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

Authors: Lu-Chuan Ceng; Cheng-Wen Liao; Chin-Tzong Pang; Ching-Feng Wen
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

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.


---

## Body

## 1. Introduction

LetC 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
(1)∥Sx-Sy∥≤L∥x-y∥,∀x,y∈C.
In particular, if L=1 then S is called a nonexpansive mapping; if L∈[0,1) then S is called a contraction. A mapping V is called strongly positive on H if there exists a constant γ¯>0 such that
(2)〈Vx,x〉≥γ¯∥x∥2,∀x∈H.LetA:C→H be a nonlinear mapping on C. We consider the following variational inequality problem (VIP): find a point x∈C such that
(3)〈Ax,y-x〉≥0,∀y∈C.
The solution set of VIP (3) is denoted by VI⁡(C,A).Letφ:C→R be a real-valued function, let A:H→H be a nonlinear mapping, and let Θ:C×C→R be a bifunction. Peng and Yao [1] introduced the following generalized mixed equilibrium problem (GMEP) of finding x∈C such that
(4)Θ(x,y)+φ(y)-φ(x)+〈Ax,y-x〉≥0,∀y∈C.
We denote the set of solutions of GMEP (4) by GMEP⁡(Θ,φ,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 Θ:C×C→R is a bifunction satisfying conditions (A1)–(A4) and φ:C→R is a lower semicontinuous and convex function with restriction (B1) or (B2), where(A1)
Θ(x,x)=0 for all x∈C;(A2)
Θ is monotone; that is, Θ(x,y)+Θ(y,x)≤0 for any x,y∈C;(A3)
Θ is upper-hemicontinuous; that is, for each x,y,z∈C,
(5)limsup⁡t→0+Θ(tz+(1-t)x,y)≤Θ(x,y);(A4)
Θ(x,·) is convex and lower semicontinuous for each x∈C;(B1)
for eachx∈H and r>0, there exists a bounded subset Dx⊂C and yx∈C such that, for any z∈C∖Dx,
(6)Θ(z,yx)+φ(yx)-φ(z)+1r〈yx-z,z-x〉<0;(B2)
C is a bounded set.Given a positive numberr>0, let Tr(Θ,φ):H→C be the solution set of the auxiliary mixed equilibrium problem; that is, for each x∈H,
(7)Tr(Θ,φ)(x):={1ry∈C:Θ(y,z)+φ(z)-φ(y)+1r〈y-x,z-y〉≥0,∀z∈C}.LetF1,F2:C→H be two mappings. Consider the following general system of variational inequalities (GSVI) [6] of finding (x*,y*)∈C×C such that
(8)〈ν1F1y*+x*-y*,x-x*〉≥0,∀x∈C,〈ν2F2x*+y*-x*,x-y*〉≥0,∀x∈C,
where ν1>0 and ν2>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 givenx¯,y¯∈C,(x¯,y¯)is a solution of the GSVI (8) if and only ifx¯is a fixed point of the mappingG:C→Cdefined by(9)Gx=PC(I-ν1F1)PC(I-ν2F2)x,∀x∈C,wherey¯=PC(I-ν2F2)x¯.In particular, if the mappingFj:C→H is ζj-inverse-strongly monotone for j=1,2, then the mapping G is nonexpansive provided νj∈(0,2ζj] for j=1,2. We denote by GSVI(G) the fixed point set of the mapping G.Letλn,1,λn,2,…,λn,N∈(0,1], n≥1. Given the nonexpansive self-mappings T1,T2,…,TN on C, for each n≥1, the mappings Un,1,Un,2,…,Un,N are defined by
(10)Un,1=λn,1T1+(1-λn,1)I,Un,2=λn,2TnUn,1+(1-λn,2)I,Un,n-1=λn-1Tn-1Un,n+(1-λn-1)I,⋮Un,N-1=λn,N-1TN-1Un,N-2+(1-λn,N-1)I,Wn:=Un,N=λn,NTNUn,N-1+(1-λn,N)I.TheWn is called the W-mapping generated by T1,…,TN and λn,1,λn,2,…,λn,N. 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}i=1N.Theorem  CGY (see [8, Theorem 3.1]).LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetΘ:C×C→Rbe a bifunction satisfying assumptions (A1)–(A4) and letφ:C→Rbe a lower semicontinuous and convex function with restriction (B1) or (B2). Let the mappingA:H→Hbeδ-inverse-strongly monotone, and let{Ti}i=1Nbe a finite family of nonexpansive mappings onHsuch thatΩ:=∩i=1NFix(Ti)∩GMEP(Θ,φ,A)≠∅. LetF:H→Hbe aκ-Lipschitzian andη-strongly monotone operator with positive constantsκ,η>0andQ:H→Hanl-Lipschitzian mapping with constantl≥0. Let0<μ<2η/κ2and0≤γl<τ, whereτ=1-1-μ(2η-μκ2). Suppose{αn}and{βn}are two sequences in(0,1),{γn}is a sequence in(0,2δ], and{λn,i}i=1Nis a sequence in[a,b]with0<a≤b<1. For everyn≥1, letWnbe theW-mapping generated byT1,…,TNandλn,1,λn,2,…,λn,N. Givenx1∈Harbitrarily, suppose the sequences{xn}and{un}are generated iteratively by(11)Θ(un,y)+φ(y)-φ(un)+〈Axn,y-un〉aaa+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnγQxn+βnxn+((1-βn)I-αnμF)Wnun,aaaaaaaaaaaaaaaaaa∀n≥1,where the sequences{αn},{βn}, and{rn}and the finite family of sequences{λn,i}i=1Nsatisfy the following conditions:(i)
lim⁡n→∞αn=0and∑n=1∞‍αn=∞;(ii)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1;(iii)
0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2δandlim⁡n→∞(rn+1-rn)=0;(iv)
lim⁡n→∞(λn+1,i-λn,i)=0fori=1,2,…,N.Then both{xn}and{un}converge strongly tox*=PΩ(I-μF+γQ)x*, which is the unique solution inΩto the VIP(12)〈(μF-γQ)x*,x*-x〉≤0,∀x∈Ω.Letf:C→R be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing f over the constraint set C(13)minimize{f(x):x∈C}.
We denote by Γ the set of minimizers of CMP (13).Next, recall some concepts. LetC 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
(14)∥Snx-Sny∥≤ℒ∥x-y∥,∀n≥1,∀x,y∈C.
Recently, Kim and Xu [9] introduced the concept of asymptotically k-strict pseudocontractive mappings in a Hilbert space as below.Definition 1.
LetC 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 {γn} if there exists a constant k∈[0,1) and a sequence {γn} in [0,∞) with lim⁡n→∞γn=0 such that
(15)∥Snx-Sny∥2≤(1+γn)∥x-y∥2+k∥x-Snx-(y-Sny)∥2,∀n≥1,∀x,y∈C.
It is important to note that every asymptoticallyk-strict pseudocontractive mapping with sequence {γn} is a uniformly ℒ-Lipschitzian mapping with ℒ=sup⁡{(k+1+(1-k)γn)/(1+k):n≥1}. Subsequently, Sahu et al. [10] considered the concept of asymptotically k-strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.Definition 2.
LetC 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 {γn} if there exist a constant k∈[0,1) and a sequence {γn} in [0,∞) with lim⁡n→∞γn=0 such that
(16)limsup⁡n→∞sup⁡x,y∈C(∥Snx-Sny∥2-(1+γn)∥x-y∥2-k∥x-Snx-(y-Sny)∥2)≤0.
Putcn:=max⁡{0,sup⁡x,y∈C(∥Snx-Sny∥2-(1+γn)∥x-y∥2-k∥x-Snx-(y-Sny)∥2)}. Then cn≥0(∀n≥1), cn→0(n→∞), and there holds the relation
(17)∥Snx-Sny∥2≤(1+γn)∥x-y∥2+k∥x-Snx-(y-Sny)∥2+cn,aaaaaaaaaaaaaaaaaaaaaaaaaaaaai∀n≥1,∀x,y∈C.In 2009, Sahu et al. [10] first established one weak convergence theorem for the following Mann-type iterative scheme:
(18)x1=x∈Cchosenarbitrary,xn+1=(1-αn)xn+αnSnxn,∀n≥1,
where 0<δ≤αn≤1-k-δ, ∑n=1∞‍αncn<∞, and ∑n=1∞‍γn<∞, and then obtained another strong convergence theorem for the following hybrid CQ iterative scheme:
(19)x1=x∈Cchosenarbitrary,yn=(1-αn)xn+αnSnxn,Cn={z∈C:∥yn-z∥2≤∥xn-z∥2+θn},Qn={z∈C:〈xn-z,x-xn〉≥0},xn+1=PCn∩Qnx,∀n≥1,
where 0<δ≤αn≤1-k, θn=cn+γnΔn, and Δn=sup⁡{∥xn-z∥2:z∈Fix⁡(S)}<∞. 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 thatH is a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. Let C 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 ωw(xn) to denote the weak ω-limit set of the sequence {xn}; that is,
(20)ωw(xn):={x∈H:xni⇀xforsomesubsequenceaaa{xni}of{xn}}.Recall that a mappingA:C→H is called(i)
monotone if(21)〈Ax-Ay,x-y〉≥0,∀x,y∈C;(ii)
η-strongly monotone if there exists a constant η>0 such that
(22)〈Ax-Ay,x-y〉≥η∥x-y∥2,∀x,y∈C;(iii)
α-inverse-strongly monotone if there exists a constant α>0 such that
(23)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C.It is obvious that ifA is α-inverse-strongly monotone, then A is monotone and 1/α-Lipschitz continuous.The metric (or nearest point) projection fromH onto C is the mapping PC:H→C which assigns to each point x∈H the unique point PCx∈C satisfying the property
(24)∥x-PCx∥=inf⁡y∈C∥x-y∥=:d(x,C).Some important properties of projections are gathered in the following proposition.Proposition 3.
For givenx∈H and z∈C,(i)
z=PCx⇔〈x-z,y-z〉≤0, ∀y∈C;(ii)
z=PCx⇔∥x-z∥2≤∥x-y∥2-∥y-z∥2, ∀y∈C;(iii)
〈PCx-PCy,x-y〉≥∥PCx-PCy∥2, ∀y∈H.Consequently, PC is nonexpansive and monotone.
IfA is an α-inverse-strongly monotone mapping of C into H, then it is obvious that A is 1/α-Lipschitz continuous. We also have that if λ≤2α, then I-λA is a nonexpansive mapping from C to H.Definition 4.
A mappingT:H→H is said to be(a)
nonexpansive if(25)∥Tx-Ty∥≤∥x-y∥,∀x,y∈H;(b)
firmly nonexpansive if2T-I is nonexpansive or, equivalently, if T is 1-inverse-strongly monotone (1-ism):
(26)〈x-y,Tx-Ty〉≥∥Tx-Ty∥2,∀x,y∈H;alternatively, T is firmly nonexpansive if and only if T can be expressed as
(27)T=12(I+S),
where S:H→H is nonexpansive; projections are firmly nonexpansive.It can be easily seen that ifT 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 mappingT: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,
(28)T≡(1-α)I+αS,
where α∈(0,1) and S:H→H is nonexpansive. More precisely, when the last equality holds, we say that T is α-averaged. Thus firmly nonexpansive mappings (in particular, projections) are 1/2-averaged mappings.Proposition 6 (see [15]).
LetT:H→H be a given mapping.(i)
Tis nonexpansive if and only if the complementI-Tis1/2-ism.(ii)
IfTisν-ism, then, forγ>0,γTisν/γ-ism.(iii)
Tis averaged if and only if the complementI-Tisν-ism for someν>1/2. Indeed, forα∈(0,1),Tisα-averaged if and only ifI-Tis1/2α-ism.Proposition 7 (see [15]).
LetS,T,V:H→H be given operators.(i)
IfT=(1-α)S+αVfor someα∈(0,1)and ifSis averaged andVis nonexpansive, thenTis averaged.(ii)
Tis firmly nonexpansive if and only if the complementI-Tis firmly nonexpansive.(iii)
IfT=(1-α)S+αVfor someα∈(0,1)and ifSis firmly nonexpansive andVis nonexpansive, thenTis averaged.(iv)
The composite of finitely many averaged mappings is averaged. That is, if each of the mappings{Ti}i=1Nis averaged, then so is the compositeT1,…,TN. In particular, ifT1isα1-averaged andT2isα2-averaged, whereα1,α2∈(0,1), then the compositeT1T2isα-averaged, whereα=α1+α2-α1α2.(v)
If the mappings{Ti}i=1Nare averaged and have a common fixed point, then(29)⋂i=1N‍Fix(Ti)=Fix(T1,…,TN).The notation Fix⁡(T) denotes the set of all fixed points of the mapping T; that is, Fix⁡(T)={x∈H:Tx=x}.Proposition 8 (see [3]).
Assume thatΘ:C×C→R satisfies (A1)–(A4) and let φ:C→R be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r>0 and x∈H, define a mapping Tr(Θ,φ):H→C as follows:
(30)Tr(Θ,φ)(x)={1rz∈C:Θ(z,y)+φ(y)-φ(z)+1r〈y-z,z-x〉≥0,∀y∈C},
for all x∈H. Then the following hold:(i)
for eachx∈H,Tr(Θ,φ)(x)≠∅;(ii)
Tr(Θ,φ)is single-valued;(iii)
Tr(Θ,φ)is firmly nonexpansive; that is, for anyx,y∈H,(31)∥Tr(Θ,φ)x-Tr(Θ,φ)y∥2≤〈Tr(Θ,φ)x-Tr(Θ,φ)y,x-y〉;(iv)
Fix(Tr(Θ,φ))=MEP⁡(Θ,φ);(v)
MEP⁡(Θ,φ)is closed and convex.We need some facts and tools in a real Hilbert spaceH which are listed as lemmas below.Lemma 9.
LetX be a real inner product space. Then there holds the following inequality:
(32)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈X.Lemma 10.
LetA:C→H be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 3(i)) implies
(33)u∈VI(C,A)⟺u=PC(u-λAu),λ>0.Lemma 11 (see [16, demiclosedness principle]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on C. Then I-T is demiclosed. That is, whenever {xn} is a sequence in C weakly converging to some x∈C 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}n=1∞ be an infinite family of nonexpansive mappings on H and let {λn}n=1∞ be a sequence of nonnegative numbers in [0,1]. For any n≥1, define a mapping Wn on H as follows:
(34)Un,n+1=I,Un,n=λnTnUn,n+1+(1-λn)I,Un,n-1=λn-1Tn-1Un,n+(1-λn-1)I,⋮Un,k=λkTkUn,k+1+(1-λk)I,Un,k-1=λk-1Tk-1Un,k+(1-λk-1)I,⋮Un,2=λ2T2Un,3+(1-λ2)I,Wn=Un,1=λ1T1Un,2+(1-λ1)I.
Such a mapping Wn is called the W-mapping generated by Tn,Tn-1,…,T1 and λn,λn-1,…,λ1.Lemma 12 (see [17, Lemma 3.2]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Tn}n=1∞ be a sequence of nonexpansive self-mappings on C such that ∩n=1∞Fix(Tn)≠∅ and let {λn} be a sequence in (0,b] for some b∈(0,1). Then, for every x∈C and k≥1, the limit lim⁡n→∞Un,kx exists, where Un,k is defined as in (34).Lemma 13 (see [17, Lemma 3.3]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Tn}n=1∞ be a sequence of nonexpansive self-mappings on C such that ∩n=1∞Fix(Tn)≠∅, and let {λn} be a sequence in (0,b] for some b∈(0,1). Then, Fix(W)=∩n=1∞Fix⁡(Tn).The following lemma can be easily proven, and, therefore, we omit the proof.Lemma 14.
LetV:H→H be a γ¯-strongly positive bounded linear operator with constant γ¯>1. Then, for γ¯-1>0,
(35)〈(V-I)x-(V-I)y,x-y〉≥(γ¯-1)∥x-y∥2,∀x,y∈H.
That is, V-I is strongly monotone with constant γ¯-1.
LetC be a nonempty closed convex subset of a real Hilbert space H. We introduce some notations. Let λ be a number in (0,1] and let μ>0. Associating with a nonexpansive mapping T:C→H, we define the mapping Tλ:C→H by
(36)Tλx:=Tx-λμF(Tx),∀x∈C,
where F:H→H is an operator such that, for some positive constants κ,η>0, F is κ-Lipschitzian and η-strongly monotone on H; that is, F satisfies the following conditions:
(37)∥Fx-Fy∥≤κ∥x-y∥,〈Fx-Fy,x-y〉≥η∥x-y∥2,
for all x,y∈H.Lemma 15 (see [18, Lemma 3.1]).
T
λ is a contraction provided 0<μ<2η/κ2; that is,
(38)∥Tλx-Tλy∥≤(1-λτ)∥x-y∥,∀x,y∈C,
where τ=1-1-μ(2η-μκ2)∈(0,1].Lemma 16 ([10, Lemma 2.5]).
LetH be a real Hilbert space. Given a nonempty closed convex subset of H and points x,y,z∈H and given also a real number a∈R, the set
(39){v∈C:∥y-v∥2≤∥x-v∥2+〈z,v〉+a}
is convex (and closed).Recall that a set-valued mappingT:D(T)⊂H→2H is called monotone if, for all x,y∈D(T), f∈Tx and g∈Ty imply
(40)〈f-g,x-y〉≥0.
A set-valued mapping T is called maximal monotone if T is monotone and (I+λT)D(T)=H for each λ>0, where I 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)∈H×H,〈f-g,x-y〉≥0 for every (y,g)∈G(T) implies f∈Tx. Let A:C→H be a monotone, k-Lipschitz-continuous mapping and let NCv be the normal cone to C at v∈C; that is,
(41)NCv={w∈H:〈v-u,w〉≥0,∀u∈C}.
Define
(42)Tv={Av+NCv,ifv∈C,∅,ifv∉C.
Then, T is maximal monotone and 0∈Tv if and only if v∈VI⁡(C,A).Lemma 17 ([10, Lemma 2.6]).
LetC be a nonempty subset of a Hilbert space H and let S:C→C be an asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {γn}. Then
(43)∥Snx-Sny∥≤11-k×(k∥x-y∥+(1+(1-k)γn)∥x-y∥2+(1-k)cn)
for all x,y∈C and n≥1.Lemma 18 ([10, Lemma 2.7]).
LetC be a nonempty subset of a Hilbert space H and let S:C→C be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {γn}. Let {xn} be a sequence in C such that ∥xn-xn+1∥→0 and ∥xn-Snxn∥→0 as n→∞. Then ∥xn-Sxn∥→0 as n→∞.Lemma 19 (demiclosedness principle [10, Proposition 3.1]).
LetC be a nonempty closed convex subset of a Hilbert space H and let S:C→C be a continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {γn}. Then I-S is demiclosed at zero in the sense that if {xn} is a sequence in C such that xn⇀x∈C and limsup⁡m→∞limsup⁡n→∞∥xn-Smxn∥=0, then (I-S)x=0.Lemma 20 ([10, Proposition 3.2]).
LetC be a nonempty closed convex subset of a Hilbert space H and let S:C→C be a continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {γn} such that Fix(S)≠∅. Then Fix(S) is closed and convex.Remark 21.
Lemmas19 and 20 give some basic properties of an asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {γn}.Lemma 22 (see [19]).
LetC be a closed convex subset of a real Hilbert space H. Let {xn} be a sequence in H and u∈H. Let q=PCu. If {xn} is such that ωw(xn)⊂C and satisfies the condition
(44)∥xn-u∥≤∥u-q∥,∀n,
then xn→q as n→∞.Lemma 23.
LetH be a real Hilbert space. Then the following hold:(a)
∥x-y∥2=∥x∥2-∥y∥2-2〈x-y,y〉 for all x,y∈H;(b)
∥λx+μy∥2=λ∥x∥2+μ∥y∥2-λμ∥x-y∥2 for all x,y∈H and λ,μ∈[0,1] with λ+μ=1;(c)
if{xn} is a sequence in H such that xn⇀x, it follows that
(45)limsup⁡n→∞∥xn-y∥2=limsup⁡n→∞∥xn-x∥2+∥x-y∥2,∀y∈H.


## 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.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let M,N be two integers. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let Θk be a bifunction from C×C to R satisfying (A1)–(A4) and let φk:C→R∪{+∞} be a proper lower semicontinuous and convex function, where k∈{1,2,…,M}. Let Bk,Ai:H→H, and Fj:C→H be μk-inverse-strongly monotone, ηi-inverse-strongly monotone, and ζj-inverse-strongly monotone, respectively, where k∈{1,2,…,M}, i∈{1,2,…,N}, 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 {γn}⊂[0,∞) such that lim⁡n→∞γn=0 and {cn}⊂[0,∞) such that lim⁡n→∞cn=0. Let V be a γ¯-strongly positive bounded linear operator with γ¯>1. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/κ2 and 0≤γl≤τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=∩k=1MGMEP(Θk,φk,Bk)∩∩i=1NVI(C,Ai)∩GSVI(G)∩Fix⁡(S)∩Γ is nonempty and bounded and that either (B1) or (B2) holds. Let 0<α≤αn≤1, k≤δn≤d<1 for all n≥1, and let {βn}, {σn} be sequences in (0,1]. Pick any x0∈H and set C1=C, x1=PC1x0. Let {xn} be a sequence generated by the following algorithm:
(46)un=TrM,n(ΘM,φM)(I-rM,nBM)TrM-1,n(ΘM-1,φM-1)×(I-rM-1,nBM-1)⋯Tr1,n(Θ1,φ1)(I-r1,nB1)xn,vn=PC(I-λN,nAN)PC(I-λN-1,nAN-1)⋯PC×(I-λ2,nA2)PC(I-λ1,nA1)un,zn=βnxn+((1-βn)I-snV)TnGvn+sn[Tnxn-σn(μF(Tnxn)-γQxn)],kn=δnzn+(1-δn)Snzn,yn=(1-αn)xn+αnkn,Cn+1={z∈Cn:∥yn-z∥2≤∥xn-z∥2+θn},xn+1=PCn+1x0,iiii∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn(here Tn is nonexpansive; sn=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L)), θn=(sn+γn)(1+γn)Δn+cn, and Δn=sup⁡{∥xn-p∥2+(∥(I-V)p∥+∥(γQ-μF)p∥)2/(γ¯-1):p∈Ω}<∞. Suppose that the following conditions are satisfied:(i)
sn∈(0,1/2) for each λn∈(0,2/L), lim⁡n→∞sn=0(⇔lim⁡n→∞λn=2/L);(ii)
{rk,n}⊂[ek,fk]⊂(0,2μk),{λi,n}⊂[ai,bi]⊂(0,2ηi), and νj∈(0,2ζj), where k∈{1,2,…,M}, i∈{1,2,…,N}, and j∈{1,2};(iii)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1. Then one has the following:(I)
{xn} converges strongly as λn→2/L(⇔sn→0) to x*=PΩx0;(II)
{xn} converges strongly as λn→2/L(⇔sn→0) to x*=PΩx0 provided ∥xn-zn∥=o(sn) and lim⁡n→∞σn=0, which is the unique solution in Ω to the VIP
(47)〈(I-V)x*,p-x*〉≤0,∀p∈Ω.Equivalently, x*=PΩ(2I-V)x*.Proof .
Since∇f is L-Lipschitzian, it follows that ∇f is 1/L-ism. By Proposition 6(ii) we know that, for λ>0,λ∇f is 1/λL-ism. So by Proposition 6(iii) we deduce that I-λ∇f is λL/2-averaged. Now since the projection PC is 1/2-averaged, it is easy to see from Proposition 7(iv) that the composite PC(I-λ∇f) is (2+λL)/4-averaged for λ∈(0,2/L). Hence we obtain that, for each n≥1, PC(I-λn∇f) is (2+λnL)/4-averaged for each λn∈(0,2/L). Therefore, we can write
(48)PC(I-λn∇f)=2-λnL4I+2+λnL4Tn=snI+(1-sn)Tn,
where Tn is nonexpansive and sn:=sn(λn)=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L). It is clear that
(49)λn⟶2L⟺sn⟶0.
As lim⁡n→∞sn=0 and 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1, we may assume, without loss of generality, that {βn}⊂[a,a^]⊂(0,1) and βn+sn∥V∥≤1 for all n≥1. Since V is a γ¯-strongly positive bounded linear operator on H, we know that
(50)∥V∥=sup⁡{〈Vu,u〉:u∈H,∥u∥=1}≥γ¯>1.
Taking into account that βn+sn∥V∥≤1 for all n≥1, we have
(51)〈((1-βn)I-snV)u,u〉=1-βn-sn〈Vu,u〉≥1-βn-sn∥V∥≥0;
that is, (1-βn)I-snV is positive. It follows that
(52)∥(1-βn)I-snV∥=sup⁡{〈((1-βn)I-snV)u,u〉:u∈H,∥u∥=1}=sup⁡{1-βn-sn〈Vu,u〉:u∈H,∥u∥=1}≤1-βn-snγ¯.
Put
(53)Δnk=Trk,n(Θk,φk)(I-rk,nBk)Trk-1,n(Θk-1,φk-1)×(I-rk-1,nBk-1)⋯Tr1,n(Θ1,φ1)(I-r1,nB1)xn
for all k∈{1,2,…,M} and n≥1 and
(54)Λni=PC(I-λi,nBi)PC(I-λi-1,nBi-1)⋯PC(I-λ1,nB1)
for all i∈{1,2,…,N}, Δn0=I, and Λn0=I, where I is the identity mapping on H. Then we have that un=ΔnMxn and vn=ΛnNun.
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
(55)〈2(xn-yn),z〉≤∥xn∥2-∥yn∥2+θn,
by Lemma 16 we know that Cn is convex for every n≥1.
First of all, let us show thatΩ⊂Cn for all n≥1. Suppose that Ω⊂Cn for some n≥1. Take p∈Ω arbitrarily. From (46) and Proposition 8(iii), we have
(56)∥un-p∥=∥TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1xn-TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1p∥≤∥(I-rM,nBM)ΔnM-1xn-(I-rM,nBM)ΔnM-1p∥≤∥ΔnM-1xn-ΔnM-1p∥⋮≤∥Δn0xn-Δn0p∥=∥xn-p∥.
Similarly, we have
(57)∥vn-p∥=∥PC(I-λN,nAN)ΛnN-1un-PC(I-λN,nAN)ΛnN-1p∥≤∥(I-λN,nAN)ΛnN-1un-(I-λN,nAN)ΛnN-1p∥≤∥ΛnN-1un-ΛnN-1p∥⋮≤∥Λn0un-Λn0p∥=∥un-p∥.
Combining (56) and (57), we have
(58)∥vn-p∥≤∥xn-p∥.
Since p=Gp=PC(I-ν1F1)PC(I-ν2F2)p, Fj is ζj-inverse-strongly monotone for j=1,2, and 0≤νj≤2ζj for j=1,2, we deduce that, for any n≥1,
(59)∥Gvn-p∥2=∥PC(I-ν1F1)PC(I-ν2F2)vn-PC(I-ν1F1)PC(I-ν2F2)p∥2≤∥(I-ν1F1)PC(I-ν2F2)vn-(I-ν1F1)PC(I-ν2F2)p∥2=∥[PC(I-ν2F2)vn-PC(I-ν2F2)p]-ν1[F1PC(I-ν2F2)vn-F1PC(I-ν2F2)p]∥2≤∥PC(I-ν2F2)vn-PC(I-ν2F2)p∥2+ν1(ν1-2ζ1)×∥F1PC(I-ν2F2)vn-F1PC(I-ν2F2)p∥2≤∥PC(I-ν2F2)vn-PC(I-ν2F2)p∥2≤∥(I-ν2F2)vn-(I-ν2F2)p∥2=∥(vn-p)-ν2(F2vn-F2p)∥2≤∥vn-p∥2+ν2(ν2-2ζ2)∥F2vn-F2p∥2≤∥vn-p∥2.
Utilizing Lemma 15, from (46), (52), (58), and (59), we obtain that
(60)∥zn-p∥=∥βn(xn-p)+((1-βn)I-snV)(TnGvn-p)+sn[Tnxn-σn(μF(Tnxn)-γQxn)-p]+sn(I-V)p((1-βn)I-snV)∥=∥βn(xn-p)+((1-βn)I-snV)(TnGvn-p)+sn[σnγ(Qxn-Qp)+(I-σnμF)Tnxn-(I-σnμF)Tnp]+sn[(I-V)p+σn(γQ-μF)p]((1-βn)I-snV)∥≤βn∥xn-p∥+∥(1-βn)I-snV∥∥TnGvn-p∥+sn[σnγ∥Qxn-Qp∥+∥(I-σnμF)Tnxn-(I-σnμF)Tnp∥]+sn∥(I-V)p+σn(γQ-μF)p∥≤βn∥xn-p∥+(1-βn-snγ¯)∥Gvn-p∥+sn[σnγl∥xn-p∥+(1-σnτ)∥xn-p∥]+sn(∥(I-V)p∥+∥(γQ-μF)p∥)≤βn∥xn-p∥+(1-βn-snγ¯)∥vn-p∥+sn(1-σn(τ-γl))∥xn-p∥+sn(∥(I-V)p∥+∥(γQ-μF)p∥)≤βn∥xn-p∥+(1-βn-snγ¯)∥xn-p∥+sn(1-σn(τ-γl))∥xn-p∥+sn(∥(I-V)p∥+∥(γQ-μF)p∥)=(1-snγ¯)∥xn-p∥+sn(1-σn(τ-γl))∥xn-p∥+sn(∥(I-V)p∥+∥(γQ-μF)p∥)≤(1-snγ¯)∥xn-p∥+sn∥xn-p∥+sn(∥(I-V)p∥+∥(γQ-μF)p∥)=(1-sn(γ¯-1))∥xn-p∥+sn(∥(I-V)p∥+∥(γQ-μF)p∥)=(1-sn(γ¯-1))∥xn-p∥+sn(γ¯-1)∥(I-V)p∥+∥(γQ-μF)p∥γ¯-1,
which hence yields
(61)∥zn-p∥2≤(1-sn(γ¯-1))∥xn-p∥2+sn(γ¯-1)×(∥(I-V)p∥+∥(γQ-μF)p∥)2(γ¯-1)2≤∥xn-p∥2+sn(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1.
By Lemma 23(b), we deduce from (46) and (61) that
(62)∥kn-p∥2=∥δn(zn-p)+(1-δn)(Snzn-p)∥2=δn∥zn-p∥2+(1-δn)∥Snzn-p∥2-δn(1-δn)∥zn-Snzn∥2≤δn∥zn-p∥2+(1-δn)×[(1+γn)∥zn-p∥2+k∥zn-Snzn∥2+cn]-δn(1-δn)∥zn-Snzn∥2=[1+γn(1-δn)]∥zn-p∥2+(1-δn)(k-δn)∥zn-Snzn∥2+(1-δn)cn≤(1+γn)∥zn-p∥2+(1-δn)(k-δn)∥zn-Snzn∥2+cn≤(1+γn)∥zn-p∥2+cn≤(1+γn)((∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1∥xn-p∥2+sn(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1)+cn.
So, from (46) and (62) we get
(63)∥yn-p∥2=∥(1-αn)(xn-p)+αn(kn-p)∥2≤(1-αn)∥xn-p∥2+αn∥kn-p∥2≤(1-αn)∥xn-p∥2+αn×[(1+γn)((∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1∥xn-p∥2+sn(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1)+cn]≤(1+γn)×(∥xn-p∥2+sn(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1)+cn=∥xn-p∥2+γn∥xn-p∥2+(1+γn)sn×(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1+cn≤∥xn-p∥2+γn(1+γn)∥xn-p∥2+sn(1+γn)×(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1+cn≤∥xn-p∥2+(γn+sn)(1+γn)∥xn-p∥2+(sn+γn)(1+γn)(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1+cn=∥xn-p∥2+(sn+γn)(1+γn)×(∥xn-p∥2+(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1)+cn≤∥xn-p∥2+(sn+γn)(1+γn)Δn+cn=∥xn-p∥2+θn,
where θn=(sn+γn)(1+γn)Δn+cn and Δn=sup⁡{∥xn-p∥2+(∥(I-V)p∥+∥(γQ-μF)p∥)2/(γ¯-1):p∈Ω}<∞. Hence p∈Cn+1. This implies that Ω⊂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→∞.
Indeed, letx*=PΩx0. From xn=PCnx0 and x*∈Ω⊂Cn, we obtain
(64)∥xn-x0∥≤∥x*-x0∥.
This implies that {xn} is bounded and hence {un},{vn},{zn},{kn}, and {yn} are also bounded. Since xn+1∈Cn+1⊂Cn and xn=PCnx0, we have
(65)∥xn-x0∥≤∥xn+1-x0∥,∀n≥1.
Therefore lim⁡n→∞∥xn-x0∥ exists. From xn=PCnx0,xn+1∈Cn+1⊂Cn, by Proposition 3(ii), we obtain
(66)∥xn+1-xn∥2≤∥x0-xn+1∥2-∥x0-xn∥2,
which implies
(67)lim⁡n→∞∥xn+1-xn∥=0.
It follows from xn+1∈Cn+1 that ∥yn-xn+1∥2≤∥xn-xn+1∥2+θn and hence
(68)∥xn-yn∥2≤2(∥xn-xn+1∥2+∥xn+1-yn∥2)≤2(∥xn-xn+1∥2+∥xn-xn+1∥2+θn)=2(2∥xn-xn+1∥2+θn).
From (67) and lim⁡n→∞θn=0, we have
(69)lim⁡n→∞∥xn-yn∥=0.
Since yn-xn=αn(kn-xn) and 0<α≤αn≤1, we have
(70)α∥kn-xn∥≤αn∥kn-xn∥=∥yn-xn∥,
which immediately leads to
(71)lim⁡n→∞∥kn-xn∥=0.
Also, utilizing Lemmas 9 and 23(b) we obtain from (46), (58), (59), and (62) that
(72)∥zn-p∥2=∥βnxn+((1-βn)I-snV)TnGvn+sn[Tnxn-σn(μF(Tnxn)-γQxn)]-p∥2=∥βn(xn-p)+(1-βn)(TnGvn-p)+sn[Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn]∥2≤∥βn(xn-p)+(1-βn)(TnGvn-p)∥2+2sn〈Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn,zn-p〉=βn∥xn-p∥2+(1-βn)∥TnGvn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn〈Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn,zn-p〉≤βn∥xn-p∥2+(1-βn)∥Gvn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn∥Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn∥×∥zn-p∥≤βn∥xn-p∥2+(1-βn)∥vn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn∥Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn∥×∥zn-p∥≤βn∥xn-p∥2+(1-βn)∥xn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn∥Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn∥×∥zn-p∥=∥xn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn∥Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn∥×∥zn-p∥,
and hence
(73)∥yn-p∥2≤(1-αn)∥xn-p∥2+αn∥kn-p∥2≤(1-αn)∥xn-p∥2+αn[(1+γn)∥zn-p∥2+cn]≤(1-αn)∥xn-p∥2+αn×[∥xn-TnGvn∥2(1+γn)×(∥xn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn∥Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn∥×∥zn-p∥∥xn-TnGvn∥2)+cn]≤(1-αn)∥xn-p∥2+αn(1+γn)×(∥xn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn∥Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn∥×∥xn-p∥2∥zn-p∥)+cn≤(1-αn)∥xn-p∥2+αn(1+γn)∥xn-p∥2-αn(1+γn)βn(1-βn)∥xn-TnGvn∥2+(1+γn)2sn×∥Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn∥×∥zn-p∥+cn≤(1+γn)∥xn-p∥2-αn(1+γn)βn(1-βn)×∥xn-TnGvn∥2+2sn(1+γn)×∥Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn∥×∥zn-p∥+cn.
So, it follows that
(74)α(1+γn)a(1-a^)∥xn-TnGvn∥2≤αn(1+γn)βn(1-βn)∥xn-TnGvn∥2≤∥xn-p∥2-∥yn-p∥2+γn∥xn-p∥2+2sn(1+γn)×∥Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn∥×∥zn-p∥+cn≤∥xn-yn∥(∥xn-p∥+∥yn-p∥)+γn∥xn-p∥2+2sn(1+γn)×∥Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn∥×∥zn-p∥+cn.
Since lim⁡n→∞sn=0, lim⁡n→∞γn=0, and lim⁡n→∞cn=0, it follows from (69) and the boundedness of {xn},{yn},{zn}, and {vn} that
(75)lim⁡n→∞∥xn-TnGvn∥=0.
Note that
(76)∥zn-xn∥=∥(1-βn)(TnGvn-xn)+sn(Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn)∥≤(1-βn)∥TnGvn-xn∥+sn∥Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn∥≤∥TnGvn-xn∥+sn∥Tnxn-σn(μF(Tnxn)-γQxn)-VTnGvn∥.
Hence, it follows from (75) and lim⁡n→∞sn=0 that
(77)lim⁡n→∞∥xn-zn∥=0.
Note that
(78)∥kn-zn∥≤∥kn-xn∥+∥xn-zn∥.
Thus, we deduce from (71) and (77) that
(79)lim⁡n→∞∥kn-zn∥=0.
Since kn-zn=(1-δn)(Snzn-zn) and k≤δn≤d<1, we have
(80)(1-d)∥Snzn-zn∥≤(1-δn)∥Snzn-zn∥=∥kn-zn∥,
which, together with (79), yields
(81)lim⁡n→∞∥Snzn-zn∥=0.
Step
3. We prove that ∥xn-un∥→0, ∥xn-vn∥→0, ∥vn-Gvn∥→0, ∥vn-PC(I-(2/L)∇f)vn∥→0, and ∥zn-Szn∥→0 as n→∞.
Indeed, from (57), (59), γ¯>1, and γl≤τ it follows that
(82)∥zn-p∥2=∥βn(xn-p)+((1-βn)I-snV)(TnGvn-p)+sn[σnγ(Qxn-Qp)+(I-σnμF)Tnxn-(I-σnμF)Tnp]+sn[(I-V)p+σn(γQ-μF)p]((1-βn)I-snV)∥2≤∥βn(xn-p)+((1-βn)I-snV)(TnGvn-p)+sn[σnγ(Qxn-Qp)+(I-σnμF)Tnxn-(I-σnμF)Tnp]((1-βn)I-snV)∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤[βn∥xn-p∥+∥(1-βn)I-snV∥∥TnGvn-p∥+sn(σnγ∥Qxn-Qp∥+∥(I-σnμF)Tnxn-(I-σnμF)Tnp∥)]2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤[βn∥xn-p∥+(1-βn-snγ¯)∥Gvn-p∥+sn(σnγl∥xn-p∥+(1-σnτ)∥xn-p∥)]2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉=[βn∥xn-p∥+(1-βn-snγ¯)∥Gvn-p∥+sn(1-σn(τ-γl))∥xn-p∥]2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤[βn∥xn-p∥+(1-βn-snγ¯)∥Gvn-p∥+sn∥xn-p∥]2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤[(βn+snγ¯)∥xn-p∥+(1-βn-snγ¯)∥Gvn-p∥]2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥Gvn-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥vn-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥un-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉.
Next let us show that
(83)lim⁡n→∞∥xn-un∥=0.
For p∈Ω, we find from (46) that
(84)∥Δnkxn-p∥2=∥Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn-Trk,n(Θk,φk)(I-rk,nBk)p∥2=∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p∥2≤∥Δnk-1xn-p∥2+rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2≤∥xn-p∥2+rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2.
By (56), (82), and (84), we obtain
(85)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥un-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×∥Δnkxn-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥xn-p∥2+rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2]+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉=∥xn-p∥2+(1-βn-snγ¯)rk,n(rk,n-2μk)×∥BkΔnk-1xn-Bkp∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉,
which immediately yields
(86)(1-a^-snγ¯)rk,n(2μk-rk,n)∥BkΔnk-1xn-Bkp∥2≤(1-βn-snγ¯)rk,n(2μk-rk,n)∥BkΔnk-1xn-Bkp∥2≤∥xn-p∥2-∥zn-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥.
Since lim⁡n→∞sn=0,{rk,n}⊂[ek,fk]⊂(0,2μk), and {xn} and {zn} are bounded sequences, it follows from (77) that
(87)lim⁡n→∞∥BkΔnk-1xn-Bkp∥=0,k=1,2,…,M.
By Proposition 8(iii) and (46), we have
(88)∥Δnkxn-p∥2=∥Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn-Trk,n(Θk,φk)(I-rk,nBk)p∥2≤〈(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p,Δnkxn-p〉=12(∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p∥2+∥Δnkxn-p∥2-∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p-(Δnkxn-p)∥2)≤12(∥Δnk-1xn-p∥2+∥Δnkxn-p∥2-∥Δnk-1xn-Δnkxn-rk,n(BkΔnk-1xn-Bkp)∥2),
which implies that
(89)∥Δnkxn-p∥2≤∥Δnk-1xn-p∥2-∥Δnk-1xn-Δnkxn-rk,n(BkΔnk-1xn-Bkp)∥2=∥Δnk-1xn-p∥2-∥Δnk-1xn-Δnkxn∥2-rk,n2∥BkΔnk-1xn-Bkp∥2+2rk,n〈Δnk-1xn-Δnkxn,BkΔnk-1xn-Bkp〉≤∥Δnk-1xn-p∥2-∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥≤∥xn-p∥2-∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥.
From (82) and (89), we have
(90)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥un-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥Δnkxn-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥xn-p∥2-∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥∥Δnk-1xn-Δnkxn∥2]+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤∥xn-p∥2-(1-βn-snγ¯)∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉,
which leads to
(91)(1-a^-snγ¯)∥Δnk-1xn-Δnkxn∥2≤(1-βn-snγ¯)∥Δnk-1xn-Δnkxn∥2≤∥xn-p∥2-∥zn-p∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥.
Since lim⁡n→∞sn=0, {rk,n}⊂[ek,fk]⊂(0,2μk), and {xn} and {zn} are bounded sequences, it follows from (77) and (87) that
(92)lim⁡n→∞∥Δnk-1xn-Δnkxn∥=0.
Hence we obtain from (92) that
(93)∥xn-un∥=∥Δn0xn-ΔnMxn∥≤∥Δn0xn-Δn1xn∥+∥Δn1xn-Δn2xn∥+⋯∥ΔnM-1xn-ΔnMxn∥⟶0asn⟶∞.
That is, (83) holds.
Next we show thatlim⁡n→∞∥AiΛniun-Aip∥=0, i=1,2,…,N. As a matter of fact, observe that
(94)∥Λniun-p∥2=∥PC(I-λi,nAi)Λni-1un-PC(I-λi,nAi)p∥2≤∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p∥2≤∥Λni-1un-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2≤∥un-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2≤∥xn-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2.
Combining (57), (82), and (94), we have
(95)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥vn-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥Λniun-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥xn-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2]+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥=∥xn-p∥2+(1-βn-snγ¯)λi,n(λi,n-2ηi)×∥AiΛni-1un-Aip∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥,
which leads to
(96)(1-a^-snγ¯)λi,n(2ηi-λi,n)∥AiΛni-1un-Aip∥2≤(1-βn-snγ¯)λi,n(2ηi-λi,n)∥AiΛni-1un-Aip∥2≤∥xn-p∥2-∥zn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥.
Since lim⁡n→∞sn=0, {λi,n}⊂[ai,bi]⊂(0,2ηi), and {xn} and {zn} are bounded sequences, it follows from (77) that
(97)lim⁡n→∞∥AiΛni-1un-Aip∥=0,i=1,2,…,N.
By Proposition 3(iii) and Lemma 23(a), we obtain
(98)∥Λniun-p∥2=∥PC(I-λi,nAi)Λni-1un-PC(I-λi,nAi)p∥2≤〈(I-λi,nAi)Λni-1un-(I-λi,nAi)p,Λniun-p〉=12(∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p∥2+∥Λniun-p∥2-∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p-(Λniun-p)∥2)≤12(∥Λni-1un-p∥2+∥Λniun-p∥2-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2)≤12(∥un-p∥2+∥Λniun-p∥2-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2)≤12(∥xn-p∥2+∥Λniun-p∥2-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2),
which implies
(99)∥Λniun-p∥2≤∥xn-p∥2-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2=∥xn-p∥2-∥Λni-1un-Λniun∥2-λi,n2∥AiΛni-1un-Aip∥2+2λi,n〈Λni-1un-Λniun,AiΛni-1un-Aip〉≤∥xn-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥.
Combining (57), (82), and (99), we have
(100)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥vn-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥Λniun-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥xn-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥∥Λni-1un-Λniun∥2]+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤∥xn-p∥2-(1-βn-snγ¯)∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉,
which yields
(101)(1-a^-snγ¯)∥Λni-1un-Λniun∥2≤(1-βn-snγ¯)∥Λni-1un-Λniun∥2≤∥xn-p∥2-∥zn-p∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥.
Since lim⁡n→∞sn=0 and {xn}, {zn}, and {un} are bounded, from (77) and (97) we get
(102)lim⁡n→∞∥Λni-1un-Λniun∥=0.
From (102) we get
(103)∥un-vn∥=∥Λn0un-ΛnNun∥≤∥Λn0un-Λn1un∥+∥Λn1un-Λn2un∥+⋯+∥ΛnN-1un-ΛnNun∥⟶0aaaaaaaaaaaaaaaaaaaasn⟶∞.
Taking into account that ∥xn-vn∥≤∥xn-un∥+∥un-vn∥, we conclude from (83) and (103) that
(104)lim⁡n→∞∥xn-vn∥=0.
On the other hand, for simplicity, we writep~=PC(I-ν2F2)p, v~n=PC(I-ν2F2)vn, and wn=Gvn=PC(I-ν1F1)v~n for all n≥1. Then
(105)p=Gp=PC(I-ν1F1)p~=PC(I-ν1F1)PC(I-ν2F2)p.
We now show thatlim⁡n→∞∥Gvn-vn∥=0; that is, lim⁡n→∞∥wn-vn∥=0. As a matter of fact, for p∈Ω, it follows from (58), (59), and (82) that
(106)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×∥Gvn-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉=(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×∥wn-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥v~n-p~∥2+ν1(ν1-2ζ1)∥F1v~n-F1p~∥2]+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥vn-p∥2+ν2(ν2-2ζ2)∥F2vn-F2p∥2+ν1(ν1-2ζ1)∥F1v~n-F1p~∥2]+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥xn-p∥2+ν2(ν2-2ζ2)∥F2vn-F2p∥2+ν1(ν1-2ζ1)∥F1v~n-F1p~∥2]+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥=∥xn-p∥2+(1-βn-snγ¯)×[ν2(ν2-2ζ2)∥F2vn-F2p∥2+ν1(ν1-2ζ1)∥F1v~n-F1p~∥2]+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥,
which immediately yields
(107)(1-a^-snγ¯)[ν2(2ζ2-ν2)∥F2vn-F2p∥2+ν1(2ζ1-ν1)∥F1v~n-F1p~∥2]≤(1-βn-snγ¯)[ν2(2ζ2-ν2)∥F2vn-F2p∥2+ν1(2ζ1-ν1)∥F1v~n-F1p~∥2]≤∥xn-p∥2-∥zn-p∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥.
Since lim⁡n→∞sn=0 and {xn} and {zn} are bounded, from (77) we get
(108)lim⁡n→∞∥F2vn-F2p∥=0,lim⁡n→∞∥F1v~n-F1p~∥=0.
Also, in terms of the firm nonexpansivity of PC and the ζj-inverse strong monotonicity of Fj for j=1,2, we obtain from νj∈(0,2ζj), j=1,2, and (59) that
(109)∥v~n-p~∥2=∥PC(I-ν2F2)vn-PC(I-ν2F2)p∥2≤〈(I-ν2F2)vn-(I-ν2F2)p,v~n-p~〉=12[∥(I-ν2F2)vn-(I-ν2F2)p∥2+∥v~n-p~∥2-∥(I-ν2F2)vn-(I-ν2F2)p-(v~n-p~)∥2]≤12[∥vn-p∥2+∥v~n-p~∥2-∥(vn-v~n)-ν2(F2vn-F2p)-(p-p~)∥2]=12[∥vn-p∥2+∥v~n-p~∥2-∥(vn-v~n)-(p-p~)∥2+2ν2〈(vn-v~n)-(p-p~),F2vn-F2p〉-ν22∥F2vn-F2p∥2],∥wn-p∥2=∥PC(I-ν1F1)v~n-PC(I-ν1F1)p~∥2≤〈(I-ν1F1)v~n-(I-ν1F1)p~,wn-p〉=12[∥(I-ν1F1)v~n-(I-ν1F1)p~∥2+∥wn-p∥2-∥(I-ν1F1)v~n-(I-ν1F1)p~-(wn-p)∥2]≤12[∥v~n-p~∥2+∥wn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1〈F1v~n-F1p~,(v~n-wn)+(p-p~)〉-ν12∥F1v~n-F1p~∥2]≤12[∥vn-p∥2+∥wn-p∥2-∥(v~n-wn)+(p-p~)∥2∥vn-p∥2+∥wn-p∥2+2ν1〈F1v~n-F1p~,(v~n-wn)+(p-p~)〉].
Thus, we have
(110)∥v~n-p~∥2≤∥vn-p∥2-∥(vn-v~n)-(p-p~)∥2+2ν2〈(vn-v~n)-(p-p~),F2vn-F2p〉-ν22∥F2vn-F2p∥2,(111)∥wn-p∥2≤∥vn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥.
Consequently, from (58), (106), and (110) it follows that
(112)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥v~n-p~∥2+ν1(ν1-2ζ1)∥F1v~n-F1p~∥2]+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥v~n-p~∥2+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥vn-p∥2-∥(vn-v~n)-(p-p~)∥2+2ν2〈(vn-v~n)-(p-p~),F2vn-F2p〉-ν22∥F2vn-F2p∥2]+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥xn-p∥2-∥(vn-v~n)-(p-p~)∥2+2ν2∥(vn-v~n)-(p-p~)∥∥F2vn-F2p∥[∥xn-p∥2-∥(vn-v~n)-(p-p~)∥2]+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥≤∥xn-p∥2-(1-βn-snγ¯)∥(vn-v~n)-(p-p~)∥2+2ν2∥(vn-v~n)-(p-p~)∥∥F2vn-F2p∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥,
which hence leads to
(113)(1-a^-snγ¯)∥(vn-v~n)-(p-p~)∥2≤(1-βn-snγ¯)∥(vn-v~n)-(p-p~)∥2≤∥xn-p∥2-∥zn-p∥2+2ν2∥(vn-v~n)-(p-p~)∥∥F2vn-F2p∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2ν2∥(vn-v~n)-(p-p~)∥∥F2vn-F2p∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥.
Since lim⁡n→∞sn=0 and {xn},{zn},{vn}, and {v~n} are bounded sequences, we conclude from (77) and (108) that
(114)lim⁡n→∞∥(vn-v~n)-(p-p~)∥=0.
Furthermore, from (58), (106), and (111) it follows that
(115)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×∥wn-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥vn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥∥(v~n-wn)+(p-p~)∥2]+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥xn-p∥2-∥(v~n-wn)+(p-p~)∥2∥xn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥]+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥≤∥xn-p∥2-(1-βn-snγ¯)×∥(v~n-wn)+(p-p~)∥2+2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥,
which hence yields
(116)(1-a^-snγ¯)∥(v~n-wn)+(p-p~)∥2≤(1-βn-snγ¯)∥(v~n-wn)+(p-p~)∥2≤∥xn-p∥2-∥zn-p∥2+2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥+2sn(∥(I-V)p∥+∥(γQ-μF)p∥)∥zn-p∥.
Since lim⁡n→∞sn=0 and {xn},{zn},{wn}, and {v~n} are bounded sequences, we conclude from (77) and (108) that
(117)lim⁡n→∞∥(v~n-wn)+(p-p~)∥=0.
Note that
(118)∥vn-wn∥≤∥(vn-v~n)-(p-p~)∥+∥(v~n-wn)+(p-p~)∥.
Hence from (114) and (117) we get
(119)lim⁡n→∞∥vn-Gvn∥=lim⁡n→∞∥vn-wn∥=0.
Observe that
(120)∥vn-Tnvn∥≤∥vn-xn∥+∥xn-TnGvn∥+∥TnGvn-Tnvn∥≤∥vn-xn∥+∥xn-TnGvn∥+∥Gvn-vn∥.
Hence, from (75), (104), and (119) we have
(121)lim⁡n→∞∥vn-Tnvn∥=0.
It is clear that
(122)∥PC(I-λn∇f)vn-vn∥=∥snvn+(1-sn)Tnvn-vn∥=(1-sn)∥Tnvn-vn∥≤∥Tnvn-vn∥,
where sn=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L). Hence we have
(123)∥PC(I-2L∇f)vn-vn∥≤∥PC(I-2L∇f)vn-PC(I-λn∇f)vn∥+∥PC(I-λn∇f)vn-vn∥≤∥(I-2L∇f)vn-(I-λn∇f)vn∥+∥PC(I-λn∇f)vn-vn∥≤(2L-λn)∥∇f(vn)∥+∥Tnvn-vn∥.
From the boundedness of {vn}, sn→0 (⇔λn→2/L), and ∥Tnvn-vn∥→0 (due to (121)), it follows that
(124)lim⁡n→∞∥vn-PC(I-2L∇f)vn∥=0.
In addition, from (67) and (77), we have
(125)∥zn+1-zn∥≤∥zn+1-xn+1∥+∥xn+1-xn∥+∥xn-zn∥⟶0aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaiasn⟶∞.
We note that
(126)∥Snzn-Sn+1zn∥≤∥Snzn-zn∥+∥zn-zn+1∥+∥zn+1-Sn+1zn+1∥+∥Sn+1zn+1-Sn+1zn∥.
From (81), (125), and Lemma 17, we obtain
(127)lim⁡n→∞∥Snzn-Sn+1zn∥=0.
In the meantime, we note that
(128)∥zn-Szn∥≤∥zn-Snzn∥+∥Snzn-Sn+1zn∥+∥Sn+1zn-Szn∥.
From (81), (127), and the uniform continuity of S, we have
(129)lim⁡n→∞∥zn-Szn∥=0.
Step
4. We prove that xn→x*=PΩx0 as n→∞.
Indeed, since{xn} is bounded, there exists a subsequence {xni} which converges weakly to some w. From (77), (83), (104), (92), and (102) we have that zni⇀w, uni⇀w, vni⇀w, Δnikxni⇀w, and Λnimuni⇀w, where k∈{1,2,…,M} and m∈{1,2,…,N}. Since S is uniformly continuous, by (129) we get lim⁡n→∞∥zn-Smzn∥=0 for any m≥1. Hence from Lemma 19, we obtain w∈Fix⁡(S). In the meantime, utilizing Lemma 11, we deduce from vni⇀w, xni⇀w, (119), and (124) that w∈GSVI(G) and w∈Fix⁡(PC(I-(2/L)∇f))=VI⁡(C,∇f)=Γ. Next we prove that w∈∩m=1NVI⁡(C,Am). Let
(130)T~mv={Amv+NCv,v∈C,∅,v∉C,
where m∈{1,2,…,N}. Let (v,u)∈G(T~m). Since u-Amv∈NCv and Λnmun∈C, we have
(131)〈v-Λnmun,u-Amv〉≥0.
On the other hand, from Λnmun=PC(I-λm,nAm)Λnm-1un and v∈C, we have
(132)〈v-Λnmun,Λnmun-(Λnm-1un-λm,nAmΛnm-1un)〉≥0,
and hence
(133)〈v-Λnmun,Λnmun-Λnm-1unλm,n+AmΛnm-1un〉≥0.
Therefore we have
(134)〈v-Λnimuni,u〉≥〈v-Λnimuni,Amv〉≥〈v-Λnimuni,Amv〉-〈v-Λnimuni,Λnimuni-Λnim-1uniλm,ni+AmΛnim-1uni〉=〈v-Λnimuni,Amv-AmΛnimuni〉+〈v-Λnimuni,AmΛnimuni-AmΛnim-1uni〉-〈v-Λnimuni,Λnimuni-Λnim-1uniλm,ni〉≥〈v-Λnimuni,AmΛnimuni-AmΛnim-1uni〉-〈v-Λnimuni,Λnimuni-Λnim-1uniλm,ni〉.
From (102) and since Am is Lipschitz continuous, we obtain that lim⁡n→∞∥AmΛnmun-AmΛnm-1un∥=0. From Λnimuni⇀w, {λi,n}⊂[ai,bi]⊂(0,2ηi), ∀i∈{1,2,…,N}, and (102), we have
(135)〈v-w,u〉≥0.
Since T~m is maximal monotone, we have w∈T~m-10 and hence w∈VI⁡(C,Am), m=1,2,…,N, which implies w∈∩m=1NVI⁡(C,Am). Next we prove that w∈∩k=1MGMEP⁡(Θk,φk,Bk). Since Δnkxn=Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn, n≥1, k∈{1,2,…,M}, we have
(136)Θk(Δnkxn,y)+φk(y)-φk(Δnkxn)+〈BkΔnk-1xn,y-Δnkxn〉+1rk,n〈y-Δnkxn,Δnkxn-Δnk-1xn〉≥0.
By (A2), we have
(137)φk(y)-φk(Δnkxn)+〈BkΔnk-1xn,y-Δnkxn〉+1rk,n〈y-Δnkxn,Δnkxn-Δnk-1xn〉≥Θk(y,Δnkxn).
Let zt=ty+(1-t)w for all t∈(0,1] and y∈C. This implies that zt∈C. Then, we have
(138)〈zt-Δnkxn,Bkzt〉≥φk(Δnkxn)-φk(zt)+〈zt-Δnkxn,Bkzt〉-〈zt-Δnkxn,BkΔnk-1xn〉-〈zt-Δnkxn,Δnkxn-Δnk-1xnrk,n〉+Θk(zt,Δnkxn)=φk(Δnkxn)-φk(zt)+〈zt-Δnkxn,Bkzt-BkΔnkxn〉+〈zt-Δnkxn,BkΔnkxn-BkΔnk-1xn〉-〈zt-Δnkxn,Δnkxn-Δnk-1xnrk,n〉+Θk(zt,Δnkxn).
By (92), we have ∥BkΔnkxn-BkΔnk-1xn∥→0 as n→∞. Furthermore, by the monotonicity of Bk, we obtain 〈zt-Δnkxn,Bkzt-BkΔnkxn〉≥0. Then, by (A4) we obtain
(139)〈zt-w,Bkzt〉≥φk(w)-φk(zt)+Θk(zt,w).
Utilizing (A1), (A4), and (139), we obtain
(140)0=Θk(zt,zt)+φk(zt)-φk(zt)≤tΘk(zt,y)+(1-t)Θk(zt,w)+tφk(y)+(1-t)φk(w)-φk(zt)≤t[Θk(zt,y)+φk(y)-φk(zt)]+(1-t)〈zt-w,Bkzt〉=t[Θk(zt,y)+φk(y)-φk(zt)]+(1-t)t〈y-w,Bkzt〉,
and hence
(141)0≤Θk(zt,y)+φk(y)-φk(zt)+(1-t)〈y-w,Bkzt〉.
Letting t→0, we have, for each y∈C,
(142)0≤Θk(w,y)+φk(y)-φk(w)+〈y-w,Bkw〉.
This implies that w∈GMEP⁡(Θk,φk,Bk) and hence w∈∩k=1MGMEP⁡(Θk,φk,Bk). Consequently, w∈∩k=1MGMEP⁡(Θk,φk,Bk)∩∩i=1NVI⁡(C,Ai)∩GSVI(G)∩Fix⁡(S)∩Γ=:Ω. This shows that ωw(xn)⊂Ω. From (64) and Lemma 22 we infer that xn→x*=PΩx0 as n→∞.
Finally, assume additionally that∥xn-zn∥=o(sn) and lim⁡n→∞σn=0. It is clear that
(143)〈(V-I)x-(V-I)y,x-y〉≥(γ¯-1)∥x-y∥2,∀x,y∈H.
So, we know that V-I is (γ¯-1)-strongly monotone with constant γ¯-1>0. In the meantime, it is easy to see that V-I is (∥V∥+1)-Lipschitzian with constant ∥V∥+1>0. Thus, there exists a unique solution x^ in Ω to the VIP
(144)〈(I-V)x^,p-x^〉≤0,∀p∈Ω.
Equivalently, x^=PΩ(2I-V)x^. Furthermore, from (58), (59), and (82) we get
(145)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×∥Gvn-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×∥vn-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×∥xn-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉=∥xn-p∥2+2sn〈(I-V)p+σn(γQ-μF)p,zn-p〉,
which hence yields
(146)〈(I-V)p+σn(γQ-μF)p,p-zn〉≤∥xn-p∥2-∥zn-p∥22sn≤∥xn-zn∥2sn(∥xn-p∥+∥zn-p∥).
Since ∥xn-zn∥=o(sn), lim⁡n→∞σn=0, lim⁡n→∞∥xn-x*∥=0, and {xn}, {zn} are bounded, we infer from (146) that
(147)〈(I-V)p,p-x*〉≤0,∀p∈Ω,
which, together with Minty's Lemma [4], implies that
(148)〈(I-V)x*,p-x*〉≤0,∀p∈Ω.
This shows that x* is a solution in Ω to the VIP (144). Utilizing the uniqueness of solutions in Ω to the VIP (144), we get x*=x^. This completes the proof.Corollary 25.
LetC 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 ∇f. Let Θ be a bifunction from C×C to R satisfying (A1)–(A4) and let φ:C→R∪{+∞} be a proper lower semicontinuous and convex function. Let B,Ai:H→H, and Fj:C→H be ζ-inverse-strongly monotone, ηi-inverse-strongly monotone, and ζj-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 {γn}⊂[0,∞) such that lim⁡n→∞γn=0 and {cn}⊂[0,∞) such that lim⁡n→∞cn=0. Let V be a γ¯-strongly positive bounded linear operator with γ¯>1. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/κ2 and 0≤γl≤τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=GMEP(Θ,φ,B)∩VI(C,A1)∩VI(C,A2)∩GSVI(G)∩Fix⁡(S)∩Γ is nonempty and bounded and that either (B1) or (B2) holds. Let 0<α≤αn≤1,k≤δn≤d<1 for all n≥1, and let {βn}, {σn} be sequences in (0,1]. Pick any x0∈H and set C1=C,x1=PC1x0. Let {xn} be a sequence generated by the following algorithm:
(149)Θ(un,y)+φ(y)-φ(un)+〈Bxn,y-un〉+1rn〈un-xn,y-un〉≥0,∀y∈C,vn=PC(I-λ2,nA2)PC(I-λ1,nA1)un,zn=βnxn+((1-βn)I-snV)TnGvn+sn[Tnxn-σn(μF(Tnxn)-γQxn)],kn=δnzn+(1-δn)Snzn,yn=(1-αn)xn+αnkn,Cn+1={z∈Cn:∥yn-z∥2≤∥xn-z∥2+θn},xn+1=PCn+1x0,∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn (here Tn is nonexpansive; sn=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L)), θn=(sn+γn)(1+γn)Δn+cn, and Δn=sup⁡{∥xn-p∥2+(∥(I-V)p∥+∥(γQ-μF)p∥)2/(γ¯-1):p∈Ω}<∞. Suppose that the following conditions are satisfied:(i)
sn∈(0,1/2) for each λn∈(0,2/L), lim⁡n→∞sn=0 (⇔lim⁡n→∞λn=2/L);(ii)
{rn}⊂[e,f]⊂(0,2ζ),{λi,n}⊂[ai,bi]⊂(0,2ηi), and νj∈(0,2ζj) for i=1,2 and j=1,2;(iii)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1.
Then one has the following:(I)
{xn} converges strongly as λn→(2/L)(⇔sn→0) to x*=PΩx0;(II)
{xn} converges strongly as λn→(2/L)(⇔sn→0) to x*=PΩx0 provided ∥xn-zn∥=o(sn) and lim⁡n→∞σn=0, which is the unique solution in Ω to the VIP(150)〈(I-V)x*,p-x*〉≤0,∀p∈Ω.Equivalently, x*=PΩ(2I-V)x*.Corollary 26.
LetC 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 ∇f. Let Θ be a bifunction from C×C to R satisfying (A1)–(A4) and let φ:C→R∪{+∞} be a proper lower semicontinuous and convex function. Let B,A:H→H, and Fj:C→H be ζ-inverse-strongly monotone, ξ-inverse-strongly monotone, and ζj-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 {γn}⊂[0,∞) such that lim⁡n→∞γn=0 and {cn}⊂[0,∞) such that lim⁡n→∞cn=0. Let V be a γ¯-strongly positive bounded linear operator with γ¯>1. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/κ2 and 0≤γl≤τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=GMEP(Θ,φ,B)∩VI(C,A)∩GSVI(G)∩Fix(S)∩Γ is nonempty and bounded and that either (B1) or (B2) holds. Let 0<α≤αn≤1, k≤δn≤d<1 for all n≥1, and let {βn}, {σn} be sequences in (0,1]. Pick any x0∈H and set C1=C, x1=PC1x0. Let {xn} be a sequence generated by the following algorithm:
(151)Θ(un,y)+φ(y)-φ(un)+〈Bxn,y-un〉+1rn〈un-xn,y-un〉≥0,∀y∈C,vn=PC(I-ρnA)un,zn=βnxn+((1-βn)I-snV)TnGvn+sn[Tnxn-σn(μF(Tnxn)-γQxn)],kn=δnzn+(1-δn)Snzn,yn=(1-αn)xn+αnkn,Cn+1={z∈Cn:∥yn-z∥2≤∥xn-z∥2+θn},xn+1=PCn+1x0,∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn (here Tn is nonexpansive; sn=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L)), θn=(sn+γn)(1+γn)Δn+cn, and Δn=sup⁡{∥xn-p∥2+(∥(I-V)p∥+∥(γQ-μF)p∥)2/(γ¯-1):p∈Ω}<∞. Suppose that the following conditions are satisfied:(i)
sn∈(0,1/2) for each λn∈(0,2/L), lim⁡n→∞sn=0(⇔lim⁡n→∞λn=2/L);(ii)
{rn}⊂[e,f]⊂(0,2ζ), {ρn}⊂[a,b]⊂(0,2ξ), and νj∈(0,2ζj) for j=1,2;(iii)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1.
Then one has the following:(I)
{xn} converges strongly as λn→(2/L)(⇔sn→0) to x*=PΩx0;(II)
{xn} converges strongly as λn→(2/L)(⇔sn→0) to x*=PΩx0 provided ∥xn-zn∥=o(sn) and lim⁡n→∞σn=0, which is the unique solution in Ω to the VIP(152)〈(I-V)x*,p-x*〉≤0,∀p∈Ω.Equivalently, x*=PΩ(2I-V)x*.Corollary 27.
LetC 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 ∇f. Let Θ be a bifunction from C×C to R satisfying (A1)–(A4) and let φ:C→R∪{+∞} be a proper lower semicontinuous and convex function. Let B,A:H→H, and Fj:C→H be ζ-inverse-strongly monotone, ξ-inverse-strongly monotone, and ζj-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 {γn}⊂[0,∞) such that lim⁡n→∞γn=0. Let V be a γ¯-strongly positive bounded linear operator with γ¯>1. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/κ2 and 0≤γl≤τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=GMEP(Θ,φ,B)∩VI(C,A)∩GSVI(G)∩Fix(S)∩Γ is nonempty and bounded and that either (B1) or (B2) holds. Let 0<α≤αn≤1, k≤δn≤d<1 for all n≥1, and let {βn},{σn} be sequences in (0,1]. Pick any x0∈H and set C1=C, x1=PC1x0. Let {xn} be a sequence generated by the following algorithm:
(153)Θ(un,y)+φ(y)-φ(un)+〈Bxn,y-un〉+1rn〈un-xn,y-un〉≥0,∀y∈C,vn=PC(I-ρnA)un,zn=βnxn+((1-βn)I-snV)TnGvn+sn[Tnxn-σn(μF(Tnxn)-γQxn)],kn=δnzn+(1-δn)Snzn,yn=(1-αn)xn+αnkn,Cn+1={z∈Cn:∥yn-z∥2≤∥xn-z∥2+θn},xn+1=PCn+1x0,∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn (here Tn is nonexpansive; sn=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L)), θn=(sn+γn)(1+γn)Δn, and Δn=sup⁡{∥xn-p∥2+(∥(I-V)p∥+∥(γQ-μF)p∥)2/(γ¯-1):p∈Ω}<∞. Suppose that the following conditions are satisfied:(i)
sn∈(0,1/2) for each λn∈(0,2/L), lim⁡n→∞sn=0 (⇔lim⁡n→∞λn=2/L);(ii)
{rn}⊂[e,f]⊂(0,2ζ),{ρn}⊂[a,b]⊂(0,2ξ), and νj∈(0,2ζj) for j=1,2;(iii)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1.
Then one has the following:(I)
{xn} converges strongly as λn→(2/L)(⇔sn→0) to x*=PΩx0;(II)
{xn} converges strongly as λn→(2/L)(⇔sn→0) to x*=PΩx0 provided ∥xn-zn∥=o(sn) and lim⁡n→∞σn=0, which is the unique solution in Ω to the VIP(154)〈(I-V)x*,p-x*〉≤0,∀p∈Ω.Equivalently, x*=PΩ(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.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let M,N be two integers. Let Θk be a bifunction from C×C to R satisfying (A1)–(A4) and let φk:C→R∪{+∞} be a proper lower semicontinuous and convex function, where k∈{1,2,…,M}. Let Bk,Ai:H→H, and Fj:C→H be μk-inverse-strongly monotone, ηi-inverse-strongly monotone, and ζj-inverse-strongly monotone, respectively, where k∈{1,2,…,M}, i∈{1,2,…,N}, and j∈{1,2}. Let {Tn}n=1∞ be a sequence of nonexpansive mappings on H and let {λn} be a sequence in (0,b] for some b∈(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 {γn}⊂[0,∞) such that lim⁡n→∞γn=0 and {cn}⊂[0,∞) such that lim⁡n→∞cn=0. Let V be a γ¯-strongly positive bounded linear operator with γ¯>1. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/κ2 and 0≤γl≤τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=∩n=1∞Fix(Tn)∩∩k=1MGMEP(Θk,φk,Bk)∩∩i=1NVI(C,Ai)∩GSVI(G)∩Fix(S) is nonempty and bounded and that either (B1) or (B2) holds. Let 0<α≤αn≤1, k≤δn≤d<1 for all n≥1, and let {βn},{ϵn}, and {σn} be sequences in (0,1]. Pick any x0∈H and set C1=C, x1=PC1x0. Let {xn} be a sequence generated by the following algorithm:
(155)un=TrM,n(ΘM,φM)(I-rM,nBM)TrM-1,n(ΘM-1,φM-1)×(I-rM-1,nBM-1)⋯Tr1,n(Θ1,φ1)(I-r1,nB1)xn,vn=PC(I-λN,nAN)PC(I-λN-1,nAN-1)⋯PC×(I-λ2,nB2)PC(I-λ1,nB1)un,zn=βnxn+((1-βn)I-ϵnV)WnGvn+ϵn[Wnxn-σn(μF(Wnxn)-γQxn)],kn=δnzn+(1-δn)Snzn,yn=(1-αn)xn+αnkn,Cn+1={z∈Cn:∥yn-z∥2≤∥xn-z∥2+θn},xn+1=PCn+1x0,∀n≥1,
where Wn is the W-mapping defined by (34), θn=(ϵn+γn)(1+γn)Δn+cn, and Δn=sup⁡{∥xn-p∥2+(∥(I-V)p∥+∥(γQ-μF)p∥)2/(γ¯-1):p∈Ω}<∞. Suppose that the following conditions are satisfied:(i)
{rk,n}⊂[ek,fk]⊂(0,2μk),{λi,n}⊂[ai,bi]⊂(0,2ηi), and νj∈(0,2ζj), where k∈{1,2,…,M}, i∈{1,2,…,N}, and j∈{1,2};(ii)
lim⁡n→∞ϵn=0 and 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1.
Then one has the following:(I)
{xn} converges strongly to x*=PΩx0;(II)
{xn} converges strongly to x*=PΩx0 provided ∥xn-zn∥=o(ϵn) and lim⁡n→∞σn=0, which is the unique solution in Ω to the VIP(156)〈(I-V)x*,p-x*〉≤0,∀p∈Ω.Equivalently, x*=PΩ(2I-V)x*.Proof.
First of all, let us show that the sequence{xn} is well defined. As lim⁡n→∞ϵn=0 and 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1, we may assume, without loss of generality, that {βn}⊂[a,a^]⊂(0,1) and βn+ϵn∥V∥≤1 for all n≥1. Utilizing the arguments similar to those in the proof of Theorem 24, we get
(157)∥(1-βn)I-ϵnV∥≤1-βn-ϵnγ¯.
Put
(158)Δnk=Trk,n(Θk,φk)(I-rk,nBk)Trk-1,n(Θk-1,φk-1)×(I-rk-1,nBk-1)⋯Tr1,n(Θ1,φ1)(I-r1,nB1)xn
for all k∈{1,2,…,M} and n≥1 and
(159)Λni=PC(I-λi,nBi)PC(I-λi-1,nBi-1)⋯PC(I-λ1,nB1)
for all i∈{1,2,…,N}, Δn0=I, and Λn0=I, where I is the identity mapping on H. Then we have that un=ΔnMxn and vn=ΛnNun.
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
(160)〈2(xn-yn),z〉≤∥xn∥2-∥yn∥2+θn,
by Lemma 16 we know that Cn is convex for every n≥1.
First of all, let us show thatΩ⊂Cn for all n≥1. Suppose that Ω⊂Cn for some n≥1. Take p∈Ω arbitrarily. Utilizing the arguments similar to those in the proof of Theorem 24 we obtain that
(161)∥un-p∥≤∥Δnkxn-Δnkp∥≤∥xn-p∥,(162)∥vn-p∥≤∥Λniun-Λnip∥≤∥un-p∥,(163)∥vn-p∥≤∥xn-p∥,(164)∥Gvn-p∥2≤∥PC(I-ν2F2)vn-PC(I-ν2F2)p∥2+ν1(ν1-2ζ1)×∥F1PC(I-ν2F2)vn-F1PC(I-ν2F2)p∥2≤∥vn-p∥2+ν2(ν2-2ζ2)∥F2vn-F2p∥2≤∥vn-p∥2,(165)∥kn-p∥2≤(1+γn)∥zn-p∥2+(1-δn)(k-δn)∥zn-Snzn∥2+cn≤(1+γn)∥zn-p∥2+cn≤(1+γn)×(∥xn-p∥2+ϵn(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1)+cn.
So, from (155) and (165) we get
(166)∥yn-p∥2≤(1-αn)∥xn-p∥2+αn∥kn-p∥2≤(1-αn)∥xn-p∥2+αn[(+εn(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1)∥xn-p∥2+εn(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1)(1+γn)×(∥xn-p∥2+ϵn(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1)+cn(+εn(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1)∥xn-p∥2+εn(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1)]≤(1+γn)((∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1∥xn-p∥2+ϵn(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1)+cn=∥xn-p∥2+γn∥xn-p∥2+(1+γn)ϵn×(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1+cn≤∥xn-p∥2+(ϵn+γn)(1+γn)×(∥xn-p∥2+(∥(I-V)p∥+∥(γQ-μF)p∥)2γ¯-1)+cn≤∥xn-p∥2+(ϵn+γn)(1+γn)Δn+cn=∥xn-p∥2+θn,
where θn=(ϵn+γn)(1+γn)Δn+cn and Δn=sup⁡{∥xn-p∥2+(∥(I-V)p∥+∥(γQ-μF)p∥)2/(γ¯-1):p∈Ω}<∞. Hence p∈Cn+1. This implies that Ω⊂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→∞.
Indeed, letx*=PΩx0. From xn=PCnx0 and x*∈Ω⊂Cn, we obtain
(167)∥xn-x0∥≤∥x*-x0∥.
This implies that {xn} is bounded and hence {un},{vn},{zn},{kn}, and {yn} are also bounded. Utilizing the arguments similar to those of (67), (75), (77), and (81) in the proof of Theorem 24 we obtain that
(168)lim⁡n→∞∥xn+1-xn∥=0,lim⁡n→∞∥xn-WnGvn∥=0,(169)lim⁡n→∞∥xn-zn∥=0,(170)lim⁡n→∞∥Snzn-zn∥=0.
Step
3. We prove that ∥xn-un∥→0, ∥xn-vn∥→0, ∥vn-Gvn∥→0, ∥vn-Wvn∥→0, and ∥zn-Szn∥→0 as n→∞.
Indeed, from (162), (164), γ¯>1, and γl≤τ, it follows that
(171)∥zn-p∥2=∥βn(xn-p)+((1-βn)I-snV)(WnGvn-p)+sn[σnγ(Qxn-Qp)+(I-σnμF)Wnxn-(I-σnμF)Wnp]+sn[(I-V)p+σn(γQ-μF)p]∥2≤[βn∥xn-p∥+∥(1-βn)I-ϵnV∥∥WnGvn-p∥+ϵn(σnγ∥Qxn-Qp∥+∥(I-σnμF)Wnxn-(I-σnμF)Wnp∥)]2+2ϵn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤[βn∥xn-p∥+(1-βn-ϵnγ¯)∥Gvn-p∥+ϵn(σnγl∥xn-p∥+(1-σnτ)∥xn-p∥)]2+2ϵn〈(I-V)p+σn(γQ-μF)p,zn-p〉=[(1-σn(τ-γl))βn∥xn-p∥+(1-βn-ϵnγ¯)∥Gvn-p∥+ϵn(1-σn(τ-γl))∥xn-p∥]2+2ϵn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤[(βn+ϵnγ¯)∥xn-p∥+(1-βn-ϵnγ¯)∥Gvn-p∥]2+2ϵn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+ϵnγ¯)∥xn-p∥2+(1-βn-ϵnγ¯)∥Gvn-p∥2+2ϵn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+ϵnγ¯)∥xn-p∥2+(1-βn-ϵnγ¯)∥vn-p∥2+2ϵn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+ϵnγ¯)∥xn-p∥2+(1-βn-ϵnγ¯)∥un-p∥2+2ϵn〈(I-V)p+σn(γQ-μF)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
(172)lim⁡n→∞∥xn-un∥=0,(173)lim⁡n→∞∥Δnk-1xn-Δnkxn∥=0,k=1,2,…,M,(174)lim⁡n→∞∥Λni-1un-Λniun∥=0,i=1,2,…,N,(175)lim⁡n→∞∥xn-vn∥=0,(176)lim⁡n→∞∥vn-Gvn∥=0,(177)lim⁡n→∞∥vn-Wnvn∥=0,(178)lim⁡n→∞∥zn-Szn∥=0.
In addition, note that
(179)∥vn-Wvn∥≤∥vn-Wnvn∥+∥Wnvn-Wvn∥.
So, from ∥vn-Wnvn∥→0 and [20, Remark 3.2] it follows that
(180)lim⁡n→∞∥vn-Wvn∥=0.
Step
4. We prove that xn→x*=PΩx0 as n→∞.
Indeed, since{xn} is bounded, there exists a subsequence {xni} which converges weakly to some w. From (169), (172), (175), (173), and (174) we have that zni⇀w,uni⇀w,vni⇀w,Δnikxni⇀w, and Λnimuni⇀w, where k∈{1,2,…,M} and m∈{1,2,…,N}. Since S is uniformly continuous, by (178) we get lim⁡n→∞∥zn-Smzn∥=0 for any m≥1. Hence, from Lemma 19, we obtain w∈Fix⁡(S). In the meantime, utilizing Lemma 11, we deduce from (176) and (180) that w∈GSVI(G) and w∈Fix⁡(W)=∩n=1∞Fix⁡(Tn) (due to Lemma 13). Hence we get w∈GSVI(G)∩∩n=1∞Fix⁡(Tn). Repeating the same arguments as in the proof of Theorem 24 we conclude that w∈∩m=1NVI⁡(C,Am) and w∈∩k=1MGMEP⁡(Θk,φk,Bk). Consequently, w∈∩n=1∞Fix⁡(Tn)∩∩k=1MGMEP⁡(Θk,φk,Bk)∩∩i=1NVI⁡(C,Ai)∩GSVI(G)∩Fix⁡(S)=:Ω. This shows that ωw(xn)⊂Ω. From (167) and Lemma 22 we infer that xn→x*=PΩx0 as n→∞.
Finally, assume additionally that∥xn-zn∥=o(ϵn) and lim⁡n→∞σn=0. It is clear that
(181)〈(V-I)x-(V-I)y,x-y〉≥(γ¯-1)∥x-y∥2,∀x,y∈H.
So, we know that V-I is (γ¯-1)-strongly monotone with constant γ¯-1>0. In the meantime, it is easy to see that V-I is (∥V∥+1)-Lipschitzian with constant ∥V∥+1>0. Thus, there exists a unique solution x^ in Ω to the VIP
(182)〈(I-V)x^,p-x^〉≤0,∀p∈Ω.
Equivalently, x^=PΩ(2I-V)x^. Furthermore, from (163), (164), and (171) we get
(183)∥zn-p∥2≤(βn+ϵnγ¯)∥xn-p∥2+(1-βn-ϵnγ¯)∥Gvn-p∥2+2ϵn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+ϵnγ¯)∥xn-p∥2+(1-βn-ϵnγ¯)∥vn-p∥2+2ϵn〈(I-V)p+σn(γQ-μF)p,zn-p〉≤(βn+ϵnγ¯)∥xn-p∥2+(1-βn-ϵnγ¯)∥xn-p∥2+2ϵn〈(I-V)p+σn(γQ-μF)p,zn-p〉=∥xn-p∥2+2ϵn〈(I-V)p+σn(γQ-μF)p,zn-p〉,
which hence yields
(184)〈(I-V)p+σn(γQ-μF)p,p-zn〉≤∥xn-p∥2-∥zn-p∥22ϵn≤∥xn-zn∥2ϵn(∥xn-p∥+∥zn-p∥).
Since ∥xn-zn∥=o(sn), lim⁡n→∞σn=0, lim⁡n→∞∥xn-x*∥=0, and {xn},{zn} are bounded, we infer from (94) that
(185)〈(I-V)p,p-x*〉≤0,∀p∈Ω,
which, together with Minty's Lemma, implies that
(186)〈(I-V)x*,p-x*〉≤0,∀p∈Ω.
This shows that x* is a solution in Ω to the VIP (182). Utilizing the uniqueness of solutions in Ω to the VIP (182), we get x*=x^. This completes the proof.Corollary 29.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C×C to R satisfying (A1)–(A4) and let φ:C→R∪{+∞} be a proper lower semicontinuous and convex function. Let B,Ai:H→H, and Fj:C→H be ζ-inverse-strongly monotone, ηi-inverse-strongly monotone, and ζj-inverse-strongly monotone, respectively, for i=1,2 and j=1,2. Let {Tn}n=1∞ be a sequence of nonexpansive mappings on H and let {λn} be a sequence in (0,b] for some b∈(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 {γn}⊂[0,∞) such that lim⁡n→∞γn=0 and {cn}⊂[0,∞) such that lim⁡n→∞cn=0. Let V be a γ¯-strongly positive bounded linear operator with γ¯>1. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/κ2 and 0≤γl≤τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=∩n=1∞Fix(Tn)∩GMEP(Θ,φ,B)∩VI(C,A1)∩VI(C,A2)∩GSVI(G)∩Fix(S) is nonempty and bounded and that either (B1) or (B2) holds. Let 0<α≤αn≤1, k≤δn≤d<1 for all n≥1, and let {βn},{ϵn}, and {σn} be sequences in (0,1]. Pick any x0∈H and set C1=C, x1=PC1x0. Let {xn} be a sequence generated by the following algorithm:
(187)Θ(un,y)+φ(y)-φ(un)+〈Bxn,y-un〉+1rn〈un-xn,y-un〉≥0,∀y∈C,vn=PC(I-λ2,nA2)PC(I-λ1,nB1)un,zn=βnxn+((1-βn)I-ϵnV)WnGvn+ϵn[Wnxn-σn(μF(Wnxn)-γQxn)],kn=δnzn+(1-δn)Snzn,yn=(1-αn)xn+αnkn,Cn+1={z∈Cn:∥yn-z∥2≤∥xn-z∥2+θn},xn+1=PCn+1x0,∀n≥1,
where Wn is the W-mapping defined by (34), θn=(ϵn+γn)(1+γn)Δn+cn, and Δn=sup⁡{∥xn-p∥2+(∥(I-V)p∥+∥(γQ-μF)p∥)2/(γ¯-1):p∈Ω}<∞. Suppose that the following conditions are satisfied:(i)
{rn}⊂[e,f]⊂(0,2ζ),{λi,n}⊂[ai,bi]⊂(0,2ηi), and νj∈(0,2ζj) for i=1,2 and j=1,2;(ii)
lim⁡n→∞ϵn=0 and 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1. Then one has the following:(I)
{xn} converges strongly to x*=PΩx0;(II)
{xn} converges strongly to x*=PΩx0 provided ∥xn-zn∥=o(ϵn) and lim⁡n→∞σn=0, which is the unique solution in Ω to the VIP(188)〈(I-V)x*,p-x*〉≤0,∀p∈Ω.Equivalently, x*=PΩ(2I-V)x*.Corollary 30.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C×C to R satisfying (A1)–(A4) and let φ:C→R∪{+∞} be a proper lower semicontinuous and convex function. Let B,A:H→H, and Fj:C→H be ζ-inverse-strongly monotone, ξ-inverse-strongly monotone, and ζj-inverse-strongly monotone, respectively, for j=1,2. Let {Tn}n=1∞ be a sequence of nonexpansive mappings on H and let {λn} be a sequence in (0,b] for some b∈(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 {γn}⊂[0,∞) such that lim⁡n→∞γn=0 and {cn}⊂[0,∞) such that lim⁡n→∞cn=0. Let V be a γ¯-strongly positive bounded linear operator with γ¯>1. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/κ2 and 0≤γl≤τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=∩n=1∞Fix(Tn)∩GMEP(Θ,φ,B)∩VI(C,A)∩GSVI(G)∩Fix(S) is nonempty and bounded and that either (B1) or (B2) holds. Let 0<α≤αn≤1, k≤δn≤d<1 for all n≥1, and let {βn},{ϵn}, and {σn} be sequences in (0,1]. Pick any x0∈H and set C1=C, x1=PC1x0. Let {xn} be a sequence generated by the following algorithm:
(189)Θ(un,y)+φ(y)-φ(un)+〈Bxn,y-un〉+1rn〈un-xn,y-un〉≥0,∀y∈C,vn=PC(I-ρnA)un,zn=βnxn+((1-βn)I-ϵnV)WnGvn+ϵn[Wnxn-σn(μF(Wnxn)-γQxn)],kn=δnzn+(1-δn)Snzn,yn=(1-αn)xn+αnkn,Cn+1={z∈Cn:∥yn-z∥2≤∥xn-z∥2+θn},xn+1=PCn+1x0,∀n≥1,
where Wn is the W-mapping defined by (34), θn=(ϵn+γn)(1+γn)Δn+cn, and Δn=sup⁡{∥xn-p∥2+(∥(I-V)p∥+∥(γQ-μF)p∥)2/(γ¯-1):p∈Ω}<∞. Suppose that the following conditions are satisfied:(i)
{rn}⊂[e,f]⊂(0,2ζ),{ρn}⊂[a,c]⊂(0,2ξ), and νj∈(0,2ζj) for j=1,2;(ii)
lim⁡n→∞ϵn=0 and 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1. Then one has the following:(I)
{xn} converges strongly to x*=PΩx0;(II)
{xn} converges strongly to x*=PΩx0 provided ∥xn-zn∥=o(ϵn) and lim⁡n→∞σn=0, which is the unique solution in Ω to the VIP(190)〈(I-V)x*,p-x*〉≤0,∀p∈Ω.Equivalently, x*=PΩ(2I-V)x*.Corollary 31.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C×C to R satisfying (A1)–(A4) and let φ:C→R∪{+∞} be a proper lower semicontinuous and convex function. Let B,A:H→H, and Fj:C→H be ζ-inverse-strongly monotone, ξ-inverse-strongly monotone, and ζj-inverse-strongly monotone, respectively, for j=1,2. Let {Tn}n=1∞ be a sequence of nonexpansive mappings on H and let {λn} be a sequence in (0,b] for some b∈(0,1). Let S:C→C be a uniformly continuous asymptotically k-strict pseudocontractive mapping for some 0≤k<1 with sequence {γn}⊂[0,∞) such that lim⁡n→∞γn=0. Let V be a γ¯-strongly positive bounded linear operator with γ¯>1. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let Q:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/κ2 and 0≤γl≤τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=∩n=1∞Fix⁡(Tn)∩GMEP⁡(Θ,φ,B)∩VI⁡(C,A)∩GSVI(G)∩Fix⁡(S) is nonempty and bounded and that either (B1) or (B2) holds. Let 0<α≤αn≤1,k≤δn≤d<1 for all n≥1, and let {βn},{ϵn}, and {σn} be sequences in (0,1]. Pick any x0∈H and set C1=C,x1=PC1x0. Let {xn} be a sequence generated by the following algorithm:
(191)Θ(un,y)+φ(y)-φ(un)+〈Bxn,y-un〉aaa+1rn〈un-xn,y-un〉≥0,∀y∈C,vn=PC(I-ρnA)un,zn=βnxn+((1-βn)I-ϵnV)WnGvn+ϵn[Wnxn-σn(μF(Wnxn)-γQxn)],kn=δnzn+(1-δn)Snzn,yn=(1-αn)xn+αnkn,Cn+1={z∈Cn:∥yn-z∥2≤∥xn-z∥2+θn},xn+1=PCn+1x0,∀n≥1,
where Wn is the W-mapping defined by (34), θn=(ϵn+γn)(1+γn)Δn, and Δn=sup⁡{∥xn-p∥2+(∥(I-V)p∥+∥(γQ-μF)p∥)2/(γ¯-1):p∈Ω}<∞. Suppose that the following conditions are satisfied:(i)
{rn}⊂[e,f]⊂(0,2ζ),{ρn}⊂[a,c]⊂(0,2ξ), and νj∈(0,2ζj) for j=1,2;(ii)
lim⁡n→∞ϵn=0 and 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1. Then one has the following:(I)
{xn} converges strongly to x*=PΩx0;(II)
{xn} converges strongly to x*=PΩx0 provided ∥xn-zn∥=o(ϵn) and lim⁡n→∞σn=0, which is the unique solution in Ω to the VIP(192)〈(I-V)x*,p-x*〉≤0,∀p∈Ω.Equivalently, x*=PΩ(2I-V)x*.Remark 32.
LetA:C→H be ξ-inverse-strongly monotone and let Fi:C→H be νj-inverse-strongly monotone for j=1,2. Let Q:C→C be a ρ-contraction with ρ∈[0,1), and let S:C→C be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense for some 0≤k<1 with sequence {γn}⊂[0,∞) such that lim⁡n→∞γn=0 and {cn}⊂[0,∞) such that lim⁡n→∞cn=0. Assume that Ω:=VI⁡(C,A)∩GSVI(G)∩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∈Ω:
(193)x1=x∈Cchoosenarbitrarily,yn=PC(xn-ρnAxn),tn=αnQxn+(1-αn)Gyn,zn=(1-μn-νn)xn+μntn+νnSntn,Cn={z∈C:∥zn-z∥2≤∥xn-z∥2+θn},Qn={z∈C:〈xn-z,x-xn〉≥0},xn+1=PCn∩Qnx,∀n≥1,
where νj∈(0,2ζj) for j=1,2,θn=(αn+γn)Δn+cn; Δn=sup⁡{∥xn-p∥2+(1+γn)/(1-ρ)∥(I-Q)p∥2:p∈Ω}<∞; {ρn} is a sequence in (0,2ξ); and {αn},{μn}, and {νn} are three sequences in [0,1] such that μn+νn≤1 for all n≥1. The authors of [11] proved that under suitable conditions {xn} converges strongly to PΩx; see [11, Theorem 3.1] for more details.
Theorem28 extends, improves, supplements, and develops [11, Theorem 3.1] in the following aspects.(i)
The problem of finding a pointp∈∩n=1∞Fix⁡(Tn)∩∩k=1MGMEP⁡(Θk,φk,Bk)∩∩i=1NVI⁡(C,Ai)∩GSVI(G)∩Fix⁡(S) in Theorem 28 is very different from the problem of finding a point p∈VI⁡(C,A)∩GSVI(G)∩Fix⁡(S) in [11, Theorem 3.1]. There is no doubt that our problem of finding a point p∈∩n=1∞Fix⁡(Tn)∩∩k=1MGMEP⁡(Θk,φk,Bk)∩∩i=1NVI⁡(C,Ai)∩GSVI(G)∩Fix⁡(S) is more general and more subtle than the problem of finding a point p∈VI⁡(C,A)∩GSVI(G)∩Fix⁡(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 Theorem28 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 Tλ-mapping (see Lemmas 12, 13, and 15) because there are the mapping Trk,n(Θk,φk), infinitely many nonexpansive mappings {Tn}n=1∞, κ-Lipschitzian and η-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 Theorem28 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.
Theorem28 also extends, improves, supplements, and develops Ceng et al. [8, Theorem 3.1] in the following aspects.(i)
The problem of finding a pointp∈∩n=1∞Fix⁡(Tn)∩∩k=1MGMEP⁡(Θk,φk,Bk)∩∩i=1NVI⁡(C,Ai)∩GSVI(G)∩Fix⁡(S) in Theorem 28 is very different from the problem of finding a point p∈∩i=1NFix⁡(Ti)∩GMEP⁡(Θ,φ,A) in Ceng et al. [8, Theorem 3.1]. Here our problem of finding a point p∈∩n=1∞Fix⁡(Tn)∩∩k=1MGMEP⁡(Θk,φk,Bk)∩∩i=1NVI⁡(C,Ai)∩GSVI(G)∩Fix⁡(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=Trn(Θ,φ)(I-rnA)xn and Wnun in [8, Theorem 3.1] are replaced by un=ΔnMxn and WnGΛnNun, respectively.(iii)
The iterative scheme in Theorem28 is very different from the iterative scheme in [8, Theorem 3.1] because the iterative scheme in Theorem 28 involves Korpelevich'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 pseudocontractive mapping in the intermediate sense (see Lemmas 17–20) because there are the SGEP (8), finitely many GMEPs, asymptotically strict pseudocontractive 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 Theorem28 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]).



---

*Source: 105928-2014-05-08.xml*