# Multistep Hybrid Iterations for Systems of Generalized Equilibria with Constraints of Several Problems

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

---

## Abstract

We first introduce and analyze one multistep iterative algorithm by hybrid shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: the generalized mixed equilibrium problem, finitely many variational inclusions, the minimization problem for a convex and continuously Fréchet differentiable functional, 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 multistep iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions.


---

## Body

## 1. Introduction

LetC be a nonempty closed convex subset of a real Hilbert space H and PC 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 V is called strongly positive on H if there exists a constant γ¯>0 such that
(1)〈Vx,x〉≥γ¯∥x∥2,∀x∈H.
A mapping S:C→H is called L-Lipschitz continuous if there exists a constant L≥0 such that
(2)∥Sx-Sy∥≤L∥x-y∥,∀x,y∈C.
In particular, if L=1, then A is called a nonexpansive mapping; if L∈[0,1), then S is called a contraction.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). We remark that VIP (3) was first discussed by Lions [1].Letφ:C→R be a real-valued function, A:H→H a nonlinear mapping, and Θ:C×C→R a bifunction. In 2008, Peng and Yao [2] 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 which includes, as special cases, the generalized equilibrium problem [3], the mixed equilibrium problem [4], and the equilibrium problem [5, 6].In [2], it is assumed that Θ:C×C→R is a bifunction satisfying conditions (H1)–(H4) and φ:C→R is a lower semicontinuous and convex function with restriction (H5), where(H1)Θ(x,x)=0 for all x∈C;(H2)Θ is monotone; that is, Θ(x,y)+Θ(y,x)≤0 for any x,y∈C;(H3)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);(H4)Θ(x,·) is convex and lower semicontinuous for each x∈C;(H5) 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.Given a positive numberr>0. Let Sr(Θ,φ):H→C be the solution set of the auxiliary mixed equilibrium problem; that is, for each x∈H,
(7)Sr(Θ,φ)(x)≔{1ry∈C:Θ(y,z)+φ(z)-φ(y)+1r〈K′(y)-K′(x),z-y〉≥0,∀z∈C}.
In particular, whenever K(x)=(1/2)∥x∥2, ∀x∈H,Sr(Θ,φ)(x) is rewritten as Tr(Θ,φ)(x). Further, if φ=0 additionally, then Tr(Θ,φ)(x) is rewritten as TrΘ(x).LetΘ1,Θ2:C×C→R be two bifunctions satisfying conditions (H1)–(H4), and let A1,A2:C→H be two nonlinear mappings. Consider the following system of generalized equilibrium problems (SGEP): find (x*,y*)∈C×C such that
(8)Θ1(x*,x)+〈A1y*,x-x*〉+1ν1〈x*-y*,x-x*〉≥0,22222222222222222222222222222222222222∀x∈C,Θ2(y*,y)+〈A2x*,y-y*〉+1ν2〈y*-x*,y-y*〉≥0,222222222222222222222222222222222222222∀y∈C,
where ν1>0 and ν2>0 are two constants. It is introduced and studied in [7]. Whenever Θ1≡Θ2≡0, the SGEP reduces to a system of variational inequalities, which is considered and studied in [8]. It is worth to mention that the system of variational inequalities is a tool to solve the Nash equilibrium problem for noncooperative games. In 2010, Ceng and Yao [7] transformed the SGEP into a fixed-point problem of the mapping G=Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2). Here, we denote the fixed point set of G by SGEP(G).Let{Tn}n=1∞ be an infinite family of nonexpansive mappings on H and {λn}n=1∞ a sequence of nonnegative numbers in [0,1]. For any n≥1, define a mapping Wn on H as follows:
(9)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.LetQ:H→H be a contraction and V a strongly positive bounded linear operator on H. Assume that φ:H→R is a lower semicontinuous and convex functional, such that Θ,Θ1,Θ2:H×H→R satisfy conditions (H1)–(H4) and that A,A1,A2:H→H are inverse-strongly monotone. Very recently, Ceng et al. [9] introduced the following hybrid extragradient-like iterative algorithm:
(10)zn=Srn(Θ,φ)(xn-rnAxn),xn+1=αn(u+γQxn)+βnxn+((1-βn)I-αn(I+μV))WnGzn,∀n≥0,
for finding a common solution of GMEP⁡ (4), SGEP (8), and the fixed point problem of an infinite family of nonexpansive mappings {Tn}n=1∞ on H, where {rn}⊂(0,∞),{αn},{βn}⊂(0,1),νk∈(0,2ζk), k=1,2, and x0,u∈H are given. The authors proved the strong convergence of the sequence {xn} to a point x*∈Ω≔∩n=1∞Fix⁡(Tn)∩GMEP⁡(Θ,φ,A)∩SGEP(G) under some suitable conditions. This point x* also solves the following optimization problem:
(OP1)min⁡x∈Ωμ2〈Vx,x〉+12∥x-u∥2-h(x),
where h:H→R is the potential function of γQ.LetB be a single-valued mapping of C into H and R a multivalued mapping with D(R)=C. Consider the following variational inclusion: find a point x∈C such that
(11)0∈Bx+Rx.
We denote by I(B,R) the solution set of the variational inclusion (11).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(12)minimize{f(x):x∈C},
which was studied in [10–13]. We denote by Γ the set of minimizers of CMP (12).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
(13)∥Snx-Sny∥≤ℒ∥x-y∥,∀n≥1,∀x,y∈C.
Recently, Kim and Xu [14] 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
(14)∥Snx-Sny∥2≤(1+γn)∥x-y∥21111111111111+k∥x-Snx-(y-Sny)∥2,11111111111122111111∀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. [15] 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
(15)limsup⁡n→∞sup⁡x,y∈C(∥Snx-Sny∥2-(1+γn)∥x-y∥211111111111-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
(16)∥Snx-Sny∥2≤(1+γn)∥x-y∥21111111111111+k∥x-Snx-(y-Sny)∥2+cn,11111111111111111111111∀n≥1,∀x,y∈C.In this paper, we first introduce and analyze one multistep iterative algorithm by hybrid shrinking projection method for finding a solution of the SGEP (8) with constraints of several problems: the GMEP (4), the CMP (12), finitely many variational inclusions, 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 multistep iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions. Our results improve and extend the corresponding results in the earlier and recent literatures.


## 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,
(17)ωw(xn)≔{x∈H:xni⇀xforsomesubsequence{xni}of{xn}}.Recall that a mappingA:C→H is called(i)
monotone if(18)〈Ax-Ay,x-y〉≥0,∀x,y∈C;(ii)
η-strongly monotone if there exists a constant η>0 such that
(19)〈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
(20)〈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
(21)∥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(22)∥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),
(23)〈x-y,Tx-Ty〉≥∥Tx-Ty∥2,∀x,y∈H;alternatively, T is firmly nonexpansive if and only if T can be expressed as
(24)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.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,
(25)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 [16]).
LetT: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  ν-ism, then for  γ>0, γT is  (ν/γ)-ism.(iii)
T is averaged if and only if the complement  I-T is  ν-ism for some  ν>1/2. Indeed, for  α∈(0,1),T is  α-averaged if and only if  I-T is  (1/2α)-ism.Proposition 7 (see [16]).
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(26)⋂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}.By using the technique in [4], we can readily obtain the following elementary result.Proposition 8 (see [9, Lemma 1 and Proposition 1]).
LetC be a nonempty closed convex subset of a real Hilbert space H and let φ:C→R be a lower semicontinuous and convex function. Let Θ:C×C→R be a bifunction satisfying the conditions (H1)–(H4). Assume that(i)
K:H→Ris strongly convex with constantσ>0and the functionx↦〈y-x,K′(x)〉is weakly upper semicontinuous for eachy∈H;(ii)
for eachx∈Handr>0, there exists a bounded subsetDx⊂Candyx∈Csuch that for anyz∈C∖Dx,(27)Θ(z,yx)+φ(yx)-φ(z)+1r〈K′(z)-K′(x),yx-z〉<0.Then the following hold:(a)
for eachx∈H,Sr(Θ,φ)(x)≠∅;(b)
Sr(Θ,φ) is single-valued;(c)
Sr(Θ,φ) is nonexpansive if K′ is Lipschitz continuous with constant ν>0 and
(28)〈K′(x1)-K′(x2),u1-u2〉≤〈K′(u1)-K′(u2),u1-u2〉,∀(x1,x2)∈H×H,whereui=Sr(Θ,φ)(xi) for i=1,2;(d)
for alls,t>0 and x∈H(29)〈K′(Ss(Θ,φ)x)-K′(St(Θ,φ)x),Ss(Θ,φ)x-St(Θ,φ)x〉≤s-ts〈K′(Ss(Θ,φ)x)-K′(x),Ss(Θ,φ)x-St(Θ,φ)x〉;(e)
Fix⁡(Sr(Θ,φ))=MEP⁡(Θ,φ);(f)
MEP⁡(Θ,φ) is closed and convex.Remark 9.
In Proposition6, whenever Θ:C×C→R is a bifunction satisfying the conditions (H1)–(H4) and K(x)=(1/2)∥x∥2, ∀x∈H, we have for any x,y∈H,
(30)∥Sr(Θ,φ)x-Sr(Θ,φ)y∥2≤〈Sr(Θ,φ)x-Sr(Θ,φ)y,x-y〉,
(Sr(Θ,φ) is firmly nonexpansive) and
(31)∥Ss(Θ,φ)x-St(Θ,φ)x∥≤|s-t|s∥Ss(Θ,φ)x-x∥,2222222222222222222222∀s,t>0,x∈H.
In this case, Sr(Θ,φ) is rewritten as Tr(Θ,φ). If, in addition, φ≡0, then Tr(Θ,φ) is rewritten as TrΘ.We need some facts and tools in a real Hilbert spaceH which are listed as lemmas below.Lemma 10.
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 11.
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 12 (see [17], 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.Lemma 13 (see [18, p. 80]).
Let{an}n=1∞,{bn}n=1∞ and {δn}n=1∞ be sequences of nonnegative real numbers satisfying the inequality
(34)an+1≤(1+δn)an+bn,∀n≥1.
If ∑n=1∞‍δn<∞ and ∑n=1∞‍bn<∞, then lim⁡n→∞an exists. If, in addition, {an}n=1∞ has a subsequence which converges to zero, then lim⁡n→∞an=0.
Recall that a Banach spaceX is said to satisfy the Opial condition [17] if for any given sequence {xn}⊂X which converges weakly to an element x∈X, there holds the inequality
(35)limsup⁡n→∞∥xn-x∥<limsup⁡n→∞∥xn-y∥,∀y∈X,y≠x.
It is well known in [17] that every Hilbert space H satisfies the Opial condition.Lemma 14 (see [19, Proposition 3.1]).
LetC be a nonempty closed convex subset of a real Hilbert space H and let {xn} be a sequence in H. Suppose that
(36)∥xn+1-p∥2≤(1+λn)∥xn-p∥2+δn,∀p∈C,n≥1,
where {λn} and {δn} are sequences of nonnegative real numbers such that ∑n=1∞‍λn<∞ and ∑n=1∞‍δn<∞. Then {PCxn} converges strongly in C.Lemma 15 (see [20]).
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
(37)∥xn-u∥≤∥u-q∥,∀n,
then xn→q as n→∞.Lemma 16.
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
(38)limsup⁡n→∞∥xn-y∥2=limsup⁡n→∞∥xn-x∥2+∥x-y∥2,∀y∈H.
Assume thatR:D(R)⊂H→2H is a maximal monotone mapping. Then, for λ>0, associated with R, the resolvent operator JR,λ can be defined as
(39)JR,λx=(I+λR)-1x,∀x∈H.
In terms of Huang [21] (see also [22]), there holds the following property for the resolvent operator JR,λ:H→D(R)¯.Lemma 17.
J
R
,
λ is single-valued and firmly nonexpansive; that is,
(40)〈JR,λx-JR,λy,x-y〉≥∥JR,λx-JR,λy∥2,∀x,y∈H.
Consequently, JR,λ is nonexpansive and monotone.Lemma 18 (see [23]).
LetR be a maximal monotone mapping with D(R)=C. Then for any given λ>0,u∈C is a solution of problem (11) if and only if u∈C satisfies
(41)u=JR,λ(u-λBu).Lemma 19 (see [22]).
LetR be a maximal monotone mapping with D(R)=C and let B:C→H be a strongly monotone, continuous, and single-valued mapping. Then for each z∈H, the equation z∈(B+λR)x has a unique solution xλ for λ>0.Lemma 20 (see [23]).
LetR be a maximal monotone mapping with D(R)=C and B:C→H a monotone, continuous, and single-valued mapping. Then (I+λ(R+B))C=H for each λ>0. In this case, R+B is maximal monotone.Lemma 21 (see [22, 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
(42){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
(43)〈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,
(44)NCv={w∈H:〈v-u,w〉≥0,∀u∈C}.
Define
(45)Tv={Av+NCv,ifv∈C,∅,ifv∉C.
Then, T is maximal monotone and 0∈Tv if and only if v∈VI(C,A); see [24].Lemma 22 (see [15, Lemma 2.6]).
LetC be a nonempty subset of a Hilbert space H and S:C→C an asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {γn}. Then
(46)∥Snx-Sny∥≤11-k((1+(1-k)γn)∥x-y∥2+(1-k)cnk∥x-y∥11111111111+(1+(1-k)γn)∥x-y∥2+(1-k)cn)
for all x,y∈C and n≥1.Lemma 23 (see [15, Lemma 2.7]).
LetC be a nonempty subset of a Hilbert space H and S:C→C 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 24 (see [15, Proposition 3.1] demiclosedness principle).
LetC be a nonempty closed convex subset of a Hilbert space H and S:C→C 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 25 (see [15, Proposition 3.2]).
LetC be a nonempty closed convex subset of a Hilbert space H and S:C→C 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.


## 3. Strong Convergence Theorem

In this section, we will introduce and analyze one multistep iterative algorithm by hybrid shrinking projection method for finding a solution of the SGEP (8) with constraints of several problems: the GMEP (4), the CMP (12), finitely many variational inclusions, 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 Korpelevich’s extragradient method, strongly positive bounded linear operator approach, viscosity approximation method, averaged mapping approach to the GPA in [16], Mann-type iteration method, and shrinking projection method. The following proposition will play a key role in the proof of the main results in this paper.Proposition CY (see [7]).LetΘ1,Θ2:C×C→Rbe two bifunctions satisfying conditions (H1)–(H4) and letAk:C→Hbeζk-inverse-strongly monotone fork=1,2. Letνk∈(0,2ζk)fork=1,2. Then,(x*,y*)∈C×Cis a solution of SGEP (8) if and only ifx*is a fixed point of the mappingG:C→Cdefined byG=Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2),wherey*=Tν2Θ2(I-ν2A2)x*. Here, we denote by SGEP(G)the fixed point set ofG.Theorem 26.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let Θ,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R a lower semicontinuous and convex functional. Let Ri:C→2H be a maximal monotone mapping and let A,Ak:H→H and Bi:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone, and ηi-inverse-strongly monotone, respectively, for k=1,2 and i=1,2,…,N. 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 and Q:H→H an l-Lipschitzian mapping with γl<γ¯. Assume that Ω≔GMEP(Θ,φ,A)∩SGEP(G)∩∩i=1NI(Bi,Ri)∩Fix⁡(S)∩Γ is nonempty and bounded, where G is defined as in Proposition CY. Let {rn} be a sequence in [0,2ζ] and {αn},{βn}, and {δn} sequences in [0,1] such that 0<α≤αn≤1 and k≤δn≤d<1, and let νk∈(0,2ζk), k=1,2, and {λi,n}⊂[ai,bi]⊂(0,2ηi), ∀i∈{1,2,…,N}. Pick any x0∈H and set C1=C,x1=PC1x0. Let {xn} be a sequence generated by the following algorithm:
(47)un=Srn(Θ,φ)(I-rnA)xn,vn=JRN,λN,n(I-λN,nBN)JRN-1,λN-1,n×(I-λN-1,nBN-1),…,JR1,λ1,n(I-λ1,nB1)un,zn=snγQxn+βnxn+((1-βn)I-snV)TnGvn,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)), and θn=(sn+γn)(1+γn)Δn+cn, Δn=sup⁡{∥xn-p∥2+∥(γQ-V)p∥2/(γ¯-γl):p∈Ω}<∞. Assume that the following conditions are satisfied:(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(48)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;(iii)
sn∈(0,1/2) for each λn∈(0,2/L), and lim⁡n→∞sn=0 (⇔lim⁡n→∞λn=2/L);(iv)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ.Assume thatSr(Θ,φ) is firmly nonexpansive. Then we have(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 that ∥xn-zn∥=o(sn), which is the unique solution in Ω to the VIP
(49)〈(γQ-V)x*,p-x*〉≤0,∀p∈Ω. Equivalently, x*=PΩ(I-V+γQ)x*.Proof.
Since∇f is L-Lipschitzian, it follows that ∇f is 1/L-ism; see [16]. 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
(50)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
(51)λn⟶2L⟺sn⟶0.
As lim⁡n→∞sn=0, 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ, we may assume, without loss of generality, that {βn}⊂[a,a^]⊂(0,1), {rn}⊂[c,c^]⊂(0,2ζ) and βn+sn∥V∥≤1 for all n≥1. Since V is a γ¯-strongly positive bounded linear operator on H, we know that
(52)∥V∥=sup⁡{〈Vu,u〉:u∈H,∥u∥=1}≥γ¯>γl.
Taking into account that βn+sn∥V∥≤1 for all n≥1, we have
(53)〈((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
(54)∥(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
(55)Λni=JRi,λi,n(I-λi,nBi)JRi-1,λi-1,n×(I-λi-1,nBi-1)⋯JR1,λ1,n(I-λ1,nB1)
for all i∈{1,2,…,N}, and Λn0=I, where I is the identity mapping on H. Then we have 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
(56)〈2(xn-yn),z〉≤∥xn∥2-∥yn∥2+θn,
by Lemma 21 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. Since p=Srn(Θ,φ)(p-rnAp), A is ζ-inverse strongly monotone and 0≤rn≤2ζ, we have, for any n≥1,
(57)∥un-p∥2=∥Srn(Θ,φ)(I-rnA)xn-Srn(Θ,φ)(I-rnA)p∥2≤∥(I-rnA)xn-(I-rnA)p∥2=∥(xn-p)-rn(Axn-Ap)∥2=∥xn-p∥2-2rn〈xn-p,Axn-Ap〉+rn2∥Axn-Ap∥2≤∥xn-p∥2-2rnζ∥Axn-Ap∥2+rn2∥Axn-Ap∥2=∥xn-p∥2+rn(rn-2ζ)∥Axn-Ap∥2≤∥xn-p∥2.
Since p=JRi,λi,n(I-λi,nBi)p, Λnip=p and Bi is ηi-inverse strongly monotone, where λi,n∈(0,2ηi), i∈{1,2,…,N}, by Lemma 17 we deduce that for each n≥1,
(58)∥vn-p∥=∥JRN,λN,n(I-λN,nBN)ΛnN-1un-JRN,λN,n(I-λN,nBN)ΛnN-1p∥≤∥(I-λN,nBN)ΛnN-1un-(I-λN,nBN)ΛnN-1p∥≤∥ΛnN-1un-ΛnN-1p∥⋮≤∥Λn0xn-Λn0p∥=∥un-p∥.
Combining (57) and (58), we have
(59)∥vn-p∥≤∥xn-p∥.
Since p=Gp=Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2)p,Ak is ζk-inverse-strongly monotone for k=1,2, and 0≤νk≤2ζk for k=1,2, we deduce that, for any n≥1,
(60)∥Gvn-p∥2=∥Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2)vn-Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2)p∥2≤∥(I-ν1A1)Tν2Θ2(I-ν2A2)vn-(I-ν1A1)Tν2Θ2(I-ν2A2)p∥2=∥[Tν2Θ2(I-ν2A2)vn-Tν2Θ2(I-ν2A2)p]-ν1[A1Tν2Θ2(I-ν2A2)vn-A1Tν2Θ2(I-ν2A2)p]∥2≤∥Tν2Θ2(I-ν2A2)vn-Tν2Θ2(I-ν2A2)p∥2+ν1(ν1-2ζ1)×∥A1Tν2Θ2(I-ν2A2)vn-A1Tν2Θ2(I-ν2A2)p∥2≤∥Tν2Θ2(I-ν2A2)vn-Tν2Θ2(I-ν2A2)p∥2≤∥(I-ν2A2)vn-(I-ν2A2)p∥2=∥(vn-p)-ν2(A2vn-A2p)∥2≤∥vn-p∥2+ν2(ν2-2ζ2)∥A2vn-A2p∥2≤∥vn-p∥2.
(This shows that G is nonexpansive.) Also, from (47), (54), (59), and (60), it follows that
(61)∥zn-p∥=∥βn(xn-p)+snγ(Qxn-Qp)+[(1-βn)I-snV]×(TnGvn-p)+sn(γQ-V)p∥≤βn∥xn-p∥+snγ∥Qxn-Qp∥+∥[(1-βn)I-snV](TnGvn-p)∥+sn∥(γQ-V)p∥≤βn∥xn-p∥+snγl∥xn-p∥+(1-βn-snγ¯)×∥TnGvn-p∥+sn∥(γQ-V)p∥≤βn∥xn-p∥+snγl∥xn-p∥+(1-βn-snγ¯)∥Gvn-p∥+sn∥(γQ-V)p∥=(βn+snγl)∥xn-p∥+(1-βn-snγ¯)∥Gvn-p∥+sn∥(γQ-V)p∥≤(βn+snγl)∥xn-p∥+(1-βn-snγ¯)∥vn-p∥+sn∥(γQ-V)p∥≤(βn+snγl)∥xn-p∥+(1-βn-snγ¯)∥xn-p∥+sn∥(γQ-V)p∥=(1-sn(γ¯-γl))∥xn-p∥+sn∥(γQ-V)p∥=(1-sn(γ¯-γl))∥xn-p∥+sn(γ¯-γl)∥(γQ-V)p∥γ¯-γl,
which hence yields
(62)∥zn-p∥2≤(1-sn(γ¯-γl))∥xn-p∥2+sn(γ¯-γl)∥(γQ-V)p∥2(γ¯-γl)2≤∥xn-p∥2+sn∥(γQ-V)p∥2γ¯-γl.
By Lemma 16(b), we deduce from (47) and (62) that
(63)∥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)(∥xn-p∥2+sn∥(γQ-V)p∥2γ¯-γl)+cn.
So, from (47) and (63) we get
(64)∥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[∥(γQ-V)p∥2γ¯-γl(1+γn)×(∥xn-p∥2+sn∥(γQ-V)p∥2γ¯-γl)+cn]≤(1+γn)(∥xn-p∥2+sn∥(γQ-V)p∥2γ¯-γl)+cn=∥xn-p∥2+γn∥xn-p∥2+(1+γn)sn∥(γQ-V)p∥2γ¯-γl+cn≤∥xn-p∥2+γn(1+γn)∥xn-p∥2+sn(1+γn)∥(γQ-V)p∥2γ¯-γl+cn≤∥xn-p∥2+(γn+sn)(1+γn)∥xn-p∥2+(sn+γn)(1+γn)∥(γQ-V)p∥2γ¯-γl+cn=∥xn-p∥2+(sn+γn)(1+γn)×(∥xn-p∥2+∥(γQ-V)p∥2γ¯-γl)+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+∥(γQ-V)p∥2/(γ¯-γl):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
(65)∥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
(66)∥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
(67)∥xn+1-xn∥2≤∥x0-xn+1∥2-∥x0-xn∥2,
which implies
(68)lim⁡n→∞∥xn+1-xn∥=0.
It follows fromxn+1∈Cn+1 that ∥yn-xn+1∥2≤∥xn-xn+1∥2+θn, and hence
(69)∥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 (68) and lim⁡n→∞θn=0, we have
(70)lim⁡n→∞∥xn-yn∥=0.
Since yn-xn=αn(kn-xn) and 0<α≤αn≤1, we have
(71)α∥kn-xn∥≤αn∥kn-xn∥=∥yn-xn∥,
which immediately leads to
(72)lim⁡n→∞∥kn-xn∥=0.
Also, utilizing Lemmas 10 and 16(b) we obtain from (47), (59), (60), and (63) that
(73)∥zn-p∥2=∥βnxn+snγQxn+[(1-βn)I-snV]TnGvn-p∥2=∥βn(xn-p)+(1-βn)(TnGvn-p)+sn(γQxn-VTnGvn)∥2≤∥βn(xn-p)+(1-βn)(TnGvn-p)∥2+2sn〈(γQxn-VTnGvn),zn-p〉=βn∥xn-p∥2+(1-βn)∥TnGvn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn〈(γQxn-VTnGvn),zn-p〉≤βn∥xn-p∥2+(1-βn)∥Gvn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn∥γQxn-VTnGvn∥∥zn-p∥≤βn∥xn-p∥2+(1-βn)∥vn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn∥γQxn-VTnGvn∥∥zn-p∥≤βn∥xn-p∥2+(1-βn)∥xn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn∥γQxn-VTnGvn∥∥zn-p∥=∥xn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn∥γQxn-VTnGvn∥∥zn-p∥,
and hence
(74)∥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[+2sn∥γQxn-VTnGvn∥∥zn-p∥∥xn-p∥2)(1+γn)(∥xn-p∥2111111111111111-βn(1-βn)∥xn-TnGvn∥2111111111111111+2sn∥γQxn-VTnGvn∥∥zn-p∥∥xn-p∥2)+cn]≤(1-αn)∥xn-p∥2+αn(1+γn)×(∥xn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn∥γQxn-VTnGvn∥∥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∥γQxn-VTnGvn∥∥zn-p∥+cn≤(1+γn)∥xn-p∥2-αn(1+γn)×βn(1-βn)∥xn-TnGvn∥2+2sn(1+γn)∥γQxn-VTnGvn∥∥zn-p∥+cn.
So, it follows that
(75)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)∥γQxn-VTnGvn∥∥zn-p∥+cn≤∥xn-yn∥(∥xn-p∥+∥yn-p∥)+γn∥xn-p∥2+2sn(1+γn)×∥γQxn-VTnGvn∥∥zn-p∥+cn.
Since lim⁡n→∞sn=0, lim⁡n→∞γn=0 and lim⁡n→∞cn=0, it follows from (70) and the boundedness of {xn}, {yn}, {zn}, and {vn} that
(76)lim⁡n→∞∥xn-TnGvn∥=0.
Note that
(77)∥zn-xn∥=∥(1-βn)(TnGvn-xn)+sn(γQxn-VTnGvn)∥≤(1-βn)∥TnGvn-xn∥+sn∥γQxn-VTnGvn∥≤∥TnGvn-xn∥+sn∥γQxn-VTnGvn∥.
Hence, it follows from (76) and lim⁡n→∞sn=0 that
(78)lim⁡n→∞∥xn-zn∥=0.
Note that
(79)∥kn-zn∥≤∥kn-xn∥+∥xn-zn∥.
Thus, we deduce from (72) and (78) that
(80)lim⁡n→∞∥kn-zn∥=0.
Since kn-zn=(1-δn)(Snzn-zn) and k≤δn≤d<1, we have
(81)(1-d)∥Snzn-zn∥≤(1-δn)∥Snzn-zn∥=∥kn-zn∥,
which together with (80), yields
(82)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 (58), (60), and γl<γ¯, it follows that
(83)∥zn-p∥2=∥βn(xn-p)+snγ(Qxn-Qp)+[(1-βn)I-snV](TnGvn-p)+sn(γQ-V)p∥2≤∥βn(xn-p)+snγ(Qxn-Qp)+[(1-βn)I-snV](TnGvn-p)∥2+2sn〈(γQ-V)p,zn-p〉≤[βn∥xn-p∥+snγ∥Qxn-Qp∥+(1-βn-snγ¯)∥TnGvn-p∥]2+2sn〈(γQ-V)p,zn-p〉≤[βn∥xn-p∥+snγl∥xn-p∥+(1-βn-snγ¯)∥Gvn-p∥]2+2sn〈(γQ-V)p,zn-p〉=[(βn+snγl)∥xn-p∥+(1-βn-snγ¯)∥Gvn-p∥]2+2sn〈(γQ-V)p,zn-p〉≤[(βn+snγ¯)∥xn-p∥+(1-βn-snγ¯)∥Gvn-p∥]2+2sn〈(γQ-V)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥Gvn-p∥2+2sn∥(γQ-V)p∥∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥vn-p∥2+2sn∥(γQ-V)p∥∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥un-p∥2+2sn∥(γQ-V)p∥∥zn-p∥.
Next let us show that
(84)lim⁡n→∞∥xn-un∥=0.
For p∈Ω, we find that
(85)∥un-p∥2=∥Srn(Θ,φ)(I-rnA)xn-Srn(Θ,φ)(I-rnA)p∥2≤∥(I-rnA)xn-(I-rnA)p∥2=∥xn-p-rn(Axn-Ap)∥2≤∥xn-p∥2+rn(rn-2ζ)∥Axn-Ap∥2.
Combining (83) and (85), we obtain
(86)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥un-p∥2+2sn∥(γQ-V)p∥∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥xn-p∥2+rn(rn-2ζ)∥Axn-Ap∥2]+2sn∥(γQ-V)p∥∥zn-p∥=∥xn-p∥2+(1-βn-snγ¯)rn(rn-2ζ)∥Axn-Ap∥2+2sn∥(γQ-V)p∥∥zn-p∥,
which immediately implies that
(87)(1-a^-snγ¯)c(2ζ-c^)∥Axn-Ap∥2≤(1-βn-snγ¯)rn(2ζ-rn)∥Axn-Ap∥2≤∥xn-p∥2-∥zn-p∥2+2sn∥(γQ-V)p∥∥zn-p∥≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2sn∥(γQ-V)p∥∥zn-p∥.
Since lim⁡n→∞sn=0 and {xn} and {zn} are bounded sequences, it follows from (78) that
(88)lim⁡n→∞∥Axn-Ap∥=0.
Furthermore, from the firm nonexpansivity of Srn(Θ,φ), we have
(89)∥un-p∥2=∥Srn(Θ,φ)(I-rnA)xn-Srn(Θ,φ)(I-rnA)p∥2≤〈(I-rnA)xn-(I-rnA)p,un-p〉=12[∥(I-rnA)xn-(I-rnA)p∥2+∥un-p∥2-∥(I-rnA)xn-(I-rnA)p-(un-p)∥2]≤12[∥xn-p∥2+∥un-p∥2-∥xn-un-rn(Axn-Ap)∥2]=12[∥xn-p∥2+∥un-p∥2-∥xn-un∥2+2rn〈Axn-Ap,xn-un〉-rn2∥Axn-Ap∥2],
which leads to
(90)∥un-p∥2≤∥xn-p∥2-∥xn-un∥2+2rn∥Axn-Ap∥∥xn-un∥.
From (83) and (90), we have
(91)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥un-p∥2+2sn∥(γQ-V)p∥∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥xn-p∥2-∥xn-un∥2+2rn∥Axn-Ap∥∥xn-un∥∥xn-p∥2]+2sn∥(γQ-V)p∥∥zn-p∥≤∥xn-p∥2-(1-βn-snγ¯)∥xn-un∥2+2rn∥Axn-Ap∥∥xn-un∥+2sn∥(γQ-V)p∥∥zn-p∥,
which hence implies that
(92)(1-a^-snγ¯)∥xn-un∥2≤(1-βn-snγ¯)∥xn-un∥2≤∥xn-p∥2-∥zn-p∥2+2rn∥Axn-Ap∥∥xn-un∥+2sn∥(γQ-V)p∥∥zn-p∥≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2rn∥Axn-Ap∥∥xn-un∥+2sn∥(γQ-V)p∥∥zn-p∥.
Since lim⁡n→∞sn=0 and {xn}, {un}, and {zn} are bounded sequences, it follows from (78) and (88) that (84) holds.
Next we show thatlim⁡n→∞∥BiΛniun-Bip∥=0, i=1,2,…,N. As a matter of fact, observe that
(93)∥Λniun-p∥2=∥JRi,λi,n(I-λi,nBi)Λni-1un-JRi,λi,n(I-λi,nBi)p∥2≤∥(I-λi,nBi)Λni-1un-(I-λi,nBi)p∥2≤∥Λni-1un-p∥2+λi,n(λi,n-2ηi)∥BiΛni-1un-Bip∥2≤∥un-p∥2+λi,n(λi,n-2ηi)∥BiΛni-1un-Bip∥2≤∥xn-p∥2+λi,n(λi,n-2ηi)∥BiΛni-1un-Bip∥2.
Combining (60), (83), and (93), we have
(94)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥vn-p∥2+2sn∥(γQ-V)p∥∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥Λniun-p∥2+2sn∥(γQ-V)p∥∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥xn-p∥2+λi,n(λi,n-2ηi)∥BiΛni-1un-Bip∥2]+2sn∥(γQ-V)p∥∥zn-p∥=∥xn-p∥2+(1-βn-snγ¯)λi,n(λi,n-2ηi)×∥BiΛni-1un-Bip∥2+2sn∥(γQ-V)p∥∥zn-p∥,
which together with {λi,n}⊂[ai,bi]⊂(0,2ηi), for all i∈{1,2,…,N}, implies that
(95)(1-a^-snγ¯)ai(2ηi-bi)∥BiΛni-1un-Bip∥2≤(1-βn-snγ¯)λi,n(λi,n-2ηi)∥BiΛni-1un-Bip∥2≤∥xn-p∥2-∥zn-p∥2+2sn∥(γQ-V)p∥∥zn-p∥≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2sn∥(γQ-V)p∥∥zn-p∥.
Since lim⁡n→∞sn=0 and {xn} and {zn} are bounded sequences, it follows from (78) that
(96)lim⁡n→∞∥BiΛni-1un-Bip∥=0,i=1,2,…,N.
By Lemma 16(a) and Lemma 17, we obtain
(97)∥Λniun-p∥2=∥JRi,λi,n(I-λi,nBi)Λni-1un-JRi,λi,n(I-λi,nBi)p∥2≤〈(I-λi,nBi)Λni-1un-(I-λi,nBi)p,Λniun-p〉=12(∥(I-λi,nBi)Λni-1un-(I-λi,nBi)p∥2+∥Λniun-p∥2-∥(I-λi,nBi)Λni-1un11111111l-(I-λi,nBi)p-(Λniun-p)∥2)≤12(∥Λni-1un-p∥2+∥Λniun-p∥21111-∥Λni-1un-Λniun-λi,n(BiΛni-1un-Bip)∥2)≤12(∥un-p∥2+∥Λniun-p∥211111-∥Λni-1un-Λniun-λi,n(BiΛni-1un-Bip)∥2)≤12(∥xn-p∥2+∥Λniun-p∥211111-∥Λni-1un-Λniun-λi,n(BiΛni-1un-Bip)∥2),
which implies
(98)∥Λniun-p∥2≤∥xn-p∥2-∥Λni-1un-Λniun-λi,n(BiΛni-1un-Bip)∥2=∥xn-p∥2-∥Λni-1un-Λniun∥2-λi,n2∥BiΛni-1un-Bip∥2+2λi,n〈Λni-1un-Λniun,BiΛni-1un-Bip〉≤∥xn-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥BiΛni-1un-Bip∥.
Combining (60), (83), and (98), we have
(99)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥vn-p∥2+2sn∥(γQ-V)p∥∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥Λniun-p∥2+2sn∥(γQ-V)p∥∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥BiΛni-1un-Bip∥∥xn-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥BiΛni-1un-Bip∥]+2sn∥(γQ-V)p∥∥zn-p∥≤∥xn-p∥2-(1-βn-snγ¯)∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥BiΛni-1un-Bip∥+2sn∥(γQ-V)p∥∥zn-p∥.
So, we conclude that
(100)(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∥∥BiΛni-1un-Bip∥+2sn∥(γQ-V)p∥∥zn-p∥≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2bi∥Λni-1un-Λniun∥∥BiΛni-1un-Bip∥+2sn∥(γQ-V)p∥∥zn-p∥.
Since lim⁡n→∞sn=0 and {xn}, {zn}, and {un} are bounded, from (78) and (96) we get
(101)lim⁡n→∞∥Λni-1un-Λniun∥=0.
From (101) we get
(102)∥un-vn∥=∥Λn0un-ΛnNun∥≤∥Λn0un-Λn1un∥+∥Λn1un-Λn2un∥+⋯+∥ΛnN-1un-ΛnNun∥⟶0asn⟶∞.
Taking into account that∥xn-vn∥≤∥xn-un∥+∥un-vn∥, we conclude from (84) and (102) that
(103)lim⁡n→∞∥xn-vn∥=0.
On the other hand, for simplicity, we writep~=Tν2Θ2(I-ν2A2)p, v~n=Tν2Θ2(I-ν2A2)vn and wn=Gvn=Tν1Θ1(I-ν1A1)v~n for all n≥1. Then
(104)p=Gp=Tν1Θ1(I-ν1A1)p~=Tν1Θ1(I-ν1A1)Tν2Θ2(I-ν2A2)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 (59), (60), and (83) that
(105)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥Gvn-p∥2+2sn∥(γQ-V)p∥∥zn-p∥=(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥wn-p∥2+2sn∥(γQ-V)p∥∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥v~n-p~∥2+ν1(ν1-2ζ1)∥A1v~n-A1p~∥2]+2sn∥(γQ-V)p∥∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥vn-p∥2+ν2(ν2-2ζ2)∥A2vn-A2p∥2+ν1(ν1-2ζ1)∥A1v~n-A1p~∥2]+2sn∥(γQ-V)p∥∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥xn-p∥2+ν2(ν2-2ζ2)∥A2vn-A2p∥2+ν1(ν1-2ζ1)∥A1v~n-A1p~∥2]+2sn∥(γQ-V)p∥∥zn-p∥=∥xn-p∥2+(1-βn-snγ¯)×[ν2(ν2-2ζ2)∥A2vn-A2p∥2+ν1(ν1-2ζ1)∥A1v~n-A1p~∥2]+2sn∥(γQ-V)p∥∥zn-p∥,
which immediately yields
(106)(1-a^-snγ¯)[ν2(2ζ2-ν2)∥A2vn-A2p∥2111111111111+ν1(2ζ1-ν1)∥A1v~n-A1p~∥2]≤(1-βn-snγ¯)[ν2(2ζ2-ν2)∥A2vn-A2p∥2111111111111111+ν1(2ζ1-ν1)∥A1v~n-A1p~∥2]≤∥xn-p∥2-∥zn-p∥2+2sn∥(γQ-V)p∥∥zn-p∥≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2sn∥(γQ-V)p∥∥zn-p∥.
Since lim⁡n→∞sn=0 and {xn} and {zn} are bounded, from (78) we get
(107)lim⁡n→∞∥A2vn-A2p∥=0,lim⁡n→∞∥A1v~n-A1p~∥=0.
Also, in terms of the firm nonexpansivity of TνkΘk and the ζk-inverse strong monotonicity of Ak for k=1,2, we obtain from νk∈(0,2ζk), k=1,2 and (60) that
(108)∥v~n-p~∥2=∥Tν2Θ2(I-ν2A2)vn-Tν2Θ2(I-ν2A2)p∥2≤〈(I-ν2A2)vn-(I-ν2A2)p,v~n-p~〉=12[∥(I-ν2A2)vn-(I-ν2A2)p∥2+∥v~n-p~∥2-∥(I-ν2A2)vn-(I-ν2A2)p-(v~n-p~)∥2]≤12[∥vn-p∥2+∥v~n-p~∥211111-∥(vn-v~n)-ν2(A2vn-A2p)-(p-p~)∥2]=12[∥vn-p∥2+∥v~n-p~∥2-∥(vn-v~n)-(p-p~)∥211111+2ν2〈(vn-v~n)-(p-p~),A2vn-A2p〉11111-ν22∥A2vn-A2p∥2],∥wn-p∥2=∥Tν1Θ1(I-ν1A1)v~n-Tν1Θ1(I-ν1A1)p~∥2≤〈(I-ν1A1)v~n-(I-ν1A1)p~,wn-p〉=12[∥(I-ν1A1)v~n-(I-ν1A1)p~∥2+∥wn-p∥2-∥(I-ν1A1)v~n-(I-ν1A1)p~-(wn-p)∥2]≤12[∥v~n-p~∥2+∥wn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1〈A1v~n-A1p~,(v~n-wn)+(p-p~)〉-ν12∥A1v~n-A1p~∥2]≤12[∥vn-p∥2+∥wn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1〈A1v~n-A1p~,(v~n-wn)+(p-p~)〉∥vn-p∥2].
Thus, we have
(109)∥v~n-p~∥2≤∥vn-p∥2-∥(vn-v~n)-(p-p~)∥2+2ν2〈(vn-v~n)-(p-p~),A2vn-A2p〉-ν22∥A2vn-A2p∥2,(110)∥wn-p∥2≤∥vn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1∥A1v~n-A1p~∥∥(v~n-wn)+(p-p~)∥.
Consequently, from (59), (105), and (109) it follows that
(111)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥v~n-p~∥2+ν1(ν1-2ζ1)∥A1v~n-A1p~∥2]+2sn∥(γQ-V)p∥∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥v~n-p~∥2+2sn∥(γQ-V)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~),A2vn-A2p〉-ν22∥A2vn-A2p∥2]+2sn∥(γQ-V)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~)∥∥A2vn-A2p∥∥xn-p∥2]+2sn∥(γQ-V)p∥∥zn-p∥≤∥xn-p∥2-(1-βn-snγ¯)∥(vn-v~n)-(p-p~)∥2+2ν2∥(vn-v~n)-(p-p~)∥∥A2vn-A2p∥+2sn∥(γQ-V)p∥∥zn-p∥,
which hence leads to
(112)(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~)∥∥A2vn-A2p∥+2sn∥(γQ-V)p∥∥zn-p∥≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2ν2∥(vn-v~n)-(p-p~)∥∥A2vn-A2p∥+2sn∥(γQ-V)p∥∥zn-p∥.
Since lim⁡n→∞sn=0 and {xn}, {zn}, {vn}, and {v~n} are bounded sequences, we conclude from (78) and (107) that
(113)lim⁡n→∞∥(vn-v~n)-(p-p~)∥=0.
Furthermore, from (59), (105), and (110) it follows that
(114)∥zn-p∥2≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×∥wn-p∥2+2sn∥(γQ-V)p∥∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥vn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1∥A1v~n-A1p~∥∥(v~n-wn)+(p-p~)∥∥vn-p∥2]+2sn∥(γQ-V)p∥∥zn-p∥≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)×[∥xn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1∥A1v~n-A1p~∥∥(v~n-wn)+(p-p~)∥∥vn-p∥2]+2sn∥(γQ-V)p∥∥zn-p∥≤∥xn-p∥2-(1-βn-snγ¯)∥(v~n-wn)+(p-p~)∥2+2ν1∥A1v~n-A1p~∥∥(v~n-wn)+(p-p~)∥+2sn∥(γQ-V)p∥∥zn-p∥,
which hence yields
(115)(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∥A1v~n-A1p~∥∥(v~n-wn)+(p-p~)∥+2sn∥(γQ-V)p∥∥zn-p∥≤∥xn-zn∥(∥xn-p∥+∥zn-p∥)+2ν1∥A1v~n-A1p~∥∥(v~n-wn)+(p-p~)∥+2sn∥(γQ-V)p∥∥zn-p∥.
Since lim⁡n→∞sn=0 and {xn},{zn},{wn}, and {v~n} are bounded sequences, we conclude from (78) and (107) that
(116)lim⁡n→∞∥(v~n-wn)+(p-p~)∥=0.
Note that(117)∥vn-wn∥≤∥(vn-v~n)-(p-p~)∥+∥(v~n-wn)+(p-p~)∥.
Hence from (113) and (116) we get
(118)lim⁡n→∞∥vn-Gvn∥=lim⁡n→∞∥vn-wn∥=0.
Observe that(119)∥vn-Tnvn∥≤∥vn-xn∥+∥xn-TnGvn∥+∥TnGvn-Tnvn∥≤∥vn-xn∥+∥xn-TnGvn∥+∥Gvn-vn∥.
Hence, from (76), (103), and (118) we have
(120)lim⁡n→∞∥vn-Tnvn∥=0.
It is clear that(121)∥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
(122)∥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 (120)), it follows that
(123)lim⁡n→∞∥vn-PC(I-2L∇f)vn∥=0.
In addition, from (68) and (78), we have
(124)∥zn+1-zn∥≤∥zn+1-xn+1∥+∥xn+1-xn∥+∥xn-zn∥⟶0asn⟶∞.
We note that(125)∥Snzn-Sn+1zn∥≤∥Snzn-zn∥+∥zn-zn+1∥+∥zn+1-Sn+1zn+1∥+∥Sn+1zn+1-Sn+1zn∥.
From (82), (124), and Lemma 22, we obtain
(126)lim⁡n→∞∥Snzn-Sn+1zn∥=0.
In the meantime, we note that(127)∥zn-Szn∥≤∥zn-Snzn∥+∥Snzn-Sn+1zn∥+∥Sn+1zn-Szn∥.
From (82), (126), and the uniform continuity of S, we have
(128)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 (78), (84), (103), and (101), we have that zni⇀w, uni⇀w, vni⇀w and Λnimuni⇀w, where m∈{1,2,…,N}. Since S is uniformly continuous, by (128) we get lim⁡n→∞∥zn-Smzn∥=0 for any m≥1. Hence from Lemma 24, we obtain w∈Fix⁡(S). In the meantime, utilizing Lemma 12, we deduce from vni⇀w, xni⇀w, (118), and (123) that w∈SGEP(G) and w∈Fix⁡(PC(I-(2/L)∇f))=VI(C,∇f)=Γ. Next we prove that w∈∩m=1NI(Bm,Rm). As a matter of fact, since Bm is ηm-inverse strongly monotone, Bm is a monotone and Lipschitz continuous mapping. It follows from Lemma 20 that Rm+Bm is maximal monotone. Let (v,g)∈G(Rm+Bm); that is, g-Bmv∈Rmv. Again, since Λnmun=JRm,λm,n(I-λm,nBm)Λnm-1un, n≥1, m∈{1,2,…,N}, we have
(129)Λnm-1un-λm,nBmΛnm-1un∈(I+λm,nRm)Λnmun;
that is,
(130)1λm,n(Λnm-1un-Λnmun-λm,nBmΛnm-1un)∈RmΛnmun.
In terms of the monotonicity ofRm, we get
(131)〈(Λnm-1un-Λnmun-λm,nBmΛnm-1un)1λm,nv-Λnmun,g-Bmv-1λm,n×(Λnm-1un-Λnmun-λm,nBmΛnm-1un)1λm,n〉≥0,
and hence
(132)〈v-Λnmun,g〉≥〈v-Λnmun,Bmv+1λm,n×(Λnm-1un-Λnmun-λm,nBmΛnm-1un)1λm,n〉=〈1λm,nv-Λnmun,Bmv-BmΛnmun+BmΛnmun-BmΛnm-1un+1λm,n(Λnm-1un-Λnmun)〉≥〈v-Λnmun,BmΛnmun-BmΛnm-1un〉+〈v-Λnmun,1λm,n(Λnm-1un-Λnmun)〉.
In particular,(133)〈v-Λnimuni,g〉≥〈v-Λnimuni,BmΛnimuni-BmΛnim-1uni〉+〈v-Λnimuni,1λm,ni(Λnim-1uni-Λnimuni)〉.
Since ∥Λnmun-Λnm-1un∥→0 (due to (90)) and ∥BmΛnmun-BmΛnm-1un∥→0 (due to the Lipschitz continuity of Bm), we conclude from Λnimuni⇀w and {λm,n}⊂[am,bm]⊂(0,2ηm) that
(134)lim⁡i→∞〈v-Λnimuni,g〉=〈v-w,g〉≥0.
It follows from the maximal monotonicity of Bm+Rm that 0∈(Rm+Bm)w; that is, w∈I(Bm,Rm). Therefore, w∈∩m=1NI(Bm,Rm).
Next, we show thatw∈GMEP⁡(Θ,φ,A). In fact, from un=Srn(Θ,φ)(I-rnA)xn, we know that
(135)Θ(un,y)+φ(y)-φ(un)+〈Axn,y-un〉+1rn〈K′(un)-K′(xn),y-un〉≥0,∀y∈C.
From (H2) it follows that
(136)φ(y)-φ(un)+〈Axn,y-un〉+1rn〈K′(un)-K′(xn),y-un〉≥Θ(y,un),∀y∈C.
Replacing n by ni, we have
(137)φ(y)-φ(uni)+〈Axni,y-uni〉+〈K′(uni)-K′(xni)rni,y-uni〉≥Θ(y,uni),kkkkkkkkkkkkkkkk222kkkkkkkkkkkkkkk∀y∈C.
Putut=ty+(1-t)w for all t∈(0,1] and y∈C. Then, from (137) we have
(138)〈ut-uni,Aut〉≥〈ut-uni,Aut〉-φ(ut)+φ(uni)-〈ut-uni,Axni〉-〈K′(uni)-K′(xni)rni,ut-uni〉+Θ(ut,uni)≥〈ut-uni,Aut-Auni〉+〈ut-uni,Auni-Axni〉-φ(ut)+φ(uni)-〈K′(uni)-K′(xni)rni,ut-uni〉+Θ(ut,uni).
Since ∥uni-xni∥→0 as i→∞, we deduce from the Lipschitz continuity of A and K′ that ∥Auni-Axni∥→0 and ∥K′(uni)-K′(xni)∥→0 as i→∞. Further, from the monotonicity of A, we have 〈ut-uni,Aut-Auni〉≥0. So, from (H4), the weakly lower semicontinuity of φ,(K′(uni)-K′(xni))/rni→0, and uni⇀w, we have
(139)〈ut-w,Aut〉≥-φ(ut)+φ(w)+Θ(ut,w),asi⟶∞.
From (H1), (H4), and (139) we also have
(140)0=Θ(ut,ut)+φ(ut)-φ(ut)≤tΘ(ut,y)+(1-t)Θ(ut,w)+tφ(y)+(1-t)φ(w)-φ(ut)=t[Θ(ut,y)+φ(y)-φ(ut)]+(1-t)[Θ(ut,w)+φ(w)-φ(w)-φ(ut)]≤t[Θ(ut,y)+φ(y)-φ(ut)]+(1-t)〈ut-w,Aut〉=t[Θ(ut,y)+φ(y)-φ(ut)]+(1-t)t〈y-w,Aut〉,
and hence
(141)0≤Θ(ut,y)+φ(y)-φ(ut)+(1-t)〈y-w,Aut〉.
Lettingt→0, we have, for each y∈C,
(142)0≤Θ(w,y)+φ(y)-φ(w)+〈Aw,y-w〉.
This implies that w∈GMEP⁡(Θ,φ,A). Consequently, w∈Ω=GMEP⁡(Θ,φ,A)∩SGEP(G)∩(∩i=1NI(Bi,Ri))∩Fix⁡(S)∩Γ. This shows that ωw(xn)⊂Ω. From (65) and Lemma 15 we infer that xn→x*=PΩx0 as n→∞.
Finally, assume additionally that∥xn-zn∥=o(sn). It is clear that
(143)〈(V-γQ)x-(V-γQ)y,x-y〉≥(γ¯-γl)∥x-y∥2,222222222222222222222222222222222222∀x,y∈H.
So, we know that V-γQ is (γ¯-γl)-strongly monotone with constant γ¯-γl>0. In the meantime, it is easy to see that V-γQ is (∥V∥+γl)-Lipschitzian with constant ∥V∥+γl>0. Thus, there exists a unique solution x^ in Ω to the VIP
(144)〈(γQ-V)x^,p-x^〉≤0,∀p∈Ω.
Equivalently, x^=PΩ(I-V+γQ)x^. Furthermore, from (59), (60), and (83) we get
(145)∥zn-p∥2≤[(βn+snγ¯)∥xn-p∥+(1-βn-snγ¯)∥Gvn-p∥]2+2sn〈(γQ-V)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥Gvn-p∥2+2sn〈(γQ-V)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥vn-p∥2+2sn〈(γQ-V)p,zn-p〉≤(βn+snγ¯)∥xn-p∥2+(1-βn-snγ¯)∥xn-p∥2+2sn〈(γQ-V)p,zn-p〉=∥xn-p∥2+2sn〈(γQ-V)p,zn-p〉,
which hence yields
(146)〈(γQ-V)p,p-zn〉≤∥xn-p∥2-∥zn-p∥22sn≤∥xn-zn∥2sn(∥xn-p∥+∥zn-p∥).
Since ∥xn-zn∥=o(sn), lim⁡n→∞∥xn-x*∥=0, and {xn}, {zn} are bounded, we infer from (146) that
(147)〈(γQ-V)p,p-x*〉≤0,∀p∈Ω,
which together with Minty’s lemma [4] implies that
(148)〈(γQ-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 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,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R a lower semicontinuous and convex functional. Let Ri:C→2H be a maximal monotone mapping and let A,Ak:H→H and Bi:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone, and ηi-inverse-strongly monotone, respectively, for k=1,2 and i=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 and Q:H→H an l-Lipschitzian mapping with γl<γ¯. Assume that Ω≔GMEP(Θ,φ,A)∩SGEP(G)∩I(B1,R1)∩I(B2,R2)∩Fix⁡(S)∩Γ is nonempty and bounded where G is defined as in Proposition CY. Let {rn} be a sequence in [0,2ζ] and {αn}, {βn}, and {δn} be sequences in [0,1] such that 0<α≤αn≤1 and k≤δn≤d<1, and let νk∈(0,2ζk), k=1,2 and {λi,n}⊂[ai,bi]⊂(0,2ηi), i=1,2. Pick any x0∈H and set C1=C, x1=PC1x0. Let {xn} be a sequence generated by the following algorithm:
(149)un=Srn(Θ,φ)(I-rnA)xn,vn=JR2,λ2,n(I-λ2,nB2)JR1,λ1,n(I-λ1,nB1)un,zn=snγQxn+βnxn+((1-βn)I-snV)TnGvn,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)), and θn=(sn+γn)(1+γn)Δn+cn, Δn=sup⁡{∥xn-p∥2+∥(γQ-V)p∥2/(γ¯-γl):p∈Ω}<∞. Assume that the following conditions are satisfied:(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(150)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;(iii)
sn∈(0,1/2) for each λn∈(0,2/L), and lim⁡n→∞sn=0(⇔lim⁡n→∞λn=2/L);(iv)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ.Assume thatSr(Θ,φ) is firmly nonexpansive. Then we have(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 that ∥xn-zn∥=o(sn), which is the unique solution in Ω to the VIP
(151)〈(γQ-V)x*,p-x*〉≤0,∀p∈Ω.Equivalently, x*=PΩ(I-V+γQ)x*.Corollary 28.
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,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R a lower semicontinuous and convex functional. Let R:C→2H be a maximal monotone mapping and let A,Ak:H→H and B:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone, and ξ-inverse-strongly monotone, respectively, for k=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 and Q:H→H be an l-Lipschitzian mapping with γl<γ¯. Assume that Ω≔GMEP(Θ,φ,A)∩SGEP(G)∩I(B,R)∩Fix(S)∩Γ is nonempty and bounded where G is defined as in Proposition CY. Let {rn} be a sequence in [0,2ζ] and {αn},{βn} and {δn} be sequences in [0,1] such that 0<α≤αn≤1 and k≤δn≤d<1, and let νk∈(0,2ζk),k=1,2 and {ρn}⊂[a,b]⊂(0,2ξ). Pick any x0∈H and set C1=C, x1=PC1x0. Let {xn} be a sequence generated by the following algorithm:
(152)un=Srn(Θ,φ)(I-rnA)xn,vn=JR,ρn(I-ρnB)un,zn=snγQxn+βnxn+((1-βn)I-snV)TnGvn,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)) and θn=(sn+γn)(1+γn)Δn+cn,Δn=sup⁡{∥xn-p∥2+∥(γQ-V)p∥2/(γ¯-γl):p∈Ω}<∞. Assume that the following conditions are satisfied:(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(153)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;(iii)
sn∈(0,1/2) for each λn∈(0,2/L), and lim⁡n→∞sn=0  (⇔lim⁡n→∞λn=(2/L));(iv)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ.
Assume thatSr(Θ,φ) is firmly nonexpansive. Then we have(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 that ∥xn-zn∥=o(sn), which is the unique solution in Ω to the VIP
(154)〈(γQ-V)x*,p-x*〉≤0,∀p∈Ω.Equivalently, x*=PΩ(I-V+γQ)x*.Corollary 29.
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,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R a lower semicontinuous and convex functional. Let R:C→2H be a maximal monotone mapping and let A,Ak:H→H and B:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone, and ξ-inverse-strongly monotone, respectively, for k=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 and Q:H→H an l-Lipschitzian mapping with γl<γ¯. Assume that Ω≔GMEP(Θ,φ,A)∩SGEP(G)∩I(B,R)∩Fix⁡(S)∩Γ is nonempty and bounded where G is defined as in Proposition CY. Let {rn} be a sequence in [0,2ζ] and {αn},{βn}, and {δn} be sequences in [0,1] such that 0<α≤αn≤1 and k≤δn≤d<1, and let νk∈(0,2ζk),k=1,2 and {ρn}⊂[a,b]⊂(0,2ξ). Pick any x0∈H and set C1=C,x1=PC1x0. Let {xn} be a sequence generated by the following algorithm:
(155)un=Srn(Θ,φ)(I-rnA)xn,vn=JR,ρn(I-ρnB)un,zn=snγQxn+βnxn+((1-βn)I-snV)TnGvn,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)), and θn=(sn+γn)(1+γn)Δn, Δn=sup⁡{∥xn-p∥2+∥(γQ-V)p∥2/(γ¯-γl):p∈Ω}<∞. Assume that the following conditions are satisfied:(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(156)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;(iii)
sn∈(0,1/2) for each λn∈(0,2/L), and lim⁡n→∞sn=0(⇔lim⁡n→∞λn=2/L);(iv)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ.
Assume thatSr(Θ,φ) is firmly nonexpansive. Then we have(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 that ∥xn-zn∥=o(sn), which is the unique solution in Ω to the VIP
(157)〈(γQ-V)x*,p-x*〉≤0,∀p∈Ω.Equivalently, x*=PΩ(I-V+γQ)x*.


## 4. Weak Convergence Theorem

In this section, we will introduce and analyze another multistep iterative algorithm involving no shrinking projection method for finding a solution of the SGEP (8) with constraints of several problems: the GMEP (4), the CMP (12), finitely many variational inclusions, and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove weak convergence theorem for the iterative algorithm under mild assumptions. This iterative algorithm is based on Korpelevich’s extragradient method, strongly positive bounded linear operator approach, viscosity approximation method, averaged mapping approach to the GPA in [16], and Mann-type iteration method.Theorem 30.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let Θ,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R a lower semicontinuous and convex functional. Let Ri:C→2H be a maximal monotone mapping and let A,Ak:H→H and Bi:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone and ηi-inverse-strongly monotone, respectively, for k=1,2 and i=1,2,…,N. 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 ∑n=1∞‍γn<∞ and {cn}⊂[0,∞) such that ∑n=1∞‍cn<∞. Let V be a γ¯-strongly positive bounded linear operator and Q:H→H an l-Lipschitzian mapping with γl<γ¯. Assume that Ω≔GMEP(Θ,φ,A)∩SGEP(G)∩∩i=1NI(Bi,Ri)∩Fix⁡(S)∩Γ is nonempty where G is defined as in Proposition CY. Let {rn} be a sequence in [0,2ζ] and {αn},{βn} and {δn} sequences in [0,1] such that 0<α≤αn≤1 and 0<k+ϵ≤δn≤d<1, and let νk∈(0,2ζk),k=1,2 and {λi,n}⊂[ai,bi]⊂(0,2ηi),∀i∈{1,2,…,N}. Pick any x1∈H and let {xn} be a sequence generated by the following algorithm:
(158)un=Srn(Θ,φ)(I-rnA)xn,vn=JRN,λN,n(I-λN,nBN)JRN-1,λN-1,n×(I-λN-1,nAN-1)⋯JR1,λ1,n(I-λ1,nB1)un,zn=snγQxn+βnxn+((1-βn)I-snV)TnGvn,kn=δnzn+(1-δn)Snzn,xn+1=(1-αn)xn+αnkn,∀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)). Assume that the following conditions are satisfied:(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(159)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;(iii)
sn∈(0,1/2) for each λn∈(0,2/L), and ∑n=1∞‍sn<∞  (⇔∑n=1∞‍(2/L-λn)<∞);(iv)
0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1 and 0<lim⁡inf⁡n→∞rn≤lim⁡sup⁡n→∞rn<2ζ.Then {xn} converges weakly to w=lim⁡n→∞PΩxn provided that Sr(Θ,φ) is firmly nonexpansive.Proof.
Since∇f is L-Lipschitzian, it follows that ∇f is (1/L)-ism; see [16]. Repeating the same arguments as in Theorem 26, we can write
(160)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
(161)∑n=1∞‍(2L-λn)<∞⟺∑n=1∞‍sn<∞.
As lim⁡n→∞sn=0, 0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1 and 0<lim⁡inf⁡n→∞rn≤lim⁡sup⁡n→∞rn<2ζ, we may assume, without loss of generality, that {βn}⊂[a,a^]⊂(0,1), {rn}⊂[c,c^]⊂(0,2ζ) and βn+sn∥V∥≤1 for all n≥1. Put
(162)Λni=JRi,λi,n(I-λi,nBi)JRi-1,λi-1,n×(I-λi-1,nBi-1)⋯JR1,λ1,n(I-λ1,nB1),
for all i∈{1,2,…,N}, and Λn0=I, where I is the identity mapping on H. Then we have vn=ΛnNun.
Take a fixedp∈Ω arbitrarily. Let us show the existence of lim⁡n→∞∥xn-p∥. Indeed, repeating the same arguments as in the proof of Theorem 26, we can obtain that
(163)∥(1-βn)I-snV∥≤1-βn-snγ¯,(164)∥un-p∥2≤∥xn-p∥2+rn(rn-2ζ)×∥Axn-Ap∥2≤∥xn-p∥2,(165)∥vn-p∥≤∥un-p∥,(166)∥Gvn-p∥2≤∥Tν2Θ2(I-ν2A2)vn-Tν2Θ2(I-ν2A2)p∥2+ν1(ν1-2ζ1)∥A1Tν2Θ2(I-ν2A2)vn1111111111111111-A1Tν2Θ2(I-ν2A2)p∥2≤∥Tν2Θ2(I-ν2A2)vn-Tν2Θ2(I-ν2A2)p∥2≤∥vn-p∥2+ν2(ν2-2ζ2)∥A2vn-A2p∥2≤∥vn-p∥2,(167)∥Λniun-p∥2≤∥xn-p∥2+λi,n(λi,n-2ηi)×∥BiΛni-1un-Bip∥2,i∈{1,2,…,N},(168)∥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+sn∥(γQ-V)p∥2γ¯-γl)+cn,(169)∥Λniun-p∥2≤∥xn-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥BiΛni-1un-Bip∥,1111111111111111111111i∈{1,2,…,N}.
Utilizing (158) and (168) we obtain
(170)∥xn+1-p∥2≤(1-αn)∥xn-p∥2+αn∥kn-p∥2≤(1-αn)∥xn-p∥2+αn[(1+γn)(∥xn-p∥2+sn∥(γQ-V)p∥2γ¯-γl)+cn]≤(1+γn)(∥xn-p∥2+sn∥(γQ-V)p∥2γ¯-γl)+cn=∥xn-p∥2+γn∥xn-p∥2+sn(1+γn)∥(γQ-V)p∥2γ¯-γl+cn.
Since ∑n=1∞‍sn<∞, ∑n=1∞‍γn<∞ and ∑n=1∞‍cn<∞, by Lemma 13 we have that lim⁡n→∞∥xn-p∥ exists. Thus {xn} is bounded and so are the sequences {un}, {vn}, {zn} and {kn}.
Also, utilizing Lemmas10 and 16(b) we obtain from (158), (164), (165), and (168) that
(171)∥zn-p∥2=∥βn(xn-p)+(1-βn)(TnGvn-p)+sn(γQxn-VTnGvn)∥2≤∥βn(xn-p)+(1-βn)(TnGvn-p)∥2+2sn〈(γQxn-VTnGvn),zn-p〉=βn∥xn-p∥2+(1-βn)∥TnGvn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn〈(γQxn-VTnGvn),zn-p〉≤βn∥xn-p∥2+(1-βn)∥xn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn∥γQxn-VTnGvn∥∥zn-p∥=∥xn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn∥γQxn-VTnGvn∥∥zn-p∥≤∥xn-p∥2+2sn∥γQxn-VTnGvn∥∥zn-p∥,
and hence
(172)∥xn+1-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[(1+γn)(∥xn-p∥2-βn(1-βn)∥xn-TnGvn∥2111111111111111+2sn∥γQxn-VTnGvn∥∥zn-p∥∥xn-p∥2)+cn]≤(1-αn)∥xn-p∥2+αn(1+γn)×(∥xn-p∥2-βn(1-βn)∥xn-TnGvn∥2+2sn∥γQxn-VTnGvn∥∥zn-p∥∥xn-p∥2)+cn≤(1+γn)∥xn-p∥2-αn(1+γn)βn(1-βn)∥xn-TnGvn∥2+2sn(1+γn)∥γQxn-VTnGvn∥∥zn-p∥+cn.
So, it follows that(173)α(1+γn)a(1-a^)∥xn-TnGvn∥2≤αn(1+γn)βn(1-βn)∥xn-TnGvn∥2≤∥xn-p∥2-∥xn+1-p∥2+γn∥xn-p∥2+2sn(1+γn)∥γQxn-VTnGvn∥∥zn-p∥+cn.
Since lim⁡n→∞sn=0, lim⁡n→∞γn=0 and lim⁡n→∞cn=0, it follows from the existence of lim⁡n→∞∥xn-p∥ and the boundedness of {xn}, {vn}, and {zn} that
(174)lim⁡n→∞∥xn-TnGvn∥=0.
Note that
(175)∥zn-xn∥=∥(1-βn)(TnGvn-xn)+sn(γQxn-VTnGvn)∥≤∥TnGvn-xn∥+sn∥γQxn-VTnGvn∥.
Hence, it follows from (171) and lim⁡n→∞sn=0 that
(176)lim⁡n→∞∥xn-zn∥=0.
In the meantime, from (168) and (171) it follows that
(177)∥xn+1-p∥2≤(1-αn)∥xn-p∥2+αn∥kn-p∥2≤(1-αn)∥xn-p∥2+αn[(1+γn)∥zn-p∥2+(1-δn)111111111×(k-δn)∥zn-Snzn∥2+cn]≤(1-αn)∥xn-p∥2+αn[(1+γn)(∥xn-p∥2+2sn∥γQxn-VTnGvn∥11111111111111111×∥zn-p∥∥xn-p∥2)+(1-δn)(k-δn)∥zn-Snzn∥2+cn]≤(1-αn)∥xn-p∥2+αn(1+γn)∥xn-p∥2+2sn(1+γn)∥γQxn-VTnGvn∥∥zn-p∥+αn(1-δn)(k-δn)∥zn-Snzn∥2+cn≤(1+γn)∥xn-p∥2+2sn(1+γn)∥γQxn-VTnGvn∥∥zn-p∥+αn(1-δn)(k-δn)∥zn-Snzn∥2+cn,
which together with 0<k+ϵ≤δn≤d<1 leads to
(178)α(1-d)ϵ∥zn-Snzn∥2≤αn(1-δn)(δn-k)∥zn-Snzn∥2≤∥xn-p∥2-∥xn+1-p∥2+γn∥xn-p∥2+2sn(1+γn)∥γQxn-VTnGvn∥∥zn-p∥+cn.
Consequently, from lim⁡n→∞sn=0, lim⁡n→∞γn=0, lim⁡n→∞cn=0, and the existence of lim⁡n→∞∥xn-p∥, we get
(179)lim⁡n→∞∥zn-Snzn∥=0.
Since kn-zn=(1-δn)(Snzn-zn), from (179) we have
(180)lim⁡n→∞∥kn-zn∥=0.
Note that
(181)∥xn+1-xn∥=αn∥kn-xn∥≤∥kn-zn∥+∥zn-xn∥.
Hence from (176) and (180) we have
(182)lim⁡n→∞∥xn+1-xn∥=0.
Repeating the same arguments as those of Step 3 in the proof of Theorem26, we can obtain that ∥xn-un∥→0, ∥xn-vn∥→0, ∥vn-Gvn∥→0, ∥vn-PC(I-(2/L)∇f)vn∥→0, ∥zn-Szn∥→0, and ∥Λni-1un-Λniun∥→0, i∈{1,2,…,N} as n→∞.
Since{xn} is bounded, there exists a subsequence {xni} of {xn} which converges weakly to w. It is easy to see that zni⇀w, uni⇀w, vni⇀w, and Λnimuni⇀w, where m∈{1,2,…,N}. Since S is uniformly continuous and ∥zn-Szn∥→0 as n→∞, we get lim⁡n→∞∥zn-Smzn∥=0 for any m≥1. Hence from Lemma 24, we obtain w∈Fix⁡(S). In the meantime, utilizing Lemma 12, we deduce from vni⇀w, xni⇀w, (118), and (123) that w∈SGEP(G) and w∈Fix⁡(PC(I-(2/L)∇f))=VI(C,∇f)=Γ. Repeating the same arguments as those of Step 4 in the proof of Theorem 26, we can conclude that w∈∩m=1NI(Bm,Rm) and w∈GMEP⁡(Θ,φ,A). Therefore, w∈GMEP⁡(Θ,φ,A)∩SGEP(G)∩∩i=1NI(Bi,Ri)∩Fix⁡(S)∩Γ≕Ω. This shows that ωw(xn)⊂Ω.
Next let us show thatωw(xn) is a single-point set. As a matter of fact, let {xnj} be another subsequence of {xn} such that xnj⇀w′. Then we get w′∈Ω. If w≠w′, from the Opial condition, we have
(183)lim⁡n→∞⁡∥xn-w∥=lim⁡i→∞⁡∥xni-w∥<lim⁡i→∞⁡∥xni-w′∥=lim⁡n→∞⁡∥xn-w′∥=lim⁡j→∞⁡∥xnj-w′∥<lim⁡j→∞⁡∥xnj-w∥=lim⁡n→∞⁡∥xn-w∥.
This attains a contradiction. So we have w=w′. Put wn=PΩxn. Since w∈Ω, we have 〈xn-wn,wn-w〉≥0. By Lemma 14, we have that {wn} converges strongly to some w~∈Ω. Since {xn} converges weakly to w, we have
(184)〈w-w~,w~-w〉≥0.
Therefore we obtain w=w~=lim⁡n→∞PΩxn. This completes the proof.Corollary 31.
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,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R be a lower semicontinuous and convex functional. Let Ri:C→2H be a maximal monotone mapping and let A,Ak:H→H and Bi:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone, and ηi-inverse-strongly monotone, respectively, for k=1,2 and i=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 ∑n=1∞‍γn<∞ and {cn}⊂[0,∞) such that ∑n=1∞‍cn<∞. Let V be a γ¯-strongly positive bounded linear operator and Q:H→H an l-Lipschitzian mapping with γl<γ¯. Assume that Ω:=GMEP(Θ,φ,A)∩SGEP(G)∩I(B1,R1)∩I(B2,R2)∩Fix(S)∩Γ is nonempty where G is defined as in Proposition CY. Let {rn} be a sequence in [0,2ζ] and {αn}, {βn}, and {δn} sequences in [0,1] such that 0<α≤αn≤1 and 0<k+ϵ≤δn≤d<1, and let νk∈(0,2ζk), k=1,2 and {λi,n}⊂[ai,bi]⊂(0,2ηi), i=1,2. Pick any x1∈H and let {xn} be a sequence generated by the following algorithm:
(185)un=Srn(Θ,φ)(I-rnA)xn,vn=JR2,λ2,n(I-λ2,nB2)JR1,λ1,n(I-λ1,nB1)un,zn=snγQxn+βnxn+((1-βn)I-snV)TnGvn,kn=δnzn+(1-δn)Snzn,xn+1=(1-αn)xn+αnkn,∀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)). Assume that the following conditions are satisfied:(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(186)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;(iii)
sn∈(0,1/2) for each λn∈(0,2/L), and ∑n=1∞‍sn<∞  (⇔∑n=1∞‍((2/L)-λn)<∞);(iv)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ.Then {xn} converges weakly to w=lim⁡n→∞PΩxn provided that Sr(Θ,φ) is firmly nonexpansive.Corollary 32.
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,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R a lower semicontinuous and convex functional. Let R:C→2H be a maximal monotone mapping and let A,Ak:H→H and B:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone, and ξ-inverse-strongly monotone, respectively, for k=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 ∑n=1∞‍γn<∞ and {cn}⊂[0,∞) such that ∑n=1∞‍cn<∞. Let V be a γ¯-strongly positive bounded linear operator and Q:H→H an l-Lipschitzian mapping with γl<γ¯. Assume that Ω≔GMEP(Θ,φ,A)∩SGEP(G)∩I(B,R)∩Fix⁡(S)∩Γ is nonempty, where G is defined as in Proposition CY. Let {rn} be a sequence in [0,2ζ] and {αn}, {βn}, and {δn} sequences in [0,1] such that 0<α≤αn≤1 and 0<k+ϵ≤δn≤d<1, and let νk∈(0,2ζk), k=1,2 and {ρn}⊂[a,b]⊂(0,2ξ). Pick any x1∈H and let {xn} be a sequence generated by the following algorithm:
(187)un=Srn(Θ,φ)(I-rnA)xn,vn=JR,ρn(I-ρnB)un,zn=snγQxn+βnxn+((1-βn)I-snV)TnGvn,kn=δnzn+(1-δn)Snzn,xn+1=(1-αn)xn+αnkn,∀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)). Assume that the following conditions are satisfied:(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(188)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;(iii)
sn∈(0,1/2) for each λn∈(0,2/L), and ∑n=1∞‍sn<∞(⇔∑n=1∞‍((2/L)-λn)<∞);(iv)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ.Then {xn} converges weakly to w=lim⁡n→∞PΩxn provided that Sr(Θ,φ) is firmly nonexpansive.Corollary 33.
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,Θ1,Θ2 be three bifunctions from C×C to R satisfying (H1)–(H4) and φ:C→R a lower semicontinuous and convex functional. Let R:C→2H be a maximal monotone mapping and let A,Ak:H→H and B:C→H be ζ-inverse strongly monotone, ζk-inverse strongly monotone, and ξ-inverse-strongly monotone, respectively, for k=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 ∑n=1∞‍γn<∞. Let V be a γ¯-strongly positive bounded linear operator and Q:H→H be an l-Lipschitzian mapping with γl<γ¯. Assume that Ω≔GMEP(Θ,φ,A)∩SGEP(G)∩I(B,R)∩Fix⁡(S)∩Γ is nonempty, where G is defined as in Proposition CY. Let {rn} be a sequence in [0,2ζ] and {αn},{βn}, and {δn} sequences in [0,1] such that 0<α≤αn≤1 and 0<k+ϵ≤δn≤d<1, and let νk∈(0,2ζk),k=1,2 and {ρn}⊂[a,b]⊂(0,2ξ). Pick any x1∈H and let {xn} be a sequence generated by the following algorithm:
(189)un=Srn(Θ,φ)(I-rnA)xn,vn=JR,ρn(I-ρnB)un,zn=snγQxn+βnxn+((1-βn)I-snV)TnGvn,kn=δnzn+(1-δn)Snzn,xn+1=(1-αn)xn+αnkn,∀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)). Assume that the following conditions are satisfied:(i)
K:H→R is strongly convex with constant σ>0 and its derivative K′ is Lipschitz continuous with constant ν>0 such that the function x↦〈y-x,K′(x)〉 is weakly upper semicontinuous for each y∈H;(ii)
for eachx∈H, there exist a bounded subset Dx⊂C and zx∈C such that for any y∉Dx,
(190)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),zx-y〉<0;(iii)
sn∈(0,1/2) for each λn∈(0,2/L), and ∑n=1∞‍sn<∞  (⇔∑n=1∞‍((2/L)-λn)<∞);(iv)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1 and 0<liminf⁡n→∞rn≤limsup⁡n→∞rn<2ζ.Then {xn} converges weakly to w=lim⁡n→∞PΩxn provided that Sr(Θ,φ) is firmly nonexpansive.



---

*Source: 637324-2014-05-08.xml*