Algorithms for the variational inequalities and fixed point problems

Research Article

Algorithms for the variational inequalities and fixed point problems

Yaqiang Liu^a, Zhangsong Yao^b,∗, Yeong-Cheng Liou^c, Li-Jun Zhu^d

aSchool of Management, Tianjin Polytechnic University, Tianjin 300387, China.

bSchool of Information Engineering, Nanjing Xiaozhuang University, Nanjing 211171, China.

cDepartment of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan and Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan.

dSchool of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021, China.

Communicated by Yeol Je Cho

Abstract

A system of variational inequality and fixed point problems is considered. Two algorithms have been constructed. Our algorithms can find the minimum norm solution of this system of variational inequality and fixed point problems. c2016 All rights reserved.

Keywords: Variational inequality, monotone mapping, nonexpansive mapping, fixed point, minimum norm.

2010 MSC: 47H05, 47H10, 47H17.

1. Introduction

Variational inequality problems were initially studied by Stampacchia [17] in 1964. Variational inequal- ities have applications in diverse disciplines such as physical, optimal control, optimization, mathematical programming, mechanics and finance, see [12], [16], [17], [29] and the references therein. The main purpose of this paper is devoted to find the minimum norm solution of some system of variational inequality and fixed point problems.

LetHbe a real Hilbert space with inner producth·,·iand normk · k, respectively. LetCbe a nonempty closed convex subset of H.

∗Corresponding author

Email addresses: [email protected](Yaqiang Liu),[email protected](Zhangsong Yao), [email protected](Yeong-Cheng Liou),[email protected](Li-Jun Zhu)

Received 2015-05-12

Definition 1.1. A mappingF:C→His called ζ-inverse strongly monotone if there exists a real number ζ >0 such that

hFx−Fy, x−yi ≥ζkFx−Fyk², ∀x, y∈C.

Definition 1.2. A mapping R : C → H is called κ-contraction, if there exists a constant κ ∈ [0,1) such thatkR(x)−R(y)k ≤κkx−yk for allx, y∈C.

Definition 1.3. A mappingN :C→C is said to be nonexpansive if kN x−N yk ≤ kx−yk,∀x, y∈C. We use F ix(N) to denote the set of fixed points ofN.

Definition 1.4. We call P roj_C : H→ C is the metric projection ifP roj_C :H→ C assigns to each point x∈Cthe unique pointP roj_Cx∈Csatisfying the property

kx−P roj_Cxk= inf

y∈C

kx−yk=:d(x,C).

LetF:C→H be a nonlinear mapping. Recall that the classical variational inequality is to findx^∗ ∈C such that

hFx^∗, x−x^∗i ≥0 for all x∈C. (1.1) The set of solutions of the variational inequality (1.1) is denoted by V I(F,C). The variational inequality problem has been extensively studied in the literature. Related works, please see, e.g. [1]-[11], [13], [15], [19]-[28], [30]-[34] and the references therein. For finding an element of F ix(N)∩V I(F,C), Takahashi and Toyoda [19] introduced the following iterative scheme:

xⁿ⁺¹=ζnxⁿ+ (1−ζn)N P roj_C(xⁿ−ηnFxⁿ), n≥0, (1.2) whereP roj_Cis the metric projection ofHontoC,{ζ_n}is a sequence in (0,1), and{η_n}is a sequence in (0,2ζ).

Takahashi and Toyoda showed that the sequence {xⁿ} converges weakly to some z ∈F ix(N)∩V I(F,C).

Consequently, Nadezhkina and Takahashi [11] and Zeng and Yao [34] proposed some so-called extragradient methods for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem.

Recently, Ceng, Wang and Yao [3] considered a general system of variational inequality of findingx^∗ ∈C such that

(hηFy^∗+x^∗−y^∗, x−x^∗i ≥0,∀x∈C,

hξGx^∗+y^∗−x^∗, x−y^∗i ≥0,∀x∈C, (SV I) whereF,G:C→Hare two nonlinear mappings,y^∗=P roj_C(x^∗−ξGx^∗),η >0 andξ >0 are two constants.

The solutions set of SVI is denoted by Ω.

If take F=G, then SVI reduces to findingx^∗∈Csuch that

(hηFy^∗+x^∗−y^∗, x−x^∗i ≥0,∀x∈C, hξFx^∗+y^∗−x^∗, x−y^∗i ≥0,∀x∈C,

which is introduced by Verma [20] (see also Verma [21]). Further, if we add up the requirement thatx^∗ =y^∗, then SVI reduces to the classical variational inequality problem (1.1). For finding an element ofF ix(N)∩Ω, Ceng, Wang and Yao [3] introduced the following relaxed extragradient method:

(yⁿ=P roj_C(xⁿ−ξGxⁿ),

xⁿ⁺¹=ζ_nu+β_nxⁿ+γ_nN P roj_C(yⁿ−ηFyⁿ), n≥0. (1.3)

They proved the strong convergence of the above method to some element inF ix(N)∩Ω.

On the other hand, in many problems, it is needed to find a solution with minimum norm. A typical example is the least-squares solution to the constrained linear inverse problem, see [14].

It is our purpose in this paper that we construct two methods, one implicit and one explicit, to find the minimum norm element inF ix(N)∩Ω; namely, the unique solution x^∗ to the quadratic minimization problem:

x^∗ = arg min

x∈F ix(N)∩Ωkxk². We obtain two strong convergence theorems.

2. Preliminaries

Let Cbe a nonempty closed convex subset ofH. The following lemmas are useful for our main results.

Lemma 2.1. Given x∈H and z∈C.

(i) z=P roj_Cx if and only if there holds the relation:

hx−z, y−zi ≤0 for all y∈C. (ii) z=P roj_Cx if and only if there holds the relation:

kx−zk² ≤ kx−yk²− ky−zk² for all y∈C. (iii) There holds the relation

hP roj_Cx−P roj_Cy, x−yi ≥ kP roj_Cx−P roj_Cyk² for all x, y∈H. Consequently, P roj_C is nonexpansive and monotone.

Lemma 2.2 ([3]). Let C be a nonempty closed convex subset of a real Hilbert space H. Let the mapping F:C→Hbe ζ-inverse strongly monotone. Then, we have

k(I−ηF)x−(I−ηF)yk² ≤ kx−yk²+η(η−2ζ)kFx−Fyk²,∀x, y∈C.

In particular, if 0≤η≤2ζ, then I−ηF is nonexpansive.

Lemma 2.3([3]). x^∗ is a solution of SVI if and only if x^∗ is a fixed point of the mappingU:C→Cdefined by

U(x) =P roj_C[P roj_C(x−ξGx)−ηFP roj_C(x−ξGx)],∀x∈C, where y^∗=P roj_C(x^∗−ξGx^∗).

In particular, if the mappings F,G : C → H are ζ-inverse strongly monotone and δ-inverse strongly monotone, respectively, then the mappingU is a nonexpansive mapping providedη ∈(0,2ζ)and ξ∈(0,2δ).

Lemma 2.4([18]). Let{xⁿ}and{yⁿ}be bounded sequences in a Banach spaceXand let{δ_n}be a sequence in [0,1] with 0<lim inf

n→∞ δ_n≤lim sup

n→∞

δ_n <1. Suppose xⁿ⁺¹ = (1−δ_n)yⁿ+δ_nxⁿ for all integers n≥0 and lim sup

n→∞

(kyⁿ⁺¹−yⁿk − kxⁿ⁺¹−xⁿk)≤0. Then, lim

n→∞kyⁿ−xⁿk= 0.

Lemma 2.5 ([10]). Let C be a closed convex subset of a real Hilbert space H and let N : C → C be a nonexpansive mapping. Then, the mapping I−N is demiclosed. That is, if {xⁿ} is a sequence in C such thatxⁿ→x^∗ weakly and (I−N)xⁿ→y strongly, then (I−N)x^∗ =y.

Lemma 2.6 ([23]). Assume {aⁿ} is a sequence of nonnegative real numbers such that aⁿ⁺¹ ≤(1−γ_n)aⁿ+δ_nγ_n,

where {γ_n} is a sequence in (0,1) and{δ_n} is a sequence such that (1) P∞

n=1γ_n=∞;

(2) lim sup_n→∞δn≤0 or P∞

n=1|δ_nγn|<∞.

Thenlimn→∞aⁿ= 0.

3. Main results

In this section we will introduce two schemes for finding the unique point x^∗ which solves the quadratic minimization

kx^∗k² = min

x∈F ix(N)∩Ωkxk². (3.1)

LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetR:C→Hbe aκ-contraction. Let the mappingsF,G:C→H beζ-inverse strongly monotone andδ-inverse strongly monotone, respectively.

Supposeη∈(0,2ζ) and ξ∈(0,2δ). Let N :C→Cbe a nonexpansive mapping.

For eacht∈(0,1), we study the following mappingT^t given by

T^tx=N P roj_C[tR(x) + (1−t)P roj_C(I−ηF)P roj_C(I−ξG)x],∀x∈C.

Since the mappings N, P roj_C, I − ηF and I − ξG are nonexpansive, we can check easily that kT^tx−T^tyk ≤[1−(1−κ)t]kx−ykwhich implies thatT^tis a contraction. Then there exists a unique fixed point x^t ofT^t inCsuch that







z^t=P roj_C(x^t−ξGx^t), y^t=P roj_C(z^t−ηFz^t),

x^t=N P roj_C[tR(x^t) + (1−t)y^t].

(3.2)

In particular, if we takeR≡0, then (3.2) reduces to







z^t=P roj_C(x^t−ξGx^t), y^t=P roj_C(z^t−ηFz^t), x^t=N P roj_C[(1−t)y^t].

(3.3)

We next prove that the implicit methods (3.2) and (3.3) both converge.

Theorem 3.1. Suppose Γ := F ix(N)∩Ω6= ∅. Then the net {x^t} generated by the implicit method (3.2) converges in norm, ast→0⁺, to the unique solution x^∗ of the following variational inequality

x^∗∈Γ, h(I−R)x^∗, x−x^∗i ≥0, x∈Γ. (3.4) In particular, if we take R = 0, then the net {x^t} defined by (3.3) converges in norm, as t → 0⁺, to the minimum norm element inΓ, namely, the unique solution x^∗ to the quadratic minimization problem:

x^∗ = arg min

x∈Γkxk². (3.5)

Proof. First, we prove that{x^t} is bounded. Takeu∈Γ. From Lemma 2.3, we haveu=N uand u=P roj_C[P roj_C(u−ξGu)−ηFP roj_C(u−ξGu)].

Putv=P roj_C(u−ξGu). Thenu=P roj_C(v−ηFv). From Lemma 2.2, we note that kz^t−vk=kP roj_C(x^t−ξGx^t)−P roj_C(u−ξGu)k ≤ kx^t−uk, and

ky^t−uk=kP roj_C(z^t−ηFz^t)−P roj_C(v−ηFv)k ≤ kz^t−vk.

It follows from (3.2) that

kx^t−uk=kN P roj_C[tR(x^t) + (1−t)y^t]−N P roj_Cuk

≤ kt(R(x^t)−u) + (1−t)(y^t−u)k

≤tkR(x^t)−R(u)k+tkR(u)−uk+ (1−t)ky^t−uk

≤tκkx^t−uk+tkR(u)−uk+ (1−t)kx^t−uk

= [1−(1−κ)t]kx^t−uk+tkR(u)−uk, that is,

kx^t−uk ≤ kR(u)−uk 1−κ .

Hence,{x^t}is bounded and so are{y^t},{z^t}and {R(x^t)}. Now we can choose a constantM >0 such that sup

t

n2kR(x^t)−ukky^t−uk+kR(x^t)−uk²,2ξkx^t−z^t−(u−v)k, 2ηkz^t−y^t+ (u−v)k,ky^t−R(x^t)k²o

≤M.

SinceFisζ-inverse strongly monotone andGisδ-inverse strongly monotone, we have from Lemma 2.2 that ky^t−uk²=k(I−ηF)z^t−(I−ηF)vk²

≤ kz^t−vk²+η(η−2ζ)kFz^t−Fvk², (3.6) and

kz^t−vk² =k(I−ξG)x^t−(I −ξG)uk²

≤ kx^t−uk²+ξ(ξ−2δ)kGx^t−Guk². (3.7) Combining (3.6) with (3.7) to get

ky^t−uk² =k(I−ηF)z^t−(I−ηF)vk²

≤ kx^t−uk²+η(η−2ζ)kFz^t−Fvk²

+ξ(ξ−2δ)kGx^t−Guk². (3.8)

From (3.2) and (3.8), we have

kx^t−uk² ≤ k(1−t)(y^t−u) +t(R(x^t)−u)k²

= (1−t)²ky^t−uk²+ 2t(1−t)hR(x^t)−u, y^t−ui+t²kR(x^t)−uk²

=ky^t−uk²+tM (3.9)

≤ kx^t−uk²+η(η−2ζ)kFz^t−Fvk² +ξ(ξ−2δ)kGx^t−Guk²+tM,

that is,

η(2ζ−η)kFz^t−Fvk²+ξ(2δ−ξ)kGx^t−Guk²≤tM →0.

Since η(2ζ−η)>0 andξ(2δ−ξ)>0, we derive

limt→0kFz^t−Fvk= 0 and lim

t→0kGx^t−Guk= 0. (3.10)

From Lemma 2.1 and (3.2), we obtain

kz^t−vk² =kP roj_C(x^t−ξGx^t)−P roj_C(u−ξGu)k²

≤ h(x^t−ξGx^t)−(u−ξGu), z^t−vi

= 1 2

k(x^t−ξGx^t)−(u−ξGu)k²+kz^t−vk²

− k(x^t−u)−ξ(Gx^t−Gu)−(z^t−v)k²

≤ 1 2

kx^t−uk²+kz^t−vk²− k(x^t−z^t)−ξ(Gx^t−Gu)−(u−v)k²

= 1 2

kx^t−uk²+kz^t−vk²− kx^t−z^t−(u−v)k² + 2ξhx^t−z^t−(u−v),Gx^t−Gui −ξ²kGx^t−Guk²

,

and

ky^t−uk=kP roj_C(z^t−ηFz^t)−P roj_C(v−ηFv)k²

≤ hz^t−ηFz^t−(v−ηFv), y^t−ui

= 1 2

kz^t−ηFz^t−(v−ηFv)k²+ky^t−uk²

− kz^t−ηFz^t−(v−ηFv)−(y^t−u)k²

≤ 1 2

kz^t−vk²+ky^t−uk²− kz^t−y^t+ (u−v)k² + 2ηhFz^t−Fv, z^t−y^t+ (u−v)i −η²kFz^t−Fvk²

≤ 1 2

kx^t−uk²+ky^t−uk²− kz^t−y^t+ (u−v)k² + 2ηhFz^t−Fv, z^t−y^t+ (u−v)i

.

Thus, we have

kz^t−vk² ≤ kx^t−uk²− kx^t−z^t−(u−v)k²+MkGx^t−Guk, (3.11) and

ky^t−uk² ≤ kx^t−uk²− kz^t−y^t+ (u−v)k²+MkFz^t−Fvk. (3.12) By (3.9) and (3.11), we have

kx^t−uk² ≤ ky^t−vk²+tM

≤ kz^t−vk²+tM

≤ kx^t−uk²− kx^t−z^t−(u−v)k²+ (kGx^t−Guk+t)M.

It follows that

kx^t−z^t−(u−v)k²≤(kGx^t−Guk+t)M.

Since kGx^t−Guk →0, we deduce that

t→0limkx^t−z^t−(u−v)k= 0. (3.13)

From (3.9) and (3.12), we have

kx^t−uk² ≤ ky^t−uk²+tM

≤ kx^t−uk²− kz^t−y^t+ (u−v)k²+ (kFz^t−Fvk+t)M.

It follows that

kz^t−y^t+ (u−v)k² ≤(kFzⁿ−Fvk+t)M, which implies that

limt→0kz^t−y^t+ (u−v)k= 0. (3.14)

Thus, combining (3.13) with (3.14), we deduce that

limt→0kx^t−y^tk= 0. (3.15)

We note that

kx^t−N y^tk=kN P roj_C[tR(x^t) + (1−t)y^t]−N P roj_Cy^tk

≤tM →0.

Hence,

kN y^t−y^tk ≤ kN y^t−x^tk+kx^t−y^tk →0.

Therefore,

kx^t−N x^tk →0. (3.16)

At the same time, from (3.2) and Lemma 2.3, we have

kx^t−U(x^t)k=kN P roj_C[tR(x^t) + (1−t)U(x^t)]−N P roj_C[U(x^t)]k

≤tM →0. (3.17)

Next we show that {x^t} is relatively norm compact as t → 0. Let {tⁿ} ⊂ (0,1) be a sequence such that tⁿ→0 asn→ ∞. Put xⁿ:=x^tn and yⁿ:=y^tn. From (3.15)-(3.17), we have

kxⁿ−yⁿk →0, kxⁿ−N xⁿk →0andkxⁿ−U(xⁿ)k →0. (3.18) From (3.2), we get

kx^t−uk² =kN P roj_C[tR(x^t) + (1−t)y^t]−N uk²

≤ ky^t−u−ty^t+tR(x^t)k²

=ky^t−uk²−2thy^t, y^t−ui+ 2thR(x^t), y^t−ui+t²ky^t−R(x^t)k²

=ky^t−uk²−2thy^t−u, y^t−ui −2thu, y^t−ui

+ 2thR(x^t)−R(u), y^t−ui+ 2thR(u), y^t−ui+t²ky^t−R(x^t)k²

≤[1−2(1−κ)t]kx^t−uk²+ 2thR(u)−u, y^t−ui +t²ky^t−R(x^t)k².

It follows that

kx^t−uk² ≤ 1

1−κhu−R(u), u−y^ti+ t

2(1−κ)ky^t−R(x^t)k²

≤ 1

1−κhu−R(u), u−y^ti+ t 2(1−κ)M.

In particular,

kxⁿ−uk² ≤ 1

1−κhu−R(u), u−yⁿi+ tⁿ

2(1−κ)M, u∈Γ. (3.19)

By the boundedness of{xⁿ}, without loss of generality, we assume that {xⁿ} converges weakly to a point x^∗ ∈C. It is clear that yⁿ →x^∗ weakly. From (3.18) we can use Lemma 2.5 to get x^∗ ∈Γ. We substitute x^∗ foru in (3.19) to get

kxⁿ−x^∗k²≤ 1

1−κhx^∗−R(x^∗), x^∗−yⁿi+ tⁿ 2(1−κ)M.

So the weak convergence of{yⁿ}tox^∗ implies thatxⁿ→x^∗ strongly. We prove the relative norm compact- ness of the net {x^t} ast→0. In (3.19), we take the limit as n→ ∞ to get

kx^∗−uk² ≤ 1

1−κhu−R(u), u−x^∗i, u∈Γ. (3.20) Which implies that x^∗ solves the following variational inequality

x^∗∈Γ, h(I−R)u, u−x^∗i ≥0, u∈Γ.

It equals to its dual variational inequality

x^∗ ∈Γ, h(I−R)x^∗, u−x^∗i ≥0, u∈Γ.

Therefore, x^∗ = (P roj_ΓR)x^∗. This shows thatx^∗ is the unique fixed point in Γ of the contraction P roj_ΓR. This is sufficient to conclude that the entire net{x^t} converges in norm tox^∗ ast→0.

Setting R= 0, then (3.20) is reduced to

kx^∗−uk²≤ hu, u−x^∗i, u∈Γ.

Equivalently,

kx^∗k²≤ hx^∗, ui, u∈Γ.

This implies that

kx^∗k ≤ kuk, u∈Γ.

Therefore, x^∗ is the minimum norm element in Γ. This completes the proof.

Below we introduce an explicit scheme for finding the minimum-norm element in Γ.

Theorem 3.2. Suppose Γ :=F ix(N)∩Ω6=∅. For given x0 ∈C arbitrarily, let the sequences {xⁿ}, {yⁿ} and {zⁿ} be generated iteratively by







zⁿ=P_C(xⁿ−ξGxⁿ), yⁿ=PC(zⁿ−ηFzⁿ),

xⁿ⁺¹=δ_nxⁿ+ (1−δ_n)N P roj_C[ζ_nR(xⁿ) + (1−ζ_n)yⁿ], n≥0,

(3.21)

where {ζ_n} and {δ_n} are two sequences in[0,1] satisfying the following conditions:

(i) lim

n→∞ζ_n= 0 and

∞

P

n=0

ζ_n=∞;

(ii) 0<lim inf

n→∞ δ_n≤lim sup

n→∞

δ_n<1.

Then the sequence {xⁿ}converges strongly tox^∗ which is the unique solution of variational inequality (3.4).

In particular, if R= 0, then x^∗ is the minimum norm element in Γ.

Proof. First, we prove that the sequences {xⁿ},{yⁿ} and {zⁿ} are bounded.

Let v=P roj_C(u−ξGu) and u=P roj_C(v−ηFv). From (3.21), we get kyⁿ−uk=kP roj_C(zⁿ−ηFzⁿ)−P roj_C(v−ηFv)k

≤ kzⁿ−vk

=kP roj_C(xⁿ−ξGxⁿ)−P roj_C(u−ξGu)k

≤ kxⁿ−uk, and

kxⁿ⁺¹−uk=kδ_n(xⁿ−u) + (1−δn)(N P roj_C[ζnR(xⁿ) + (1−ζn)yⁿ]−u)k

≤δnkxⁿ−uk+ (1−δn)kζ_n(R(xⁿ)−u) + (1−ζn)(yⁿ−u)k

≤δ_nkxⁿ−uk+ (1−δ_n)[ζ_nkR(xⁿ)−R(u)k+ζ_nkR(u)−uk+ (1−ζ_n)kyⁿ−uk]

≤δ_nkxⁿ−uk+ (1−δ_n)[ζ_nκkxⁿ−uk+ζ_nkR(u)−uk+ (1−ζ_n)kxⁿ−uk]

= [1−(1−κ)(1−δ_n)ζ_n]kxⁿ−uk+ζ_n(1−δ_n)kR(u)−uk

≤max{kxⁿ−uk,kR(u)−uk 1−κ }.

By induction, we obtain, for alln≥0,

kxⁿ−uk ≤max

kx₀−uk,kR(u)−uk 1−κ

.

Hence, {xⁿ} is bounded. Consequently, we deduce that {yⁿ} ,{zⁿ}, {R(xⁿ)}, {Fzⁿ} and {Gxⁿ} are all bounded. LetM >0 is a constant such that

sup

n

n

kyⁿk+kR(xⁿ)k,2kyⁿ−R(xⁿ)kkyⁿ−uk+kyⁿ−R(xⁿ)k²,

(kxⁿ−uk+kxⁿ⁺¹−uk),2ξkxⁿ−zⁿ−(u−v)k,2ηkzⁿ−yⁿ+ (u−v)k, 2ηkFzⁿ−Fvkkzⁿ−yⁿ+ (u−v)k,2ξkxⁿ−zⁿ−(u−v)kkGxⁿ−Gukko

≤M.

Next we show lim

n→∞kyⁿ−N yⁿk= 0.

Define xⁿ⁺¹=δnxⁿ+ (1−δn)uⁿ for all n≥0. It follows from (3.21) that

kuⁿ⁺¹−uⁿk=kN P roj_C[ζ_n+1R(xⁿ⁺¹) + (1−ζ_n+1)yⁿ⁺¹]−N P roj_C[ζ_nR(xⁿ) + (1−ζ_n)yⁿ]k

≤ kζ_n+1R(xⁿ⁺¹) + (1−ζn+1)yⁿ⁺¹−ζnR(xⁿ)−(1−ζn)yⁿk

≤ kyⁿ⁺¹−yⁿk+ζ_n+1(kyⁿ⁺¹k+kR(xⁿ⁺¹)k) +ζ_n(kyⁿk+kR(xⁿ)k)

≤ kP roj_C(zⁿ⁺¹−ηFzⁿ⁺¹)−P roj_C(zⁿ−ηFzⁿ)k+M(ζn+1+ζn)

≤ kzⁿ⁺¹−zⁿk+M(ζ_n+1+ζ_n)

=kP roj_C(xⁿ⁺¹−ξGxⁿ⁺¹)−P roj_C(xⁿ−ξGxⁿ)k+M(ζn+1+ζn)

≤ kxⁿ⁺¹−xⁿk+M(ζ_n+1+ζ_n).

This together with (i) imply that lim sup

n→∞

kuⁿ⁺¹−uⁿk − kxⁿ⁺¹−xⁿk

≤0.

Hence by Lemma 2.4, we get lim

n→∞kuⁿ−xⁿk= 0. Consequently,

n→∞lim kxⁿ⁺¹−xⁿk= lim

n→∞(1−δn)kuⁿ−xⁿk= 0.

By the convexity of the normk · k, we have

kxⁿ⁺¹−uk²=kδ_n(xⁿ−u) + (1−δn)(uⁿ−u)k²

≤δ_nkxⁿ−uk²+ (1−δ_n)kuⁿ−uk²

≤δnkxⁿ−uk²+ (1−δn)kyⁿ−u−ζn(yⁿ−R(xⁿ))k²

≤δ_nkxⁿ−uk²+ (1−δ_n)kyⁿ−uk²+ζ_nM. (3.22) From Lemma 2.2 and (3.21), we have

kyⁿ−uk²≤ kzⁿ−vk²+η(η−2ζ)kFzⁿ−Fvk²

≤ kxⁿ−uk²+ξ(ξ−2δ)kGxⁿ−Guk²+η(η−2ζ)kFzⁿ−Fvk². (3.23) Substituting (3.23) into (3.22), we have

kxⁿ⁺¹−uk² ≤δnkxⁿ−uk²+ (1−δn)[kxⁿ−uk²+ξ(ξ−2δ)kGxⁿ−Guk² +η(η−2ζ)kFzⁿ−Fvk²] +ζ_nM

=kxⁿ−uk²+ (1−δn)ξ(ξ−2δ)kGxⁿ−Guk² + (1−δ_n)η(η−2ζ)kFzⁿ−Fvk²+ζ_nM.

Therefore,

(1−δ_n)η(2ζ−η)kFzⁿ−Fvk²+ (1−δ_n)ξ(2δ−ξ)kGxⁿ−Guk²

≤ kxⁿ−uk − kxⁿ⁺¹−uk+ζ_nM

≤(kxⁿ−uk+kxⁿ⁺¹−uk)kxⁿ−xⁿ⁺¹k+ζnM

≤(kxⁿ−xⁿ⁺¹k+ζ_n)M.

Since lim inf

n→∞ (1−δn)η(2ζ−η)>0, lim inf

n→∞ (1−δn)ξ(2δ−ξ)>0,kxⁿ−xⁿ⁺¹k →0 and ζn→0, we derive

n→∞lim kFzⁿ−Fvk= 0 and lim

n→∞kGxⁿ−Guk= 0.

From Lemma 2.1 and (3.21), we obtain

kzⁿ−vk² =kP_C(xⁿ−ξGxⁿ)−PC(u−ξGu)k²

≤ h(xⁿ−ξGxⁿ)−(u−ξGu), zⁿ−vi

= 1 2

k(xⁿ−ξGxⁿ)−(u−ξGu)k²+kzⁿ−vk²− k(xⁿ−u)−ξ(Gxⁿ−Gu)−(zⁿ−v)k²

≤ 1 2

kxⁿ−uk²+kzⁿ−vk²− k(xⁿ−zⁿ)−ξ(Gxⁿ−Gu)−(u−v)k²

= 1 2

kxⁿ−uk²+kzⁿ−vk²− kxⁿ−zⁿ−(u−v)k² + 2ξhxⁿ−zⁿ−(u−v),Gxⁿ−Gui −ξ²kGxⁿ−Guk²

,

and

kyⁿ−uk=kP roj_C(zⁿ−ηFzⁿ)−P roj_C(v−ηFv)k²

≤ hzⁿ−ηFzⁿ−(v−ηFv), yⁿ−ui

= 1 2

kzⁿ−ηFzⁿ−(v−ηFv)k²+kyⁿ−uk²

− kzⁿ−ηFzⁿ−(v−ηFv)−(yⁿ−u)k²

≤ 1 2

kzⁿ−vk²+kyⁿ−uk²− kzⁿ−yⁿ+ (u−v)k² + 2ηhFzⁿ−Fv, zⁿ−yⁿ+ (u−v)i −η²kFzⁿ−Fvk²

≤ 1 2

kxⁿ−uk²+kyⁿ−uk²− kzⁿ−yⁿ+ (u−v)k² + 2ηhFzⁿ−Fv, zⁿ−yⁿ+ (u−v)i

.

Thus, we deduce

kzⁿ−vk²≤ kxⁿ−uk²− kxⁿ−zⁿ−(u−v)k² + 2ξkxⁿ−zⁿ−(u−v)kkGxⁿ−Guk

≤ kxⁿ−uk²− kxⁿ−zⁿ−(u−v)k²+MkGxⁿ−Guk (3.24) and

kyⁿ−uk² ≤ kxⁿ−uk²− kzⁿ−yⁿ+ (u−v)k²+MkFzⁿ−Fvk. (3.25) By (3.22) and (3.24), we have

kxⁿ⁺¹−uk²≤δ_nkxⁿ−uk²+ (1−δ_n)kyⁿ−uk²+ζ_nM

≤δnkxⁿ−uk²+ (1−δn)kzⁿ−vk²+ζnM

≤δ_nkxⁿ−uk²+ (1−δ_n)[kxⁿ−uk²− kxⁿ−zⁿ−(u−v)k² +MkGxⁿ−Guk] +ζnM

≤ kxⁿ−uk²−(1−δn)kxⁿ−zⁿ−(u−v)k²+ (kGxⁿ−Guk+ζn)M.

It follows that

(1−δ_n)kxⁿ−zⁿ−(u−v)k²≤(kxⁿ⁺¹−xⁿk+kGxⁿ−Guk+ζ_n)M.

Since lim inf

n→∞ (1−δn)>0, ζn→0,kxⁿ⁺¹−xⁿk →0 and kGxⁿ−Guk →0, we deduce that

n→∞lim kxⁿ−zⁿ−(u−v)k= 0. (3.26)

From (3.22) and (3.25), we have

kxⁿ⁺¹−uk²≤δnkxⁿ−uk²+ (1−δn)kyⁿ−uk²+ζnM

≤δ_nkxⁿ−uk²+ (1−δ_n)[kxⁿ−uk²− kzⁿ−yⁿ+ (u−v)k² +MkFzⁿ−Fvk] +ζ_nM

≤ kxⁿ−uk²−(1−δn)kzⁿ−yⁿ+ (u−v)k²+ (kFzⁿ−Fvk+ζn)M.

It follows that

(1−δ_n)kzⁿ−yⁿ+ (u−v)k² ≤(kxⁿ⁺¹−xⁿk+kFzⁿ−Fvk+ζ_n)M,

which implies that

n→∞lim kzⁿ−yⁿ+ (u−v)k= 0. (3.27)

Thus, from (3.26) and (3.27), we deduce that

n→∞lim kxⁿ−yⁿk= 0.

Hence,

kN yⁿ−uⁿk=kN P roj_Cyⁿ−N P roj_C[ζnR(xⁿ) + (1−ζn)yⁿ]k ≤ζnM →0.

Therefore,

kN yⁿ−yⁿk ≤ kN yⁿ−uⁿk+kuⁿ−xⁿk+kxⁿ−yⁿk →0.

Next we prove

lim sup

n→∞

hx^∗−R(x^∗), x^∗−yⁿi ≤0, wherex^∗ =P_ΓR(x^∗).

Indeed, we can choose a subsequence {y_ni} of {yⁿ} such that lim sup

n→∞

hx^∗−R(x^∗), x^∗−yⁿi= lim

i→∞hx^∗−R(x^∗), x^∗−ynii.

Without loss of generality, we may further assume thatyni →zweakly, then it is clear thatz∈Γ. Therefore, lim sup

n→∞

hx^∗−R(x^∗), x^∗−yⁿi= lim

i→∞hx^∗−R(x^∗), x^∗−zi ≤0.

From (3.21), we have

kxⁿ⁺¹−x^∗k²≤δ_nkxⁿ−x^∗k²+ (1−δ_n)kζ_n(R(xⁿ)−x^∗) + (1−ζ_n)(yⁿ−x^∗)k²

≤δnkxⁿ−x^∗k²+ (1−δn)[(1−ζn)²kyⁿ−x^∗k²

+ 2ζ_n(1−ζ_n)hR(xⁿ)−x^∗, yⁿ−x^∗i+ζ_n²kR(xⁿ)−x^∗k²]

=δnkxⁿ−x^∗k²+ (1−δn)[(1−ζn)²kyⁿ−x^∗k² + 2ζn(1−ζn)hR(xⁿ)−R(x)^∗, yⁿ−x^∗i

+ 2ζ_n(1−ζ_n)hR(x^∗)−x^∗, yⁿ−x^∗i+ζ_n²kR(xⁿ)−x^∗k²]

≤[1−2(1−κ)(1−δn)ζn]kxⁿ−x^∗k²

+ 2ζ_n(1−ζ_n)(1−δ_n)hR(x^∗)−x^∗, yⁿ−x^∗i+ (1−δ_n)ζ_n²M

= (1−γⁿ))kxⁿ−x^∗k²+δⁿγⁿ,

whereγⁿ= 2(1−κ)(1−δn)ζn and δⁿ/= ^(1−ζ_1−κⁿ⁾hR(x)^∗−x^∗, yⁿ−x^∗i+_2(1−κ)^ζnM . It is clear that

∞

P

n=0

γⁿ=∞ and lim sup

n→∞ δ ≤0. Hence, all conditions of Lemma 2.6 are satisfied. Therefore, we immediately deduce that xⁿ→x^∗.

Finally, if we takeR= 0, by the similar argument as that Theorem 3.1, we deduce immediately that x^∗ is a solution of (3.5). This completes the proof.

Acknowledgment

Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230- 005-CC3. Li-Jun Zhu was supported in part by NNSF of China (61362033).

References

[1] J. Y. Bello Cruz, A. N. Iusem,Convergence of direct methods for paramonotone variational inequalities, Comput.

Optim. Appl.,46(2010), 247–263. 1

[2] A. Cabot,Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimiza- tion, SIAM J. Optim.,15(2005), 555–572.

[3] L. C. Ceng, C. Wang, J. C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res.,67(2008), 375–390. 1, 2.2, 2.3

[4] Y. Censor, A. Gibali, S. Reich,The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl.,148(2011), 318–335.

[5] S. S. Chang, H. W. Joseph Lee, C. K. Chan, J. A. Liu,A new method for solving a system of generalized nonlinear variational inequalities in Banach spaces, Appl. Math. Comput.,217(2011), 6830–6837.

[6] R. Chen, Y. Su, H. K. Xu,Regularization and iteration methods for a class of monotone variational inequalities, Taiwanese J. Math.,13(2009), 739–752.

[7] B. S. He, Z. H. Yang, X. M. Yuan,An approximate proximal-extragradient type method for monotone variational inequalities, J. Math. Anal. Appl.,300(2004), 362–374.

[8] G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, `Ekonom. i Mat.

Metody,12(1976), 747–756.

[9] P. Li, S. M. Kang, L. J. Zhu,Visco-resolvent algorithms for monotone operators and nonexpansive mappings, J.

Nonlinear Sci. Appl.,7(2014), 325–344.

[10] X. Lu, H. K. Xu, X. Yin,Hybrid methods for a class of monotone variational inequalities, Nonlinear Anal.,71 (2009), 1032–1041. 2.5

[11] N. Nadezhkina, W. Takahashi,Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl.,128(2006), 191–201. 1, 1

[12] M. A. Noor,Some developments in general variational inequalities, Appl. Math. Comput.,152(2004), 199–277.

1

[13] J. W. Peng, J. C. Yao,Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems, Math. Comput. Modeling,49(2009), 1816–1828. 1 [14] A. Sabharwal, L. C. Potter,Convexly constrained linear inverse problems: iterative least-squares and regulariza-

tion, IEEE Trans. Signal Process.,46(1998), 2345–2352. 1

[15] P. Saipara, P. Chaipunya, Y. J. Cho, P. Kumam, On strong and ∆-convergence of modified S-iteration for uniformly continuous total asymptotically nonexpansive mappings in CAT(κ) spaces, J. Nonlinear Sci. Appl.,8 (2015), 965–975. 1

[16] P. Shi,Equivalence of variational inequalities with Wiener-Hopf equations, Proc. Amer. Math. Soc.,111(1991), 339–346. 1

[17] G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C.R. Acad. Sci. Paris, 258 (1964), 4413–4416. 1

[18] T. Suzuki,Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl.,2005(2005), 103–123. 2.4

[19] W. Takahashi, M. Toyoda,Weak convergence theorems for nonexpansive mappings and monotone mappings, J.

Optim. Theory Appl.,118(2003), 417–428. 1

[20] R. U. Verma,On a new system of nonlinear variational inequalities and associated iterative algorithms, Math.

Sci. Res. Hot-line,3(1999), 65–68. 1

[21] R. U. Verma,Iterative algorithms and a new system of nonlinear quasivariational inequalities, Adv. Nonlinear Var. Inequal.,4(2001), 117–124. 1

[22] U. Witthayarat, Y. J. Cho, P. Kumam, Approximation algorithm for fixed points of nonlinear operators and solutions of mixed equilibrium problems and variational inclusion problems with applications, J. Nonlinear Sci.

Appl.,5(2012), 475–494.

[23] H. K. Xu,An iterative approach to quadratic optimization, J. Optimi. Theory Appl.,116(2003), 659–678. 2.6 [24] H. K. Xu,A variable Krasnoselski-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems,

22(2006), 2021–2034.

[25] H. K. Xu,Viscosity method for hierarchical fixed point approach to variational inequalities, Taiwanese J. Math., 14(2010), 463–478.

[26] H. K. Xu, T. H. Kim, Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim.

Theory Appl.,119(2003), 185–201.

[27] I. Yamada, N. Ogura,Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings, Numer. Funct. Anal. Optim.,25(2005), 619–655.

[28] I. Yamada, N. Ogura, N. Shirakawa,A numerically robust hybrid steepest descent method for the convexly con- strained generalized inverse problems, Inverse Problems, Image Analysis, and Medical Imaging, Contemp. Math., 313(2002), 269–305. 1

[29] J. C. Yao,Variational inequalities with generalized monotone operators, Math. Oper. Res.,19(1994), 691–705. 1

[30] Y. Yao, Y. C. Liou, Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems, Inverse Problems,24(2008), 501–508. 1

[31] Y. Yao, Y. C. Liou, J. C. Yao, An iterative algorithm for approximating convex minimization problem, Appl.

Math. Comput.,188(2007), 648–656.

[32] Y. Yao, J. C. Yao,On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math.

Comput.,186(2007), 1551–1558.

[33] Z. Yao, L. J. Zhu, S. M. Kang, Y. C. Liou,Iterative algorithms with perturbations for Lipschitz pseudocontractive mappings in Banach spaces, J. Nonlinear Sci. Appl.,8(2015), 935–943.

[34] L. C. Zeng, J. C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J. Math.,10(2006), 1293–1303. 1, 1