# A General Iterative Scheme Based on Regularization for Solving Equilibrium and Constrained Convex Minimization Problems

Authors: Ming Tian
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

The present paper is divided into two parts. First, we introduce implicit and explicit iterative schemes based on the regularization for solving equilibrium and constrained convex minimization problems. We establish results on the strong convergence of the sequences generated by the proposed schemes to a common solution of minimization and equilibrium problem. Such a point is also a solution of a variational inequality. In the second part, as applications, we apply the algorithm to solve split feasibility problem and equilibrium problem.


---

## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Let C be a nonempty closed convex subset of H. Let ϕ be a bifunction of C×C into ℝ, where ℝ is the set of real numbers. Consider the equilibrium problem (EP) which is to find z∈C such that
(1)ϕ(z,y)≥0,∀y∈C.We denoted the set of solutions of EP byEP(ϕ). Given a mapping  T:C→H, let ϕ(x,y)=〈Tx,y-x〉 for all x,y∈C; then z∈EP(ϕ) if and only if 〈Tz,y-z〉≥0 for all y∈C; that is, z is a solution of the variational inequality. Numerous problems in physics, optimizations and economics reduce to find a solution of (1). Some methods have been proposed to solve the equilibrium problem; see for instance see [1–8] and the references therein.Some composite iterative algorithms were proposed by many authors for finding the common solution of equilibrium problem and fixed point problem. Next, we list some main results as follows.With some appropriate assumptions, Ceng et al. [9] established the following iterative scheme:  x1∈H and
(2)ϕ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnun+(1-αn)Sun,∀n∈ℕ.
Under certain conditions, the sequences {xn} and {un} converge weakly to an element of EP(ϕ)∩F(S).For finding an element ofEP(ϕ)∩F(S), S. Takahashi and W. Takahashi [10] introduced the following iterative scheme by the viscosity approximation method in a Hilbert space: x1∈H and
(3)ϕ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnh(xn)+(1-αn)Sun,∀n∈ℕ.
Under suitable conditions, some strong convergence theorems are obtained.In 2009, Liu [11] introduced two iterative schemes by the general iterative method for finding an element of EP(ϕ)∩F(S), where S:C→H is a k-strictly pseudocontraction nonself mapping in the setting of a real Hilbert space. Let {xn} be a sequence generated by
(4)ϕ(un,y)+1λn〈y-un,un-xn〉≥0,∀y∈C,yn=βnun+(1-βn)Sun,xn=αnγh(xn)+(I-αnB)yn,∀n∈ℕ,
and x1∈H arbitrarily,
(5)ϕ(un,y)+1λn〈y-un,un-xn〉≥0,∀y∈C,yn=βnun+(1-βn)Sun,xn+1=αnγh(xn)+(I-αnB)yn,∀n∈ℕ,
where B is a strongly positive bounded linear operator on H. Under some assumptions, the strong convergence theorems are obtained.In 2012, based on the concept of the shrinking projection method, Reich and Sabach [12] consider the following algorithm for finding the common solution of finite equilibrium problems in a reflexive Banach space
(6)x0∈X,Q0i=X,i=1,2,…,N,yni=Resλnigif(xn+eni),Qn+1i={z∈Qni:〈∇f(xn+eni)-∇f(yni),z-yni〉≤0},Qn+1:=∩i=1NQn+1i,xn+1=projQn+1f(x0),n=0,1,2,….
Under some consumption, the sequence  {xn}n∈ℕ  converges strongly to ProjEf(x0).The gradient-projection algorithm is a classical power method for solving constrained convex optimization problems and has been studied by many authors (see [13–26] and the reference therein). The method has recently been applied to solve split feasibility problems which find applications in image reconstructions and the intensity modulated radiation therapy (see [27–34]).Consider the problem of minimizingf over the constraint set C (assuming C is a nonempty closed and convex subset of a real Hilbert space H). The main results we all know about the gradient projection are that if f:H→ℝ is a convex and continuously Fréchet differentiable functional, the gradient-projection algorithm generates a sequence {xn}n=0∞ determined by the gradient of f and the metric projection onto C. Under the condition that f has a Lipschitz continuous and strongly monotone gradient, the sequence {xn}n=0∞ can be strongly convergent to a minimizer of f in C. If the gradient of f is only assumed to be inverse strongly monotone, then {xn}n=0∞ can only be weakly convergent if H is infinite-dimensional.Recently, Xu [35] gave an operator-oriented approach as an alternative to the gradient-projection method and to the relaxed gradient-projection algorithm, namely, an averaged mapping approach. He also presented two modifications of gradient-projection algorithms which are shown to have strong convergence.On the other hand, regularization, in particular the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems [36]. The disadvantage is the weak convergence of the method RGPA for the regularization problem under some conditions.The purpose of the paper is to study the iterative method for finding the common solution of an equilibrium problem and a constrained convex minimization problem. Based on the Viscosity method [18], we combine the RGPA and averaged mapping approaches to propose implicit and explicit composite iterative methods for finding the common element of the set of solutions of an equilibrium problem and the solution set of a constrained convex minimization problem and also to prove some strong convergence theorems.


## 2. Preliminaries

Throughout the paper, we assume thatH is a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively, and C  is a nonempty closed convex subset of H. The set of fixed points of a mapping T is denoted by Fix(T); that is, Fix(T)={x∈H:Tx=x}. We write xn⇀x to indicate that the sequence {xn} converges weakly to x. The fact that the sequence {xn} converges strongly to x is denoted by xn→x. The following definition and results are needed in the subsequent sections.Recall that a mappingV:H→H is said to be L-Lipschitzian if
(7)∥Vx-Vy∥≤L∥x-y∥,∀x,y∈H,
where L>0 is a constant. In particular, if L∈[0,1), then V is called a contraction on H; if L=1, then V is called a nonexpansive mapping on H. V is called firmly nonexpansive if 2V-I is nonexpansive, or equivalently,  〈x-y,Vx-Vy〉≥∥Vx-Vy∥2,for all x,y∈H. Alternatively, T is firmly nonexpansive if and only if T can be expressed as T=(1/2)(I+S), where S:H→H is nonexpansive.In 1978, Baillon et al. [37] defined the concept of averaged mapping which is used very frequently now.Definition 1 (see [37]).
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,
(8)T=(1-α)I+αS,
where α is a number in (0,1) and S:H→H is nonexpansive. More precisely, when (8) holds, we say that T is α-averaged. Clearly, a firmly nonexpansive mapping (in particular, projection) is a (1/2)-averaged map.Proposition 2 (see [28, 38]).
For given operatorsS,T,V:H→H one has the following.(i) If
T
=
(
1
-
α
)
S
+
α
V
for some
α
∈
(
0,1
)
and if
S
is averaged and
V
is nonexpansive, then
T
is averaged.(ii)
T
is firmly nonexpansive if and only if the complement
I
-
T
is firmly nonexpansive.(iii) If
T
=
(
1
-
α
)
S
+
α
V
for some
α
∈
(
0,1
)
and if S is firmly nonexpansive and
V
is nonexpansive, then
T
is averaged.Recall that 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
(9)∥x-PCx∥=infy∈C∥x-y∥=:d(x,C).In 1984, Goebel and Reich [39] discussed the properties of the nearest point projection.Lemma 3 (see [39]).
For givenx∈H one has the following:(i)
z=PCx if and only if
(10)〈x-z,y-z〉≤0,∀y∈C;(ii)
z=PCx if and only if
(11)∥x-z∥2≤∥x-y∥2-∥y-z∥2,∀y∈C;(iii)
(12)〈PCx-PCy,x-y〉≥∥PCx-PCy∥2,∀x,y∈H.
Consequently,PC is nonexpansive and monotone.Lemma 4.
The following inequality holds in a Hilbert spaceX(13)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈X.Lemma 5 (see [40]).
In a Hilbert spaceH, one has
(14)∥λx+(1-λ)y∥2=λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)∥x-y∥2∀x,y∈H,λ∈[0,1].Lemma 6 (Demiclosedness Principle [40]).
LetH be a Hilbert space, K a closed convex subset of H, and T:K→K a nonexpansive mapping with Fix(T)≠∅; if {xn} is a sequence in K weakly converging to x and if {(I-T)xn} converges strongly to y, then (I-T)x=y; in particular if y=0 then x∈Fix(T).Definition 7.
A nonlinear operatorT whose domain D(T)⊆H and range R(T)⊆H is said to be(i)
monotone if(15)〈x-y,Tx-Ty〉≥0,∀x,y∈D(T);(ii)
β-strongly monotone if there exists β>0 such that
(16)〈x-y,Tx-Ty〉≥β∥x-y∥2,∀x,y∈D(T);(iii)
ν-inverse strongly monotone (for short, ν-ism) if there exists ν>0 such that
(17)〈x-y,Tx-Ty〉≥ν∥Tx-Ty∥2,∀x,y∈D(T).Proposition 8 (see [28]).
LetT:H→H be an operator from H to itself.(i)
Tis nonexpansive if and only if the complementI-Tis(1/2)-ism.(ii)
IfTisν-ism, then forγ>0,γTis(ν/γ)-ism.(iii)
Tis averaged if and only if the complementI-Tisν-ism for someν>1/2.
Indeed, forα∈(0,1),Tisα-averaged if and only ifI-Tis(1/2α)-ism.Lemma 9 (see [18]).
Assume that{an} is a sequence of nonnegative real numbers such that
(18)an+1≤(1-γn)an+γnδn,n≥0,
where {γn} is a sequence in (0,1) and {δn} is a sequence in ℝ such that(i)
∑n=1∞γn=∞;(ii)
limsupn→∞δn≤0or∑n=1∞γn|δn|<∞.Thenlimn→∞an=0.In order to solve the equilibrium problem for a bifunctionϕ:C×C→ℝ, let us assume that ϕ satisfies the following conditions:(A1)
ϕ(x,x)=0, for all x∈C;(A2)
ϕ is monotone; that is, ϕ(x,y)+ϕ(y,x)≤0,for all x,y∈C;(A3)
For allx,y,z∈C,
(19)limt↓0ϕ(tz+(1-t)x,y)≤ϕ(x,y);(A4)
for each fixedx∈C, the function y↦ϕ(x,y) is convex and lower semicontinuous.Let us recall the following lemmas which will be useful for our paper.Lemma 10 (see [28]).
Letϕ be a bifunction from C×C into ℝ satisfying (A1), (A2), (A3), and (A4). Then, for any r>0 and x∈H, there exists z∈C such that
(20)ϕ(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.
Further, if  Trx={z∈C:ϕ(z,y)+(1/r)〈y-z,z-x〉≥0,∀y∈C}, then the following holds:(1)
Tr is single-valued;(2)
Tr is firmly nonexpansive; that is,
(21)∥Trx-Try∥2≤〈Trx-Try,x-y〉,∀x,y∈H;(3)
F(Tr)=EP(ϕ);(4)
EP(ϕ) is closed and convex.


## 3. Main Results

We now look at the constrained convex minimization problem:(22)minx∈Cf(x),
where C is a closed and convex subset of a Hilbert space H and f:C→ℝ is a real-valued convex function. If f is Fréchet differentiable, then the gradient-projection algorithm (GPA) generates a sequence {xn}n=0∞ according to the recursive formula
(23)xn+1=ProjC(I-γ∇f)(xn),n≥0,
or more generally,
(24)xn+1=ProjC(I-γn∇f)(xn),n≥0,
where, in both (23) and (24), the initial guess x0 is taken from C arbitrarily and the parameters γ or γn are positive real numbers.As a matter of fact, it is known that, if∇f fails to be strongly monotone and is only 1/L-ism; namely, there is constant L>0 such that
(25)〈∇f(x)-∇f(y),x-y〉≥1L∥∇f(x)-∇f(y)∥2,x,y∈C,
under some assumption for γ or γn, then algorithms (23) and (24) can still converge in the weak topology.Now, consider the regularized minimization problem(26)minx∈Cfα(x):=f(x)+α2∥x∥2,
where α>0 is the regularization parameter, and again f is convex with 1/L-ism continuous gradient ∇f.It is obvious that there exists a unique pointxα∈C such that xα is the unique fixed point of the mapping
(27)Vα:=ProjC(I-γ∇fα)=ProjC(I-γ(∇f+αI)).We can prove that{xα}⇀x*, where x* is a solution of the constrained convex minimization problem.Throughout the rest of this paper, assume that the minimization problem (22) is consistent and let U denote its solution set; we always assume that h is a contraction of C into H with coefficient ρ∈(0,1); let {Gβn} be a sequence of mappings defined as Lemma 3 and define a mapping Tn:C→C by
(28)PC(I-γ∇fαn)=λnI+(1-λn)Tnλn:=2-γ(L+αn)4
Consider the following mapping Qn on H defined by
(29)Qnx=θnh(x)+(1-θn)TnGβnx,x∈H,n∈N,
where θn∈(0,1); then by Lemmas 3 and 10(30)∥Qnx-Qny∥≤θnρ∥x-y∥+(1-θn)∥x-y∥=(1-(1-ρ)θn).Since0<1-(1-ρ)θn<1, it follows that Qn is a contraction. Therefore, by the Banach contraction principle, Qn has a unique fixed point xnh∈H such that
(31)xnh=θnh(xnh)+(1-θn)TnGβnxnh.For simplicity, we will writexn for xnh provided that no confusion occurs. Next, we prove the convergence of {xn} while we claim the existence of the q∈U⋂EP(ϕ) which solves the variational inequality
(32)〈(I-h)q,p-q〉≥0,∀p∈U⋂‍EP(ϕ).


### 3.1. Convergence of the Implicit Scheme

Proposition 11.
If0<γ<2/L, 0<α<2/γ-L,  ∇f is (1/L)-ism, for all x∈C,
(33)ProjC(I-γ∇fα)=(1-μα)I+μαTα,ProjC(I-γ∇f)=(1-μ)I+μT,
where μα=(2+γ(L+α))/4,  μ=(2+γL)/4, then
(34)∥Tαx-Tx∥≤αM(x),
where
(35)M(x)=γ(5∥x∥+∥Tx∥).Proof.
One here(36)∥ProjC(I-γ∇fα)x-ProjC(I-γ∇fα)x∥=∥(μ-μα)x+μαTαx-μTx∥≤∥(I-γ∇fα)x-(I-γ∇f)x∥=γ∥∇fα(x)-∇f(x)∥=αγ∥x∥,
then
(37)∥μα(Tαx)-μTx∥≤|μ-μα|∥x∥+αγ∥x∥,∥Tαx-Tx∥≤αγ(5∥x∥+∥Tx∥)2+γ(L+α)≤αM(x),
where, M(x)=γ(5∥x∥+∥Tx∥).Theorem 12.
LetC be a nonempty closed convex subset of a real Hilbert space H and h:C→H a contraction with ρ∈(0,1), U∩EP(ϕ)≠∅, and ϕ a bifunction from C×C into ℝ satisfying  (A1), (A2), (A3), and  (A4). Let {xn} be sequence generated by
(38)ϕ(un,y)+1βn〈y-un,un-xn〉≥0,∀y∈C,xn=θnh(xn)+(1-θn)Tn(un),
where, Pc[I-γ∇fαn]=λnI+(1-λn)Tn, 0<γ<2/L and λn=(2-γ(L+αn))/4.(i)
{βn}⊂(0,+∞)liminfn→∞βn>0;(ii)
{θn}⊂(0,1), limn→∞θn=0;(iii)
αn=o(θn).
Then,{xn} converges strongly to a point q∈U∩EP(ϕ) which solves the variational inequality (32).Proof.
Pick anyp∈U∩EP(ϕ),  un=Gβnxn,   p=Gβnp; then we have
(39)∥un-p∥≤∥xn-p∥,
(noting Tp=p)
(40)∥xn-p∥=∥θnh(xn)+(1-θn)Tn(un)-p∥=∥θn(h(xn)-h(p))+θn(h(p)-p)+(1-θn)(Tn(un)-p)∥≤θnρ∥xn-p∥+θn∥h(p)-p∥+(1-θn)[∥un-p∥+∥Tn(p)-T(p)∥]≤(1-(1-ρ)θn)∥xn-p∥+θn∥h(p)-p∥+αnM1;
hence,
(41)∥xn-p∥≤11-ρ[∥h(p)-p∥+M1].
So, {xn} is bounded. Next, we claim that ∥xn-un∥→0.
Takep∈U∩EP(ϕ); by Lemma 3, we have
(42)∥un-p∥2=∥Tβnxn-Tβnp∥2≤〈xn-p,un-p〉=12(∥xn-p∥2+∥un-p∥2-∥un-xn∥2).
It follows that
(43)∥un-p∥2≤∥xn-p∥2-∥xn-un∥2,∥xn-p∥2=∥θn(h(xn)-p)+(1-θn)(Tnun-p)∥2≤(1-θn)2[∥Tnun-Tnp+Tnp-Tp∥2]+2θn〈h(xn)-p,xn-p〉≤(1-θn)2×[∥un-p∥2+2∥un-p∥×∥Tnp-Tp∥+∥Tnp-Tp∥2]+2θn〈h(xn)-p,xn-p〉≤(1-θn)2×([∥xn-p∥2-∥xn-un∥2]+2∥un-p∥∥Tnp-Tp∥+∥Tnp-Tp∥2)+2θnρ∥xn-p∥2+2θn〈h(p)-p,xn-p〉.
So, ∥xn-un∥→0. Next, we show that ∥xn-Tnxn∥→0,
(44)∥xn-Tnxn∥=∥xn-Tnun+Tnun-Tnxn∥≤θn∥h(xn)-Tnun∥+∥un-xn∥⟶0.∥un-Tnun∥=∥un-xn+xn-Tnxn+Tnxn-Tnun∥⟶0.
Observe that
(45)∥PC(I-γ∇fαn)un-un∥=∥λnun+(1-λn)Tnun-un∥=(1-λn)∥Tnun-un∥≤∥Tnun-un∥.
Hence,
(46)∥PC(I-γ∇f)un-un∥≤∥PC(I-γ∇f)un-PC(I-γ∇fαn)un∥+∥PC(I-γ∇fαn)un-un∥≤γαn∥un∥+∥Tnun-un∥⟶0.
So,
(47)limn→∞∥PC(I-γ∇f)un-un∥=0.
Since{un} is bounded, there exists {uni} such that {uni}⇀q. Since C is closed and convex, C is weakly closed. So, we have q∈C. Let us show that q∈U. Assume that q∈-U. Since uni⇀q and q≠Tq, it follows from the Opial’s condition that
(48)liminfn→∞∥uni-q∥<liminfn→∞∥uni-Tq∥≤liminfn→∞(∥uni-Tuni∥+∥Tuni-Tq∥)≤liminfn→∞∥uni-q∥.
This is a contradiction. So, we get q∈U.
Next, we show thatq∈EP(ϕ). Since un=Gβnxn, for any y∈C, we obtain
(49)ϕ(un,y)+1βn〈y-un,un-xn〉≥0.
From (A2), we have
(50)1βn〈y-un,un-xn〉≥ϕ(y,un).
Replacing n with ni, we have
(51)〈y-uni,uni-xniβni〉≥ϕ(y,uni).
Since (uni-xni)/βni→0 and uni⇀q, it follows from (A4) that 0≥ϕ(y,q), for all y∈C. Let zt=ty+(1-t)q for all t∈(0,1] and y∈C. Then, we have zt∈C and hence ϕ(zt,q)≤0. Thus, from (A1) and (A4) we have
(52)0=ϕ(zt,zt)≤tϕ(zt,y)+(1-t)ϕ(zt,q)≤tϕ(zt,y)
and hence 0≤ϕ(zt,y). From (A3), we have 0≤ϕ(q,y) for all y∈C and hence q∈EP(ϕ). Therefore, q∈U∩EP(ϕ).
On the other hand,(53)xn-q=θn(h(xn)-q)+(1-θn)(Tnun-q),(54)∥xn-q∥2=θn〈h(xn)-q,xn-q〉+(1-θn)〈Tnun-q,xn-q〉=θn〈h(xn)-h(q),xn-q〉+(1-θn)〈Tnun-Tnq,xn-q〉+(1-θn)〈Tnq-Tq,xn-q〉+θn〈h(q)-q,xn-q〉≤θnρ∥xn-q∥2+(1-θn)∥xn-q∥2+θn〈h(q)-q,xn-q〉+(1-θn)αnM2∥xn-q∥;
then xnj→q if xnj⇀q.
Next, we prove thatq solves the VI (problem):
(55)(I-h)(xn)=-1θn(I-TnQβn)xn+(I-TnQβn)xn.
Note that
(56)〈(I-h)q,q-p〉=limj→∞〈(I-h)(xnj),xnj-p〉=limj→∞[-1θnj〈(I-TnQβnj)xnj-(I-TnjQβnj)p,xnj-p〉-1θnj〈(I-TnjQβnj)p,xnj-p〉+〈(I-TnjQβnj)xnj,xnj-p〉-1θnj〈(I-TnQβnj)xnj]≤0.


### 3.2. Convergence of the Explicit Scheme

Theorem 13.
LetC be a nonempty closed convex subset of a real Hilbert space H,  h:C→H a contraction with ρ∈(0,1),  U∩EP(ϕ)≠∅, and ϕ a bifunction from C×C into ℝ satisfying (A1), (A2), (A3), and  (A4). Let {xn} be sequence generated by x1∈H and
(57)ϕ(un,y)+1βn〈y-un,un-xn〉≥0,∀y∈C,xn+1=θnh(xn)+(1-θn)Tnun,
where Pc[I-γ∇fαn]=λnI+(1-λn)Tn, 0<γ<2/L and  λn=(2-γ(L+αn))/4,  un=Gβnxn. Let {αn} and {θn} satisfy the following conditions:(i)
{θn}⊂(0,1), limn→∞θn=0, ∑n=1∞θn=∞, ∑n=1∞‍|θn+1-θn|<∞;(ii)
αn=o(θn)∑n=1∞|αn+1-αn|<∞;(iii)
{βn}∈(0,+∞), limn→∞βn>0  and  ∑n=1∞|βn+1-βn|<∞.
Then,{xn} and {un} converge strongly to a point q∈U∩EP(ϕ) which solves the variational inequality (32).Proof.
First we prove that{xn} is bounded.
Taking anyp∈U∩EP(ϕ), we have
(58)un=Gβnxn,p=Gβnp.
So, ∥un-p∥≤∥xn-p∥,
(59)∥xn+1-p∥=∥θnh(xn)+(1-θn)Tn(un)-p∥=∥θn(h(xn)-h(p))+θn(h(p)-p)+(1-θn)(Tn(un)-p)∥≤θnρ∥xn-p∥+θn∥h(p)-p∥+(1-θn)[∥xn-p∥+∥Tn(p)-T(p)∥]≤(1-(1-ρ)θn)∥xn-p∥+θn∥h(p)-p∥+αnM3≤max{∥xn-p∥;11-ρ[∥h(p)-p∥+M3]}.
So, {xn} is bounded.
Next we prove that∥xn+1-xn∥→0,
(60)∥xn+1-xn∥=∥[θnh(xn)+(1-θn)Tnun]-[θn-1h(xn-1)+(1-θn-1)Tn-1un-1]∥=∥θn(h(xn)-h(xn-1))+(1-θn)(Tnun-Tnun-1)+(θn-θn-1)(h(xn-1)-Tnun-1)+(1-θn-1)(Tnun-1-Tn-1un-1)∥≤θnρ∥xn-xn-1∥+|θn-θn-1|[∥h(xn-1)∥+∥Tnun-1∥]+(1-θn)∥un-un-1∥+(1-θn-1)∥Tnun-1-Tn-1un-1∥.
From un+1=Gβn+1xn+1 and un=Gβnxn, we note that
(61)ϕ(un+1,y)+1βn+1〈y-un+1,un+1-xn+1〉≥0,∀y∈C,(62)ϕ(un,y)+1βn〈y-un,un-xn〉≥0,∀y∈C.
Putting y=un in (61) and y=un+1 in (62), we have
(63)ϕ(un+1,un)+1βn+1〈un-un+1,un+1-xn+1〉≥0,∀y∈C,ϕ(un,un+1)+1βn〈un+1-un,un-xn〉≥0,∀y∈C.
So, from (A2), we have
(64)〈un+1-un,un-xnβn-un+1-xn+1βn+1〉≥0,
and hence
(65)〈un+1-un,un-un+1+un+1-xn-βnβn+1(un+1-xn+1)〉≥0.
Sincelimn→∞βn>0, without loss of generality, let us assume that there exists a real number a such that βn>a>0 for all n∈ℕ. Thus, we have
(66)∥un+1-un∥2≤〈un+1-un,xn+1-xn+(1-βnβn+1)(un+1-xn+1)〉≤∥un+1-un∥×{∥xn+1-xn∥+|1-βnβn+1|∥un+1-xn+1∥};
thus,
(67)∥un+1-un∥≤∥xn+1-xn∥+1a|βn+1-βn|M4,
where M4=sup{∥un-xn∥:n∈ℕ},
(68)∥Tnun-1-Tn-1un-1∥=∥4PC(I-γ∇fαn)-[2-γ(L+αn)]I2+γ(L+αn)un-1-4PC(I-γ∇fαn-1)-[2-γ(L+αn-1)]I2+γ(L+αn-1)un-1∥≤∥4PC(I-γ∇fαn)2+γ(L+αn)un-1-4PC(I-γ∇fαn-1)2+γ(L+αn-1)un-1∥+∥-[2-γ(L+αn)]2+γ(L+αn)un-1+[2-γ(L+αn-1)]2+γ(L+αn-1)un-1∥=∥(4[2+γ(L+αn-1)]PC(I-γ∇fαn)un-1-4[2+γ(L+αn)]PC(I-γ∇fαn-1)un-1)×([2+γ(L+αn)][2+γ(L+αn-1)])-1∥+4γ|αn-αn-1|∥un-1∥[2+γ(L+αn)][2+γ(L+αn-1)]=∥4γ(αn-1-αn)PC(I-γ∇fαn)un-1[2+γ(L+αn)][2+γ(L+αn-1)]+(PC(I-γ∇fαn-1)un-1]4(2+γ(L+αn))×[PC(I-γ∇fαn)un-1-PC(I-γ∇fαn-1)un-1])×([2+γ(L+αn)][2+γ(L+αn-1)])-14γ(αn-1-αn)PC(I-γ∇fαn)un-1[2+γ(L+αn)][2+γ(L+αn-1)]∥+4γ|αn-αn-1|∥un-1∥[2+γ(L+αn)][2+γ(L+αn-1)]≤4γ|αn-1-αn|∥PC(I-γ∇fαn)un-1∥[2+γ(L+αn)][2+γ(L+αn-1)]+4γ|αn-1-αn|[2+γ(L+αn)]∥un-1∥[2+γ(L+αn)][2+γ(L+αn-1)]+4γ|αn-αn-1|∥un-1∥[2+γ(L+αn)][2+γ(L+αn-1)]≤|αn-1-αn|[γ∥PC(I-γ∇fαn)un-1∥+5γ∥un-1∥]≤M5|αn-1-αn|.
So(69)∥xn+1-xn∥≤[1-(1-ρ)θn]∥xn-xn-1∥+M6[|θn-θn-1|+|αn-αn-1|+|βn-βn-1|].
So
(70)lim∥xn+1-xn∥=0,lim∥un+1-un∥=0.
Next, we prove that ∥xn-Tnxn∥→0,
(71)∥xn+1-p∥2=∥θn(h(xn)-p)+(1-θn)(Tnun-p)∥2≤(1-θn)2[∥Tnun-Tnp+Tnp-Tp∥2]+2θn〈h(xn)-p,xn-p〉≤(1-θn)2×[∥un-p∥2+2∥un-p∥×∥Tnp-Tp∥+∥Tnp-Tp∥2]+2θn〈h(xn)-p,xn-p〉≤(1-θn)2×([∥xn-p∥2-∥xn-un∥2]+2∥un-p∥∥Tnp-Tp∥+∥Tnp-Tp∥2)+2θnρ∥xn-p∥2+2θn〈h(p)-p,xn-p〉.
So, ∥xn-un∥→0,
(72)∥xn-Tnxn∥=∥xn-xn+1+xn+1-Tnun+Tnun-Tnxn∥≤θn∥h(xn)-Tnun∥+∥xn-xn+1∥+∥un-xn∥⟶0,∥un-Tnun∥=∥un-xn+xn-Tnxn+Tnxn-Tnun∥⟶0.
Observe that
(73)∥PC(I-γ∇fαn)un-un∥=∥λnun+(1-λn)Tnun-un∥=(1-λn)∥Tnun-un∥≤∥Tnun-un∥.
Hence,
(74)∥PC(I-γ∇f)un-un∥≤∥PC(I-γ∇f)un-PC(I-γ∇fαn)un∥+∥PC(I-γ∇fαn)un-un∥≤γαn∥un∥+∥Tnun-un∥⟶0.
So,
(75)limn→∞∥PC(I-γ∇f)un-un∥=0.
Since{un} is bounded, there exists {uni} such that {uni}⇀q. Since C is closed and convex, C is weakly closed. So, we have q∈C. Let us show that q∈U. Assume that q∈-U. Since uni⇀q and q≠Tq, it follows from the Opial’s condition that
(76)liminfn→∞∥uni-q∥<liminfn→∞∥uni-Tq∥≤liminfn→∞(∥uni-Tuni∥+∥Tuni-Tq∥)≤liminfn→∞∥uni-q∥.
This is a contradiction. So, we get q∈U.
Next We show thatq∈EP(ϕ). Since un=Gβnxn, for any y∈C, we obtain
(77)ϕ(un,y)+1βn〈y-un,un-xn〉≥0.
From (A2), we have
(78)1βn〈y-un,un-xn〉≥ϕ(y,un).
Replacing n with ni, we have
(79)〈y-uni,uni-xniβni〉≥ϕ(y,uni).
Since (uni-xni)/βni→0 and uni⇀q, it follows from (A4) that 0≥ϕ(y,q), for all y∈C. Let zt=ty+(1-t)q for all t∈(0,1] and y∈C. Then, we have zt∈C and hence ϕ(zt,q)≤0. Thus, from (A1) and (A4), we have
(80)0=ϕ(zt,zt)≤tϕ(zt,y)+(1-t)ϕ(zt,q)≤tϕ(zt,y)
and hence 0≤ϕ(zt,y). From (A3), we have 0≤ϕ(q,y) for all y∈C and hence q∈EP(ϕ). Therefore, q∈U∩EP(ϕ).
We assume that;xni⇀x~, then x~∈EP(ϕ)∩U,
(81)limsupn→∞〈(I-h)q,q-xn〉=〈(I-h)q,q-x~〉≤0.
Finally, we prove that xn→q,
(82)∥xn+1-q∥2=∥θn(h(xn)-q)+(1-θn)(Tnun-Tq)∥2=∥θn(h(xn)-h(q))+(1-θn)(Tnun-Tq)+θn(h(q)-q)∥2≤∥θn(h(xn)-h(q))+(1-θn)(Tnun-Tq)∥2+2θn〈h(q)-q,xn+1-q〉≤θn∥(h(xn)-h(q))∥2+(1-θn)×∥(Tnun-Tq)∥2+2θn〈h(q)-q,xn+1-q〉≤θnρ2∥xn-q∥2+(1-θn)×∥(Tnun-Tnq+Tnq-Tq)∥2+2θn〈h(q)-q,xn+1-q〉≤θnρ2∥xn-q∥2+(1-θn)[∥xn-q∥+αnM7]2+2θn〈h(q)-q,xn+1-q〉≤[1-(1-ρ)θn]∥xn-q∥2+(1-θn)[2αnM7∥xn-q∥+αn2M72]+2θn〈h(q)-q,xn+1-q〉∥xn+1-q∥2=(1-βn)∥xn-q∥2+βnδn.βn=(1-ρ)θn,δn=11-ρ[αnθn(1-θn)2M7∥xn-q∥+(1-θn)M72αn2θn+2〈h(q)-q,xn+1-q〉M72αn2θn],
by Lemma 9 and limn→∞βn=0, ∑n=0∞βn=∞; limsupn→∞δn≤0, then xn→q.


## 3.1. Convergence of the Implicit Scheme

Proposition 11.
If0<γ<2/L, 0<α<2/γ-L,  ∇f is (1/L)-ism, for all x∈C,
(33)ProjC(I-γ∇fα)=(1-μα)I+μαTα,ProjC(I-γ∇f)=(1-μ)I+μT,
where μα=(2+γ(L+α))/4,  μ=(2+γL)/4, then
(34)∥Tαx-Tx∥≤αM(x),
where
(35)M(x)=γ(5∥x∥+∥Tx∥).Proof.
One here(36)∥ProjC(I-γ∇fα)x-ProjC(I-γ∇fα)x∥=∥(μ-μα)x+μαTαx-μTx∥≤∥(I-γ∇fα)x-(I-γ∇f)x∥=γ∥∇fα(x)-∇f(x)∥=αγ∥x∥,
then
(37)∥μα(Tαx)-μTx∥≤|μ-μα|∥x∥+αγ∥x∥,∥Tαx-Tx∥≤αγ(5∥x∥+∥Tx∥)2+γ(L+α)≤αM(x),
where, M(x)=γ(5∥x∥+∥Tx∥).Theorem 12.
LetC be a nonempty closed convex subset of a real Hilbert space H and h:C→H a contraction with ρ∈(0,1), U∩EP(ϕ)≠∅, and ϕ a bifunction from C×C into ℝ satisfying  (A1), (A2), (A3), and  (A4). Let {xn} be sequence generated by
(38)ϕ(un,y)+1βn〈y-un,un-xn〉≥0,∀y∈C,xn=θnh(xn)+(1-θn)Tn(un),
where, Pc[I-γ∇fαn]=λnI+(1-λn)Tn, 0<γ<2/L and λn=(2-γ(L+αn))/4.(i)
{βn}⊂(0,+∞)liminfn→∞βn>0;(ii)
{θn}⊂(0,1), limn→∞θn=0;(iii)
αn=o(θn).
Then,{xn} converges strongly to a point q∈U∩EP(ϕ) which solves the variational inequality (32).Proof.
Pick anyp∈U∩EP(ϕ),  un=Gβnxn,   p=Gβnp; then we have
(39)∥un-p∥≤∥xn-p∥,
(noting Tp=p)
(40)∥xn-p∥=∥θnh(xn)+(1-θn)Tn(un)-p∥=∥θn(h(xn)-h(p))+θn(h(p)-p)+(1-θn)(Tn(un)-p)∥≤θnρ∥xn-p∥+θn∥h(p)-p∥+(1-θn)[∥un-p∥+∥Tn(p)-T(p)∥]≤(1-(1-ρ)θn)∥xn-p∥+θn∥h(p)-p∥+αnM1;
hence,
(41)∥xn-p∥≤11-ρ[∥h(p)-p∥+M1].
So, {xn} is bounded. Next, we claim that ∥xn-un∥→0.
Takep∈U∩EP(ϕ); by Lemma 3, we have
(42)∥un-p∥2=∥Tβnxn-Tβnp∥2≤〈xn-p,un-p〉=12(∥xn-p∥2+∥un-p∥2-∥un-xn∥2).
It follows that
(43)∥un-p∥2≤∥xn-p∥2-∥xn-un∥2,∥xn-p∥2=∥θn(h(xn)-p)+(1-θn)(Tnun-p)∥2≤(1-θn)2[∥Tnun-Tnp+Tnp-Tp∥2]+2θn〈h(xn)-p,xn-p〉≤(1-θn)2×[∥un-p∥2+2∥un-p∥×∥Tnp-Tp∥+∥Tnp-Tp∥2]+2θn〈h(xn)-p,xn-p〉≤(1-θn)2×([∥xn-p∥2-∥xn-un∥2]+2∥un-p∥∥Tnp-Tp∥+∥Tnp-Tp∥2)+2θnρ∥xn-p∥2+2θn〈h(p)-p,xn-p〉.
So, ∥xn-un∥→0. Next, we show that ∥xn-Tnxn∥→0,
(44)∥xn-Tnxn∥=∥xn-Tnun+Tnun-Tnxn∥≤θn∥h(xn)-Tnun∥+∥un-xn∥⟶0.∥un-Tnun∥=∥un-xn+xn-Tnxn+Tnxn-Tnun∥⟶0.
Observe that
(45)∥PC(I-γ∇fαn)un-un∥=∥λnun+(1-λn)Tnun-un∥=(1-λn)∥Tnun-un∥≤∥Tnun-un∥.
Hence,
(46)∥PC(I-γ∇f)un-un∥≤∥PC(I-γ∇f)un-PC(I-γ∇fαn)un∥+∥PC(I-γ∇fαn)un-un∥≤γαn∥un∥+∥Tnun-un∥⟶0.
So,
(47)limn→∞∥PC(I-γ∇f)un-un∥=0.
Since{un} is bounded, there exists {uni} such that {uni}⇀q. Since C is closed and convex, C is weakly closed. So, we have q∈C. Let us show that q∈U. Assume that q∈-U. Since uni⇀q and q≠Tq, it follows from the Opial’s condition that
(48)liminfn→∞∥uni-q∥<liminfn→∞∥uni-Tq∥≤liminfn→∞(∥uni-Tuni∥+∥Tuni-Tq∥)≤liminfn→∞∥uni-q∥.
This is a contradiction. So, we get q∈U.
Next, we show thatq∈EP(ϕ). Since un=Gβnxn, for any y∈C, we obtain
(49)ϕ(un,y)+1βn〈y-un,un-xn〉≥0.
From (A2), we have
(50)1βn〈y-un,un-xn〉≥ϕ(y,un).
Replacing n with ni, we have
(51)〈y-uni,uni-xniβni〉≥ϕ(y,uni).
Since (uni-xni)/βni→0 and uni⇀q, it follows from (A4) that 0≥ϕ(y,q), for all y∈C. Let zt=ty+(1-t)q for all t∈(0,1] and y∈C. Then, we have zt∈C and hence ϕ(zt,q)≤0. Thus, from (A1) and (A4) we have
(52)0=ϕ(zt,zt)≤tϕ(zt,y)+(1-t)ϕ(zt,q)≤tϕ(zt,y)
and hence 0≤ϕ(zt,y). From (A3), we have 0≤ϕ(q,y) for all y∈C and hence q∈EP(ϕ). Therefore, q∈U∩EP(ϕ).
On the other hand,(53)xn-q=θn(h(xn)-q)+(1-θn)(Tnun-q),(54)∥xn-q∥2=θn〈h(xn)-q,xn-q〉+(1-θn)〈Tnun-q,xn-q〉=θn〈h(xn)-h(q),xn-q〉+(1-θn)〈Tnun-Tnq,xn-q〉+(1-θn)〈Tnq-Tq,xn-q〉+θn〈h(q)-q,xn-q〉≤θnρ∥xn-q∥2+(1-θn)∥xn-q∥2+θn〈h(q)-q,xn-q〉+(1-θn)αnM2∥xn-q∥;
then xnj→q if xnj⇀q.
Next, we prove thatq solves the VI (problem):
(55)(I-h)(xn)=-1θn(I-TnQβn)xn+(I-TnQβn)xn.
Note that
(56)〈(I-h)q,q-p〉=limj→∞〈(I-h)(xnj),xnj-p〉=limj→∞[-1θnj〈(I-TnQβnj)xnj-(I-TnjQβnj)p,xnj-p〉-1θnj〈(I-TnjQβnj)p,xnj-p〉+〈(I-TnjQβnj)xnj,xnj-p〉-1θnj〈(I-TnQβnj)xnj]≤0.


## 3.2. Convergence of the Explicit Scheme

Theorem 13.
LetC be a nonempty closed convex subset of a real Hilbert space H,  h:C→H a contraction with ρ∈(0,1),  U∩EP(ϕ)≠∅, and ϕ a bifunction from C×C into ℝ satisfying (A1), (A2), (A3), and  (A4). Let {xn} be sequence generated by x1∈H and
(57)ϕ(un,y)+1βn〈y-un,un-xn〉≥0,∀y∈C,xn+1=θnh(xn)+(1-θn)Tnun,
where Pc[I-γ∇fαn]=λnI+(1-λn)Tn, 0<γ<2/L and  λn=(2-γ(L+αn))/4,  un=Gβnxn. Let {αn} and {θn} satisfy the following conditions:(i)
{θn}⊂(0,1), limn→∞θn=0, ∑n=1∞θn=∞, ∑n=1∞‍|θn+1-θn|<∞;(ii)
αn=o(θn)∑n=1∞|αn+1-αn|<∞;(iii)
{βn}∈(0,+∞), limn→∞βn>0  and  ∑n=1∞|βn+1-βn|<∞.
Then,{xn} and {un} converge strongly to a point q∈U∩EP(ϕ) which solves the variational inequality (32).Proof.
First we prove that{xn} is bounded.
Taking anyp∈U∩EP(ϕ), we have
(58)un=Gβnxn,p=Gβnp.
So, ∥un-p∥≤∥xn-p∥,
(59)∥xn+1-p∥=∥θnh(xn)+(1-θn)Tn(un)-p∥=∥θn(h(xn)-h(p))+θn(h(p)-p)+(1-θn)(Tn(un)-p)∥≤θnρ∥xn-p∥+θn∥h(p)-p∥+(1-θn)[∥xn-p∥+∥Tn(p)-T(p)∥]≤(1-(1-ρ)θn)∥xn-p∥+θn∥h(p)-p∥+αnM3≤max{∥xn-p∥;11-ρ[∥h(p)-p∥+M3]}.
So, {xn} is bounded.
Next we prove that∥xn+1-xn∥→0,
(60)∥xn+1-xn∥=∥[θnh(xn)+(1-θn)Tnun]-[θn-1h(xn-1)+(1-θn-1)Tn-1un-1]∥=∥θn(h(xn)-h(xn-1))+(1-θn)(Tnun-Tnun-1)+(θn-θn-1)(h(xn-1)-Tnun-1)+(1-θn-1)(Tnun-1-Tn-1un-1)∥≤θnρ∥xn-xn-1∥+|θn-θn-1|[∥h(xn-1)∥+∥Tnun-1∥]+(1-θn)∥un-un-1∥+(1-θn-1)∥Tnun-1-Tn-1un-1∥.
From un+1=Gβn+1xn+1 and un=Gβnxn, we note that
(61)ϕ(un+1,y)+1βn+1〈y-un+1,un+1-xn+1〉≥0,∀y∈C,(62)ϕ(un,y)+1βn〈y-un,un-xn〉≥0,∀y∈C.
Putting y=un in (61) and y=un+1 in (62), we have
(63)ϕ(un+1,un)+1βn+1〈un-un+1,un+1-xn+1〉≥0,∀y∈C,ϕ(un,un+1)+1βn〈un+1-un,un-xn〉≥0,∀y∈C.
So, from (A2), we have
(64)〈un+1-un,un-xnβn-un+1-xn+1βn+1〉≥0,
and hence
(65)〈un+1-un,un-un+1+un+1-xn-βnβn+1(un+1-xn+1)〉≥0.
Sincelimn→∞βn>0, without loss of generality, let us assume that there exists a real number a such that βn>a>0 for all n∈ℕ. Thus, we have
(66)∥un+1-un∥2≤〈un+1-un,xn+1-xn+(1-βnβn+1)(un+1-xn+1)〉≤∥un+1-un∥×{∥xn+1-xn∥+|1-βnβn+1|∥un+1-xn+1∥};
thus,
(67)∥un+1-un∥≤∥xn+1-xn∥+1a|βn+1-βn|M4,
where M4=sup{∥un-xn∥:n∈ℕ},
(68)∥Tnun-1-Tn-1un-1∥=∥4PC(I-γ∇fαn)-[2-γ(L+αn)]I2+γ(L+αn)un-1-4PC(I-γ∇fαn-1)-[2-γ(L+αn-1)]I2+γ(L+αn-1)un-1∥≤∥4PC(I-γ∇fαn)2+γ(L+αn)un-1-4PC(I-γ∇fαn-1)2+γ(L+αn-1)un-1∥+∥-[2-γ(L+αn)]2+γ(L+αn)un-1+[2-γ(L+αn-1)]2+γ(L+αn-1)un-1∥=∥(4[2+γ(L+αn-1)]PC(I-γ∇fαn)un-1-4[2+γ(L+αn)]PC(I-γ∇fαn-1)un-1)×([2+γ(L+αn)][2+γ(L+αn-1)])-1∥+4γ|αn-αn-1|∥un-1∥[2+γ(L+αn)][2+γ(L+αn-1)]=∥4γ(αn-1-αn)PC(I-γ∇fαn)un-1[2+γ(L+αn)][2+γ(L+αn-1)]+(PC(I-γ∇fαn-1)un-1]4(2+γ(L+αn))×[PC(I-γ∇fαn)un-1-PC(I-γ∇fαn-1)un-1])×([2+γ(L+αn)][2+γ(L+αn-1)])-14γ(αn-1-αn)PC(I-γ∇fαn)un-1[2+γ(L+αn)][2+γ(L+αn-1)]∥+4γ|αn-αn-1|∥un-1∥[2+γ(L+αn)][2+γ(L+αn-1)]≤4γ|αn-1-αn|∥PC(I-γ∇fαn)un-1∥[2+γ(L+αn)][2+γ(L+αn-1)]+4γ|αn-1-αn|[2+γ(L+αn)]∥un-1∥[2+γ(L+αn)][2+γ(L+αn-1)]+4γ|αn-αn-1|∥un-1∥[2+γ(L+αn)][2+γ(L+αn-1)]≤|αn-1-αn|[γ∥PC(I-γ∇fαn)un-1∥+5γ∥un-1∥]≤M5|αn-1-αn|.
So(69)∥xn+1-xn∥≤[1-(1-ρ)θn]∥xn-xn-1∥+M6[|θn-θn-1|+|αn-αn-1|+|βn-βn-1|].
So
(70)lim∥xn+1-xn∥=0,lim∥un+1-un∥=0.
Next, we prove that ∥xn-Tnxn∥→0,
(71)∥xn+1-p∥2=∥θn(h(xn)-p)+(1-θn)(Tnun-p)∥2≤(1-θn)2[∥Tnun-Tnp+Tnp-Tp∥2]+2θn〈h(xn)-p,xn-p〉≤(1-θn)2×[∥un-p∥2+2∥un-p∥×∥Tnp-Tp∥+∥Tnp-Tp∥2]+2θn〈h(xn)-p,xn-p〉≤(1-θn)2×([∥xn-p∥2-∥xn-un∥2]+2∥un-p∥∥Tnp-Tp∥+∥Tnp-Tp∥2)+2θnρ∥xn-p∥2+2θn〈h(p)-p,xn-p〉.
So, ∥xn-un∥→0,
(72)∥xn-Tnxn∥=∥xn-xn+1+xn+1-Tnun+Tnun-Tnxn∥≤θn∥h(xn)-Tnun∥+∥xn-xn+1∥+∥un-xn∥⟶0,∥un-Tnun∥=∥un-xn+xn-Tnxn+Tnxn-Tnun∥⟶0.
Observe that
(73)∥PC(I-γ∇fαn)un-un∥=∥λnun+(1-λn)Tnun-un∥=(1-λn)∥Tnun-un∥≤∥Tnun-un∥.
Hence,
(74)∥PC(I-γ∇f)un-un∥≤∥PC(I-γ∇f)un-PC(I-γ∇fαn)un∥+∥PC(I-γ∇fαn)un-un∥≤γαn∥un∥+∥Tnun-un∥⟶0.
So,
(75)limn→∞∥PC(I-γ∇f)un-un∥=0.
Since{un} is bounded, there exists {uni} such that {uni}⇀q. Since C is closed and convex, C is weakly closed. So, we have q∈C. Let us show that q∈U. Assume that q∈-U. Since uni⇀q and q≠Tq, it follows from the Opial’s condition that
(76)liminfn→∞∥uni-q∥<liminfn→∞∥uni-Tq∥≤liminfn→∞(∥uni-Tuni∥+∥Tuni-Tq∥)≤liminfn→∞∥uni-q∥.
This is a contradiction. So, we get q∈U.
Next We show thatq∈EP(ϕ). Since un=Gβnxn, for any y∈C, we obtain
(77)ϕ(un,y)+1βn〈y-un,un-xn〉≥0.
From (A2), we have
(78)1βn〈y-un,un-xn〉≥ϕ(y,un).
Replacing n with ni, we have
(79)〈y-uni,uni-xniβni〉≥ϕ(y,uni).
Since (uni-xni)/βni→0 and uni⇀q, it follows from (A4) that 0≥ϕ(y,q), for all y∈C. Let zt=ty+(1-t)q for all t∈(0,1] and y∈C. Then, we have zt∈C and hence ϕ(zt,q)≤0. Thus, from (A1) and (A4), we have
(80)0=ϕ(zt,zt)≤tϕ(zt,y)+(1-t)ϕ(zt,q)≤tϕ(zt,y)
and hence 0≤ϕ(zt,y). From (A3), we have 0≤ϕ(q,y) for all y∈C and hence q∈EP(ϕ). Therefore, q∈U∩EP(ϕ).
We assume that;xni⇀x~, then x~∈EP(ϕ)∩U,
(81)limsupn→∞〈(I-h)q,q-xn〉=〈(I-h)q,q-x~〉≤0.
Finally, we prove that xn→q,
(82)∥xn+1-q∥2=∥θn(h(xn)-q)+(1-θn)(Tnun-Tq)∥2=∥θn(h(xn)-h(q))+(1-θn)(Tnun-Tq)+θn(h(q)-q)∥2≤∥θn(h(xn)-h(q))+(1-θn)(Tnun-Tq)∥2+2θn〈h(q)-q,xn+1-q〉≤θn∥(h(xn)-h(q))∥2+(1-θn)×∥(Tnun-Tq)∥2+2θn〈h(q)-q,xn+1-q〉≤θnρ2∥xn-q∥2+(1-θn)×∥(Tnun-Tnq+Tnq-Tq)∥2+2θn〈h(q)-q,xn+1-q〉≤θnρ2∥xn-q∥2+(1-θn)[∥xn-q∥+αnM7]2+2θn〈h(q)-q,xn+1-q〉≤[1-(1-ρ)θn]∥xn-q∥2+(1-θn)[2αnM7∥xn-q∥+αn2M72]+2θn〈h(q)-q,xn+1-q〉∥xn+1-q∥2=(1-βn)∥xn-q∥2+βnδn.βn=(1-ρ)θn,δn=11-ρ[αnθn(1-θn)2M7∥xn-q∥+(1-θn)M72αn2θn+2〈h(q)-q,xn+1-q〉M72αn2θn],
by Lemma 9 and limn→∞βn=0, ∑n=0∞βn=∞; limsupn→∞δn≤0, then xn→q.


## 4. Application of the Iterative Method

Next, we give an application of Theorem13 to the split feasibility problem (say SFP, for short) which was introduced by Censor and Elfving [27],
(83)findx∈C,suchthatAx∈Q,
where C and Q are nonempty closed convex subsets of Hilbert space H1 and H2, respectively. A:H1→H2 is a bounded linear operator.It is clear thatx* is a solution to the split feasibility problem (83) if and only if x*∈C and Ax*-PQAx*=0.We define the proximity functionf by
(84)f(x)=12∥Ax-PQAx∥2,
and consider the convex optimization problem
(85)minx∈Cf(x)=minx∈C12∥Ax-PQAx∥2.Then,x* solves the split feasibility problem (83) if and only if x* solves the minimization (85) with the minimization equal to 0. Byrne [28] introduced the so-called CQ algorithm to solve the (SFP),
(86)xn+1=PC(I-μA*(I-PQ)A)xn,n≥0,
where 0<μ<2/∥A*A∥=2/∥A∥2.He obtained that the sequence{xn} generated by (86) converges weakly to a solution of the (SFP).Now we consider the regularization technique; let(87)fα(x)=12∥Ax-PQAx∥2+α2∥x∥2.
Applying Theorem 13, we obtain the following result.Theorem 14.
Assume that the split problem (83) is consistent. Let C be a nonempty closed convex subset of a real Hilbert space H,  h:C→H a contraction with ρ∈(0,1), U∩EP(ϕ)≠∅, and ϕ be a bifunction from C×C into ℝ satisfying (A1), (A2), (A3), and (A4). Let {xn} be sequence generated by x1∈H and
(88)ϕ(un,y)+1βn〈y-un,un-xn〉≥0,∀y∈C,xn+1=θnh(xn)+(1-θn)Tnun,
where
(89)PC[I-μ(A*(I-PQ)A+αnI)]=λnI+(1-λn)Tn,λn=2-μ(∥A∥2+αn)4,
where un=Gβnxn; let {αn}, {θn} satisfy the following conditions:(i)
{θn}⊂(0,1), limn→∞θn=0,∑n=1∞‍θn=∞,∑n=1∞‍|θn+1-θn|<∞,0<μ<2/∥A*A∥=2/∥A∥2;(ii)
αn=o(θn)∑n=1∞‍|αn+1-αn|<∞;(iii)
{βn}∈(0,+∞),limn→∞βn>0  and  ∑n=1∞‍|βn+1-βn|<∞.
Then,{xn} and {un} converge strongly to a point q∈U∩EP(ϕ) which solves the variational inequality (32).Proof.
By the definition of the functionfα, we have
(90)∇fα(x)=A*(I-PQ)Ax+αx,
and ∇fα is 1/(∥A∥2+α)-ism,
(91)∥∇fα(x)-∇fα(y)∥≤∥A∥2+α;
then, due to Theorem 13, we have the conclusion immediately.



---

*Source: 583710-2013-07-29.xml*