# Hybrid Algorithms for Solving Variational Inequalities, Variational Inclusions, Mixed Equilibria, and Fixed Point Problems

Authors: Lu-Chuan Ceng; Adrian Petrusel; Mu-Ming Wong; Jen-Chih Yao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We present a hybrid iterative algorithm for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed hybrid iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters. Here, our hybrid algorithm is based on Korpelevič’s extragradient method, hybrid steepest-descent method, and viscosity approximation method.


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## Body

## 1. Introduction

Throughout this paper, we assume thatH is a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, C is a nonempty closed convex subset of H, and PC is the metric projection of H onto C. Let S:C→C be a self-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 A:C→H is called ρ-Lipschitzian if there exists a constant ρ≥0 such that
(1)∥Ax-Ay∥≤ρ∥x-y∥,∀x,y∈C.
In particular, if ρ=1, then A is called a nonexpansive mapping [1]; if ρ∈[0,1), then A is called ρ-contraction.Recall that a mappingA:C→H is called(i)
η-strongly monotone if there exists a constant η>0 such that
(2)〈Ax-Ay,x-y〉≥η∥x-y∥2,∀x,y∈C;(ii)
α-inverse strongly monotone if there exists a constant α>0 such that
(3)〈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. In addition, a mapping V is called strongly positive on H if there exists a constant μ>0 such that
(4)〈Vx,x〉≥μ∥x∥2,∀x∈H.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. In 2008, Peng and Yao [2] introduced the following generalized mixed equilibrium problem (GMEP) of finding x∈C such that
(5)Θ(x,y)+φ(y)-φ(x)+〈Ax,y-x〉≥0,∀y∈C.
We denote the set of solutions of GMEP (5) by GMEP⁡(Θ,φ,A). The GMEP (5) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games, and others. The GMEP is further considered and studied in [3–8].We present some special cases of GMEP (5) as follows.Ifφ=0, then GMEP (5) reduces to the generalized equilibrium problem (GEP) which is to find x∈C such that
(6)Θ(x,y)+〈Ax,y-x〉≥0,∀y∈C,
which was introduced and studied by S. Takahashi and W. Takahashi [9]. The set of solutions of GEP is denoted by GEP⁡(Θ,A).IfA=0, then GMEP (5) reduces to the mixed equilibrium problem (MEP) which is to find x∈C such that
(7)Θ(x,y)+φ(y)-φ(x)≥0,∀y∈C,
which was considered and studied in [10]. The set of solutions of MEP is denoted by MEP⁡(Θ,φ).Ifφ=0, A=0, then GMEP (5) reduces to the equilibrium problem (EP) which is to find x∈C such that
(8)Θ(x,y)≥0,∀y∈C.
The set of solutions of EP is denoted by EP⁡(Θ). It is worth pointing out that the EP is a unified model of several problems, for instance, variational inequality problems, optimization problems, saddle point problems, complementarity problems, fixed point problems, Nash equilibrium problems, and so forth.Throughout this paper, it is assumed as in [2] 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,
(9)lim⁡sup⁡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,
(10)Θ(z,yx)+φ(yx)-φ(z)+1r〈yx-z,z-x〉<0,(B2)
C is a bounded set.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). In particular, if B=R=0, then I(B,R)=C. If B=0, then problem (11) becomes the inclusion problem introduced by Rockafellar [11]. It is known that problem (11) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria, and game theory.In 1998, Huang [12] studied problem (11) in the case where R is maximal monotone and B is strongly monotone and Lipschitz continuous with D(R)=C=H. Subsequently, Zeng et al. [13] further studied problem (11) in the case which is more general than Huang’s one [12]. Moreover, the authors [13] obtained the same strong convergence conclusion as in Huang’s result [12]. In addition, the authors also gave the geometric convergence rate estimate for approximate solutions. Also, various types of iterative algorithms for solving variational inclusions have been further studied and developed; for more details, refer to [14–17] and the references therein.On the other hand, consider the following variational inequality problem (VIP): find a point x∈C such that
(12)〈Ax,y-x〉≥0,∀y∈C.
The solution set of VIP (12) is denoted by VI⁡(C,A).In 1976, Korpelevič [18] proposed an iterative algorithm for solving the VIP (12) in Euclidean space Rn:
(13)yn=PC(xn-τAxn),xn+1=PC(xn-τAyn),∀n≥0,
with τ>0 being a given number, which is known as the extragradient method (see also [19]). The literature on the VIP is vast and Korpelevič’s extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [2, 5, 6, 8, 17, 20–28] and references therein, to name but a few.VIP (12) was first discussed by Lions [29] and now is well known; there are a lot of different approaches towards solving VIP (12) in finite-dimensional and infinite-dimensional spaces, and the research is intensively continued. VIP (12) has many applications in computational mathematics, mathematical physics, operations research, mathematical economics, optimization theory, and other fields; see, for example, [30–33]. It is well known that, if A is a strongly monotone and Lipschitz-continuous mapping on C, then VIP (12) has a unique solution. Not only the existence and uniqueness of solutions are important topics in the study of VIP (12) but also how to actually find a solution of VIP (12) is important.LetC be a nonempty closed convex subset of a real Banach space X. Let {Tn}n=1∞ be an infinite family of nonexpansive self-mappings on C and let {λn}n=0∞ be a sequence of nonnegative numbers in [0,1]. For any n≥1, define a self-mapping Wn on C as follows:
(14)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; see [34].In 2008, Ceng and Yao [35] introduced and analyzed the following relaxed viscosity approximation method for finding a common fixed point of an infinite family of nonexpansive mappings in a strictly convex and reflexive Banach space.Theorem 1 (see [35, Theorem 3.2]).
LetX be a strictly convex and reflexive Banach space with a uniformly Gateaux differentiable norm, let C be a nonempty closed convex subset of X, let {Tn}n=1∞ be an infinite family of nonexpansive self-mappings on C such that the common fixed point set ∩n=1∞Fix⁡(Tn)≠∅, and let f:C→C be a ρ-contraction with the contraction coefficient ρ∈(1/2,1). Let {λ}n=1∞ be a sequence of positive numbers in (0,b] for some b∈(0,1). For any given x1∈C, let {xn}n=1∞ be the iterative sequence defined by
(15)xn+1=(1-αn-βn)xn+αnf(yn)+βnWnyn,yn=(1-γn)xn+γnWnxn,∀n≥1,
where {αn}n=1∞ and {βn}n=1∞ are two sequences in (0,1) with αn+βn≤1(n≥1), {γn}n=1∞ is a sequence in [0,1], and Wn is the W-mapping defined by (14). Assume that(i)
lim⁡n→∞αn=0, ∑n=0∞‍αn=∞ and 0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1,(ii)
lim⁡n→∞|γn+1-γn|=0 and lim⁡sup⁡n→∞γn<1. Then there hold the following:(I)
lim⁡n→∞∥xn+1-xn∥=0;(II)
the sequence{xn}n=1∞ converges strongly to some q∈∩n=1∞Fix⁡(Tn), provided lim⁡n→∞γn=0 and βn≡β for some fixed β∈(0,1), which is the unique solution of the VIP:
(16)〈(I-f)q,J(q-p)〉,∀p∈∩n=1∞Fix⁡(Tn),whereJ is the normalized duality mapping of X.
Furthermore, letλn,1,λn,2,…,λn,N∈(0,1], n≥1. Given the nonexpansive mappings S1,S2,…,SN on H, Atsushiba and Takahashi [36] defined, for each n≥1, mappings Un,1,Un,2,…,Un,N by
(17)Un,1=λn,1S1+(1-λn,1)I,Un,2=λn,2SnUn,1+(1-λn,2)I,Un,n-1=λn-1Tn-1Un,n+(1-λn-1)I,⋮Un,N-1=λn,N-1SN-1Un,N-2+(1-λn,N-1)I,Wn:=Un,N=λn,NSNUn,N-1+(1-λn,N)I.
TheWn is called the W-mapping generated by S1,…,SN and λn,1,λn,2,…,λn,N. Note that the nonexpansivity of Si implies the nonexpansivity of Wn.
In 2008, Colao et al. [37] introduced and studied an iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space H. Subsequently, combining Yamada’s hybrid steepest-descent method [38] and Colao et al.’s hybrid viscosity approximation method [37], Ceng et al. [7] proposed and analyzed the following hybrid iterative method for finding a common element of the set of solutions of GMEP⁡ (5) and the set of fixed points of a finite family of nonexpansive mappings {Si}i=1N.Theorem 2 (see [7, Theorem 3.1]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ:C×C→R be a bifunction satisfying assumptions (A1)–(A4) and let φ:C→R be a lower semicontinuous and convex function with restriction (B1) or (B2). Let the mapping A:H→H be δ-inverse strongly monotone, and let {Si}i=1N be a finite family of nonexpansive mappings on H such that ∩i=1NFix⁡(Si)∩GMEP⁡(Θ,φ,A)≠∅. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with constants κ,η>0 and f:H→H a ρ-Lipschitzian mapping with constant ρ≥0. Let 0<μ<2η/κ2 and 0≤γρ<τ, where τ=1-1-μ(2η-μκ2). Suppose that {αn} and {βn} are two sequences in (0,1), {γn} is a sequence in (0,2δ], and {λn,i}i=1N is a sequence in [a,b] with 0<a≤b<1. For every n≥1, let Wn be the W-mapping generated by S1,…,SN and λn,1,λn,2,…,λn,N. Given x1∈H arbitrarily, suppose that the sequences {xn} and {un} are generated iteratively by
(18)Θ(un,y)+φ(y)-φ(un)+〈Axn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnγf(xn)+βnxn+((1-βn)I-αnμF)Wnun,mmmmmmmmmmmmmmmmmmmmm∀n≥1,
where the sequences {αn}, {βn}, {rn} and the finite family of sequences {λn,i}i=1N satisfy the following conditions:(i)
lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞;(ii)
0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1;(iii)
0<lim⁡inf⁡n→∞rn≤lim⁡sup⁡n→∞rn<2δ and lim⁡n→∞(rn+1-rn)=0;(iv)
lim⁡n→∞(λn+1,i-λn,i)=0 for all i=1,2,…,N. Then, both {xn} and {un} converge strongly to x*∈∩i=1NFix⁡(Si)∩GMEP⁡(Θ,φ,A), where x*=P∩i=1NFix⁡(Si)∩GMEP⁡(Θ,φ,A)(I-μF+γf)x* is a unique solution of the variational inequality:
(19)mmmmii〈(μF-γf)x*,x*-x〉≤0,∀x∈∩i=1NFix⁡(Si)∩GMEP⁡(Θ,φ,A).
On the other hand, wheneverC=H a real Hilbert space, Yao et al. [4] very recently introduced and analyzed an iterative algorithm for finding a common element of the set of solutions of GMEP⁡ (5), the set of solutions of the variational inclusion (11), and the set of fixed points of an infinite family of nonexpansive mappings.Theorem 3 (see [4, Theorem 3.2]).
Letφ:H→R be a lower semicontinuous and convex function and let Θ:H×H→R be a bifunction satisfying conditions (A1)–(A4) and (B1). Let V be a strongly positive bounded linear operator with coefficient μ>0 and let R:H→2H be a maximal monotone mapping. Let the mappings A,B:H→H be α-inverse strongly monotone and β-inverse strongly monotone, respectively. Let f:H→H be a ρ-contraction. Let r>0, γ>0, and λ>0 be three constants such that r<2α, λ<2β, and 0<γ<μ/ρ. Let {λ}n=1∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and {Tn}n=1∞ an infinite family of nonexpansive self-mappings on H such that Ω:=∩n=1∞Fix⁡(Tn)∩GMEP⁡(Θ,φ,A)∩I(B,R)≠∅. For arbitrarily given x1∈H, let the sequence {xn} be generated by
(20)Θ(un,y)+φ(y)-φ(un)+〈y-un,Axn〉+1r〈y-un,un-xn〉≥0,∀y∈H,xn+1=αnγf(xn)+βnxn+[(1-βn)I-αnV]×WnJR,λ(un-λBun),∀n≥1,
where {αn}, {βn} are two real sequences in [0,1] and Wn is the W-mapping defined by (14) (with X=H and C=H). Assume that the following conditions are satisfied:(C1)
lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞;(C2)
0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1. Then, the sequence {xn} converges strongly to x*∈Ω, where x*=PΩ(γf(x*)+(I-V)x*) is a unique solution of the VIP:
(21)〈(γf-V)x*,y-x*〉≤0,∀y∈Ω.Motivated and inspired by the above facts, in this paper, we introduce and analyze a hybrid iterative algorithm for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of the variational inclusion (11) in a real Hilbert space. Furthermore, it is proven that the proposed hybrid iterative algorithm is strongly convergent under some mild conditions imposed on algorithm parameters. Here our hybrid algorithm is based on Korpelevič’s extragradient method, hybrid steepest-descent method, and viscosity approximation method. The results obtained in this paper improve and extend the corresponding results announced by many others.


## 2. Preliminaries

LetH be 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,
(22)ωw(xn):={x∈H:xni⇀xmimmmmmforsomesubsequence{xni}of{xn}}.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
(23)∥x-PCx∥=inf⁡y∈C∥x-y∥=:d(x,C).Some important properties of projections are gathered in the following proposition.Proposition 4 (see [31, 39]).
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 (α-ism) of C into H, then it is obvious that A is 1/α-Lipschitzian. We also have that, for all u,v∈C and λ>0,
(24)∥(I-λA)u-(I-λA)v∥2=∥(u-v)-λ(Au-Av)∥2=∥u-v∥2-2λ〈Au-Av,u-v〉+λ2∥Au-Av∥2≤∥u-v∥2+λ(λ-2α)∥Au-Av∥2.
So, if λ≤2α, then I-λA is a nonexpansive mapping from C to H.
It is also easy to see that a projectionPC is 1-ism. Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.
A set-valued mappingR:D(R)⊂H→2H is called monotone if, for all x,y∈D(R), f∈R(x) and g∈R(y) imply
(25)〈f-g,x-y〉≥0.
A set-valued mapping R is called maximal monotone if R is monotone and (I+λR)D(R)=H for each λ>0, where I is the identity mapping of H. We denote by G(R) the graph of R. It is known that a monotone mapping R is maximal if and only if, for (x,f)∈H×H, 〈f-g,x-y〉≥0 for every (y,g)∈G(R) implies f∈R(x).
LetA:C→H be a monotone, k-Lipschitzian mapping and let NCv be the normal cone to C at v∈C; that is,
(26)NCv={w∈H:〈v-u,w〉≥0,∀u∈C}.
Defining
(27)Tv={Av+NCv,ifv∈C,∅,ifv∉C,
then, T is maximal monotone and 0∈Tv if and only if v∈VI(C,A); see [11].
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
(28)JR,λx=(I+λR)-1x,∀x∈H.
In terms of Huang [12] (see also [13]), there holds the following property for the resolvent operator JR,λ:H→D(R)¯.Lemma 5.
J
R
,
λ is single-valued and firmly nonexpansive; that is,
(29)〈JR,λx-JR,λy,x-y〉≥∥JR,λx-JR,λy∥2,∀x,y∈H.
Consequently, JR,λ is nonexpansive and monotone.Lemma 6.
LetC be a nonempty closed convex subset of H and A:C→H a monotone mapping. In the context of the VIP (12), there holds the following relation:
(30)u∈VI⁡(C,A)⟺u=PC(u-λAu),forsomeλ>0.Lemma 7 (see [17]).
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
(31)u=JR,λ(u-λBu).Lemma 8 (see [13]).
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 9 (see [17]).
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 10 ([40], see also [10]).
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:
(32)Tr(Θ,φ)(x)={1rz∈C:Θ(z,y)+φ(y)-φ(z)mm+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 any x,y∈H,
(33)∥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.Proposition 11 (see [5, Proposition 2.1]).
LetC, H, Θ, φ, and Tr(Θ,φ) be as in Lemma 10. Then the following inequality holds:
(34)∥Ts(Θ,φ)x-Tt(Θ,φ)x∥2≤s-ts〈Ts(Θ,φ)x-Tt(Θ,φ)x,Ts(Θ,φ)x-x〉,
for all s,t>0 and x∈H.Remark 12.
From the conclusion of Proposition11, it immediately follows that
(35)∥Ts(Θ,φ)x-Tt(Θ,φ)x∥≤|s-t|s∥Ts(Θ,φ)x-x∥
for all s,t>0 and x∈H.Lemma 13 (see [41]).
Let{xn} and {zn} be bounded sequences in a Banach space X and {βn} a sequence in [0,1] with
(36)0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1.
Suppose that xn+1=(1-βn)zn+βnxn for each n≥1 and
(37)lim⁡sup⁡n→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0.
Then lim⁡n→∞∥zn-xn∥=0.We have the following crucial lemmas concerning theW-mapping defined by (14).Lemma 14 (see [42, Lemma 3.2]).
LetC be a nonempty closed convex subset of a strictly convex Banach space X. Let {Tn}n=1∞ be a sequence of nonexpansive self-mappings on C such that ∩n=1∞Fix⁡(Tn)≠∅ and let {λn}n=1∞ be a sequence of positive numbers in (0,b] for some b∈(0,1). Then, for every x∈C and k≥1, the limit lim⁡n→∞Un,kx exists.Lemma 15 (see [42, Lemma 3.3]).
LetC be a nonempty closed convex subset of a strictly convex Banach space X. Let {Tn}n=1∞ be a sequence of nonexpansive self-mappings on C such that ∩n=1∞Fix⁡(Tn)≠∅ and let {λn}n=1∞ be a sequence of positive numbers in (0,b] for some b∈(0,1). Then, Fix⁡(W)=∩n=1∞Fix⁡(Tn).Remark 16.
Using Lemma14, we can define the mapping W:C→C as follows:
(38)Wx=lim⁡n→∞Wnx=lim⁡n→∞Un,1x,∀x∈C.
Such a W is called the W-mapping generated by the sequences {Tn}n=1∞ and {λn}n=1∞. As pointed out in [43], if {xn} is a bounded sequence in C, then we have
(39)lim⁡n→∞∥Wxn-Wnxn∥=0.
Throughout this paper, we always assume that {λn}n=1∞ is a sequence of positive numbers in (0,b] for some b∈(0,1).Lemma 17 (see [44]).
Let{sn} be a sequence of nonnegative numbers satisfying the conditions
(40)sn+1≤(1-αn)sn+αnβn,∀n≥1,
where {αn} and {βn} are sequences of real numbers such that(i)
{αn}⊂[0,1] and ∑n=1∞‍αn=∞, or equivalently,
(41)∏n=1∞‍(1-αn):=lim⁡n→∞∏k=1n‍(1-αk)=0,(ii)
lim⁡sup⁡n→∞βn≤0, or ∑n=1∞‍|αnβn|<∞. Then lim⁡n→∞sn=0.Lemma 18 (see [39, demiclosedness principle]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on C with Fix⁡(T)≠∅. 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.The following lemma is an immediate consequence of an inner product.Lemma 19.
In a real Hilbert spaceH, there holds the inequality
(42)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H.
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→C, we define the mapping Tλ:C→H by
(43)Tλx:=Tx-λμF(Tx),∀x∈C,
where F:C→H is an operator such that, for some positive constants κ,η>0, F is κ-Lipschitzian and η-strongly monotone on C; that is, F satisfies the conditions
(44)∥Fx-Fy∥≤κ∥x-y∥,〈Fx-Fy,x-y〉≥η∥x-y∥2
for all x,y∈C.Lemma 20 (see [44, Lemma 3.1]).
T
λ is a contraction provided 0<μ<2η/κ2; that is,
(45)∥Tλx-Tλy∥≤(1-λτ)∥x-y∥,∀x,y∈C,
where τ=1-1-μ(2η-μκ2)∈(0,1].Remark 21.
(i) SinceF is κ-Lipschitzian and η-strongly monotone on C, we get 0<η≤κ. Hence, whenever 0<μ<2η/κ2, we have
(46)0≤(1-μη)2=1-2μη+μ2η2≤1-2μη+μ2κ2<1-2μη+2ηκ2μκ2=1,
which implies
(47)0<1-1-2μη+μ2κ2≤1.
So, τ=1-1-μ(2η-μκ2)∈(0,1].
(ii) In Lemma20, put F=(1/2)I and μ=2. Then we know that κ=η=1/2, 0<μ=2<2η/κ2=4 and
(48)τ=1-1-μ(2η-μκ2)=1-1-2(2×12-2×(12)2)=1.


## 3. A Strong Convergence Theorem

In this section, we will prove a strong convergence theorem for a hybrid iterative algorithm for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of the variational inclusion (11) in a real Hilbert space.Theorem 22.
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:H→H and Ai:C→H be μk-inverse strongly monotone and ηi-inverse strongly monotone, respectively, where k∈{1,2,…,M}, i∈{1,2,…,N}. Let F:C→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0 and let f:H→H be a ρ-Lipschitzian mapping with constant ρ≥0. Let R:C→2H be a maximal monotone mapping and let the mapping B:C→H be β-inverse strongly monotone. Let 0<λ<2β, 0<μ<2η/κ2, and 0≤γρ<τ, where τ=1-1-μ(2η-μκ2). Let {λn}n=1∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and {Tn}n=1∞ an infinite family of nonexpansive self-mappings on C such that Ω:=∩k=1MGMEP⁡(Θk,φk,Bk)∩∩i=1NVI⁡(C,Ai)∩∩n=1∞Fix⁡(Tn)∩I(B,R)≠∅. For arbitrarily given x1∈H, let the sequence {xn} be generated by
(49)un=TrM,n(ΘM,φM)(I-rM,nBM)TrM,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,xn+1=αnγf(xn)+βnxn+[(1-βn)I-αnμF]×WnJR,λ(vn-λBvn),∀n≥1,
where {αn}, {βn} are two real sequences in [0,1] and Wn is the W-mapping defined by (14). Assume that the following conditions are satisfied:(i)
lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞;(ii)
0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1;(iii)
{λi,n}⊂[ai,bi]⊂(0,2ηi) and lim⁡n→∞|λi,n+1-λi,n|=0 for all i∈{1,2,…,N};(iv)
{rk,n}⊂[ek,fk]⊂(0,2μk) and lim⁡n→∞|rk,n+1-rk,n|=0 for all k∈{1,2,…,M}. Assume that either (B1) or (B2) holds. Then the sequence {xn} converges strongly to x*∈Ω, where x*=PΩ(I-μF+γf)x* is a unique solution of the VIP:
(50)〈(μF-γf)x*,x*-y〉≤0,∀y∈Ω.Proof.
LetQ=PΩ. Note that F:C→H is a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0 and f:H→H is a ρ-Lipschitzian mapping with constant ρ≥0. Then, we have
(51)∥(I-μF)x-(I-μF)y∥2=∥x-y∥2-2μ〈x-y,Fx-Fy〉+μ2∥Fx-Fy∥2≤(1-2μη+μ2κ2)∥x-y∥2=(1-τ)2∥x-y∥2,
where τ=1-1-μ(2η-μκ2), and hence
(52)∥Q(I-μF+γf)x-Q(I-μF+γf)y∥≤∥(I-μF+γf)x-(I-μF+γf)y∥≤∥(I-μF)x-(I-μF)y∥+γ∥f(x)-f(y)∥≤(1-τ)∥x-y∥+γρ∥x-y∥=(1-(τ-γρ))∥x-y∥,∀x,y∈C.
Since 0≤γρ<τ≤1, it is known that 1-(τ-γρ)∈[0,1). Therefore, Q(I-μF+γf) is a contraction of C into itself, which implies that there exists a unique element x*∈C such that x*=Q(I-μF+γf)x*=PΩ(I-μF+γf)x*.
We divide the remainder of the proof into several steps.
Step  1. Let us show that {xn} is bounded.
Indeed, taking into account the control conditions (i) and (ii), we may assume, without loss of generality, thatαn≤1-βn for all n≥1. Put
(53)Δnk=Trk,n(Θk,φk)(I-rk,nBM)×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,
(54)Λni=PC(I-λi,nAi)PC(I-λi-1,nAi-1)×⋯PC(I-λ2,nA2)PC(I-λ1,nA1)
for all i∈{1,2,…,N} and n≥1, and Δn0=Λn0=I, where I is the identity mapping on H. Then we have that un=ΔnMxn and vn=ΛnNun. Take p∈Ω arbitrarily. Then from (24) and Lemma 10 we have
(55)∥un-p∥=∥TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1xnmmm-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
(56)∥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 (55) and (56), we have
(57)∥vn-p∥≤∥xn-p∥.
Since the mapping B:C→H is β-inverse strongly monotone with 0<λ<2β, we have
(58)∥(I-λB)x-(I-λB)y∥2≤∥x-y∥2+λ(λ-2β)∥Bx-By∥2.
It is clear that, if 0≤λ≤2β, then I-λB is nonexpansive. Set yn=JR,λ(vn-λBvn) for each n≥1. It follows that
(59)∥yn-p∥=∥JR,λ(vn-λBvn)-JR,λ(p-λBp)∥≤∥(vn-λBvn)-(p-λBp)∥≤∥vn-p∥,
which, together with (57), yields
(60)∥yn-p∥≤∥xn-p∥.
Utilizing Lemma 20, from (49) we obtain
(61)∥xn+1-p∥=∥αn(γf(xn)-μFp)+βn(xn-p)+((1-βn)I-αnμF)mm×Wnyn-((1-βn)I-αnμF)Wnp∥≤αn∥γf(xn)-μFp∥+βn∥xn-p∥+∥((1-βn)I-αnμF)Wnyn-((1-βn)I-αnμF)Wnp∥=αn∥γf(xn)-μFp∥+βn∥xn-p∥+(1-βn)×∥(I-αn1-βnμF)Wnyn-(I-αn1-βnμF)Wnp∥≤(1-βn)(1-αnτ1-βn)∥yn-p∥+βn∥xn-p∥+αn∥γf(xn)-μFp∥=(1-βn-αnτ)∥yn-p∥+βn∥xn-p∥+αn∥γf(xn)-μFp∥≤(1-αnτ)∥xn-p∥+αnγ∥f(xn)-f(p)∥+αn∥γf(p)-μFp∥≤(1-αnτ)∥xn-p∥+αnγρ∥xn-p∥+αn∥γf(p)-μFp∥=(1-αn(τ-γρ))∥xn-p∥+αn∥γf(p)-μFp∥=(1-αn(τ-γρ))∥xn-p∥+αn(τ-γρ)∥γf(p)-μFp∥τ-γρ≤max⁡{∥xn-p∥,∥γf(p)-μFp∥τ-γρ}.
By induction, we get
(62)∥xn-p∥≤max⁡{∥x1-p∥,∥γf(p)-μFp∥τ-γρ},∀n≥1.
Therefore, {xn} is bounded and hence {un}, {vn}, {yn}, {Wnyn}, {FWnyn}, and {f(xn)} are also bounded.
Step  2. Let us show that ∥xn+1-xn∥→0 as n→∞.
Indeed, definexn+1=βnxn+(1-βn)zn for each n≥1. Then from the definition of zn we obtain
(63)zn+1-zn=xn+2-βn+1xn+11-βn+1-xn+1-βnxn1-βn=αn+1γf(xn+1)+[(1-βn+1)I-αn+1μF]Wn+1yn+11-βn+1-αnγf(xn)+[(1-βn)I-αnμF]Wnyn1-βn=αn+11-βn+1γf(xn+1)-αn1-βnγf(xn)+Wn+1yn+1-Wnyn+αn1-βnμFWnyn-αn+11-βn+1μFWn+1yn+1=αn+11-βn+1[γf(xn+1)-μFWn+1yn+1]+αn1-βn[μFWnyn-γf(xn)]+Wn+1yn+1-Wn+1yn+Wn+1yn-Wnyn.
It follows that
(64)∥zn+1-zn∥-∥xn+1-xn∥≤αn+11-βn+1(γ∥f(xn+1)∥+μ∥FWn+1yn+1∥)+αn1-βn(μ∥FWnyn∥+γ∥f(xn)∥)+∥Wn+1yn+1-Wn+1yn∥+∥Wn+1yn-Wnyn∥-∥xn+1-xn∥≤αn+11-βn+1(γ∥f(xn+1)∥+μ∥FWn+1yn+1∥)+αn1-βn(μ∥FWnyn∥+γ∥f(xn)∥)+∥Wn+1yn-Wnyn∥+∥yn+1-yn∥-∥xn+1-xn∥.
Utilizing (14) and the nonexpansivity of Ti and Un,i, we obtain that for each n≥1(65)∥Wn+1yn-Wnyn∥=∥λ1T1Un+1,1yn-λ1T1Un,1yn∥≤λ1∥Un+1,1yn-Un,1yn∥=λ1∥λ2T2Un+1,2yn-λ2T2Un,2yn∥≤λ1λ2∥Un+1,2yn-Un,2yn∥⋮≤(∏i=1n‍λi)∥Un+1,n+1yn-Un,n+1yn∥≤M~∏i=1n‍λi,
where sup⁡n≥1∥Un+1,n+1yn-Un,n+1yn∥≤M~ for some M~>0. Note that
(66)∥yn+1-yn∥=∥JR,λ(vn+1-λBvn+1)-JR,λ(vn-λBvn)∥≤∥(vn+1-λBvn+1)-(vn-λBvn)∥≤∥vn+1-vn∥.
Note that
(67)∥vn+1-vn∥=∥Λn+1Nun+1-ΛnNun∥=∥PC(I-λN,n+1AN)Λn+1N-1un+1m-PC(I-λN,nAN)ΛnN-1un∥≤∥PC(I-λN,n+1AN)Λn+1N-1un+1m-PC(I-λN,nAN)Λn+1N-1un+1∥+∥PC(I-λN,nAN)Λn+1N-1un+1mm-PC(I-λN,nAN)ΛnN-1un∥≤∥(I-λN,n+1AN)Λn+1N-1un+1-(I-λN,nAN)Λn+1N-1un+1∥+∥(I-λN,nAN)Λn+1N-1un+1-(I-λN,nAN)ΛnN-1un∥≤|λN,n+1-λN,n|∥ANΛn+1N-1un+1∥+∥Λn+1N-1un+1-ΛnN-1un∥≤|λN,n+1-λN,n|∥ANΛn+1N-1un+1∥+|λN-1,n+1-λN-1,n|×∥AN-1Λn+1N-2un+1∥+∥Λn+1N-2un+1-ΛnN-2un∥⋮≤|λN,n+1-λN,n|∥ANΛn+1N-1un+1∥+|λN-1,n+1-λN-1,n|×∥AN-1Λn+1N-2un+1∥+⋯+|λ1,n+1-λ1,n|×∥A1Λn+10un+1∥+∥Λn+10un+1-Λn0un∥≤M~0∑i=1N‍|λi,n+1-λi,n|+∥un+1-un∥,
where sup⁡n≥1{∑i=1N‍∥AiΛn+1i-1un+1∥}≤M~0 for some M~0>0. Also, utilizing Lemma 10 and Proposition 11 we deduce that
(68)∥un+1-un∥=∥Δn+1Mxn+1-ΔnMxn∥=∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1m-TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1xn∥≤∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1m-TrM,n(ΘM,φM)(I-rM,nBM)Δn+1M-1xn+1∥+∥TrM,n(ΘM,φM)(I-rM,nBM)Δn+1M-1xn+1mm-TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1xn∥≤∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1m-TrM,n(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1∥+∥TrM,n(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1mm-TrM,n(ΘM,φM)(I-rM,nBM)Δn+1M-1xn+1∥+∥(I-rM,nBM)Δn+1M-1xn+1-(I-rM,nBM)ΔnM-1xn∥≤|rM,n+1-rM,n|rM,n+1∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1mmmmmmmm-(I-rM,n+1BM)Δn+1M-1xn+1TrM,n+1(ΘM,φM)∥+|rM,n+1-rM,n|∥BMΔn+1M-1xn+1∥+∥Δn+1M-1xn+1-ΔnM-1xn∥=|rM,n+1-rM,n|[∥BMΔn+1M-1xn+1∥+1rM,n+1mmmmmmmm×∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1mmmmmmmmmi-(I-rM,n+1BM)Δn+1M-1xn+1TrM,n+1(ΘM,φM)∥1rM,n+1]+∥Δn+1M-1xn+1-ΔnM-1xn∥⋮≤|rM,n+1-rM,n|[∥BMΔn+1M-1xn+1∥+1rM,n+1mmmmmmmm×∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1mmmmmmmimm-(I-rM,n+1BM)Δn+1M-1xn+1TrM,n+1(ΘM,φM)∥1rM,n+1]+⋯+|r1,n+1-r1,n|[∥B1Δn+10xn+1∥+1r1,n+1mmmmmmmimmm×∥Tr1,n+1(Θ1,φ1)(I-r1,n+1B1)Δn+10xn+1mmmmmmmmmmmm-(I-r1,n+1B1)Δn+10xn+1TrM,n+1(ΘM,φM)∥1r1,n+1]+∥Δn+10xn+1-Δn0xn∥≤M~1∑k=1M‍|rk,n+1-rk,n|+∥xn+1-xn∥,
where M~1>0 is a constant such that for each n≥1(69)∑k=1M‍[∥BkΔn+1k-1xn+1∥mm+1rk,n+1∥Trk,n+1(Θk,φk)(I-rk,n+1Bk)Δn+1k-1xn+1mmmmimm-(I-rk,n+1Bk)Δn+1k-1xn+1TrM,n+1(ΘM,φM)∥]≤M~1.
Combining (64)–(68), we get
(70)∥zn+1-zn∥-∥xn+1-xn∥≤αn+11-βn+1(γ∥f(xn+1)∥+μ∥FWn+1yn+1∥)+αn1-βn(μ∥FWnyn∥+γ∥f(xn)∥)+∥Wn+1yn-Wnyn∥+∥yn+1-yn∥-∥xn+1-xn∥≤αn+11-βn+1(γ∥f(xn+1)∥+μ∥FWn+1yn+1∥)+αn1-βn(μ∥FWnyn∥+γ∥f(xn)∥)+M~∏i=1n‍λi+∥vn+1-vn∥-∥xn+1-xn∥≤αn+11-βn+1(γ∥f(xn+1)∥+μ∥FWn+1yn+1∥)+αn1-βn(μ∥FWnyn∥+γ∥f(xn)∥)+M~∏i=1n‍λi+M~0∑i=1N‍|λi,n+1-λi,n|+∥un+1-un∥-∥xn+1-xn∥≤αn+11-βn+1(γ∥f(xn+1)∥+μ∥FWn+1yn+1∥)+αn1-βn(μ∥FWnyn∥+γ∥f(xn)∥)+M~∏i=1n‍λi+M~0∑i=1N‍|λi,n+1-λi,n|+M~1∑k=1M‍|rk,n+1-rk,n|.
Consequently, it follows from (70), {λn}⊂(0,b], and conditions (i)–(iv) that
(71)lim⁡sup⁡n→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0.
Hence, by Lemma 13 we have
(72)lim⁡n→∞∥zn-xn∥=0.
Consequently
(73)lim⁡n→∞∥xn+1-xn∥=lim⁡n→∞(1-βn)∥zn-xn∥=0.
Step  3. Let us show that ∥Bvn-Bp∥→0, ∥BkΔnk-1xn-Bkp∥→0, and ∥AiΛni-1un-Aip∥→0, k∈{1,2,…,M}, i∈{1,2,…,N}.
Indeed, we can rewrite (49) as follows:
(74)xn+1=αn(γf(xn)-μFWnyn)+βn(xn-Wnyn)+Wnyn.
It follows that
(75)∥xn-Wnyn∥≤∥xn-xn+1∥+∥xn+1-Wnyn∥≤∥xn-xn+1∥+αn∥γf(xn)-μFWnyn∥+βn∥xn-Wnyn∥;
that is,
(76)∥xn-Wnyn∥≤11-βn∥xn-xn+1∥+αn1-βn∥γf(xn)-μFWnyn∥.
This, together with αn→0 and (73), implies that
(77)lim⁡n→∞∥xn-Wnyn∥=0.
Also, from (24) it follows that for all i∈{1,2,…,N} and k∈{1,2,…,M}(78)∥vn-p∥=∥ΛnNun-p∥2≤∥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,∥un-p∥=∥ΔnMxn-p∥2≤∥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.
So, from (57), (58), and (78), it follows that
(79)∥yn-p∥2=∥JR,λ(vn-λBvn)-JR,λ(p-λBp)∥2≤∥(vn-λBvn)-(p-λBp)∥2≤∥vn-p∥2+λ(λ-2β)∥Bvn-Bp∥2≤∥un-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2+λ(λ-2β)∥Bvn-Bp∥2≤∥xn-p∥2+rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2+λ(λ-2β)∥Bvn-Bp∥2.
By (49) and Lemma 20, we obtain
(80)∥xn+1-p∥2=∥αn(γf(xn)-μFp)+βn(xn-Wnyn)m+(I-αnμF)Wnyn-(I-αnμF)p∥2≤∥βn(xn-Wnyn)+(I-αnμF)Wnyn-(I-αnμF)p∥2+2αn〈γf(xn)-μFp,xn+1-p〉≤[βn∥xn-Wnyn∥m+∥(I-αnμF)Wnyn-(I-αnμF)Wnp∥]2+2αn∥γf(xn)-μFp∥∥xn+1-p∥≤[βn∥xn-Wnyn∥+(1-αnτ)∥yn-p∥]2+2αn∥γf(xn)-μFp∥∥xn+1-p∥=(1-αnτ)2∥yn-p∥2+βn2∥xn-Wnyn∥2+2(1-αnτ)βn∥yn-p∥∥xn-Wnyn∥+2αn∥γf(xn)-μFp∥∥xn+1-p∥.
From (79) and (80), it follows that
(81)∥xn+1-p∥2≤∥yn-p∥2+βn2∥xn-Wnyn∥2+2(1-αnτ)βn∥yn-p∥×∥xn-Wnyn∥+2αn∥γf(xn)-μFp∥∥xn+1-p∥≤∥xn-p∥2+rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2+λ(λ-2β)×∥Bvn-Bp∥2+βn2∥xn-Wnyn∥2+2(1-αnτ)βn∥yn-p∥∥xn-Wnyn∥+2αn∥γf(xn)-μFp∥∥xn+1-p∥,
and so
(82)rk,n(2μk-rk,n)∥BkΔnk-1xn-Bkp∥2+λi,n(2ηi-λi,n)×∥AiΛni-1un-Aip∥2+λ(2β-λ)∥Bvn-Bp∥2≤∥xn-p∥2-∥xn+1-p∥2+βn2∥xn-Wnyn∥2+2(1-αnτ)βn∥yn-p∥∥xn-Wnyn∥+2αn∥γf(xn)-μFp∥∥xn+1-p∥≤(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+βn2∥xn-Wnyn∥2+2(1-αnτ)βn∥yn-p∥×∥xn-Wnyn∥+2αn∥γf(xn)-μFp∥∥xn+1-p∥.
Since {λi,n}⊂[ai,bi]⊂(0,2ηi) and {rk,n}⊂[ek,fk]⊂(0,2μk) for all i∈{1,2,…,N} and k∈{1,2,…,M}, by (73), (77), and (82) we conclude immediately that
(83)lim⁡n→∞∥Bvn-Bp∥=0,lim⁡n→∞∥AiΛni-1un-Aip∥=0,lim⁡n→∞∥BkΔnk-1xn-Bkp∥=0,
for all i∈{1,2,…,N} and k∈{1,2,…,M}.
Step  4. Let us show that ∥xn-Wxn∥→0.
Indeed, by Lemma10 (iii) we obtain that for each k∈{1,2,…,M}(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,Δnkxn-p〉=12(∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p∥2+∥Δnkxn-p∥2mmm-∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p-(Δnkxn-p)∥2)≤12(∥Δnk-1xn-p∥2+∥Δnkxn-p∥2mm-∥Δnk-1xn-Δnkxn-rk,n(BkΔnk-1xn-Bkp)∥2),
which implies that
(85)∥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∥.
Also, by Proposition 4 (iii), we obtain that for each i∈{1,2,…,N}(86)∥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∥2mm-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2)≤12(∥un-p∥2+∥Λniun-p∥2mm-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2),
which implies that
(87)∥Λniun-p∥2≤∥un-p∥2-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2=∥un-p∥2-∥Λni-1un-Λniun∥2-λi,n2∥AiΛni-1un-Aip∥2+2λi,n〈Λni-1un-Λniun,AiΛni-1un-Aip〉≤∥un-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥.
Since JR,λ is 1-inverse strongly monotone, we have
(88)∥yn-p∥2=∥JR,λ(vn-λBvn)-JR,λ(p-λBp)∥2≤〈vn-λBvn-(p-λBp),yn-p〉=12(∥vn-λBvn-(p-λBp)∥2+∥yn-p∥2mii-∥vn-λBvn-(p-λBp)-(yn-p)∥2)≤12(∥vn-p∥2+∥yn-p∥2-∥vn-yn-λ(Bvn-Bp)∥2)=12(∥vn-p∥2+∥yn-p∥2-∥vn-yn∥2i+2λ〈Bvn-Bp,vn-yn〉-λ2∥Bvn-Bp∥2),
which implies that
(89)∥yn-p∥2≤∥vn-p∥2-∥vn-yn∥2+2λ〈Bvn-Bp,vn-yn〉.
Thus, from (85)–(89) we get
(90)∥yn-p∥2≤∥vn-p∥2-∥vn-yn∥2+2λ∥Bvn-Bp∥∥vn-yn∥=∥ΛnNun-p∥2-∥vn-yn∥2+2λ∥Bvn-Bp∥∥vn-yn∥≤∥Λniun-p∥2-∥vn-yn∥2+2λ∥Bvn-Bp∥∥vn-yn∥≤∥un-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥-∥vn-yn∥2+2λ∥Bvn-Bp∥∥vn-yn∥=∥ΔnMxn-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥-∥vn-yn∥2+2λ∥Bvn-Bp∥∥vn-yn∥≤∥Δnkxn-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥-∥vn-yn∥2+2λ∥Bvn-Bp∥∥vn-yn∥≤∥xn-p∥2-∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥×∥BkΔnk-1xn-Bkp∥-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥-∥vn-yn∥2+2λ∥Bvn-Bp∥∥vn-yn∥.
Substituting (90) into (80), we have
(91)∥xn+1-p∥2≤(1-αnτ)2∥yn-p∥2+βn2∥xn-Wnyn∥2+2(1-αnτ)βn∥yn-p∥∥xn-Wnyn∥+2αn∥γf(xn)-μFp∥∥xn+1-p∥≤(1-αnτ)2{∥xn-p∥2-∥Δnk-1xn-Δnkxn∥2mmmmmm+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥mmmmmm-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥mmmmmm×∥AiΛni-1un-Aip∥-∥vn-yn∥2mmmmmm+2λ∥Bvn-Bp∥∥vn-yn∥∥Δnk-1xn-Δnkxn∥∥Δnk-1xn-Δnkxn∥2}+βn2∥xn-Wnyn∥2+2(1-αnτ)βn∥yn-p∥∥xn-Wnyn∥+2αn∥γf(xn)-μFp∥∥xn+1-p∥≤∥xn-p∥2-(1-αnτ)2∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥-(1-αnτ)2∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥×∥AiΛni-1un-Aip∥-(1-αnτ)2∥vn-yn∥2+2λ∥Bvn-Bp∥∥vn-yn∥+βn2∥xn-Wnyn∥2+2(1-αnτ)βn∥yn-p∥∥xn-Wnyn∥+2αn∥γf(xn)-μFp∥∥xn+1-p∥;
that is,
(92)(1-αnτ)2×[∥Δnk-1xn-Δnkxn∥2+∥Λni-1un-Λniun∥2+∥vn-yn∥2]≤∥xn-p∥2-∥xn+1-p∥2+2rk,n∥Δnk-1xn-Δnkxn∥×∥BkΔnk-1xn-Bkp∥+2λi,n∥Λni-1un-Λniun∥×∥AiΛni-1un-Aip∥+2λ∥Bvn-Bp∥∥vn-yn∥+βn2∥xn-Wnyn∥2+2(1-αnτ)βn∥yn-p∥∥xn-Wnyn∥+2αn∥γf(xn)-μFp∥∥xn+1-p∥≤(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+2λ∥Bvn-Bp∥∥vn-yn∥+βn2∥xn-Wnyn∥2+2(1-αnτ)βn∥yn-p∥∥xn-Wnyn∥+2αn∥γf(xn)-μFp∥∥xn+1-p∥.
So, from αn→0, (73), (77), and (83) we immediately get
(93)lim⁡n→∞∥vn-yn∥=0,lim⁡n→∞∥Λni-1un-Λniun∥,lim⁡n→∞∥Δnk-1xn-Δnkxn∥=0,
for all i∈{1,2,…,N} and k∈{1,2,…,M}. Note that
(94)∥xn-un∥=∥Δn0xn-ΔnMxn∥≤∥Δn0xn-Δn1xn∥+∥Δn1xn-Δn2xn∥+⋯+∥ΔnM-1xn-ΔnMxn∥,(95)∥un-vn∥=∥Λn0un-ΛnNun∥≤∥Λn0un-Λn1un∥+∥Λn1un-Λn2un∥+⋯+∥ΛnN-1un-ΛnNun∥.
Thus, from (93) we have
(96)lim⁡n→∞∥xn-un∥=0,lim⁡n→∞∥un-vn∥=0.
It is easy to see that as n→∞(97)∥xn-yn∥≤∥xn-un∥+∥un-vn∥+∥vn-yn∥⟶0.
Also, observe that
(98)∥Wnyn-yn∥≤∥Wnyn-xn∥+∥xn-un∥+∥un-vn∥+∥vn-yn∥.
Hence, we have
(99)lim⁡n→∞∥Wnyn-yn∥=0.
Since
(100)∥Wyn-yn∥≤∥Wyn-Wnyn∥+∥Wnyn-yn∥,
it follows from Remark 16 that
(101)lim⁡n→∞∥Wyn-yn∥=0.
This, together with ∥xn-yn∥→0, implies that
(102)lim⁡n→∞∥xn-Wxn∥=0.
Step  5. Let us show that limsup⁡n→∞〈(γf-μF)x*,xn-x*〉≤0 where x*=PΩ(I-μF+γf)x*.
Indeed, as previously noted, it is known thatPΩ(I-μF+γf) is contractive and so PΩ(I-μF+γf) has a unique fixed point, denoted by x*∈C. This implies that x*=PΩ(I-μF+γf)x*.
First, we show thatωw(xn)⊂Ω. As a matter of fact, we note that there exists a subsequence {xnj} of {xn} such that
(103)lim⁡sup⁡n→∞〈(γf-μF)x*,xn-x*〉=lim⁡j→∞〈(γf-μF)x*,xnj-x*〉.
Since {xnj} is bounded, there exists a subsequence {xnji} of {xnj} which converges weakly to w. Without loss of generality, we may assume that xnj⇀w. Note that ∥Wxn-xn∥→0. Then, by the demiclosedness principle for nonexpansive mappings, we obtain w∈Fix⁡(W)=∩n=1∞Fix⁡(Tn). Furthermore, from (93) and (96), we have that Δnikxni⇀w, Λnimuni⇀w, uni⇀w, vni⇀w, and yni⇀w, where k∈{1,2,…,M} and m∈{1,2,…,N}.
Now we prove thatw∈∩m=1NVI⁡(C,Am). Let
(104)Tmv={Amv+NCv,v∈C,∅,v∉C,
where m∈{1,2,…,N}. Let (v,u)∈G(Tm). Since u-Amv∈NCv and Λnmun∈C, we have
(105)〈v-Λnmun,u-Amv〉≥0.
On the other hand, from Λnmun=PC(I-λm,nAm)Λnm-1un and v∈C, we have
(106)〈v-Λnmun,Λnmun-(Λnm-1un-λm,nAmΛnm-1un)〉≥0,
and hence
(107)〈v-Λnmun,Λnmun-Λnm-1unλm,n+AmΛnm-1un〉≥0.
Therefore, we have
(108)〈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 (93) and since Am is Lipschitzian, we obtain that lim⁡n→∞∥AmΛnmun-AmΛnm-1un∥=0. From Λnimuni⇀w, {λm,n}⊂[am,bm]⊂(0,2ηm), ∀m∈{1,2,…,N}, and (93), we have
(109)〈v-w,u〉≥0.
Since Tm is maximal monotone, we have w∈Tm-10 and hence w∈VI⁡(C,Am), m=1,2,…,N, which implies that w∈∩m=1NVI⁡(C,Am).
Next we prove thatw∈∩k=1MGMEP⁡(Θk,φk,Bk). Since Δnkxn=Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn, n≥1, k∈{1,2,…,M}, we have
(110)Θk(Δnkxn,y)+φk(y)-φk(Δnkxn)+〈BkΔnk-1xn,y-Δnkxn〉+1rk,n〈y-Δnkxn,Δnkxn-Δnk-1xn〉≥0.
By (A2), we have
(111)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
(112)〈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 (93) and the fact that Bk is Lipschitzian, we have ∥BkΔnkxn-BkΔnk-1xn∥→0 as n→∞. In addition, by the monotonicity of Bk, we obtain 〈zt-Δnkxn,Bkzt-BkΔnkxn〉≥0. Then, by (A4) and (112) we obtain
(113)〈zt-w,Bkzt〉≥φk(w)-φk(zt)+Θk(zt,w).
Utilizing (A1), (A4), and (113), we obtain
(114)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
(115)0≤Θk(zt,y)+φk(y)-φk(zt)+(1-t)〈y-w,Bkzt〉.
Letting t→0, we have, for each y∈C,
(116)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).
Further, we prove thatw∈I(B,R). In fact, since B is β-inverse strongly monotone, B is monotone and Lipschitzian. It follows from Lemma 9 that R+B is maximal monotone. Let (v,g)∈G(R+B); that is, g-Bv∈Rv. Again, since yni=JR,λ(vni-λBvni), we have vni-λBvni∈(I+λR)yni; that is, (1/λ)(vni-yni-λBvni)∈Ryni. By virtue of the monotonicity of R, we have
(117)〈v-yni,g-Bv-1λ(vni-yni-λBvni)〉≥0,
and so
(118)〈v-yni,g〉≥〈v-yni,Bv+1λ(vni-yni-λBvni)〉=〈v-yni,Bv-Byni+Byni-Bvni+1λ(vni-yni)〉≤〈v-yni,Byni-Bvni〉+〈v-yni,1λ(vni-yni)〉.
Since ∥vn-yn∥→0, ∥Bvn-Byn∥→0, and yni⇀w, we have
(119)lim⁡ni→∞〈v-yni,g〉=〈v-w,g〉≥0.
It follows from the maximal monotonicity of B+R that 0∈(R+B)w; that is, w∈I(B,R). Therefore, w∈Ω. This shows that ωw({xn})⊂Ω. Consequently, it follows from (103) that
(120)lim⁡sup⁡n→∞〈(γf-μF)x*,xn-x*〉=lim⁡j→∞〈(γf-μF)x*,xnj-x*〉=〈(γf-μF)x*,w-x*〉≤0.
Step  6. Let us show that xn→x* as n→∞.
Indeed, from (49) and Lemma 20 it follows that
(121)∥xn+1-x*∥2=∥αn(γf(xn)-μFx*)+βn(xn-x*)ii+((1-βn)I-αnμF)Wnyn-((1-βn)I-αnμF)Wnx*∥2≤∥βn(xn-x*)+((1-βn)I-αnμF)Wnynm-((1-βn)I-αnμF)Wnx*∥2+2αn〈γf(xn)-μFx*,xn+1-x*〉≤[∥((1-βn)I-αnμF)Wnyn-((1-βn)I-αnμF)Wnx*∥m+βn∥xn-x*∥]2+2αnγ〈f(xn)-f(x*),xn+1-x*〉+2αn〈(γf-μF)x*,xn+1-x*〉=[(1-βn)∥(I-αn1-βnμF)Wnyn-(I-αn1-βnμF)x*∥m+βn∥xn-x*∥αn1-βn]2+2αnγ〈f(xn)-f(x*),xn+1-x*〉+2αn〈(γf-μF)x*,xn+1-x*〉≤[(1-βn)(1-αnτ1-βn)∥yn-x*∥+βn∥xn-x*∥]2+2αnγρ∥xn-x*∥∥xn+1-x*∥+2αn〈(γf-μF)x*,xn+1-x*〉≤[(1-βn-αnτ)∥xn-x*∥+βn∥xn-x*∥]2+2αnγρ∥xn-x*∥∥xn+1-x*∥+2αn〈(γf-μF)x*,xn+1-x*〉≤(1-αnτ)2∥xn-x*∥2+αnγρ[∥xn-x*∥2+∥xn+1-x*∥2]+2αn〈(γf-μF)x*,xn+1-x*〉,
which immediately yields
(122)∥xn+1-x*∥2≤1-2αnτ+αn2τ2+αnγρ1-αnγρ∥xn-x*∥2+2αn1-αnγρ〈(γf-μF)x*,xn+1-x*〉=[1-2(τ-γρ)αn1-αnγρ]∥xn-x*∥2+(αnτ)21-αnγρ∥xn-x*∥2+2αn1-αnγρ〈(γf-μF)x*,xn+1-x*〉≤[1-2(τ-γρ)αn1-αnγρ]∥xn-x*∥2+2(τ-γρ)αn1-αnγρ×{αnτ2M~22(τ-γρ)+1τ-γρ〈(γf-μF)x*,xn+1-x*〉}=(1-δn)∥xn-x*∥2+δnσn,
where
(123)M~2=sup⁡{∥xn-x*∥2:n≥1},δn=2(τ-γρ)αn1-αnγρ,σn=αnτ2M~22(τ-γρ)+1τ-γρ〈(γf-μF)x*,xn+1-x*〉.
It is easy to see that ∑n=1∞‍δn=∞ and lim⁡sup⁡n→∞σn≤0. Hence, by Lemma 17 we conclude that the sequence {xn} converges strongly to x*. This completes the proof.From Theorem22, we can readily derive the following.Corollary 23.
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:H→H and Ai:C→H be ζ-inverse strongly monotone and ηi-inverse strongly monotone, respectively, where i=1,2. Let F:C→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0 and let f:H→H be a ρ-Lipschitzian mapping with constant ρ≥0. Let R:C→2H be a maximal monotone mapping and let the mapping B:C→H be β-inverse strongly monotone. Let 0<λ<2β, 0<μ<2η/κ2, and 0≤γρ<τ, where τ=1-1-μ(2η-μκ2). Let {λn}n=1∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and let {Tn}n=1∞ be an infinite family of nonexpansive self-mappings on C such that Ω:=GMEP⁡(Θ,φ,ℬ)∩VI⁡(C,A1)∩VI⁡(C,A2)∩∩n=1∞Fix⁡(Tn)∩I(B,R)≠∅. For arbitrarily given x1∈H, let the sequence {xn} be generated by
(124)Θ(un,y)+φ(y)-φ(un)+〈ℬxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,vn=PC(I-λ2,nA2)PC(I-λ1,nA1)un,xn+1=αnγf(xn)+βnxn+[(1-βn)I-αnμF]×WnJR,λ(vn-λBvn),∀n≥1,
where {αn},{βn} are two real sequences in [0,1] and Wn is the W-mapping defined by (14). Assume that the following conditions are satisfied:(i)
lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞;(ii)
0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1;(iii)
{λi,n}⊂[ai,bi]⊂(0,2ηi) and lim⁡n→∞|λi,n+1-λi,n|=0 for i=1,2;(iv)
{rn}⊂[e,f]⊂(0,2ζ) and lim⁡n→∞|rn+1-rn|=0. Assume that either (B1) or (B2) holds. Then the sequence {xn} converges strongly to x*∈Ω, where x*=PΩ(I-μF+γf)x* is a unique solution of the VIP:
(125)〈(μF-γf)x*,x*-y〉≤0,∀y∈Ω.Corollary 24.
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:H→H and 𝒜:C→H be ζ-inverse strongly monotone and ξ-inverse strongly monotone, respectively. Let F:C→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0 and let f:H→H be a ρ-Lipschitzian mapping with constant ρ≥0. Let R:C→2H be a maximal monotone mapping and let the mapping B:C→H be β-inverse strongly monotone. Let 0<λ<2β, 0<μ<2η/κ2, and 0≤γρ<τ, where τ=1-1-μ(2η-μκ2). Let {λn}n=1∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and let {Tn}n=1∞ be an infinite family of nonexpansive self-mappings on C such that Ω:=GMEP⁡(Θ,φ,ℬ)∩VI⁡(C,𝒜)∩∩n=1∞Fix⁡(Tn)∩I(B,R)≠∅. For arbitrarily given x1∈H, let the sequence {xn} be generated by
(126)Θ(un,y)+φ(y)-φ(un)+〈ℬxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,vn=PC(I-ρn𝒜)un,xn+1=αnγf(xn)+βnxn+[(1-βn)I-αnμF]×WnJR,λ(vn-λBvn),∀n≥1,
where {αn}, {βn} are two real sequences in [0,1] and Wn is the W-mapping defined by (14). Assume that the following conditions are satisfied:(i)
lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞;(ii)
0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1;(iii)
{ρn}⊂[a,b]⊂(0,2ξ) and lim⁡n→∞|ρn+1-ρn|=0;(iv)
{rn}⊂[e,f]⊂(0,2ζ) and lim⁡n→∞|rn+1-rn|=0. Assume that either (B1) or (B2) holds. Then the sequence {xn} converges strongly to x*∈Ω, where x*=PΩ(I-μF+γf)x* is a unique solution of the VIP:
(127)〈(μF-γf)x*,x*-y〉≤0,∀y∈Ω.Corollary 25.
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 𝒜:C→H be ξ-inverse strongly monotone and f:H→H a ρ-contractive mapping with constant ρ∈[0,1). Let R:C→2H be a maximal monotone mapping and let the mapping B:C→H be β-inverse strongly monotone. Let 0<λ<2β. Let {λn}n=1∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and let {Tn}n=1∞ be an infinite family of nonexpansive self-mappings on C such that Ω:=MEP⁡(Θ,φ)∩VI⁡(C,𝒜)∩∩n=1∞Fix⁡(Tn)∩I(B,R)≠∅. For arbitrarily given x1∈H, let the sequence {xn} be generated by
(128)Θ(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,mmmmmmmmmmmmmmmmmmmmmm∀y∈C,vn=PC(I-ρn𝒜)un,xn+1=αnf(xn)+βnxn+(1-αn-βn)WnJR,λ(vn-λBvn),mmmmmmmmmmmmmmmmmmmmmmmmmm∀n≥1,
where {αn}, {βn} are two real sequences in [0,1] and Wn is the W-mapping defined by (14). Assume that the following conditions are satisfied:(i)
lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞;(ii)
0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1;(iii)
{ρn}⊂[a,b]⊂(0,2ξ) and lim⁡n→∞|ρn+1-ρn|=0;(iv)
{rn}⊂[e,f]⊂(0,∞) and lim⁡n→∞|rn+1-rn|=0. Assume that either (B1) or (B2) holds. Then the sequence {xn} converges strongly to x*∈Ω, where x*=PΩf(x*) is a unique solution of the VIP:
(129)〈(I-f)x*,x*-y〉≤0,∀y∈Ω.Proof.
In Corollary24, put ℬ=0, F=(1/2)I, μ=2, and γ=1. Then from Remark 16 (ii), we get τ=1. Moreover, for {rn}⊂[e,f]⊂(0,∞), we can choose a positive constant ζ>0 such that {rn}⊂[e,f]⊂(0,2ζ). It is easy to see that ℬ is ζ-inverse strongly monotone. In addition, for the contraction f:H→H, we have 0≤γρ<τ. Hence, all the conditions of Corollary 24 are satisfied. Thus, in terms of Corollary 24, we obtain the desired result.Corollary 26.
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 𝒜:C→H be ξ-inverse strongly monotone. Let F:C→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0 and let f:H→H be a ρ-Lipschitzian mapping with constant ρ≥0. Let R:C→2H be a maximal monotone mapping and let the mapping B:C→H be β-inverse strongly monotone. Let 0<λ<2β, 0<μ<2η/κ2, and 0≤γρ<τ, where τ=1-1-μ(2η-μκ2). Let {λn}n=1∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and let {Tn}n=1∞ be an infinite family of nonexpansive self-mappings on C such that Ω:=MEP⁡(Θ,φ)∩VI⁡(C,𝒜)∩∩n=1∞Fix⁡(Tn)∩I(B,R)≠∅. For arbitrarily given x1∈H, let the sequence {xn} be generated by
(130)Θ(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,mmmmmmmmmmmmmmmmmmmmmm∀y∈C,vn=PC(I-ρn𝒜)un,xn+1=αnγf(xn)+βnxn+[(1-βn)I-αnμF]×WnJR,λ(vn-λBvn),∀n≥1,
where {αn}, {βn} are two real sequences in [0,1] and Wn is the W-mapping defined by (14). Assume that the following conditions are satisfied:(i)
lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞;(ii)
0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1;(iii)
{ρn}⊂[a,b]⊂(0,2ξ) and lim⁡n→∞|ρn+1-ρn|=0;(iv)
{rn}⊂[e,f]⊂(0,∞) and lim⁡n→∞|rn+1-rn|=0. Assume that either (B1) or (B2) holds. Then the sequence {xn} converges strongly to x*∈Ω, where x*=PΩ(I-μF+γf)x* is a unique solution of the VIP:
(131)〈(μF-γf)x*,x*-y〉≤0,∀y∈Ω.Corollary 27.
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:H→H and 𝒜:C→H be ζ-inverse strongly monotone and ξ-inverse strongly monotone, respectively. Let F:C→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0 and let f:H→H be a ρ-Lipschitzian mapping with constant ρ≥0. Let R:C→2H be a maximal monotone mapping and let the mapping B:C→H be β-inverse strongly monotone. Let 0<λ<2β, 0<μ<2η/κ2, and 0≤γρ<τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=GMEP⁡(Θ,φ,ℬ)∩VI⁡(C,𝒜)∩I(B,R)≠∅. For arbitrarily given x1∈H, let the sequence {xn} be generated by
(132)Θ(un,y)+φ(y)-φ(un)+〈ℬxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,vn=PC(I-ρn𝒜)un,xn+1=αnγf(xn)+βnxn+[(1-βn)I-αnμF]×JR,λ(vn-λBvn),∀n≥1,
where {αn}, {βn} are two real sequences in [0,1]. Assume that the following conditions are satisfied:(i)
lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞;(ii)
0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1;(iii)
{ρn}⊂[a,b]⊂(0,2ξ) and lim⁡n→∞|ρn+1-ρn|=0;(iv)
{rn}⊂[e,f]⊂(0,2ζ) and lim⁡n→∞|rn+1-rn|=0. Assume that either (B1) or (B2) holds. Then the sequence {xn} converges strongly to x*∈Ω, where x*=PΩ(I-μF+γf)x* is a unique solution of the VIP:
(133)〈(μF-γf)x*,x*-y〉≤0,∀y∈Ω.Corollary 28.
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). Let ℬ:H→H and 𝒜:C→H be ζ-inverse strongly monotone and ξ-inverse strongly monotone, respectively. Let F:C→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0 and let f:H→H be a ρ-Lipschitzian mapping with constant ρ≥0. Let R:C→2H be a maximal monotone mapping and let the mapping B:C→H be β-inverse strongly monotone. Let 0<λ<2β, 0<μ<2η/κ2, and 0≤γρ<τ, where τ=1-1-μ(2η-μκ2). Let {λn}n=1∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and let {Tn}n=1∞ be an infinite family of nonexpansive self-mappings on C such that Ω:=GEP⁡(Θ,ℬ)∩VI⁡(C,𝒜)∩∩n=1∞Fix⁡(Tn)∩I(B,R)≠∅. For arbitrarily given x1∈H, let the sequence {xn} be generated by
(134)Θ(un,y)+〈ℬxn,y-un〉+1rn〈y-un,un-xn〉≥0,mmmmmmmmmmmmmmmmmmmmmm∀y∈C,vn=PC(I-ρn𝒜)un,xn+1=αnγf(xn)+βnxn+[(1-βn)I-αnμF]×WnJR,λ(vn-λBvn),∀n≥1,
where {αn}, {βn} are two real sequences in [0,1] and Wn is the W-mapping defined by (14). Assume that the following conditions are satisfied:(i)
lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞;(ii)
0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1;(iii)
{ρn}⊂[a,b]⊂(0,2ξ) and lim⁡n→∞|ρn+1-ρn|=0;(iv)
{rn}⊂[e,f]⊂(0,2ζ) and lim⁡n→∞|rn+1-rn|=0. Assume that either (B1) or (B2) holds. Then the sequence {xn} converges strongly to x*∈Ω, where x*=PΩ(I-μF+γf)x* is a unique solution of the VIP:
(135)〈(μF-γf)x*,x*-y〉≤0,∀y∈Ω.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). Let ℬ:H→H and 𝒜:C→H be ζ-inverse strongly monotone and ξ-inverse strongly monotone, respectively. Let F:C→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0 and let f:H→H be a ρ-Lipschitzian mapping with constant ρ≥0. Let R:C→2H be a maximal monotone mapping and let the mapping B:C→H be β-inverse strongly monotone. Let 0<λ<2β, 0<μ<2η/κ2, and 0≤γρ<τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=GEP⁡(Θ,ℬ)∩VI⁡(C,𝒜)∩I(B,R)≠∅. For arbitrarily given x1∈H, let the sequence {xn} be generated by
(136)Θ(un,y)+〈ℬxn,y-un〉+1rn〈y-un,un-xn〉≥0,mmmmmmmmmmmmmmmmmmmmmm∀y∈C,vn=PC(I-ρn𝒜)un,xn+1=αnγf(xn)+βnxn+[(1-βn)I-αnμF]×JR,λ(vn-λBvn),∀n≥1,
where {αn}, {βn} are two real sequences in [0,1]. Assume that the following conditions are satisfied:(i)
lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞;(ii)
0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1;(iii)
{ρn}⊂[a,b]⊂(0,2ξ) and lim⁡n→∞|ρn+1-ρn|=0;(iv)
{rn}⊂[e,f]⊂(0,2ζ) and lim⁡n→∞|rn+1-rn|=0. Assume that either (B1) or (B2) holds. Then the sequence {xn} converges strongly to x*∈Ω, where x*=PΩ(I-μF+γf)x* is a unique solution of the VIP:
(137)〈(μF-γf)x*,x*-y〉≤0,∀y∈Ω.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). Let ℬ:H→H and 𝒜:C→H be ζ-inverse strongly monotone and ξ-inverse strongly monotone, respectively. Let f:H→H be a ρ-contractive mapping with constant ρ∈[0,1). Let R:C→2H be a maximal monotone mapping and let the mapping B:C→H be β-inverse strongly monotone. Let 0<λ<2β. Assume that Ω:=GEP⁡(Θ,ℬ)∩VI⁡(C,𝒜)∩I(B,R)≠∅. For arbitrarily given x1∈H, let the sequence {xn} be generated by
(138)Θ(un,y)+〈ℬxn,y-un〉+1rn〈y-un,un-xn〉≥0,mmmmmmmmmmmmmmmmmmmmmm∀y∈C,vn=PC(I-ρn𝒜)un,xn+1=αnf(xn)+βnxn+(1-αn-βn)JR,λ(vn-λBvn),mmmmmmmmmmmmmmmmiimmmmmmmm∀n≥1,
where {αn}, {βn} are two real sequences in [0,1]. Assume that the following conditions are satisfied:(i)
lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞;(ii)
0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1;(iii)
{ρn}⊂[a,b]⊂(0,2ξ) and lim⁡n→∞|ρn+1-ρn|=0;(iv)
{rn}⊂[e,f]⊂(0,2ζ) and lim⁡n→∞|rn+1-rn|=0. Assume that either (B1) or (B2) holds. Then the sequence {xn} converges strongly to x*∈Ω, where x*=PΩf(x*) is a unique solution of the VIP:
(139)〈(I-f)x*,x*-y〉≤0,∀y∈Ω.Remark 31.
Theorem22 extends, improves, and supplements [4, Theorem 3.2] in the following aspects.(i)
The problem of finding a pointp∈∩k=1MGMEP⁡(Θk,φk,Bk)∩∩i=1NVI⁡(C,Ai)∩∩n=1∞Fix⁡(Tn)∩I(B,R) in Theorem 22 is very different from the problem of finding a point p∈GMEP⁡(Θ,φ,A)∩∩n=1∞Fix⁡(Tn)∩I(B,R) in [4, Theorem 3.2] (i.e., Theorem 3 in this paper). There is no doubt that the problem of finding a point p∈∩k=1MGMEP⁡(Θk,φk,Bk)∩∩i=1NVI⁡(C,Ai)∩∩n=1∞Fix⁡(Tn)∩I(B,R) is more general and more subtle than the problem of finding a point p∈GMEP⁡(Θ,φ,A)∩∩n=1∞Fix⁡(Tn)∩I(B,R) in [4, Theorem 3.2].(ii)
If, in Corollary24, C=H, 𝒜=0, rn=r>0 (∀n≥1), μF=V is a strongly positive bounded linear operator, and f is a contraction, then Corollary 24 reduces essentially to [4, Theorem 3.2]. This shows that Theorem 22 includes [4, Theorem 3.2] as a special case.(iii)
The iterative scheme in [4, Algorithm 3.1] is extended to develop the iterative scheme in Theorem 22 by virtue of Korpelevič’s extragradient method and hybrid steepest-descent method [38]. The iterative scheme in Theorem 22 is more advantageous and more flexible than the iterative scheme in [4, Algorithm 3.1] because it involves solving four problems: a finite family of GMEPs, a finite family of VIPs, the variational inclusion (11), and the fixed point problem of an infinite family of nonexpansive self-mappings.(iv)
The iterative scheme in Theorem22 is very different from the iterative scheme in [4, Algorithm 3.1] because the iterative scheme in Theorem 22 involves Korpelevič’s extragradient method and hybrid steepest-descent method.(v)
The proof of Theorem22 combines the proof for viscosity approximation method in [4, Theorem 3.2], the proof for Korpelevič’s extragradient method in [8, Theorem 3.1], and the proof for hybrid steepest-descent method in [44, Theorem 3.1].



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*Source: 208717-2014-02-27.xml*