# Iterative Methods of Weak and Strong Convergence Theorems for the Split Common Solution of the Feasibility Problems, Generalized Equilibrium Problems, and Fixed Point Problems

Authors: Shahram Rezapour; Yuanheng Wang; Seyyed Hasan Zakeri
Journal: Journal of Mathematics
 (2020)

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## Abstract

The purpose of this paper is to introduce the extragradient methods for solving split feasibility problems, generalized equilibrium problems, and fixed point problems involved in nonexpansive mappings and pseudocontractive mappings. We establish the results of weak and strong convergence under appropriate conditions. As applications of our three main theorems, when the mappings and their domains take different types of cases, we can obtain nine iterative approximation theorems and corollas on fixed points, variational inequality solutions, and equilibrium points.


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## Body

## 1. Introduction

LetH1 and H2 be two real Hilbert spaces, and let C and Q be two nonempty closed and convex subsets of H1 and H2, respectively. Let A:H1⟶H2 be a bounded linear operator with its adjoint A∗. The split feasibility problem SFP is to find a point x such that(1)x∈C,Ax∈Q.We denote the solution set of the split feasibility problemSFP by(2)Ω=x∈C:Ax∈Q=C∩A−1Q.Problem (1) was first introduced by Censor and Elfving [1] in the finite-dimensional spaces and further has been studied by many researchers (see, for example, [2–6]) and the references therein. To solve the SFP, Byrne [2, 7] first introduced the so-called CQ algorithm as follows:(3)x0∈H1,xn+1=PCI−λA∗I−PQAxn,∀n≥0,where 0<λ<2/ρA∗A, PC denotes the projection onto C, and ρA∗A is the spectral radius of the self-adjoint operator A∗A. Many authors continue to study the CQ algorithm in its various forms (see, for example, [8–14]). The CQ algorithm can be viewed from two different but equivalent ways: optimization and fixed point [6]. From the view of optimization point, x∗∈Ω in (2) if and only if x∗ is a solution of the following minimization problem with zero optimal value minx∈Cfx≔1/2Ax−PQAx2, where f is a differentiable convex function and has a Lipschitz gradient given by ∇fx=A∗I−PQA, with Lipschitz constant L=ρA∗A. Thus, x∗ solves the SFP if and only if x∗ solves the variational inequality problem of finding x∗∈C such that ∇fx∗,y−x∗≥0 for all y∈C.Xu [6] considered the following Tikhonov regularized problem:(4)minx∈Cfαx≔12Ax−PQAx2+12αx2,where α>0 is the regularization parameter. We observe that the gradient(5)∇fαx=∇fx+αI=A∗I−PQA+αI,is α+A2-Lipschitz continuous and α-strongly monotone. The fixed point approach method to solve the SFP is based on the following observations. Let λ>0, and assume that x∗∈Ω. Then, Ax∗∈Q, which implies that I−PQAx∗=0, and thus, λA∗I−PQAx∗=0. Hence, we have the fixed point equation I−λA∗I−PQAx∗=x∗. Requiring that x∗∈C, we consider the fixed point equation(6)PCI−λ∇fx∗=PCI−λA∗I−PQAx∗=x∗.In [6], it is proved that the solutions of fixed point equation (6) are precisely the solutions of the SFP.LetA:C⟶H be a nonlinear mapping and F be a bifunction from C×C to ℝ, where ℝ is the set of real numbers. The generalized equilibrium problem is to find x∗∈C such that Fx∗,y+Ax∗,y−x∗≥0,∀y∈C. The set of solutions is denoted by GEPF,A. If A=0, then GEPF,A is denoted by EPF. If Fx,y=0 for all x,y∈C, then GEPF,A is denoted by VIC,A=x∗∈C:Ax∗,y−x∗≥0,∀y∈C. This is the set of solutions of the variational inequality for A (see, for example, [15–21]). If C=H, then VIH,A=A−10 where A−10=x∈H:Ax=0.In 2008, Takahashi and Takahashi [15] have suggested the following iterative method. Let xn be a sequence generated by(7)x1∈C,Fyn,y+Axn,y−yn+1rny−yn,yn−xn≥0,∀y∈C,xn+1=anxn+1−anTβnu+1−βnyn,∀n≥1.Under some appropriate conditions, they proved that the sequencexn converges strongly to a point PFT∩GEPF,Au.Motivated and inspired by the above works, we will investigate the weak and strong convergence methods for solving the split feasibility problems, generalized equilibrium problems, and fixed point problems involved in nonexpansive mappings and pseudocontractive mappings. As applications of our three main theorems, when the mappings and their domains take different types of cases, we can obtain nine iterative approximation theorems and corollaries on fixed points, variational inequality solutions, and equilibrium points. So, our results in this paper generalize and improve upon the corresponding modern results of many other authors.


## 2. Preliminaries

LetH be a real Hilbert space with the inner product ⋅,⋅ and norm ⋅ and C be a nonempty, closed, and convex subset of H. Recall that a mapping A:C⟶H is said to be monotone if Au−Av,u−v≥0 for all u,v∈C [18, 19]. A mapping A is said to be α-strongly monotone whenever there exists a positive real number α such that Au−Av,u−v≥αu−v2 for all u,v∈C. A mapping A is said to be α-inverse strongly monotone if there exists a positive real number α such that Au−Av,u−v≥αAu−Av2 for all u,v∈C. Recall that the classical variational inequality problem, which we denote by VIC,A, is to find x∈C such that Ax,y−x≥0, for all y∈C [16, 17]. It is well known that, for any x∈H, there exists a unique nearest point in C, denoted by PCx, such that x−PCx=infy∈Cx−y=:dx,C. It is well known that PC is a nonexpansive and monotone mapping from H onto C and satisfy the following:(1)
x−PCx,z−PCx≤0 for all x∈H,z∈C(2)
x−z2≥x−PCx2+z−PCx2 for all x∈H,z∈C(3)
The relationPCx−PCz,x−z≥PCx−PCz2 holds for all z,x∈HLetA be a monotone mapping of C into H. In the context of the variational inequality problem, it is easy to see from (2) that(8)p∈VIC,A⇔p=PCp−λAp,∀λ>0.For solving the equilibrium problem, we assume thatF satisfies the following conditions:(i)
A1Fx,x=0 for all x∈C(ii)
A2F is monotone, that is, Fx,y+Fy,x≤0 for all x,y∈C(iii)
A3 for each x,y,z∈C, limt⟶0Ftz+1−tx,y≤Fx,y(iv)
A4 for each x∈C, the function y⟶Fx,y is convex and lower semicontinuousIfFx,y=Ax,y−x for every x,y∈C, we see that the equilibrium problem is reduced to the variational inequality problem.Lemma 1 (see [22]).
LetC be a nonempty, closed, and convex subset of H, and let F be a bifunction from C×C to ℝ satisfying A1−A4. For r>0 and x∈H, consider the mapping Tr:H⟶C defined by(9)Trx=z∈C:Fz,y+1ry−z,z−x≥0,∀y∈C.Then,Trx≠∅ for all x∈H, Tr is single-valued, EPF is closed and convex, FTr=EPF, and Tr is firmly nonexpansive, that is, Trx−Try2≤Trx−Try,x−y for all x,y∈H.Lemma 2 (see [23]).
LetC be a nonempty, closed, and convex subset of H, F be a bifunction from C×C to ℝ satisfying A1−A4, ad AF be a multivalue mapping from H into itself defined by AFx=z∈C:Fz,y≤y−x,z,∀y∈C whenever x∈C and AFx=∅ otherwise. Then, AF is a maximal monotone operator with the domain Trx=I+rAF−1x, for all x∈H and r>0.Definition 1.
LetT:H⟶H be a nonlinear operator.(1)
T is said to be L-Lipschitz whenever there exists L≥0 such that Tu−Tv≤Lu−v,∀u,v∈H. If L=1, we call T is nonexpansive, and T is said to be a contraction if L<1.(2)
T is said to be firmly nonexpansive if 2T−I is nonexpansive and I is the identity mapping, or equivalently, Tu−Tv,u−v≥Tu−Tv2,∀u,v∈H. Alternatively, T is firmly nonexpansive if and only if T can be expressed as T=1/2I+S, where S:H⟶H is nonexpansive.(3)
T is said to be α-averaged nonexpansive mapping, if there exists a nonexpansive mapping S, such that T=1−αI+αS, where α∈0,1. Thus, firmly nonexpansive mappings are 1/2-averaged mapping.(4)
T is said to be pseudocontractive if and only if Tu−Tv2≤u−v2+I−Tu−I−Tv2,∀u,v∈H.(5)
T is said to be k-strictly pseudocontractive if and only if there exists 0≤k<1, such that(10)Tu−Tv2≤u−v2+kI−Tu−I−Tv2,∀u,v∈H.Remark 1 (see [2]).
LetT:C⟶C be a given mapping:(i)
T is nonexpansive if and only if the complement I−T is 1/2-inverse strongly monotone.(ii)
IfT is α-inverse strongly monotone, then for γ>0,γT is α/γ-inverse strongly monotone.(iii)
T is averaged if and only if the complement I−T is α-inverse strongly monotone for some α>1/2. Indeed, for α∈0,1,T is α-averaged if and only if I−T is 1/2α-inverse strongly monotone.
We denote byFT the set of fixed points of T. Note that every α-inverse strongly monotone mapping T is Lipschitz and Tu−Tv≤1/αu−v. Every nonexpansive mapping is a k-strictly pseudocontractive mapping and every k-strictly pseudocontractive mapping is pseudocontractive. Assume that T:C⟶C is a strictly pseudocontractive. If A=I−T, we easily find that A is 1−k/2-inverse strongly monotone and FT=VIC,A. Note that T is pseudocontractive if and only if A=I−T is monotone, and FT=A−10=x∈H:Ax=0. There are a lot works associated with the fixed point algorithms for nonexpansive mappings and pseudocontractive mappings (see, for example, [24–28]).
A set-valued mappingT:H⟶2H is called monotone if for all x,y∈H,f∈Tx, and h∈Ty imply x−y,f−h≥0. A monotone mapping T:H⟶2H is maximal if the graph GT of T is not properly contained in the graph of any other monotone mappings. Also, a monotone mapping T:H⟶2H is maximal if and only if, for x,f∈H×H,x−y,f−h≥0 for every y,h∈GT implies f∈Tx. Let A:C⟶H be an inverse strongly monotone mapping and let NCu be the normal cone to C at u∈C, i.e., NCu=v∈H:u−w,v≥0,∀w∈C. Define(11)Tu≔Au+NCu,u∈C,∅,u∉C.
It is known thatT is maximal monotone and 0∈Tu if and only if u∈VIC,A [29, 30].Lemma 3 (see [8]).
LetC and Q be nonempty, closed, and convex subsets of real Hilbert spaces H1 and H2, respectively, and let A:H1⟶H2 be a bounded linear operator and f:H1⟶ℝ be a continuous differentiable function. If α>0 and λ∈0,1/A2, then(1)
∇fαx=∇fx+αI=A∗I−PQA+αI is 1/α+A2-inverse strongly monotone mapping(2)
I−λ∇fα is λα+A2/2-averaged(3)
PCI−λ∇fα is ζ-averaged, with ζ=2+λα+A2/4(4)
PCI−λ∇fα is nonexpansiveLemma 4 (see [31]).
LetH be a real Hilbert space, C be a closed convex subset of H, and T:C⟶C be a continuous pseudocontractive mapping. Then,(i)
FT is a closed convex subset of C(ii)
I−T is demiclosed at zero, i.e., if xn is a sequence in C such that xn⟶x and Txn−xn⟶0; as n⟶∞, then x=Tx.Lemma 5 (see [32]).
LetH be a real Hilbert space. Then, for all xj∈H and aj∈0,1, for j=1,2,3 such that a1+a2+a3=1, the following equality holds:(12)a1x1+a2x2+a3x32=a1x12+a2x22+a3x32−∑1≤i,j≤3aiajxi−xj2.Lemma 6 (see [33]).
LetC be a nonempty closed and convex subset of a real Hilbert space H and T:C⟶C be a nonexpansive mapping. Then, I−T is demiclosed at zero.Lemma 7 (see [34]).
Letxn and γn be sequences of nonnegative real numbers satisfying xn+1≤xn+γn. If ∑n=0∞γn converges, then limn⟶∞xn exists.Lemma 8 (see [35]).
LetC be a nonempty closed convex subset of a real Hilbert space H and let T:C⟶C be a k-strictly pseudocontraction with a fixed point. Define S:C⟶C by Sx=ax+1−aTx for each x∈C. Then, as a∈k,1,S is nonexpansive such that FS=FT.Lemma 9 (see [36]).
Letxn be a sequence of nonnegative real numbers satisfying xn+1≤1−βnxn+βnγn+αn, where βn⊂0,1 and γn is a sequence such that ∑n=0∞βn=∞, limsupn⟶∞γn≤0 or ∑n=0∞γnβn<∞, and ∑n=0∞αn<∞ where αn≥0. Then, limn⟶∞xn=0.Lemma 10 (see [37]).
Letxn, εn, and αn be the sequences in 0,∞ such that(13)xn+1≤xn+εnxn−xn−1+αn,∀n≥0,∑n=0∞αn<∞, and there exists a real number ε with 0≤εn≤ε<1 for all n≥0. Then, the following holds:(i)
∑n=0∞xn−xn−1+<∞, where t+=maxt,0(ii)
There existsx∗∈0,∞ such that limn⟶∞xn=x∗Lemma 11 (see [31]).
LetH be a real Hilbert space. Then, for any given x,y∈H, the following inequality holds: x+y2≤x2+2y,x+y.


## 3. Weak and Strong Convergence Results

Now, we are ready to state and prove some of our main results in this section.Theorem 1.
Assume thatC and Q are 2 nonempty, closed, and convex subsets of real Hilbert spaces H1 and H2, respectively. Let A:H1⟶H2 be a bounded linear operator,f:H1⟶ℝ be a continuous differentiable function, F be a bifunction from C×C to ℝ satisfying A1−A4, M be an α-inverse strongly monotone mapping from C into H1, S:C⟶C be a nonexpansive mapping, and T:C⟶C be a strictly pseudocontractive mapping with constant k such that Γ=FT∩FS∩Ω∩GEPF,M≠∅. Let xn, yn, zn, and vn be sequences generated by the following extragradient algorithm:(14)x0=x∈C,Fvn,y+Mxn,y−vn+1rny−vn,vn−xn≥0,∀y∈C,zn=PCI−λn∇fαnvn,yn=PCvn−λn∇fαnzn,xn+1=anx0+bnxn+cn1−βnSyn+βnTnzn,∀n≥0,where Tn=1−γnI+γnT and γn∈k,1. Suppose the following conditions are satisfied:(a)
∑n=0∞αn<∞, limn⟶∞an=0, ∑n=1∞an=∞(b)
βn⊂β1∗,β2∗ for some β1∗,β2∗∈0,1(c)
λn⊂e,d for some e,d∈0,1/A2(d)
0<an≤a′<1,0<b≤bn≤b′<1,0<c≤cn≤c′<1 and an+bn+cn=1,(e)
0<q1≤rn≤q2<2α
Then,xn converges strongly to the point u=PΓx0 provided limn⟶∞xn+1−xn=0.Proof.
For any fixedu∈Γ, we find that u=PCI−λ∇fu for λ∈0,1/A2 and Su=u. We see from Lemma 8 that Tn is nonexpansive and FTn=FT. It is observed that vn can be rewritten as vn=Trnxn−rnMxn,n≥0. From condition (e) and Lemma 1, we have(15)vn−u2=Trnxn−rnMxn−u2=Trnxn−rnMxn−Trnu−rnMu2≤xn−rnMxn−u−rnMu2=xn−u2+rnrn−2αMxn−Mu2≤xn−u2.
From (14), (15), and Lemma 3, it follows that(16)zn−u=PCI−λn∇fαnvn−PCI−λn∇fu≤PCI−λn∇fαnvn−PCI−λn∇fαnu+PCI−λn∇fαnu−PCI−λn∇fu≤vn−u+I−λn∇fαnu−I−λn∇fu≤vn−u+λnαnu≤xn−u+λnαnu.
By the property of metric projection, we have(17)yn−u2≤vn−λn∇fαnzn−u2−vn−λn∇fαnzn−yn2≤vn−u2−vn−yn2+2λn∇fαnzn,u−yn≤vn−u2−vn−yn2+2λn∇fαnzn−∇fαnu,u−zn+2λn∇fαnu,u−zn+2λn∇fαnzn,zn−yn≤vn−u2−vn−yn2+2λn∇fαnu,u−zn+2λn∇fαnzn,zn−yn=vn−u2−vn−yn2+2λnαnI+∇fu,u−zn+2λn∇fαnzn,zn−yn≤vn−u2−vn−yn2+2λnαnu,u−zn+2λn∇fαnzn,zn−yn=vn−u2−vn−zn2−2vn−zn,zn−yn−zn−yn2+2λnαnu,u−zn+∇fαnzn,zn−yn=vn−u2−vn−zn2+2vn−λn∇fαnzn−zn,yn−zn+2λnαnu,u−zn−zn−yn2.
Furthermore, by the property of metric projection, we have(18)vn−λn∇fαnzn−zn,yn−zn=vn−λn∇fαnvn−zn,yn−zn+λn∇fαnvn−λn∇fαnzn,yn−zn≤λn∇fαnvn−λn∇fαnzn,yn−zn≤λn∇fαnvn−∇fαnznyn−zn≤λnαn+A2vn−znyn−zn.
Hence, we have(19)yn−u2≤vn−u2−vn−zn2+2vn−λn∇fαnzn−zn,yn−zn−zn−yn2+2λnαnu,u−zn≤vn−u2−vn−zn2+2λnαn+A2vn−znyn−zn−zn−yn2+2λnαnuu−zn≤vn−u2−vn−zn2+λn2αn+A22vn−zn2+yn−zn2−zn−yn2+2λnαnuu−zn=vn−u2+λn2αn+A22−1vn−zn2+2λnαnuu−zn≤vn−u2+2λnαnuu−zn≤vn−u2+2λnαnuvn−u+λnαnu≤vn−u2+4λnαnuvn−u+4λn2αn2u2=vn−u+2λnαnu2.
So, from (15), we obtain(20)yn−u2≤xn−u+2λnαnu2.
We find from (14) and (16) and the last inequality that(21)xn+1−u=anx0+bnxn+cn1−βnSyn+βnTnzn−u≤anx0−u+bnxn−u+cn1−βnSyn−u+βnTnzn−u≤anx0−u+bnxn−u+cn1−βnyn−u+βnzn−u≤anx0−u+bnxn−u+cn1−βnxn−u+2λnαnu+cnβnxn−u+λnαnu≤anx0−u+1−anxn−u+2λnαnu≤maxx0−u,xn−u+2λnαnu⋮≤x0−u+2du∑i=0∞αi.
Consequently, from condition (a), we deduce thatxn is bounded and so there exist the sequences zn, vn, and yn. Put tn=1−βnSyn+βnTnzn for all n≥0. We find from (15), (16), (19), and Lemma 5 that(22)tn−u2=1−βnSyn+βnTnzn−u2≤1−βnSyn−u2+βnTnzn−u2−βn1−βnSyn−Tnzn2≤1−βnyn−u2+βnzn−u2−βn1−βnSyn−Tnzn2≤1−βnxn−u2+λn2αn+A22−1vn−zn2+rnrn−2αMxn−Mu2+2λnαnuzn−u+βnxn−u+λnαnu2−βn1−βnSyn−Tnzn2≤xn−u2+λn2αn+A22−1vn−zn2+rnrn−2αMxn−Mu2+2λnαnuzn−u+2λnαnuxn−u+λn2αn2u2−βn1−βnSyn−Tnzn2.
From (14) and the last inequality, we conclude that(23)xn+1−u2=anx0+bnxn+cntn−u2≤anx0−u2+bnxn−u2+cntn−u2−bncnxn−tn2≤anx0−u2+bnxn−u2−bncnxn−tn2+cnxn−u2+λn2αn+A22−1vn−zn2+rnrn−2αMxn−Mu2+2λnαnuzn−u+2λnαnuxn−u+λn2αn2u2−βn1−βnSyn−Tnzn2≤anx0−u2+1−anxn−u2−bncnxn−tn2+cnλn2αn+A22−1vn−zn2+2λnαnuzn−u+2λnαnuxn−u+rnrn−2αMxn−Mu2+λn2αn2u2−cnβn1−βnSyn−Tnzn2≤anx0−u2+xn−u2−bncnxn−tn2+cnλn2αn+A22−1vn−zn2+2λnαnuzn−u+2λnαnuxn−u+rnrn−2αMxn−Mu2+λn2αn2u2−cnβn1−βnSyn−Tnzn2.
This yields that(24)c1−d2αn+A22vn−zn2+cbxn−tn2+rn2α−rnMxn−Mu2+β11−β2cTnzn−Syn2≤cn1−λn2αn+A22vn−zn2+cnbnxn−tn2+rn2α−rnMxn−Mu2+cnβn1−βnTnzn−Syn2≤anx0−u2+xn−u2−xn+1−u2+2λnαnuzn−u+xn−u+λnαnu.
Sincelimn⟶∞xn+1−xn=0, we have(25)xn−u2−xn+1−u2≤xn−u−xn+1−uxn−u+xn+1−u≤xn+1−xnxn−u+xn+1−u⟶0,asn⟶∞.
From (97) and the condition (a)–(d), we also obtain(26)limn⟶∞Tnzn−Syn=limn⟶∞Mxn−Mu=limn⟶∞xn−tn=limn⟶∞vn−zn=0.
It is observe that(27)yn−zn=PCvn−λn∇fαnzn−PCvn−λn∇fαnvn≤vn−λn∇fαnzn−vn−λn∇fαnvn=λn∇fαnzn−∇fαnvn≤λnαn+A2zn−vn⟶0,asn⟶∞.
Using Lemma1 and (14), we have(28)vn−u2=Trnxn−rnMxn−Trnu−rnMu2≤xn−rnMxn−u−rnMu,vn−u=12xn−rnMxn−u−rnMu2+12vn−u2−12xn−rnMxn−u−rnMu−vn−u2≤12xn−u2+vn−u2−xn−vn−2rnMxn−Mu2=12xn−u2+vn−u2−xn−vn2+2rnxn−vn,Mxn−Mu−rn2Mxn−Mu2.
It follows that(29)vn−u2≤xn−u2−xn−vn2+2rnxn−vn,Mxn−Mu.
From (19) and (29), we find that(30)tn−u2=1−βnSyn+βnTnzn−u2≤1−βnSyn−u2+βnTnzn−u2≤1−βnyn−u2+βnzn−u2≤1−βnxn−u2+2λnαnuu−zn+βnxn−u2−xn−vn2+2rnxn−vn,Mxn−Mu≤xn−u2+2λnαnuu−zn+βn−xn−vn2+2rnxn−vn,Mxn−Mu.
From (14) and the last inequality, we conclude that(31)xn+1−u2=anx0+bnxn+cntn−u2≤anx0−u2+bnxn−u2+cntn−u2≤anx0−u2+bnxn−u2+cnxn−u2+2λnαnuu−zn+βn−xn−vn2+2rnxn−vn,Mxn−Mu≤anx0−u2+1−anxn−u2−cnβnxn−vn2+cn2λnαnuu−zn+2rnxn−vn,Mxn−Mu.
This yields that(32)cnβnxn−vn2≤anx0−u2+1−anxn−u2−xn+1−u2+cn2λnαnuu−zn+2rnxn−vn,Mxn−Mu.
It follows from condition (a) andlimn⟶∞xn−xn+1=limn⟶∞Mxn−Mu=0 that(33)limn⟶∞xn−vn=0.
Sincexn−zn≤xn−vn+vn−zn, Tnzn−zn≤Tnzn−xn+xn−zn, Tnzn−xn≤Tnzn−tn+tn−xn, Tnzn−tn=1−βnTnzn−Syn, we obtain Tnzn−tn⟶0 as n⟶∞. Note that 1−βn>0. This implies that(34)limn⟶∞Tzn−zn=0.
Also, fromyn−xn≤yn−zn+zn−xn, Syn−xn≤Syn−tn+tn−xn, Syn−tn=βnSyn−Tnzn, and Syn−yn≤Syn−xn+xn−yn, we get(35)limn⟶∞Syn−yn=0.
Since∇f=A∗I−PQA is Lipschitz continuous, we obtain limn⟶∞∇fzn−∇fyn=0.
Next, we show that(36)limsupn⟶∞x0−u,xn−u≤0,where u=PΓx0. To show it, choose a subsequence xnk of xn such that(37)limsupn⟶∞x0−u,xn−u=limk⟶∞x0−u,xnk−u.
Sincexnk is bounded, there exists a subsequence xnkj of xnk, converges weakly to x∗. Without loss of generality, we assume that xnk⟶x∗. Since xn−vn⟶0, xn−yn⟶0,xn−zn⟶0 as n⟶∞, we obtain that ynk⇀x∗,vnk⇀x∗,znk⇀x∗. Since ynk⊂C and C is closed and convex, we obtain x∗∈C. First, we show that x∗∈FT∩FS. Then, from (34), (35), Lemma 6, and Lemma 4, we have that x∗∈FT∩FS. We now show that x∗∈GEPF,M. By vn=Trnxn−rnMxn, we know that(38)Fvn,y+Mxn,y−vn+1rny−vn,vn−xn≥0,∀y∈C.
It follows fromA2 that(39)Mxn,y−vn+1rny−vn,vn−xn≥Fy,vn,∀y∈C.
Hence,(40)Mxnj,y−vnj+y−vnj,vnj−xnjrnj≥Fy,vnj,∀y∈C.
Fort with 0<t≤1 and y∈C, let vt=ty+1−tx∗. Since y∈C and x∗∈C, we obtain vt∈C. So, from (74), we have(41)vt−vnj,Mvt≥vt−vnj,Mvt−vt−vnj,Mxnj−vt−vnj,vnj−xnjrnj+Fvt,vnj=vt−vnj,Mvt−Mvnj+vt−vnj,Mvnj−Mxnj−vt−vnj,vnj−xnjrnj+Fvt,vnj.
Sincevnj−xnj⟶0, we have Mvnj−Mxnj⟶0. Furthermore, from the inverse strongly monotonicity of M, we have vt−vnj,Mvt−Mvnj≥0. It follows from condition A4 and vnj−xnj/rnj⟶0 and vnj⇀x∗, we have(42)vt−x∗,Mvt≥Fvt,x∗,as j⟶∞. From A1 and A4, we have(43)0=Fvt,vt≤tFvt,y+1−tFvt,x∗≤tFvt,y+1−tvt−x∗,Mvt=tFvt,y+1−tty−x∗,Mvt,and hence,(44)0≤Fvt,y+1−ty−x∗,Mvt.
Lettingt⟶0, we have, for each y∈C,(45)Fx∗,y+y−x∗,Mx∗≥0.
This implies thatx∗∈GEPF,M. Next, we show that x∗∈Ω (1). Let(46)T′p≔∇fp+NCp,p∈C,∅,p∉C.
Then,T′ is maximal monotone and 0∈T′p if and only if p∈VIC,∇f [29]. Let GT′ be the graph of T′, let p,v∈GT′. Then, we have v∈T′p=∇fp+NCp and hence v−∇fp∈NCp. Therefore, we have p−w,v−∇fp≥0 for all w∈C. By the property of metric projection, from yn=PCvn−λn∇fαnzn and p∈C, we have p−yn,yn−vn−λn∇fαnzn≥0, and hence,(47)p−yn,yn−vnλn+∇fαnzn≥0.
Fromp−w,v−∇fp≥0 for all w∈C and ynk∈C, we have(48)p−ynk,v≥p−ynk,∇fp≥p−ynk,∇fp−p−ynk,ynk−vnkλnk+∇fαnznk≥p−ynk,∇fp−p−ynk,ynk−vnkλnk+∇fznk−αnkp−ynk,znk=p−ynk,∇fp−∇fynk−p−ynk,ynk−vnkλnk−αnkp−ynk,znk+p−ynk,∇fynk−∇fznk≥p−ynk,∇fynk−∇fznk−p−ynk,ynk−vnkλnk−αnkp−ynk,znk.
Thus, we obtainp−x∗,v≥0 as k⟶∞. Since T′ is maximal monotone, we have x∗∈T′−10, and hence, x∗∈VIC,∇f. This implies x∗∈Ω. This implies that x∗∈Γ. Thanks to (37), we arrive at(49)limsupn⟶∞x0−u,xn−u=limk⟶∞x0−u,xnk−u=x0−u,x∗−u≤0.
Next, we show thatxn⟶u as n⟶∞. Observe that(50)tn−u=1−βnSyn−u+βnTnzn−u≤1−βnSyn−u+βnTnzn−u≤1−βnyn−u+βnzn−u≤1−βnxn−u+2λnαnu+βnxn−u+λnαnu≤xn−u+2λnαnu.
With the help of (14), we obtain(51)xn+1−u2=anx0+bnxn+cntn−u,xn+1−u≤anx0−u,xn+1−u+bnxn−u+cntn−u,xn+1−u≤anx0−u,xn+1−u+bnxn−u+cntn−uxn+1−u≤anx0−u,xn+1−u+bnxn−u+cnxn−u+2λnαnuxn+1−u≤anx0−u,xn+1−u+1−anxn−u+2λnαnuxn+1−u≤anx0−u,xn+1−u+2λnαnuxn+1−u+1−an2xn−u2+xn+1−u2,which implies that(52)xn+1−u2≤1−anxn−u2+2anx0−u,xn+1−u+4λnαnuxn+1−u.
It follows from condition (a) and Lemma9 that(53)limn⟶∞xn−u=0.
Therefore, fromxn−zn⟶0, xn−yn⟶0, we can conclude that xn, zn, vn, and yn converge strongly to the same point u=PΓx0. The proof is complete.In the following, we will discuss the weak convergence of the sequence of the new iteration.Theorem 2.
Assume thatC and Q are 2 nonempty, closed, and convex subsets of real Hilbert spaces H1 and H2, respectively. Let A:H1⟶H2 be a bounded linear operator andf:H1⟶ℝ be a continuous differentiable function. Assume that C and Q are 2 nonempty, closed, and convex subsets of real Hilbert spaces H1 and H2, respectively. Let Ai:H1⟶H2 bounded linear operators, fi:H1⟶ℝ be a continuous differentiable function, i=1,2, and F be a bifunction from C×C to ℝ satisfying A1−A4, M be an α-inverse strongly monotone mapping from C into H1, S:C⟶C be a nonexpansive mapping, and T:C⟶C be a strictly pseudocontractive mapping with constant k such that Γ=FT∩FS∩GEPF,M∩∩i=12Ωi≠∅. Suppose xn and zn are sequences generated by the following extragradient algorithm:(54)x0=x∈C,zn=γnPCxn−λn∇f1αnxn+1−γnPCxn−λn∇f2snxn,Fvn,y+Mzn,y−vn+1rny−vn,vn−zn≥0,∀y∈C,xn+1=anxn+bn1−βnzn+βnTzn+cn1−δnvn+δnSvn,∀n≥0.
Suppose the following conditions are satisfied:(a)
∑n=0∞αn<∞, ∑n=0∞sn<∞(b)
βn⊂k,r for some r,k∈0,1, λn⊂e,d for some e,d∈0,1/A2(c)
γn⊂t,m for some t,m∈0,1, δn⊂δ1∗,δ2∗ for some δ1∗,δ2∗∈0,1(d)
0<a≤an≤a′<1, 0<b≤bn≤b′<1 and 0<c≤cn≤c′<1 and an+bn+cn=1(e)
0<q1≤rn≤q2<2α
Then,xn converges weakly to an element u∈Γ.Proof.
For any fixedu∈Γ, we find that u=PCI−λ∇fu for λ∈0,1/A2 and Su=u. Let yn=PCI−λn∇f1αnxn, tn=PCI−λn∇f2snxn, and Tn=1−βnI+βnT. We see from Lemma 8 that Tn is nonexpansive and FTn=FT. From (54) and Lemma 3, it follows that(55)yn−u≤PCI−λn∇f1αnxn−PCI−λn∇f1αnu+PCI−λn∇f1αnu−PCI−λn∇f1u≤xn−u+I−λn∇f1αnu−I−λn∇f1u≤xn−u+λnαnu.
In a similar way, we have(56)tn−u≤PCI−λn∇f2snxn−PCI−λn∇f2snu+PCI−λn∇f2snu−PCI−λn∇f2u≤xn−u+I−λn∇f2snu−I−λn∇f2u≤xn−u+λnsnu.
This implies that(57)zn−u≤γnyn−u+1−γntn−u≤γnxn−u+λnαnu+1−γnxn−u+λnsnu≤xn−u+λnuαn+sn.
Observe thatvn can be rewritten as vn=Trnzn−rnMzn,n≥0. From (e) and Lemma 1, we have(58)vn−u2=Trnzn−rnMzn−u2=Trnzn−rnMzn−Trnu−rnMu2≤zn−rnMzn−u−rnMu2=zn−u2+rnrn−2αMzn−Mu2≤zn−u2≤xn−u+λnuαn+sn.
We find from (54) and the last inequality that(59)xn+1−u≤anxn−u+bnTnzn−u+cn1−δnvn−u+δnSvn−u≤anxn−u+1−anzn−u≤anxn−u+1−anxn−u+λnuαn+sn≤xn−u+λnuαn+sn.
Consequently, from condition (a) and Lemma7, we deduce that, for every u∈Γ, limn⟶∞xn−u exists and the sequences xn andzn are bounded. It follows from (55), (56), and Lemma 5 that(60)zn−u2≤γnyn−u2+1−γntn−u2−γn1−γnyn−tn2≤γnxn−u+λnαnu2+1−γnxn−u+λnsnu2−γn1−γnyn−tn2≤γn2xn−u2+2λn2αn2u2+1−γn2xn−u2+2λn2sn2u2−γn1−γnyn−tn2≤xn−u2+2λn2u2αn2+sn2−γn1−γnyn−tn2.
LetSnvn=1−δnvn+δnSvn. We find from (54), (58), and Lemma 5 and the last inequality that(61)xn+1−u2=anxn+bnTnzn+cn1−δnvn+δnSvn−u2≤anxn−u2+bnTnzn−u2+cn1−δnvn−u2+δnSvn−u2−1−δnδnvn−Svn2−anbnxn−Tnzn2−ancnxn−Snvn2≤anxn−u2+1−anzn−u2+rnrn−2αMzn−Mu2−cn1−δnδnvn−Svn2−anbnxn−Tnzn2−ancnxn−Snvn2≤anxn−u2+1−anxn−u2+2λn2u2αn2+sn2−γn1−γnyn−tn2−cn1−δnδnvn−Svn2−anbnxn−Tnzn2−ancnxn−Snvn2+rnrn−2αMzn−Mu2≤xn−u2+2λn2u2αn2+sn2−1−anγn1−γnyn−tn2−cn1−δnδnvn−Svn2−anbnxn−Tnzn2−ancnxn−Snvn2+rnrn−2αMzn−Mu2.
From conditions (b)–(e) and (61), we also obtain(62)1−a′t1−myn−tn2+abxn−Tnzn2+c1−δ2δ1vn−Svn2+acxn−Snvn2+rn2α−rnMzn−Mu2≤1−anγn1−γnyn−tn2+cn1−δnδnvn−Svn2+anbnxn−Tnzn2+ancnxn−Snvn2+rn2α−rnMzn−Mu2≤xn−u2−xn+1−u2+2λn2u2αn2+sn2.
Sincelimn⟶∞xn−u exists and ∑n=0∞αn+sn<∞, we see that(63)limn⟶∞vn−Svn=limn⟶∞xn−Tnzn=limn⟶∞yn−tn=limn⟶∞xn−Snvn=limn⟶∞Mzn−Mu=0.
Sincexn+1−xn≤bnxn−Tnzn+cnxn−Snvn and zn−yn≤yn−tn,zn−tn≤yn−tn, it follows that(64)limn⟶∞xn+1−xn=limn⟶∞yn−zn=limn⟶∞tn−zn=0.
Using Lemma1 and (58), we have(65)vn−u2=Trnzn−rnMzn−Trnu−rnMu2≤zn−rnMzn−u−rnMu,vn−u=12zn−rnMzn−u−rnMu2+12vn−u2−12zn−rnMzn−u−rnMu−vn−u2≤12zn−u2+vn−u2−zn−vn−2rnMzn−Mu2=12zn−u2+vn−u2−zn−vn2+2rnzn−vn,Mzn−Mu−rn2Mxn−Mu2.
It follows that(66)vn−u2≤zn−u2−zn−vn2+2rnzn−vn,Mzn−Mu.
We find from (54) and (66) that(67)xn+1−u2=anxn+bnTnzn+cn1−δnvn+δnSvn−u2≤anxn−u2+bnTnzn−u2+cn1−δnvn−u2+δnSvn−u2≤anxn−u2+1−anzn−u2−cnzn−vn2+2rnzn−vn,Mzn−Mu≤anxn−u2+1−anxn−u2+2λn2u2αn2+sn2−cnzn−vn2+2rnzn−vn,Mzn−Mu≤xn−u2+1−an2λn2u2αn2+sn2−cnzn−vn2+2rnzn−vn,Mzn−Mu.
This yields that(68)zn−vn2≤xn−u2−xn+1−u2+2λn2u2αn2+sn2+2rnzn−vn,Mzn−Mu.
It follows from condition (a),limn⟶∞Mzn−Mu=0, and limn⟶∞xn−u exists that(69)limn⟶∞zn−vn=0.
Also, fromTnzn−zn≤Tnzn−xn+xn−zn, vn−xn≤vn−Snvn+xn−Snvn, and vn−Snvn≤vn−Svn, we get(70)limn⟶∞Tnzn−zn=limn⟶∞vn−xn=0.
Note thatxn−yn≤xn−zn+zn−yn,xn−tn≤xn−zn+zn−tn,βnTzn−zn=Tnzn−zn. This implies that(71)limn⟶∞xn−yn=limn⟶∞xn−tn=limn⟶∞Tzn−zn=0.
Since∇f=A∗I−PQA is Lipschitz continuous, we obtain limn⟶∞∇fxn−∇fyn=0.
Sincexn is bounded, there exists a subsequence xnk of xn such that it converges weakly to some x∗. Since xn−yn⟶0,xn−zn⟶0, and xn−vn⟶0 as n⟶∞, we obtain that ynk⇀x∗,znk⇀x∗, and vnk⇀x∗. Since ynk⊂C and C is closed and convex, we obtain x∗∈C. First, we show that x∗∈FT∩FS. Then, from (63), (71), Lemma 6, and Lemma 4, we have that x∗∈FT∩FS. We now show x∗∈GEPF,M. By vn=Trnzn−rnMzn, we know that(72)Fvn,y+Mzn,y−vn+1rny−vn,vn−zn≥0,∀y∈C.
It follows fromA2 that(73)Mzn,y−vn+1rny−vn,vn−xn≥Fy,vn,∀y∈C.
Hence,(74)Mznj,y−vnj+y−vnj,vnj−znjrnj≥Fy,vnj,∀y∈C.
Fort with 0<t≤1 and y∈C, let vt=ty+1−tx∗. Since y∈C and x∗∈C, we obtain vt∈C. So, from (74), we have(75)vt−vnj,Mvt≥vt−vnj,Mvt−vt−vnj,Mznj−vt−vnj,vnj−znjrnj+Fvt,vnj=vt−vnj,Mvt−Mvnj+vt−vnj,Mvnj−Mznj−vt−vnj,vnj−znjrnj+Fvt,vnj.
Sincevnj−znj⟶0, we have Mvnj−Mznj⟶0. Furthermore, from the inverse strongly monotonicity of M, we have vt−vnj,Mvt−Mvnj≥0. It follows from A4 and vnj−znj/rnj⟶0 and vnj⇀x∗, and we have(76)vt−v,Mvt≥Fvt,x∗,as j⟶∞. From A1 and A4, we have(77)0=Fvt,vt≤tFvt,y+1−tFvt,x∗≤tFvt,y+1−tvt−x∗,Mvt=tFvt,y+1−tty−x∗,Mvt,and hence,(78)0≤Fvt,y+1−ty−x∗,Mvt.
Lettingt⟶0, we have, for each y∈C,(79)Fx∗,y+y−x∗,Mx∗≥0.
This implies thatx∗∈GEPF,M. Next, we show that x∗∈∩i=12Ωi (1). For i=1,2, let(80)Ti′p≔∇fip+NCp,p∈C,∅,p∉C.
Then,Ti′ is maximal monotone and 0∈Ti′p if and only if p∈VIC,∇fi [29]. Let GTi′ be the graph of Ti′, and p,v∈GTi′. Then, we have v∈Ti′p=∇fip+NCp, and hence, v−∇fip∈NCp. Therefore, we have p−w,v−∇fip≥0 for all w∈C. By the property of metric projection, from yn=PCxn−λn∇f1αnxn and p∈C, we have p−yn,yn−xn−λn∇f1αnxn≥0, and hence,(81)p−yn,yn−xnλn+∇f1αnxn≥0.
Fromp−w,v−∇f1p≥0 for all w∈C and ynk∈C, we have(82)p−ynk,v≥p−ynk,∇f1p≥p−ynk,∇f1p−p−ynk,ynk−xnkλnk+∇f1αnxnk≥p−ynk,∇f1p−p−ynk,ynk−xnkλnk+∇f1xnk−αnkp−ynk,xnk=p−ynk,∇f1p−∇f1ynk−p−ynk,ynk−xnkλnk−αnkp−ynk,xnk+p−ynk,∇f1ynk−∇f1xnk≥p−ynk,∇f1ynk−∇f1xnk−p−ynk,ynk−xnkλnk−αnkp−ynk,xnk.
Thus, we obtainp−x∗,v≥0 as k⟶∞. Since T1′ is maximal monotone, we have x∗∈T1′−10, and hence, x∗∈VIC,∇f1. Similarly, we have x∗∈VIC,∇f2. This implies x∗∈Ωi for i=1,2. This implies that x∗∈Γ. Therefore, from xn−zn⟶0, we can conclude that xn, zn, and vn converge weakly to a point u∈Γ. The proof is complete.Theorem 3.
LetC and Q be nonempty, closed, and convex subsets of real Hilbert spaces H1 and H2, respectively. Let A:H1⟶H2 be a bounded linear operator, S:C⟶C be a nonexpansive map, and T:C⟶C be a strictly pseudocontractive mapping with constant k such that Γ=FT∩FS∩Ω≠∅. Suppose xn, vn, and zn are sequences generated by the following extragradient algorithm:(83)x0=x∈C,vn=xn+εnxn−xn−1,zn=PCI−λn∇fαnvn,xn+1=anxn+bn1−βnzn+βnTzn+cnSPCvn−λn∇fαnzn,∀n≥0.
Suppose the following conditions are satisfied:(a)
∑n=0∞αn<∞, ∑n=0∞bn<∞(b)
βn⊂k,r for some r,k∈0,1(c)
λn⊂e,d for some e,d∈0,1/A2(d)
0<a≤an≤a′<1,0<bn≤b′<1,0<c≤cn≤c′<1 and an+bn+cn=1(e)
εn⊂0,ε and ε∈0,1, ∑n=0∞εnxn−xn−1<∞
Then,xn converges weakly to an element u∈Γ.Proof.
For any fixedu∈Γ, we find that u=PCI−λ∇fu for λ∈0,1/A2 and Su=u. Putting Tn=1−βnI+βnT, we see from Lemma 8 that Tn is nonexpansive and FTn=FT. We observe that(84)vn−u=xn+εnxn−xn−1−u≤xn−u+εnxn−xn−1.
From (83) and Lemma 3, it follows that(85)zn−u=PCI−λn∇fαnvn−PCI−λn∇fu≤PCI−λn∇fαnvn−PCI−λn∇fαnu+PCI−λn∇fαnu−PCI−λn∇fu≤vn−u+I−λn∇fαnu−I−λn∇fu≤vn−u+λnαnu.
Putyn=PCvn−λn∇fαnzn for all n≥0. Then, by property of metric projection, we have(86)yn−u2≤vn−λn∇fαnzn−u2−vn−λn∇fαnzn−yn2≤vn−u2−vn−yn2+2λn∇fαnzn,u−yn≤vn−u2−vn−yn2+2λn∇fαnzn−∇fαnu,u−zn+2λn∇fαnu,u−zn+∇fαnzn,zn−yn≤vn−u2−vn−yn2+2λn∇fαnu,u−zn+2λn∇fαnzn,zn−yn≤vn−u2−vn−yn2+2λnαnI+∇fu,u−zn+2λn∇fαnzn,zn−yn≤vn−u2−vn−yn2+2λnαnu,u−zn+2λn∇fαnzn,zn−yn=vn−u2−vn−zn2−2vn−zn,zn−yn−zn−yn2+2λnαnu,u−zn+∇fαnzn,zn−yn=vn−u2−vn−zn2+2vn−λn∇fαnzn−zn,yn−zn+2λnαnu,u−zn−zn−yn2.
Furthermore, by property of metric projection, we have(87)vn−λn∇fαnzn−zn,yn−zn=vn−λn∇fαnvn−zn,yn−zn+λn∇fαnvn−λn∇fαnzn,yn−zn≤λn∇fαnvn−λn∇fαnzn,yn−zn≤λn∇fαnvn−∇fαnznyn−zn≤λnαn+A2vn−znyn−zn.
Hence, we have(88)yn−u2≤vn−u2−vn−zn2+2vn−λn∇fαnzn−zn,yn−zn−zn−yn2+2λnαnu,u−zn≤vn−u2−vn−zn2+2λnαn+A2vn−znyn−zn−zn−yn2+2λnαnuu−zn≤vn−u2−vn−zn2+λn2αn+A22vn−zn2+yn−zn2−zn−yn2+2λnαnuu−zn=vn−u2+λn2αn+A22−1vn−zn2+2λnαnuu−zn≤vn−u2+2λnαnuu−zn≤vn−u2+2λnαnuvn−u+λnαnu≤vn−u2+4λnαnuvn−u+4λn2αn2u2=vn−u+2λnαnu2.
We find from (83), (84), and (85) and the last inequality that(89)xn+1−u=anxn+bnTnzn+cnSPCvn−λn∇fαnzn−u≤anxn−u+bnTnzn−u+cnSyn−u≤anxn−u+bnzn−u+cnyn−u≤anxn−u+bnvn−u+λnαnu+cnvn−u+2λnαnu≤anxn−u+1−anvn−u+2λnαnu≤xn−u+εnxn−xn−1+2λnαnu.
Consequently, from conditions (a) and (e) and Lemma10, we deduce that, for every u∈Γ, limn⟶∞xn−u exists and the sequences xn,zn, and yn are bounded. We find from (83), (84), (85), (88), Lemma 5, and Lemma 11 that(90)xn+1−u2=anxn+bnTnzn+cnSPCvn−λn∇fαnzn−u2≤anxn−u2+bnTnzn−u2+cnSyn−u2−anbnxn−Tnzn2−ancnxn−Syn2≤anxn−u2+bnzn−u2+cnyn−u2−anbnxn−Tnzn2−ancnxn−Syn2≤anxn−u2+bnvn−u+λnαnu2+cnvn−u2+λn2αn+A22−1vn−zn2−anbnxn−Tnzn2−ancnxn−Syn2+2cnλnαnuzn−u≤anxn−u2+bn2vn−u2+2λn2αn2u2+cnvn−u2+λn2αn+A22−1vn−zn2+2λnαnuzn−u−anbnxn−Tnzn2−ancnxn−Syn2≤1+bnxn−u2+2bnλn2αn2u2+22bn+cnεnxn−xn−1,vn−u+cnλn2αn+A22−1vn−zn2+αnλn2u2+zn−u2−anbnxn−Tnzn2−ancnxn−Syn2≤1+bnxn−u2+2bnλn2αn2u2+22bn+cnεnxn−xn−1vn−u+cnλn2αn+A22−1vn−zn2+αnλn2u2+vn−u+λnαnu2−anbnxn−Tnzn2−ancnxn−Syn2≤1+bnxn−u2+2bnλn2αn2u2+22bn+cnεnxn−xn−1vn−u+cnλn2αn+A22−1vn−zn2+αnλn2u2+2vn−u2+2λn2αn2u2−anbnxn−Tnzn2−ancnxn−Syn2≤1+bn+2αnxn−u2+cnλn2αn+A22−1xn−zn2+αnλn2u21+2bnαn+2αn2−anbnxn−Tnzn2−ancnxn−Syn2+22bn+cn+2αnεnxn−xn−1vn−u.
From conditions (b) and (d), we obtain(91)c1−d2αn+A22vn−zn2+abnxn−Tnzn2+acxn−Syn2≤cn1−λn2αn+A22vn−zn2+anbnxn−Tnzn2+ancnxn−Syn2≤1+bn+2αnxn−u2−xn+1−u2+αnλn2u21+2αn2+22bn+cn+2αnεnxn−xn−1vn−u.
From conditions (a) and (e), we also obtain(92)limn⟶∞xn−Syn=limn⟶∞xn−Tnzn=limn⟶∞vn−zn=0.
By the definition ofvn and (e), we have(93)limn⟶∞vn−xn=limn⟶∞εnxn−xn−1=0.
This implies that(94)zn−xn≤zn−vn+vn−xn⟶0,asn⟶∞.
It is observe that(95)yn−zn=PCvn−λn∇fαnzn−PCvn−λn∇fαnvn≤vn−λn∇fαnzn−vn−λn∇fαnvn=λn∇fαnzn−∇fαnvn≤λnαn+A2zn−vn⟶0,asn⟶∞.
Also, fromTnzn−zn≤Tnzn−xn+xn−zn, yn−xn≤yn−zn+zn−xn, and yn−vn≤yn−zn+zn−vn, we get(96)limn⟶∞Tnzn−zn=limn⟶∞yn−xn=limn⟶∞yn−vn=0.
Note thatSyn−yn≤Syn−xn+xn−yn,βnTzn−zn=Tnzn−zn. This implies that(97)limn⟶∞Syn−yn=limn⟶∞Tzn−zn=0.
Since∇f=A∗I−PQA is Lipschitz continuous, we obtain limn⟶∞∇fzn−∇fyn=0.
Since,xn is bounded, there exists a subsequence xnk of xn such that it converges weakly to some x∗. Since xn−yn⟶0,xn−zn⟶0, and xn−vn⟶0 as n⟶∞, we obtain that ynk⇀x∗,znk⇀x∗, and vnk⇀x∗. Since ynk⊂C and C is closed and convex, we obtain x∗∈C. First, we show that x∗∈FT∩FS. Then, from (97), Lemma 6, and Lemma 4, we have that x∗∈FT∩FS. We now show x∗∈Ω (1). Let(98)T′p≔∇fp+NCp,p∈C,∅,p∉C.
Then,T′ is maximal monotone and 0∈T′p if and only if p∈VIC,∇f [29]. Let GT′ be the graph of T′, and let p,v∈GT′. Then, we have v∈T′p=∇fp+NCp, and hence, v−∇fp∈NCp. Therefore, we have p−w,v−∇fp≥0 for all w∈C. By property of metric projection, from yn=PCvn−λn∇fαnzn and p∈C, we have p−yn,yn−vn−λn∇fαnzn≥0, and hence,(99)p−yn,yn−vnλn+∇fαnzn≥0.
Fromp−w,v−∇fp≥0 for all w∈C and ynk∈C, we have(100)p−ynk,v≥p−ynk,∇fp≥p−ynk,∇fp−p−ynk,ynk−vnkλnk+∇fαnkznk≥p−ynk,∇fp−p−ynk,ynk−vnkλnk+∇fznk−αnkp−ynk,znk=p−ynk,∇fp−∇fynk−p−ynk,ynk−vnkλnk−αnkp−ynk,znk+p−ynk,∇fynk−∇fznk≥p−ynk,∇fynk−∇fznk−p−ynk,ynk−vnkλnk−αnkp−ynk,znk.
Thus, we obtainp−x∗,v≥0 as k⟶∞. Since T′ is maximal monotone, we have x∗∈T′−10, and hence, x∗∈VIC,∇f. This implies that x∗∈Ω. This implies that x∗∈Γ. Therefore, from xn−zn⟶0 and xn−vn⟶0, we can conclude that xn, zn, and vn converge weakly to a point u∈Γ. The proof is complete.


## 4. Applications

If, in Theorem3 and Theorem 1, we assume that C=H1, then we can get the following theorems.Theorem 4.
LetH1 and H2 be real Hilbert spaces, Ai:H1⟶H2 be a bounded linear operator, for i=1,2, S:H1⟶H1 be a nonexpansive mapping, and T:H1⟶H1 a strictly pseudocontractive mapping with constant k such that Γ=FT∩FS∩i=12∇fi−10≠∅. Suppose xn and zn are sequences generated by the following extragradient algorithm:(101)x0=x∈C,zn=γnxn−λn∇f1αnxn+1−γnxn−λn∇f2snxn,xn+1=anxn+bn1−βnzn+βnTzn+cn1−δnzn+δnSzn,∀n≥0.
If conditionsa−d are satisfied, then xn converges weakly to an element u∈Γ.Proof.
We have∇fi−10=VIH1,∇fi for i=1,2 and PH1=I; by Theorem 3, we obtain the desired result.Theorem 5.
LetH1 and H2 be real Hilbert spaces, A:H1⟶H2 be a bounded linear operator, S:H1⟶H1 be a nonexpansive map, and T:H1⟶H1 be a strictly pseudocontractive mapping with constants k such that Γ=FT∩FS∩∇f−10≠∅. Suppose xn, vn, and zn are sequences generated by the following extragradient algorithm:(102)x0=x∈C,vn=xn+εnxn−xn−1,zn=I−λn∇fαnvn,xn+1=anxn+bn1−βnzn+βnTzn+cnSvn−λn∇fαnzn,∀n≥0.
If conditionsa−e are satisfied, then xn converges weakly to an element u∈Γ.Proof.
We have∇f−10=VIH1,∇f and PH1=I; by Theorem 3, we obtain the desired result.Theorem 6.
LetH1 and H2 be real Hilbert spaces, A:H1⟶H2 be a bounded linear operator, S:H1⟶H1 be a nonexpansive map, and T:H1⟶H1 be a strictly pseudocontractive mapping with constant k such that Γ=FT∩FS∩∇f−10≠∅. Suppose xn, yn, and zn are sequences generated by the following extragradient algorithm:(103)x0=x∈C,zn=I−λn∇fαnxn,yn=xn−λn∇fαnzn,xn+1=anx0+bnxn+cn1−βnSyn+βnTnzn,∀n≥0,where Tn=1−γnI+γnT and γn⊂k,1. If conditions a−d are satisfied, then xn converges strongly to the point u=PΓx0.Proof.
We have∇f−10=VIH1,∇f and PH1=I; by Theorem 1, we obtain the desired result.LetB:H⟶2H be a maximal monotone mapping. Then, for any x∈H and r>0, consider JrBx=y∈H:x=y+rBy. Likewise, a JrB is called the resolvent of B and is denoted by JrB=I+rB−1.Theorem 7.
LetH1 and H2 be real Hilbert spaces, Bi:H1⟶2H1 be maximal monotone mappings, for i=1,2, Ai:H1⟶H2 be bounded linear operators, for i=1,2, JrBi be the resolvents of Bi for each r>0, and T:H1⟶H1 be a strictly pseudocontractive mapping with constant k such that Γ=FT∩FS∩Bi−10∩∇fi−10≠∅. Suppose xn and zn are sequences generated by the following extragradient algorithm:(104)x0=x∈C,zn=γnJrB1xn−λn∇f1αnxn+1−γnJrB2xn−λn∇f2snxn,xn+1=anxn+bn1−βnzn+βnTzn+cnzn,∀n≥0.
If conditionsa−d are satisfied, then xn converges weakly to an element u∈Γ.Proof.
We haveFJrBi=Bi−10, ∇fi−10=VIH1,∇fi for i=1,2 and PH1=I; by Theorem 3, we obtain the desired result.Theorem 8.
LetH1 and H2 be real Hilbert spaces, A:H1⟶H2 be a bounded linear operator, B:H1⟶2H1 be a maximal monotone mapping, JrB be the resolvent of B for each r>0, and T:H1⟶H1 be a strictly pseudocontractive mapping with constant k such that Γ=FT∩B−10∩∇f−10≠∅. Suppose xn, vn, and zn are sequences generated by the following extragradient algorithm:(105)x0=x∈C,vn=xn+εnxn−xn−1,zn=I−λn∇fαnvn,xn+1=anxn+bn1−βnzn+βnTzn+cnJrBvn−λn∇fαnzn,∀n≥0.
If conditionsa−e are satisfied, then xn converges weakly to an element u∈Γ.Proof.
We haveFJrB=B−10, ∇f−10=VIH1,∇f, and PH1=I; by Theorem 3, we obtain the desired result.Theorem 9.
LetH1 and H2 be real Hilbert spaces, A:H1⟶H2 be a bounded linear operator, B:H1⟶2H1 be a maximal monotone mapping, JrB be the resolvent of B for each r>0, and T:H1⟶H1 be a strictly pseudocontractive mapping with constant k such that Γ=FT∩B−10∩∇f−10≠∅. Suppose xn, yn, and zn are sequences generated by the following extragradient algorithm:(106)x0=x∈C,zn=I−λn∇fαnxn,yn=JrBxn−λn∇fαnzn,xn+1=anx0+bnxn+cn1−βnyn+βnTnzn,∀n≥0,where Tn=1−γnI+γnT and γn⊂k,1. If conditions a−d are satisfied, then xn converges strongly to the point u=PΓx0.Proof.
We haveFJrB=B−10, ∇f−10=VIH1,∇f, and PH1=I; by Theorem 1, we obtain the desired result.If in Theorems3 and 1 we assume that T is nonexpansive, then we have that T is strictly pseudocontractive with k=1, and hence, we get the following corollaries.Corollary 1.
LetC and Q be nonempty, closed, and convex subsets of real Hilbert spaces H1 and H2, respectively. Let Ai:H1⟶H2 be bounded linear operators for i=1,2, S:C⟶C be a nonexpansive mapping, and T:C⟶C be a nonexpansive mapping such that Γ=FT∩FS∩i=1,22Ωi≠∅. Suppose xn and zn are sequences generated by the following extragradient algorithm:(107)x0=x∈C,zn=γnPCxn−λn∇f1αnxn+1−γnPCxn−λn∇f2snxn,xn+1=anxn+bn1−βnzn+βnTzn+cn1−δnzn+δnSzn,∀n≥0.
If conditionsa−d are satisfied, then xn converges weakly to an element u∈Γ.Corollary 2.
LetC and Q be nonempty, closed, and convex subsets of real Hilbert spaces H1 and H2, respectively. Let A:H1⟶H2 be a bounded linear operator, S:C⟶C be a nonexpansive mapping, and T:C⟶C be a nonexpansive mapping such that Γ=FT∩FS∩Ω≠∅. Suppose that xn, vn, and zn are sequences generated by the following extragradient algorithm:(108)x0=x∈C,vn=xn+εnxn−xn−1,zn=PCI−λn∇fαnvn,xn+1=anxn+bn1−βnzn+βnTzn+cnSPCvn−λn∇fαnzn,∀n≥0.
If conditionsa−e are satisfied, then xn converges weakly to an element u∈Γ.Corollary 3.
LetC and Q be nonempty, closed, and convex subsets of real Hilbert spaces H1 and H2, respectively. Let A:H1⟶H2 be a bounded linear operator, S:C⟶C be a nonexpansive map, and T:C⟶C be nonexpansive such that Γ=FT∩FS∩Ω≠∅. Let xn and zn be sequences generated by the following extragradient algorithm:(109)x0=x∈C,zn=PCI−λn∇fαnxn,yn=PCxn−λn∇fαnzn,xn+1=anx0+bnxn+cn1−βnSyn+βnTnzn,∀n≥0,where Tn=1−γnI+γnT and γn⊂k,1. If conditions a−d are satisfied, then xn converges strongly to the point u=PΓx0.



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