point p e n™1Fi'x(TtJ) n n^1 GMEP(0fc, <pfc, Bfc) n n^1 VI(C, A;) n GSVI(G) n Fi'x(S) is more general and more subtle than the problem of finding a point p e VI(C, A) n GSVI(G) n Fi'x(S) in [11, Theorem 3.1].

(ii) The iterative scheme in [11, Theorem 3.1] is extended to develop the iterative scheme in Theorem 28 by virtue of Cai and Bu iterative algorithm in [21, Theorem 3.1] and Ceng et al. iterative one in [8, Theorem 3.1]. The iterative scheme in Theorem 28 is more advantageous and more flexible than the iterative scheme in [11, Theorem 3.1] because it involves solving four problems: the GSVI (8), finitely many GMEPs, finitely many VIPs, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings on H.

(iii) The iterative scheme in Theorem 28 is very different from the iterative scheme in [11, Theorem 3.1] because the iterative scheme in our theorem (Theorem 28) involves hybrid steepest-descent method in [7], strongly positive bounded linear operator technique, finitely many GMEPs, finitely many VIPs, and infinitely many nonexpansive mappings. The proof in [11, Theorem 3.1] makes use of Proposition CWY and the properties of asymptotically strict pseudocontractive mapping in the intermediate sense (see Lemmas 17-20). However, the proof of Theorem 28 depends on not only Proposition CWY and Lemmas 17-20 but also Proposition 8, the properties of strongly positive bounded linear operator V, and the ones of the W-mapping Wn and TA-mapping (see Lemmas 12,13, and 15) because there are the mapping T(®k'ft\ infinitely many nonexpansive mappings {Ги}™1, к-Lipschitzian and ^-strongly monotone operator F, and strongly positive bounded linear operator V appearing in the iterative scheme of our theorem (Theorem 28).

(iv) The proof of Theorem 28 combines Cai and Bu convergence analysis for their iterative algorithm to solve finitely many GMEPs, finitely many VIPs, and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense

(see [21, Theorem 3.1]); the convergence analysis for the W-mapping approach to fixed points of infinitely many nonexpansive mappings and strongly positive bounded linear operator technique; and Ceng, Guu, and Yao convergence analysis for hybrid iterative method (see [11, Theorem 3.1]).

Remark 33. Theorem 28 also extends, improves, supplements, and develops Ceng et al. [8, Theorem 3.1] in the following aspects.

(i) The problem of finding a point p e n™1Fi'x(TtJ) n nf=1 GMEP(0fc, <pfc, Bfc) n n^1 VI(C, A;) n GSVI(G) n Fi'x(S) in Theorem 28 is very different from the problem of finding a point p e n^1Fi'x(Ti) n GMEP(0, <p, A) in Ceng et al. [8, Theorem 3.1]. Here our problem of finding a point p e n™1Fi'x(TtJ) n nf=1 GMEP(0fc, fk, Bfc) n n^1 VI(C, A;) n GSVI(G) n Fi'x(S) is put forth after one GMEP; finitely many nonexpansive mappings in their problem are replaced by finitely many GMEPs and infinitely many nonexpansive mappings, respectively; and the GSVI (8), finitely many VIPs, and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense are added to their problem.

(ii) The iterative scheme in [8, Theorem 3.1] is extended to develop the iterative scheme in our theorem (Theorem 28) by virtue of Korpelevich's extragradient method [22], shrinking projection method, Mann iterative method, and strongly positive bounded linear operator technique. The iterative scheme in Theorem 28 is put forth after un = Т^'^Ц - гиА)хи and w„m„ in [8, Theorem 3.1] are replaced by un = A^fxn and W„GA^м„, respectively.

(iii) The iterative scheme in Theorem 28 is very different from the iterative scheme in [8, Theorem 3.1] because the iterative scheme in Theorem 28 involves Korpele-vich's extragradient method [22], shrinking projection method, Mann iterative method, and strongly positive bounded linear operator technique. The proof of [8, Theorem 3.1] makes use of Proposition 8. However, the proof of Theorem 28 depends on not only Proposition 8 but also Proposition CWY, the properties of strongly positive bounded linear operator, and the ones of asymptotically strict pseudo-contractive mapping in the intermediate sense (see Lemmas 17-20) because there are the SGEP (8), finitely many GMEPs, asymptotically strict pseudo-contractive mapping S in the intermediate sense, and the strongly positive bounded linear operator V appearing in the iterative scheme of our theorem (Theorem 28).

(iv) The proof of Theorem 28 involves the convergence analysis for Korpelevich's extragradient method to solve the SGEP (8), finitely many GMEPs, and finitely many VIPs; the convergence analysis for the W-mapping approach to fixed points of infinitely many nonexpansive mappings and strongly positive

bounded linear operator technique; and Ceng, Guu, and Yao convergence analysis for viscosity approximation method and hybrid steepest-descent method (see [23, Theorem 4.2] and [8, Theorem 3.1]).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), the Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and the Ph.D. Program Foundation of Ministry of Education of China (20123127110002). This work was supported partly by the National Science Council of the Republic of China. This research was partially supported by a grant from NSC.

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