R E S E A R C H
Open Access
Implicit and explicit iterative algorithms
for hierarchical variational inequality in
uniformly smooth Banach spaces
Lu-Chuan Ceng
1, Yung-Yih Lur
2and Ching-Feng Wen
3,4**Correspondence: [email protected]
3Center for Fundamental Science; and Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung, 80702, Taiwan 4Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung, 80702, Taiwan Full list of author information is available at the end of the article
Abstract
The purpose of this paper is to solve the hierarchical variational inequality with the constraint of a general system of variational inequalities in a uniformly convex and 2-uniformly smooth Banach space. We introduce implicit and explicit iterative algorithms which converge strongly to a unique solution of the hierarchical variational inequality problem. Our results improve and extend the corresponding results announced by some authors.
MSC: 49J30; 47H09; 47J20; 49M05
Keywords: system of variational inequalities; hierarchical variational inequality; nonexpansive mapping; fixed point; implicit and explicit iterative algorithms; global convergence
1 Introduction
LetXbe a real Banach space with its topological dualX∗, andCbe a nonempty closed convex subset ofX. LetT:C→Xbe a nonlinear mapping onC. We denote byFix(T) the set of fixed points ofT and by R the set of all real numbers. A mappingT:C→Xis calledL-Lipschitz continuous if there exists a constantL≥ such that
Tx–Ty ≤Lx–y, ∀x,y∈C.
In particular, ifL= thenTis called a nonexpansive mapping; ifL∈[, ) thenTis called a contraction.
The normalized dual mappingJ:X→X∗is defined as
J(x) :=ϕ∈X∗: x,ϕ=x=ϕ, ∀x∈X, ()
where ·,·denotes the generalized duality pairing; see, e.g., [] for further details. LetU:={x∈X:x= }be the unit sphere ofX. Then the spaceXis said to
• have a Gâteaux differentiable norm if the limitlimt→+(x+ty–x)/texists for
eachx,y∈U.
• have a uniformly Gâteaux differentiable norm if the limit is attained uniformly for x∈U.
• be strictly convex if and only if forx,y∈Uwithx=y, we have
( –λ)x+λy< , ∀λ∈(, ).
We note that ifXis smooth, then the normalized duality mapping is single-valued; and if the norm ofXis uniformly Gâteaux differentiable, then the normalized duality mapping is norm to weak star uniformly continuous on every bounded subset ofX(see []). In the sequel, we shall denote byjthe single-valued normalized duality mapping.
LetXbe a smooth Banach space. LetA,B:C→Xbe two nonlinear mappings andλ,μ be two positive real numbers. The general system of variational inequalities (GSVI, for short) is to find (x∗,y∗)∈C×Csuch that
⎧ ⎨ ⎩
λAy∗+x∗–y∗,j(x–x∗) ≥, ∀x∈C,
μBx∗+y∗–x∗,j(x–y∗) ≥, ∀x∈C. ()
The equivalence between GSVI () and the fixed point problem of some nonexpansive mapping defined on a Banach space is established in Yao et al. []. The authors [] intro-duced and analyzed implicit and explicit iterative algorithms for solving GSVI () by using this equivalence, and they proved the strong convergence of the sequences generated by the proposed algorithms. Subsequently, Ceng et al. [] proposed and analyzed an implicit algorithm of Mann’s type and another explicit algorithm of Mann’s type for solving GSVI ().
IfX is a real Hilbert space, then GSVI () was introduced and studied by Ceng et al. []. In this case, forA=B, it was considered by Verma [] (see also []). Further, in this case, whenx∗=y∗, problem () reduces to the following classical variational inequality (in short, VI) of findingx∗∈Csuch that
Ax∗,x–x∗ ≥, ∀x∈C. ()
This problem is a fundamental problem in the variational analysis, in particular, in the optimization theory and mechanics; see, e.g., [–] and the references therein. A large number of algorithms for solving this problem are essentially projection algorithms that employ projections onto the feasible setCof the VI, or onto some related set, so as to iter-atively reach a solution. In particular, Korpelevich [] proposed extragradient method for solving the VI in the Euclidean space. This method further has been improved by several researchers; see, e.g., [, , ] and the references therein.
In the case of a Banach space setting, that is, ifA=Bandx∗=y∗, the VI is defined as
Ax∗,jx–x∗ ≥, ∀x∈C. ()
To overcome this drawback in a Hilbert space, where the retraction is a metric projection, in Yamada [], the feasible set is assumed to be the common fixed points of a finite family of nonexpansive mappings, and an explicit hybrid steepest-descent method is introduced. In this case, the variational inequality defined on such a feasible set is also called a hier-archical variational inequality (in short, HVI). Yamada’s method is then extended to solve more complex problems involving finite or infinite nonexpansive mappings (one can re-fer to, e.g., [, ] and the rere-ferences therein). Zeng and Yao [] introduced an implicit method that converges weakly to a solution of a variational inequality, containing a Lips-chitz continuous and strongly monotone mapping in a Hilbert spaceH, where the feasible set is the common fixed points of a finite family of nonexpansive mappings onH. Ceng et al. [] extended this result from nonexpansive mappings to Lipschitz pseudocontractive mappings and strictly pseudocontractive mappings onH. Recently, Buong and Anh [] modified Yamada’s result and proposed a strongly convergent implicit method.
In this paper, we are going to solve the hierarchical variational inequality with the constraint of a general system of variational inequalities in a uniformly convex and -uniformly smooth Banach space. We introduce implicit and explicit iterative algorithms for finding a solution of the problem and derive the strong convergence of the proposed algorithms to a unique solution of the problem. Our results improve and extend the corresponding results announced by some others, e.g., Ceng et al. [] and Buong and Phuong [].
2 Preliminaries
LetXbe a real Banach space with the dual spaceX∗. For simplicity, the norms ofXand
X∗are denoted by the symbol · . LetXbe a nonempty closed convex subset of a real Banach spaceX. We writexnx(respectively,xn→x) to indicate that the sequence{xn}
converges weakly (respectively, strongly) tox. A mappingJ:X→X∗, defined by
J(x) =ϕ∈X∗: x,ϕ=ϕandϕ=x,
is called the normalized duality mapping ofX. We know thatJ(tx) =tJ(x) for allt> and
x∈X, andJ(–x) = –J(x).
LetU:={x∈X:x= }. A Banach spaceXis said to be uniformly convex if for each ε∈(, ], there existsδ> such that for anyx,y∈U,x+y> –δ⇒ x–y<ε. It is known that a uniformly convex Banach space is reflexive and strictly convex. Also, it is known that if a Banach spaceXis reflexive, thenXis strictly convex if and only ifX∗is smooth as well asXis smooth if and only ifX∗is strictly convex.
Proposition ([]) Let X be a smooth and uniformly convex Banach space,and let r> .
Then there exists a strictly increasing, continuous and convex function g: [, r]→R,
g() = such that
gx–y≤ x– x,j(y) +y, ∀x,y∈Br,
Here we define a functionρ: [,∞)→[,∞) called the modulus of smoothness ofXas follows:
ρ(τ) =sup
x+y+x–y– :x,y∈X,x= ,y=τ
.
It is known thatXis uniformly smooth if and only iflimτ→+ρ(τ)/τ = . Letqbe a fixed real number with <q≤. Then a Banach spaceXis said to beq-uniformly smooth if there exists a constantc> such thatρ(τ)≤cτqfor allτ > . For further details on the
geometry of Banach spaces, we refer to [, ] and the references therein. Takahashi et al. [] reminded us of the fact that no Banach space isq-uniformly smooth forq> . So, in this paper, we focus on only a -uniformly smooth Banach space as in [].
Lemma ([]) Let q be a given real number with <q≤,and let X be a q-uniformly smooth Banach space.Then
x+yq≤ xq+qy,Jq(x) + κyq, ∀x,y∈X,
whereκis the q-uniformly smooth constant of X and Jqis the generalized duality mapping from X intoX∗defined by
Jq(x) =ϕ∈X∗: x,ϕ=xq,ϕ=xq–, ∀x∈X.
LetDbe a subset ofC, and letbe a mapping ofCintoD. Thenis said to be sunny if
(x) +tx–(x)=(x),
whenever(x) +t(x–(x))∈Cforx∈Candt≥. A mappingofCinto itself is called a retraction if=. If a mappingofCinto itself is a retraction, then(z) =zfor each
z∈R(), whereR() is the range of. A subsetDofCis called a sunny nonexpansive retract ofCif there exists a sunny nonexpansive retraction fromContoD.
Lemma ([]) Let C be a nonempty closed convex subset of a smooth Banach space X,
D be a nonempty subset of C andbe a retraction of C onto D.Then the following are equivalent:
(i) is sunny and nonexpansive;
(ii) (x) –(y)≤ x–y,j((x) –(y)),∀x,y∈C; (iii) x–(x),j(y–(x)) ≤,∀x∈C,y∈D.
It is well known that ifXis a Hilbert space, then a sunny nonexpansive retractionC
coincides with the metric projection fromXontoC. LetCbe a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach spaceX, and letTbe a non-expansive mapping ofCinto itself with the fixed point setFix(T)=∅. Then the setFix(T) is a sunny nonexpansive retract ofC; see, e.g., [].
whereGSVI(C,A,B)is the set of fixed points of the mapping G:=C(I–λA)C(I–μB) and y∗=C(x∗–μBx∗).
Proposition ([]) Let C be a nonempty closed convex subset of a real-uniformly smooth Banach space X.Let the mappings A,B:C→X beα-inverse-strongly accretive andβ-inverse-strongly accretive,respectively.Then
(I–λA)x– (I–λA)y≤ x–y+ λκλ–αAx–Ay
and
(I–μB)x– (I–μB)y≤ x–y+ μκμ–βBx–By.
In particular,if≤λ≤κα and≤μ≤ β
κ,then I–λA and I–μB are nonexpansive.
Lemma ([]) Let C be a nonempty closed convex subset of a real-uniformly smooth Ba-nach space X.LetCbe a sunny nonexpansive retraction from X onto C.Let the mappings A,B:C→X beα-inverse-strongly accretive andβ-inverse-strongly accretive,respectively.
Let the mapping G:C→C be defined as G:=C(I–λA)C(I–μB).If≤λ≤κα and ≤μ≤κβ,then G:C→C is nonexpansive.
Let C be a nonempty closed convex subset of a uniformly convex and -uniformly
smooth Banach spaceX. LetCbe a sunny nonexpansive retraction fromXontoC. Let
the mappingsA,B:C→Xbeα-inverse-strongly accretive andβ-inverse-strongly accre-tive, respectively. LetF:C→Xbeδ-strongly accretive andζ-strictly pseudocontractive withδ+ζ> . Assume thatλ∈(,κα) andμ∈(,
β
κ), whereκis the -uniformly smooth constant of X(see Lemma ). Very recently, in order to solve GSVI (), Ceng et al. [] introduced an implicit algorithm of Mann’s type.
Algorithm ([]) For each t∈(, ),choose a numberθt∈(, )arbitrarily.The net{xt} is generated by the implicit method
xt=tC(I–λA)C(I–μB)xt+(–t)C(I–θtF)C(I–λA)C(I–μB)xt, ∀t∈(, ),
where xtis a unique fixed point of the contraction
Wt=tC(I–λA)C(I–μB) + ( –t)C(I–θtF)C(I–λA)C(I–μB).
It was proven in [] that the net{xt}converges in norm, ast→+, to the unique solution
x∗∈GSVI(C,A,B) to the following VI:
Fx∗,jx–x∗ ≥, ∀x∈GSVI(C,A,B), ()
Algorithm ([]) For arbitrarily given x∈C,let the sequence{xk}be generated iteratively by
xk+=βkxk+γkC(I–λA)C(I–μB)xk
+ ( –βk–γk)C(I–λkF)C(I–λA)C(I–μB)xk,
where{λk},{βk}and{γk}are three sequences in[, ]such thatβk+γk≤,∀k≥.
On the other hand, a mappingFwith domainD(F) and rangeR(F) inXis called
(a) accretive if for eachx,y∈D(F), there existsj(x–y)∈J(x–y)such that
Fx–Fy,j(x–y) ≥,
whereJis the normalized duality mapping;
(b) δ-strongly accretive if for eachx,y∈D(F), there existsj(x–y)∈J(x–y)such that
Fx–Fy,j(x–y) ≥δx–y for someδ∈(, );
(c) α-inverse-strongly accretive if for eachx,y∈D(F), there existsj(x–y)∈J(x–y)
such that
Fx–Fy,j(x–y) ≥αFx–Fy for someα∈(, );
(d) ζ-strictly pseudocontractive if for eachx,y∈D(F), there existsj(x–y)∈J(x–y)
such that
Fx–Fy,j(x–y) ≤ x–y–ζx–y– (Fx–Fy) for someζ∈(, ).
It is easy to see that () can be rewritten as (see [])
(I–F)x– (I–F)y,j(x–y) ≥ζ(I–F)x– (I–F)y, ()
whereIdenotes the identity mapping ofX. Clearly, ifFisζ-strictly pseudocontractive with ζ = , then it is said to be pseudocontractive. It is not hard to find that every nonexpansive mapping is pseudocontractive.
LetCbe a nonempty closed convex subset of a smooth Banach spaceXand{Ti}∞i=be an infinite family of nonexpansive self-mappings onC. Then we setF:=∞i=Fix(Ti). In , Buong and Phuong [] considered the following HVI withC=X: findx∗∈Fsuch that
Fx∗,jx–x∗ ≥, ∀x∈F. ()
In the case whereX=H, a Hilbert space, we haveJ=I, and hence problem () reduces to the HVI: findx∗∈Fsuch that
Assume thatF=Ni=Fix(Ti) is the set of common fixed points of a family ofN nonex-pansive mappingsTionH, andFis anL-Lipschitz continuous andη-strongly monotone mapping, i.e.,Fx–Fy ≤Lx–yand Fx–Fy,x–y ≥ηx–yfor allx,y∈H. Zeng and
Yao [] introduced the following implicit iteration: for an arbitrarily initial pointx∈H,
the sequence{xk}∞k=is generated as follows:
xk=βkxk–+ ( –βk)
T[k]xk–λkμF(T[k]xk)
, ∀k≥, ()
where T[n]=TnmodN, for integern≥, with the mod function taking values in the set
{, , . . . ,N}. They proved the following result.
Theorem ([]) Let H be a real Hilbert space,and let F:H→H be a mapping such that, for some positive constants L and η, F is L-Lipschitz continuous and η-strongly monotone.Let{Ti}Ni= be N nonexpansive mappings on H such thatF:Ni=Fix(Ti)=∅.
Let μ∈(, η/L), x
∈H, {λk}∞k= ⊂[, ) and{βk}∞k=⊂(, ) satisfying the conditions
∞
k=λk<∞,and let a≤βk≤b,k≥,for some a,b∈(, ).Then the sequence{xk}∞k=, defined by(),converges weakly to x∗∈F.
Recently, for deriving the strong convergence and weakening the condition onλk, Buong
and Anh [] proposed the following implicit iteration method:
xt=Ttxt, Tt:=TtTNt · · ·Tt,t∈(, ), ()
where the sequence{Tt
i}Ni=is defined by
⎧ ⎨ ⎩
Titx:= ( –βi
t)x+βtiTix, i= , . . . ,N, Tt
y:= (I–λtμF)y, x,y∈H,
()
and proved that the nets{xt}, defined by ()-(), converge strongly to an elementx∗in (). Under the assumptions thatN= ,Xis a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm, andT is a continuous pseudocontractive mapping, Ceng et al. [] proved the following result.
Theorem ([]) Let F be aδ-strongly accretive andζ-strictly pseudocontractive mapping withδ+ζ > ,and let T be a continuous and pseudocontractive mapping on X,which is a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm such thatF:=Fix(T)=∅.For each t∈(, ),choose a numberμt∈(, )arbitrarily, and let{zt}be defined by
zt=t(I–μtF)zt+ ( –t)Tzt. ()
In [], Takahashi introduced the followingW-mapping which is generated byTk,Tk–,
. . . ,Tand real numbersαk,αk–, . . . ,αas follows:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
Uk,k+=I,
Uk,k=αkTkUk,k++ ( –αk)I, Uk,k–=αk–Tk–Uk,k+ ( –αk–)I,
· · ·
Uk,=αTUk,+ ( –α)I,
Wk=Uk,=αTUk,+ ( –α)I.
()
Kikkawa and Takahashi [] considered the following strongly convergent implicit meth-od:
Skx=
–
k
Ux+
kf(x), and Ux=k→∞limWkx=k→∞lim Uk,x. ()
However, the method () is very difficult to be grasped due to the limit mappingU. Motivated by methods () and (), Buong and Phuong [] considered two implicit methods by introducing a mappingVkdefined by
Vk=Vk, Vki=TiTi+· · ·Tk, Ti= ( –αi)I+αiTi, i= , , . . . ,k, ()
where
αi∈(, ) and ∞
i=
αi<∞. ()
In both methods, the iteration sequence{xk}∞k=is defined, respectively, by
xk=Vk(I–λkF)xk, ∀k≥, ()
and
xk=γk(I–λkF)xk+ (I–γk)Vkxk, ∀k≥, ()
whereλk andγk are the positive parameters satisfying some additional conditions. The
strong convergence theorems for methods () and () are also established. We will make use of the following well-known results.
Lemma Let X be a real normed linear space.Then the following inequality holds: x+y≤ x+ y,j(x+y), ∀x,y∈X,∀j(x+y)∈J(x+y).
Lemma ([]) Let C be a nonempty closed convex subset of a real smooth Banach space X.Assume that the mapping F:C→X is accretive and weakly continuous along segments
(that is,F(x+ty)F(x)as t→).Then the variational inequality
x∗∈C, Fx∗,jx–x∗ ≥, ∀x∈C
is equivalent to the following Minty type variational inequality:
x∗∈C, F(x),jx–x∗ ≥, ∀x∈C.
Lemma ([]) Let X be a real smooth Banach space and F:C→X be a mapping.
(a) IfFisζ-strictly pseudocontractive,thenFis Lipschitz continuous with constant +ζ. (b) IfFisδ-strongly accretive andζ-strictly pseudocontractive withδ+ζ> ,thenI–F
is contractive with constant
–δ ζ ∈(, ).
(c) IfFisδ-strongly accretive andζ-strictly pseudocontractive withδ+ζ > ,then for any fixed numberλ∈(, ),I–λFis contractive with constant –λ( –
–δ
ζ )∈(, ).
3 Iterative algorithms and convergence criteria
In this section, we study iterative methods for computing approximate solutions of the HVI (for an infinite family of nonexpansive mappings) with a GSVI constraint. We intro-duce implicit and explicit iterative algorithms for solving such a problem. We show the strong convergence theorems for the sequences generated by the proposed algorithms.
The following lemmas will be used to prove our main results in the sequel.
Lemma ([]) Let C be a nonempty closed convex subset of a strictly convex Banach space X,and let{Ti}ki=,k≥,be k nonexpansive self-mappings on C such that the set of common fixed pointsF:=ki=Fix(Ti)= ∅.Let a,b andαi,i= , , . . . ,k,be real numbers such that
<a≤αi≤b< ,and let Vkbe a mapping defined by()for all k≥.ThenFix(Vk) =F.
Lemma ([]) Let C be a nonempty closed convex subset of a Banach space X,and let
{Ti}∞i=be an infinite family of nonexpansive self-mappings on C such that the set of common fixed pointsF:=∞i=Fix(Ti)=∅.Let Vk be a mapping defined by(),and letαisatisfy
().Then,for each x∈C and i≥,limk→∞Vkix exists.
Remark
(i) We can define the mappings
V∞i x:= lim k→∞V
i
kx and Vx:=V∞x=k→∞lim Vkx, ∀x∈C.
(ii) It can be readily seen from the proof of Lemma that ifDis a nonempty and bounded subset ofC, then the following holds:
lim k→∞supx∈DV
i
kx–V∞i x= , ∀i≥.
In particular, wheneveri= , we have
lim
Lemma ([]) Let C be a nonempty closed convex subset of a strictly convex Banach space X,and let{Ti}∞i=be an infinite family of nonexpansive self-mappings on C such that the set of common fixed pointsF:=∞i=Fix(Ti)=∅.Letαisatisfy the first condition in(). ThenFix(V) =F.
Lemma ([]) Let{xn}and{zn}be bounded sequences in a Banach space X,and let
{αk}be a sequence in[, ]such that
<lim inf
k→∞ αk≤lim supk→∞ αk< .
Suppose that xk+=αkxk+ ( –αk)zk,∀k≥,and
lim sup k→∞
zk+–zk–xk+–xk
≤.
Thenlimk→∞zk–xk= .
Lemma ([]) Assume that{ak}is a sequence of nonnegative real numbers such that ak+≤( –γk)ak+γkδk, ∀k≥,
where{γk}is a sequence in[, ]and{δk}is a sequence inRsuch that (i) ∞k=γk=∞;
(ii) lim supk→∞δk≤or
∞
k=|γkδk|<∞. Thenlimk→∞ak= .
Now, we are in a position to prove the following main results.
Theorem Let C be a nonempty closed convex subset of a uniformly convex and -uniformly smooth Banach space X.LetCbe a sunny nonexpansive retraction from X onto C.Let the mappings A,B:C→X beα-inverse-strongly accretive andβ-inverse-strongly ac-cretive,respectively.Let F:C→X beδ-strongly accretive andζ-strictly pseudocontractive withδ+ζ > .Assume thatλ∈(,κα)andμ∈(,
β
κ)whereκis the-uniformly smooth
constant of X.Let{Ti}∞i=be an infinite family of nonexpansive self-mappings on C such that
F:=∞i=Fix(Ti)∩GSVI(C,A,B)=∅.Let{Vk}∞k=be defined by()and().Let{xk}∞k= be defined by
xk=γkC(I–λkF)C(I–λA)C(I–μB)Vkxk
+ ( –γk)C(I–λA)C(I–μB)Vkxk, ∀k≥,
where{γk}and{λk}are sequences in(, ]such thatγk→andλk→as k→ ∞.Then
{xk}∞k=converges strongly to a unique solution x∗∈Fto the following VI:
Fx∗,jx–x∗ ≥, ∀x∈F. ()
Proof Let the mappingG:C→Cbe defined asG:=C(I–λA)C(I–μB), where <λ<
α
κ and <μ< β
the implicit iterative scheme can be rewritten as
xk=γkC(I–λkF)GVkxk+ ( –γk)GVkxk, ∀k≥. ()
Consider the mappingUkx=γkC(I–λkF)GVkx+ ( –γk)GVkx,∀x∈C. From Lemmas
and (c), it follows that for eachx,y∈C,
Ukx–Uky =γk
C(I–λkF)GVkx–C(I–λkF)GVky
+ ( –γk)(GVkx–GVky)
≤γk(I–λkF)GVkx– (I–λkF)GVky+ ( –γk)GVkx–GVky
≤γk( –λkτ)GVkx–GVky+ ( –γk)GVkx–GVky
= ( –γkλkτ)GVkx–GVky
≤( –γkλkτ)x–y,
whereτ= –
–δ
ζ ∈(, ) (by usingδ+ζ> ). Due toγkλkτ ∈(, ),Ukis a contraction ofCinto itself. Hence, by Banach’s contraction principle, there exists a unique element
xk∈Csatisfying ().
Next, we divide the rest of the proof into several steps.
Step . We show that{xk}∞k= is bounded. Indeed, take an arbitrarily givenp∈F. Then we haveVkp=pandGp=p. Hence, by Lemma (c) we get
xk–p =γkC(I–λkF)GVkxk+ ( –γk)GVkxk–p
≤γkC(I–λkF)GVkxk–p
+ ( –γk)GVkxk–p
≤γk(I–λkF)GVkxk–p
+ ( –γk)GVkxk–p
=γk(I–λkF)GVkxk– (I–λkF)p–λkF(p)
+ ( –γk)GVkxk–p
≤γk
( –λkτ)GVkxk–p+λkF(p)
+ ( –γk)GVkxk–p
≤γk
( –λkτ)GVkxk–p+λkτ–F(p)
+ ( –γk)GVkxk–p
= ( –γkλkτ)GVkxk–p+γkλkτ–F(p)
≤( –γkλkτ)xk–p+γkλkτ–F(p). ()
Therefore,xk–p ≤ F(p)/τ, which also leads to the boundedness of{xk}∞k=. So, the sequences {Vkxk}∞k=,{yk}∞k= and{F(yk)}∞k=, whereyk=GVkxk, are also bounded. Since
γk→ ask→ ∞, and the following relation holds
xk–GVkxk=γkC(I–λkF)GVkxk–GVkxk
≤γk(I–λkF)GVkxk–GVkxk
=γkλkF(yk)≤γkF(yk),
Step .We show thatzk–Gzk → ask→ ∞, wherezk=Vkxkfor allk≥. Indeed, for simplicity, putq=C(p–μBp) anduk=C(zk–μBzk). Thenyk=Gzk=C(uk–λAuk)
for allk≥. From Lemma , we have
uk–q =C(zk–μBzk) –C(p–μBp)
≤zk–p–μ(Bzk–Bp)
≤ zk–p– μβ–κμBzk–Bp
≤ xk–p– μ
β–κμBzk–Bp, ()
and
yk–p=C(uk–λAuk) –C(q–λAq)
≤uk–q–λ(Auk–Aq)
≤ uk–q– λα–κλAuk–Aq. ()
Substituting () for (), we obtain
yk–p≤ xk–p– μβ–κμBzk–Bp
– λα–κλAuk–Aq. ()
From () and (), we have
xk–p≤( –γkλkτ)Gzk–p+γkλkτ–F(p)
≤ Gzk–p+γkτ–F(p)
≤ xk–p– μβ–κμBzk–Bp
– λα–κλAuk–Aq+γk F(p)
τ ,
which immediately yields
μβ–κμBzk–Bp+ λα–κλAuk–Aq≤γk F(p)
τ .
So, fromλ∈(,κα),μ∈(, β
κ) andγk→ ask→ ∞, we deduce that
lim
k→∞Bzk–Bp= and k→∞limAuk–Aq= . ()
Utilizing Proposition and Lemma , we have
uk–q =C(zk–μBzk) –C(p–μBp)
≤(zk–μBzk) – (p–μBp),j(uk–q)
≤
zk–p+uk–q–gzk–uk– (p–q)
+μBp–Bzkuk–q
≤
xk–p+uk–q–gzk–uk– (p–q)
+μBp–Bzkuk–q,
which implies that
uk–q≤ xk–p–gzk–uk– (p–q)
+ μBp–Bzkuk–q. ()
In the same way, we derive
yk–p=C(uk–λAuk) –C(q–λAq)
≤uk–λAuk– (q–λAq),j(yk–p)
=uk–q,j(yk–p) +λAq–Auk,j(yk–p)
≤
uk–q+yk–p–guk–yk+ (p–q)
+λAq–Aukyk–p,
which implies that
yk–p≤ uk–q–guk–yk+ (p–q)+ λAq–Aukyk–p. ()
Substituting () for (), we get
yk–p≤ xk–p–gzk–uk– (p–q)–guk–yk+ (p–q)
+ μBp–Bzkuk–q+ λAq–Aukyk–p. ()
So, from () and () it follows that
xk–p≤( –γkλkτ)Gzk–p+γkλkτ–F(p)
≤ yk–p+γk F(p)
τ
≤ xk–p–gzk–uk– (p–q)–guk–yk+ (p–q)
+ μBp–Bzkuk–q+ λAq–Aukyk–p+γk
F(p)
τ ,
which hence leads to
gzk–uk– (p–q)+guk–yk+ (p–q)
≤μBp–Bzkuk–q+ λAq–Aukyk–p+γk
F(p)
From (),γk→ ask→ ∞, and the boundedness of{uk}and{yk}, we deduce that
lim
k→∞gzk–uk– (p–q)= and k→∞lim guk–yk+ (p–q)= .
Utilizing the properties ofgandg, we conclude that
lim
k→∞zk–uk– (p–q)= and k→∞limuk–yk+ (p–q)= . ()
From (), we get
zk–yk ≤zk–uk– (p–q)+uk–yk+ (p–q)→ ask→ ∞.
That is,
lim
k→∞zk–Gzk=k→∞limzk–yk= . ()
This together withxk–GVkxk → implies that
lim
k→∞xk–yk= and k→∞limxk–Vkxk=k→∞lim xk–zk= . ()
Step .We show thatωw(xk)⊂F, where
ωw(xk) =
x∈C:xkixfor some subsequence{xki}of{xk}
.
Indeed, we first claim thatxk–Vxk → ask→ ∞. It is easy to see from Remark (ii)
that ifDis a nonempty and bounded subset ofC, then, forε> , there existsk>isuch
that for allk>k
sup x∈D
Vkix–V∞i x≤ε.
TakingD={xk:k≥}andi= , we have
Vkxk–Vxk ≤sup
x∈DVkx–Vx ≤ε.
So, it follows that
lim
k→∞Vkxk–Vxk= . ()
Noting that
GVxk–Vxk ≤ GVxk–GVkxk+GVkxk–Vkxk+Vkxk–Vxk
≤ Vxk–Vkxk+Gzk–zk+Vkxk–Vxk
from (), () and () we obtain that
lim
k→∞GVxk–Vxk= . ()
Also, noting thatxk–Vxk ≤ xk–Vkxk+Vkxk–Vxk, from () and () we get
lim
k→∞xk–Vxk= . ()
In addition, observing that
xk–Gxk ≤ xk–zk+zk–Gzk+Gzk–Gxk
≤ xk–zk+zk–Gzk+zk–xk
= xk–zk+zk–Gzk,
from () and () we get
lim
k→∞xk–Gxk= . ()
SinceX is reflexive, there exists at least a weak convergence subsequence of {xk}, and
henceωw(xk)=∅. Take an arbitraryp∈ωw(xk). Then there exists a subsequence{xki}of
{xk}such thatxkip. SinceVk is nonexpansive for allk≥,V is a nonexpansive self-mapping onC. Also, sinceXis uniformly convex andVandGare two nonexpansive self-mappings onC, utilizing Lemma we know from () and () thatp∈GSVI(C,A,B) and
p∈Fix(V) =∞i=Fix(Ti) (due to Lemma ). Consequently,p∈Fix(V) =∞i=Fix(Ti)∩
GSVI(C,A,B) =:F. This shows thatωw(xk)⊂F. Step .We show thatωw(xk) =ωs(xk), where
ωs(xk) =
x∈C:xki→xfor some subsequence{xki}of{xk}
.
Indeed, by Lemma , we haveGVkxk–z ≤ xk–zfor any fixedz∈F, and hence
xk–z =γkC(I–λkF)GVkxk+ ( –γk)GVkxk–z
=γk
C(I–λkF)GVkxk– (I–λkF)GVkxk,j(xk–z)
+λk(I–F)GVkxk+ ( –λk)GVkxk–z,j(xk–z)
+ ( –γk)
GVkxk–z,j(xk–z)
=γk
C(I–λkF)GVkxk– (I–λkF)GVkxk,j
C(I–λkF)GVkxk–z
+C(I–λkF)GVkxk– (I–λkF)GVkxk,j(xk–z)
–jC(I–λkF)GVkxk–z
+λk(I–F)GVkxk+ ( –λk)GVkxk–z,j(xk–z)
+ ( –γk)
≤γkC(I–λkF)GVkxk– (I–λkF)GVkxkj(xk–z)
–jC(I–λkF)GVkxk–z
+λk(I–F)GVkxk+ ( –λk)GVkxk–z,j(xk–z)
+ ( –γk)
GVkxk–z,j(xk–z)
≤γkC(I–λkF)GVkxk– (I–λkF)GVkxkj(xk–z)
–jC(I–λkF)GVkxk–z
+λk
(I–F)GVkxk–z,j(xk–z) + ( –λk)xk–z
+ ( –γk)xk–z
≤γkC(I–λkF)GVkxk–GVkxk
+λkF(GVkxk)j(xk–z) –j
C(I–λkF)GVkxk–z
+γkλk
(I–F)GVkxk–z,j(xk–z) + ( –γkλk)xk–z
≤γkλkF(GVkxk)j(xk–z) –j
C(I–λkF)GVkxk–z
+γkλk
(I–F)GVkxk– (I–F)z–F(z),j(xk–z) + ( –γkλk)xk–z.
Therefore, by Lemma (b) we get
xk–z≤( –τ)xk–z–F(z),j(xk–z)
+ F(xk)j(xk–z) –j
C(I–λkF)xk–z,
which immediately leads to
xk–z≤
τ
F(z),j(z–xk)
+ F(GVkxk)j(xk–z) –jC(I–λkF)GVkxk–z, ∀z∈F, ()
whereτ= –
–δ
ζ ∈(, ). Note that
(xk–z) –C(I–λkF)GVkxk–z≤xk– (I–λkF)yk
≤ xk–yk+λkF(yk).
Since the uniform smoothness of X guarantees the uniform continuity of j on every
nonempty bounded subset ofX, we deduce from (),λk→ and the boundedness of
{yk}that
lim
k→∞j(xk–z) –j
C(I–λkF)GVkxk–z= .
Now, take an arbitraryp∈ωw(xk). Then there exists a subsequence{xki}of{xk} such
forxkandpforzin () to get
xki–p
≤
τ
F(p),j(p–xki)
+ F(yki)j(xki–p) –j
C(I–λkiF)yki–p. ()
Consequently, the weak convergence of{xki}toptogether with () actually implies that xki→pasi→ ∞, and hencep∈ωs(xk). This shows thatωw(xk) =ωs(xk).
Step .We show that eachp∈ωs(xk) solves the variational inequality (). Indeed, take
an arbitraryp∈ωs(xk). Then there exists a subsequence{xki}of{xk}such thatxki→pas
i→ ∞. According to Steps and , we know thatp∈ωs(xk) (=ωw(xk)⊂F). Replacingxk
in () withxki, and noticing thatxki→p, we have the Minty type variational inequality
F(z),j(z–p) ≤, ∀z∈F,
which is equivalent to the variational inequality (see Lemma )
F(p),j(p–z) ≤, ∀z∈F. ()
That is,p∈Fis a solution of ().
Step .We show that{xk}converges strongly to a unique solution inFto VI (). Indeed, we first claim that the solution set of () is a singleton. As a matter of fact, assume that
¯
p∈Fis also a solution of (). Then we have
F(¯p),j(¯p–p) ≤.
From (), we have
F(p),j(p–p¯) ≤.
So, by theδ-strong accretiveness ofF, we have
F(¯p),j(¯p–p) +F(p),j(p–p¯) ≤
⇒F(¯p) –F(p),j(p¯–p) ≤ ⇒δ¯p–p≤.
Therefore,p¯=p. In summary, we have shown that each cluster point of{xk}(ask→ ∞)
equalsp. Consequently,xk→pask→ ∞.
Theorem Let C,X,C,A,B,F,{Ti}∞i=,F,δ,ζ,λandμbe as in Theorem.Let{Vk}∞k=be defined by()and().For arbitrarily given x∈C,let{xk}∞k=be defined by
xk+=βkxk+γkC(I–λA)C(I–μB)Vkxk
+ ( –βk–γk)C(I–λkF)C(I–λA)C(I–μB)Vkxk, ∀k≥,
(i) limk→∞λk= and
∞
k=λk=∞; (ii) limk→∞( γk+
–βk+– γk
–βk) = ;
(iii) <lim infk→∞βk≤lim supk→∞(βk+γk) < .
Then{xk}∞k=converges strongly to a unique solution x∗∈F.
Proof Let the mappingG:C→Cbe defined asG:=C(I–λA)C(I–μB), where <λ<
α
κ and <μ< β
κ. In terms of Lemma we know thatG:C→Cis nonexpansive. Then the explicit iterative scheme can be rewritten as
xk+=βkxk+γkGVkxk+ ( –βk–γk)C(I–λkF)GVkxk, ∀k≥. ()
Next, we divide the rest of the proof into several steps.
Step .We show that{xk}∞k= is bounded. Indeed, take an arbitrarily givenp∈F. Then
we haveVkp=pandGp=p. Hence, by Lemma (c) we get
xk+–p=βkxk+γkGVkxk+ ( –βk–γk)C(I–λkF)GVkxk–p
≤βkxk–p+γkGVkxk–p+ ( –βk–γk)C(I–λkF)GVkxk–p
≤βkxk–p+γkxk–p
+ ( –βk–γk)(I–λkF)GVkxk– (I–λkF)p–λkF(p)
≤(βk+γk)xk–p+ ( –βk–γk)(I–λkF)GVkxk– (I–λkF)p
+λk( –βk–γk)F(p)
≤(βk+γk)xk–p+ ( –βk–γk)( –λkτ)GVkxk–p
+λk( –βk–γk)F(p)
≤ – ( –βk–γk)λkτ
xk–p+ ( –βk–γk)λkτ
F(p)
τ .
By induction, we conclude that
xk+–p ≤max
x–p, F(p)
τ
.
Therefore,{xk}∞k=is bounded. So, the sequences{zk}∞k=,{yk}∞k=and{F(yk)}∞k=, wherezk= Vkxkandyk=GVkxk, are also bounded.
Step .We show thatxk+–xk → andxk–yk → ask→ ∞. Indeed, observe that
yk+–yk=GVk+xk+–GVkxk
≤ Vk+xk+–Vkxk
≤ Vk+xk+–Vk+xk+Vk+xk–Vkxk
≤ xk+–xk+VkTk+xk–Vkxk
≤ xk+–xk+Tk+xk–xk
Setxk+=βkxk+ ( –βk)wkfor allk≥. Thenwk=γkyk+(–βk––γkβ)kC(I–λkF)yk. Note that
C(I–λk+F)yk+–C(I–λkF)yk
≤(I–λk+F)yk+– (I–λkF)yk
=yk+–yk–λk+F(yk+) +λkF(yk)
≤ yk+–yk+λk+F(yk+)+λkF(yk)
≤ xk+–xk+λk+F(yk+)+λkF(yk)+αk+xk–Tk+xk.
Hence
wk+–wk
=γk+yk++ ( –βk+–γk+)C(I–λk+F)yk+ –βk+
–γkyk+ ( –βk–γk)C(I–λkF)yk –βk
≤ γk+
–βk+ yk+–
γk
–βk yk
+( –βk+–γk+) –βk+
C(I–λk+F)yk+–
( –βk–γk)
–βk
C(I–λkF)yk
≤ γk+
–βk+
– γk
–βk
yk++
γk
–βk|
yk+–yk
+( –βk+–γk+) –βk+
–( –βk–γk) –βk
C(I–λk+F)yk+
+( –βk–γk) –βk
C(I–λk+F)yk+–C(I–λkF)yk
≤ γk+
–βk+
– γk
–βk
yk++C(I–λk+F)yk+
+ γk
–βk|
xk+–xk+αk+xk–Tk+xk
+( –βk–γk) –βk
xk+–xk+λk+F(yk+)+λkF(yk)+αk+xk–Tk+xk
≤ γk+
–βk+
– γk
–βk
yk++C(I–λk+F)yk+
+xk+–xk+λk+F(yk+)+λkF(yk)+αk+
xk+Tk+xk
.
Since{xk},{yk}and{F(yk)}are bounded, we have that{yk++C(I–λk+F)yk+}and
{xk+Tk+xk}are bounded. So it follows fromαk→ and conditions (i) and (ii) that
lim sup k→∞
wk+–wk–xk+–xk
≤.
Hence, by Lemma , we getwk–xk → ask→ ∞. Consequently,
lim
We also note that
wk–yk=γkyk+ ( –βk–γk)C(I–λkF)yk –βk
–yk
= –βk–γk –βk
C(I–λkF)yk–yk
≤C(I–λkF)yk–Cyk
≤λkF(yk)→ ask→ ∞.
It follows that
lim
k→∞xk–yk= . ()
Step .We show thatzk–Gzk → andxk–Vkxk → ask→ ∞. Indeed, for
sim-plicity, putq=C(p–μBp) anduk=C(zk–μBzk). Thenyk=Gzk=C(uk–λAuk) for
allk≥. Repeating the same arguments as those of (), we obtain
yk–p≤ xk–p– μβ–κμBzk–Bp– λα–κλAuk–Aq. ()
Observe that
xk+–p≤βkxk–p+γkGVkxk–p+ ( –βk–γk)C(I–λkF)GVkxk–p
≤βkxk–p+γkGVkxk–p
+ ( –βk–γk)(I–λkF)GVkxk– (I–λkF)p–λkF(p)
≤βkxk–p+γkGVkxk–p
+ ( –βk–γk)
( –λkτ)GVkxk–p+λkF(p)
≤βkxk–p+γkGVkxk–p
+ ( –βk–γk)
( –λkτ)GVkxk–p+λkτ–F(p)
≤βkxk–p+γkGVkxk–p
+ ( –βk–γk)GVkxk–p+λkτ–F(p)
=βkxk–p+ ( –βk)GVkxk–p+λkτ–F(p)
, ()
which together with () implies that
xk+–p≤βkxk–p+ ( –βk)
xk–p– μβ–κμBzk–Bp
– λα–κλAuk–Aq+λkτ–F(p)
=xk–p– ( –βk)
μβ–κμBzk–Bp
+λα–κλAuk–Aq
+λkτ–F(p)
So, it follows that
( –βk)
μβ–κμBzk–Bp+λ
α–κλAuk–Aq
≤ xk–p–xk+–p+λkτ–F(p)
≤ xk–xk+
xk–p+xk+–p
+λkτ–F(p)
. ()
Sinceλ∈(,κα),μ∈(, β
κ) andλk→ ask→ ∞, we deduce from () and condition
(iii) that
lim
k→∞Bzk–Bp= and k→∞limAuk–Aq= . ()
Repeating the same arguments as those of (), we get
yk–p≤ xk–p–gzk–uk– (p–q)–guk–yk+ (p–q)
+ μBp–Bzkuk–q+ λAq–Aukyk–p. ()
Combining () and (), we have
xk+–p≤βkxk–p+ ( –βk)
xk–p–gzk–uk– (p–q)
–guk–yk+ (p–q)+ μBp–Bzkuk–q
+λAq–Aukyk–p
+λkτ–F(p)
≤ xk–p– ( –βk)
gzk–uk– (p–q)+guk–yk+ (p–q)
+ μBp–Bzkuk–q+ λAq–Aukyk–p+λkτ–F(p),
which immediately leads to
( –βk)
gzk–uk– (p–q)+guk–yk+ (p–q)
≤ xk–p–xk+–p
+ μBp–Bzkuk–q+ λAq–Aukyk–p+λkτ–F(p)
≤ xk–xk+
xk–p+xk+–p
+ μBp–Bzkuk–q+ λAq–Aukyk–p+λkτ–F(p)
.
Sinceλk→ ask→ ∞, and{uk}and{yk}are bounded, we deduce from (), () and
condition (iii) that
lim
k→∞gzk–uk– (p–q)= and k→∞lim guk–yk+ (p–q)= .
Utilizing the properties ofgandg, we conclude that
lim
From (), we get
zk–yk ≤zk–uk– (p–q)+uk–yk+ (p–q)→ ask→ ∞.
That is,
lim
k→∞zk–Gzk=k→∞limzk–yk= . ()
This together with () implies that
lim
k→∞xk–Vkxk=k→∞limxk–zk= . ()
Step .We show thatωw(xk)⊂F, where
ωw(xk) =
x∈C:xkixfor some subsequence{xki}of{xk}
.
Indeed, repeating the same arguments as those of () and () in the proof of Theorem , we obtain that
lim
k→∞xk–Vxk= and k→∞lim xk–Gxk= . ()
Utilizing Lemma and Lemma and the nonexpansivity ofV andG, we conclude that ωw(xk)⊂F.
Step .We show thatlim supk→∞ F(x∗),j(x∗–vk) ≤, wherevk=C(I–λkF)ykfor all k≥ andx∗∈Fis the unique solution of VI (). Indeed, we first take a subsequence{vki}
of{vk}such that
lim sup k→∞
Fx∗,jx∗–vk =lim i→∞
Fx∗,jx∗–vki .
We may also assume thatvkiz. Note that
vk–xk=C(I–λkF)yk–xk
≤(I–λkF)yk–xk≤ yk–xk+λkF(yk)→ ask→ ∞. ()
Combining () withωw(xk)⊂F(due to Step ), we getz∈F. So, it follows from VI ()
that
lim sup k→∞
Fx∗,jx∗–vk =lim i→∞
Fx∗,jx∗–vki
=Fx∗,jx∗–z ≤.
Sincevk=C(I–λkF)ykfor allk≥, according to Lemma (iii), we have
(I–λkF)yk–C(I–λkF)yk,j
From (), we have
vk–x∗
=C(I–λkF)yk–x∗,j
vk–x∗
=C(I–λkF)yk– (I–λkF)yk,j
vk–x∗
+(I–λkF)yk–x∗,j
vk–x∗
≤(I–λkF)yk–x∗,jvk–x∗
=(I–λkF)yk– (I–λkF)x∗,j
vk–x∗ +λk
Fx∗,jx∗–vk
≤( –λkτ)yk–x∗vk–x∗+λk
Fx∗,jx∗–vk
≤
( –λkτ)
yk–x∗
+ vk–x
∗
+λk
Fx∗,jx∗–vk ,
which immediately yields
vk–x∗≤( –λkτ)yk–x∗+ λk
Fx∗,jx∗–vk
≤( –λkτ)xk–x∗
+ λk
Fx∗,jx∗–vk . ()
Step .We show thatxk→x∗ask→ ∞. Indeed, from () and (), we have
xk+–x∗
≤βkxk–x∗
+γkyk–x∗
+ ( –βk–γk)vk–x∗
≤βkxk–x∗
+γkxk–x∗
+ ( –βk–γk)
( –λkτ)xk–x∗
+ λk
Fx∗,jx∗–vk
= –λk( –βk–γk)τxk–x∗
+λk( –βk–γk)τ·
τ
Fx∗,jx∗–vk . ()
Since∞k=λk=∞,lim supk→∞(βk+γk) < andτ= –
–δ
ζ ∈(, ), we get
∞
k=
λk( –βk–γk)τ =∞.
Taking into accountlim supk→∞ F(x∗),j(x∗–vk) ≤, we can apply Lemma to relation
() and conclude thatxk→x∗ask→ ∞.
Acknowledgements
This research was partially supported by the Innovation Program of Shanghai Municipal Education Commission (15ZZ068), Ph.D. Program Foundation of Ministry of Education of China (20123127110002) and Program for Outstanding Academic Leaders in Shanghai City (15XD1503100). This research was partially supported by the grant MOST 106-2115-M-037-001, and the grant from Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Taiwan.
Competing interests
The authors declare that they have no competing interests. Authors’ contributions
Author details
1Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China.2Department of Industrial Management, Vanung University, Taoyuan, Taiwan.3Center for Fundamental Science; and Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung, 80702, Taiwan.4Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung, 80702, Taiwan.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 2 April 2017 Accepted: 18 September 2017
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