# Composite Iterative Algorithms for Variational Inequality and Fixed Point Problems in Real Smooth and Uniformly Convex Banach Spaces

Authors: Lu-Chuan Ceng; Ching-Feng Wen
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We introduce composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space. These composite iterative algorithms are based on Korpelevich's extragradient method and viscosity approximation method. We first consider and analyze a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.


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## Body

## 1. Introduction

LetX be a real Banach space whose dual space is denoted by X*. The normalized duality mapping J:X→2X* is defined by
(1)J(x)={x*∈X*:〈x,x*〉=∥x∥2=∥x*∥2},∀x∈X,
where 〈·,·〉 denotes the generalized duality pairing. It is an immediate consequence of the Hahn-Banach theorem that J(x) is nonempty for each x∈X. Let C be a nonempty, closed, and convex subset of X. A mapping T:C→C is called nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for every x,y∈C. The set of fixed points of T is denoted by Fix(T). We use the notation ⇀ to indicate the weak convergence and the one → to indicate the strong convergence. A mapping A:C→X is said to be accretive if for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(2)〈Ax-Ay,j(x-y)〉≥0.
It is said to be α-strongly accretive if for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(3)〈Ax-Ay,j(x-y)〉≥α∥x-y∥2,
for some α∈(0,1). The mapping is called β-inverse strongly-accretive if for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(4)〈Ax-Ay,j(x-y)〉≥β∥Ax-Ay∥2,
for some β>0 and is said to be λ-strictly pseudocontractive if for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(5)〈Ax-Ay,j(x-y)〉≤∥x-y∥2-λ∥x-y-(Ax-Ay)∥2
for some λ∈(0,1).LetU={x∈X:∥x∥=1} denote the unite sphere of X. A Banach space X is said to be uniformly convex if for each ϵ∈(0,2], there exists δ>0 such that for all x,y∈U,
(6)∥x-y∥≥ϵ⟹∥x+y∥2≤1-δ.
It is known that a uniformly convex Banach space is reflexive and strict convex. A Banach space X is said to be smooth if the limit
(7)limt→0∥x+ty∥-∥x∥t
exists for all x,y∈U; in this case, X is also said to have a Gâteaux differentiable norm. X is said to have a uniformly Gâteaux differentiable norm if for each y∈U, the limit is attained uniformly for x∈U. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for x,y∈U. The norm of X is said to be the Fréchet differential if for each x∈U, this limit is attained uniformly for y∈U. In addition, we define a function ρ:[0,∞)→[0,∞) called the modulus of smoothness of X as follows:
(8)ρ(τ)=sup{12(∥x+y∥+∥x-y∥)-1:x,y∈X,∥x∥=1,∥y∥=τ12}.
It is known that X is uniformly smooth if and only if limτ→0ρ(τ)/τ=0. Let q be a fixed real number with 1<q≤2. Then a Banach space X is said to be q-uniformly smooth if there exists a constant c>0 such that ρ(τ)≤cτq for all τ>0. As pointed out in [1], no Banach space is q-uniformly smooth for q>2. In addition, it is also known that J is single-valued if and only if X is smooth, whereas if X is uniformly smooth, then the mapping J is norm-to-norm uniformly continuous on bounded subsets of X. If X has a uniformly Gâteaux differentiable norm, then the duality mapping J is norm-to-weak* uniformly continuous on bounded subsets of X.Very recently, Cai and Bu [2] considered the following general system of variational inequalities (GSVI) in a real smooth Banach space X, which involves finding (x*,y*)∈C×C such that
(9)〈μ1B1y*+x*-y*,J(x-x*)〉≥0,∀x∈C,〈μ2B2x*+y*-x*,J(x-y*)〉≥0,∀x∈C,
where C is a nonempty, closed, and convex subset of X, B1, and B2:C→X are two nonlinear mappings, and μ1 and μ2 are two positive constants. Here the set of solutions of GSVI (9) is denoted by GSVI(C,B1,B2). In particular, if X=H, a real Hilbert space, then GSVI (9) reduces to the following GSVI of finding (x*,y*)∈C×C such that
(10)〈μ1B1y*+x*-y*,x-x*〉≥0,∀x∈C,〈μ2B2x*+y*-x*,x-y*〉≥0,∀x∈C,
which μ1 and μ2 are two positive constants. The set of solutions of problem (10) is still denoted by GSVI(C,B1,B2). It is clear that the problem (10) covers as special case the classical variational inequality problem (VIP) of finding x*∈C such that
(11)〈Ax*,x-x*〉≥0,∀x∈C.
The solution set of the VIP (11) is denoted by VI(C,A).Recently, Ceng et al. [3] transformed problem (10) into a fixed point problem in the following way.Lemma 1 (see [3]).
For givenx-,y-∈C,(x-,y-) is a solution of problem (10) if and only if x- is a fixed point of the mapping G:C→C defined by
(12)G(x)=PC[PC(x-μ2B2x)-μ1B1PC×(x-μ2B2x)],∀x∈C,
where y-=PC(x--μ2B2x-) and PC is the the projection of H onto C.In particular, if the mappingsBi:C→H is βi-inverse strongly monotone for i=1,2, then the mapping G is nonexpansive provided μi∈(0,2βi) for i=1,2.LetC be a nonempty, closed, and convex subset of a real smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C, and let f:C→C be a contraction with coefficient ρ∈(0,1). In this paper we introduce composite implicit and explicit iterative algorithms for solving GSVI (9) and the common fixed point problem of an infinite family {Sn} of nonexpansive mappings of C into itself. These composite iterative algorithms are based on Korpelevich’s extragradient method [4] and viscosity approximation method [5]. Let the mapping G be defined by
(13)G(x)∶=ΠC(I-μ1B1)ΠC(I-μ2B2)x,∀x∈C.
We first propose a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space X:
(14)yn=αnf(yn)+(1-αn)SnG(xn),xn+1=βnxn+γnyn+δnSnG(yn),∀n≥0,
where Bi:C→X is αi-inverse-strongly accretive with 0<μi<αi/κ2 for i=1,2 and {αn}, {βn}, {γn}, and {δn} are the sequences in (0,1) such that βn+γn+δn=1 for all n≥0. It is proven that under appropriate conditions, {xn} converges strongly to q∈F=⋂i=0∞‍Fix(Si)∩Ω, which solves the following VIP:
(15)〈q-f(q),J(q-p)〉≤0,∀p∈F.
On the other hand, we also propose another composite explicit iterative algorithm in a uniformly convex Banach space X with a uniformly Gateaux differentiable norm:
(16)yn=αnG(xn)+(1-αn)SnG(xn),xn+1=βnf(xn)+γnyn+δnSnG(yn),∀n≥0,
where Bi:C→X is λi-strictly pseudocontractive and αi-strongly accretive with αi+λi≥1 for i=1,2 and {αn}, {βn}, {γn}, and {δn} are the sequences in (0,1) such that βn+γn+δn=1 for all n≥0. It is proven that under mild conditions, {xn} also converges strongly to q∈F=⋂i=0∞‍Fix(Si)∩Ω, which solves the VIP (15). The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.


## 2. Preliminaries

We list some lemmas that will be used in the sequel. Lemma2 can be found in [6]. Lemma 3 is an immediate consequence of the subdifferential inequality of the function (1/2)∥·∥2.Lemma 2.
Let{sn} be a sequence of nonnegative real numbers satisfying
(17)sn+1≤(1-αn)sn+αnβn+γn,∀n≥0,
where {αn}, {βn}, and {γn} satisfy the conditions:(i)
{αn}⊂[0,1] and ∑n=0∞‍αn=∞,(ii)
limsupn→∞βn≤0,(iii)
γn≥0, ∀n≥0, and ∑n=0∞‍γn<∞.Then limsupn→∞sn=0.Lemma 3.
In a smooth Banach spaceX, there holds the inequality
(18)∥x+y∥2≤∥x∥2+2〈y,J(x+y)〉,∀x,y∈X.Lemma 4 (see [7]).
Let{xn} and {zn} be bounded sequences in a Banach space X, and let {αn} be a sequence in [0,1] which satisfies the following condition:
(19)0<liminfn→∞αn≤limsupn→∞αn<1.
Suppose that xn+1=αnxn+(1-αn)zn, ∀n≥0, and limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0. Then limn→∞∥zn-xn∥=0.LetD be a subset of C, and let Π be a mapping of C into D. Then Π is said to be sunny if
(20)Π[Π(x)+t(x-Π(x))]=Π(x),
whenever Π(x)+t(x-Π(x))∈C for x∈C and t≥0. A mapping Π of C into itself is called a retraction if Π2=Π. If a mapping Π of C into itself is a retraction, then Π(z)=z for every z∈R(Π) where R(Π) is the range of Π. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D. The following lemma concerns the sunny nonexpansive retraction.Lemma 5 (see [8]).
LetC be a nonempty, closed, and convex subset of a real smooth Banach space X. Let D be a nonempty subset of C. Let Π be a retraction of C onto D. Then the following are equivalent:(i)
Π is sunny and nonexpansive;(ii)
∥Π(x)-Π(y)∥2≤〈x-y,J(Π(x)-Π(y))〉, ∀x,y∈C;(iii)
〈x-Π(x),J(y-Π(x))〉≤0, ∀x∈C, y∈D.It is well known that ifX=H a Hilbert space, then a sunny nonexpansive retraction ΠC is coincident with the metric projection from X onto C; that is, ΠC=PC. If C is a nonempty, closed, and convex subset of a strictly convex and uniformly smooth Banach space X and if T:C→C is a nonexpansive mapping with the fixed point set Fix(T)≠∅, then the set Fix(T) is a sunny nonexpansive retract of C.Lemma 6 (see [9]).
Given a numberr>0. A real Banach space X is uniformly convex if and only if there exists a continuous strictly increasing function g:[0,∞)→[0,∞), g(0)=0, such that
(21)∥λx+(1-λ)y∥2≤λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)g(∥x-y∥)
for all λ∈[0,1] and x,y∈X such that ∥x∥≤r and ∥y∥≤r.Lemma 7 (see [10]).
LetC be a nonempty, closed, and convex subset of a Banach space X. Let S0,S1,…, be a sequence of mappings of C into itself. Suppose that ∑n=1∞‍sup{∥Snx-Sn-1x∥:x∈C}<∞. Then for each y∈C, {Sny} converges strongly to some point of C. Moreover, let S be a mapping of C into itself defined by Sy=limn→∞Sny for all y∈C. Then limn→∞sup{∥Sx-Snx∥:x∈C}=0.LetC be a nonempty, closed, and convex subset of a Banach space X, and let T:C→C be a nonexpansive mapping with Fix(T)≠∅. As previous, let ΞC be the set of all contractions on C. For t∈(0,1) and f∈ΞC, let xt∈C be the unique fixed point of the contraction x↦tf(x)+(1-t)Tx on C; that is,
(22)xt=tf(xt)+(1-t)Txt.Lemma 8 (see [11, 12]).
LetX be a uniformly smooth Banach space or a reflexive and strictly convex Banach space with a uniformly Gateaux differentiable norm. Let C be a nonempty, closed, and convex subset of X, let T:C→C be a nonexpansive mapping with Fix(T)≠∅, and f∈ΞC. Then the net {xt} defined by xt=tf(xt)+(1-t)Txt converges strongly to a point in Fix(T). If we define a mapping Q:ΞC→Fix(T) by Q(f):=s-limt→0xt, ∀f∈ΞC, then Q(f) solves the VIP:
(23)〈(I-f)Q(f),J(Q(f)-p)〉≤0,∀f∈ΞC,p∈Fix(T).Lemma 9 (see [13]).
LetC be a nonempty, closed, and convex subset of a strictly convex Banach space X. Let {Tn}n=0∞ be a sequence of nonexpansive mappings on C. Suppose that ⋂n=0∞‍Fix(Tn) is nonempty. Let {λn} be a sequence of positive numbers with ∑n=0∞λn=1. Then a mapping S on C defined by Sx=∑n=0∞‍λnTnx for x∈C is defined well, nonexpansive, and Fix(S)=⋂n=0∞‍Fix(Tn) holds.


## 3. Implicit Iterative Schemes

In this section, we introduce our implicit iterative schemes and show the strong convergence theorems. We will use the following useful lemmas in the sequel.Lemma 10 (see [2, Lemma 2.8]).
LetC be a nonempty, closed, and convex subset of a real 2-uniformly smooth Banach space X. Let the mapping Bi:C→X be αi-inverse-strongly accretive. Then, one has
(24)∥(I-μiBi)x-(I-μiBi)y∥2≤∥x-y∥2+2μi(μiκ2-αi)×∥Bix-Biy∥2,∀x,y∈C,
for i=1,2, where μi>0. In particular, if 0<μi≤αi/κ2, then I-μiBi is nonexpansive for i=1,2.Lemma 11 (see [2, Lemma 2.9]).
LetC be a nonempty, closed, and convex subset of a real 2-uniformly smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C. Let the mapping Bi:C→X be αi-inverse-strongly accretive for i=1,2. Let G:C→C be the mapping defined by
(25)G(x)=ΠC[ΠC(x-μ2B2x)-μ1B1ΠC(x-μ2B2x)],∀x∈C.
If  0<μi≤αi/κ2 for i=1,2, then G:C→C is nonexpensive.Lemma 12 (see [2, Lemma 2.10]).
LetC be a nonempty, closed, and convex subset of a real 2-uniformly smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C. Let B1,B2:C→X be two nonlinear mappings. For given x*,y*∈C,(x*,y*) is a solution of GSVI (9) if and only if x*=ΠC(y*-μ1B1y*) where y*=ΠC(x*-μ2B2x*).Remark 13.
By Lemma12, we observe that
(26)x*=ΠC[ΠC(x*-μ2B2x*)-μ1B1ΠC(x*-μ2B2x*)],
which implies that x* is a fixed point of the mapping G.We now state and prove our first result on the implicit iterative scheme.Theorem 14.
LetC be a nonempty, closed, and convex subset of a uniformly convex and 2-uniformly smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C. Let the mapping Bi:C→X be αi-inverse-strongly accretive for i=1,2. Let f:C→C be a contraction with coefficient ρ∈(0,1). Let {Sn}n=0∞ be an infinite family of nonexpansive mappings of C into itself such that F=⋂n=0∞‍Fix(Sn)∩Ω≠∅, where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2). For arbitrarily given x0∈C, let {xn} be the sequence generated by
(27)yn=αnf(yn)+(1-αn)SnG(xn),xn+1=βnxn+γnyn+δnSnG(yn),∀n≥0,
where 0<μi<αi/κ2 for i=1,2 and {αn}, {βn}, {γn}, and {δn} are the sequences in (0,1) such that βn+γn+δn=1, ∀n≥0. Suppose that the following conditions hold:(i)
limn→∞αn=0 and ∑n=0∞‍αn=∞,(ii)
0<liminfn→∞βn≤limsupn→∞(βn+γn)<1 and liminfn→∞γn>0,(iii)
limn→∞|γn/(1-βn)-γn-1/(1-βn-1)|=0.Assume that ∑n=1∞‍supx∈D∥Snx-Sn-1x∥<∞ for any bounded subset D of C, and let S be a mapping of C into itself defined by Sx=limn→∞Snx for all x∈C. Suppose that Fix(S)=⋂n=0∞‍Fix(Sn). Then {xn} converges strongly to q∈F, which solves the following VIP:
(28)〈q-f(q),J(q-p)〉≤0,∀p∈F.Proof.
Take a fixedp∈F arbitrarily. Then by Lemma 12, we know that p=G(p) and p=Snp for all n≥0. Moreover, by Lemma 11, we have
(29)∥yn-p∥=∥αn(f(yn)-p)+(1-αn)(SnG(xn)-p)∥≤αn∥f(yn)-f(p)∥+αn∥f(p)-p∥+(1-αn)∥SnG(xn)-p∥≤αnρ∥yn-p∥+αn∥f(p)-p∥+(1-αn)∥G(xn)-p∥≤αnρ∥yn-p∥+αn∥f(p)-p∥+(1-αn)∥xn-p∥,
which hence implies that
(30)∥yn-p∥≤(1-1-ρ1-αnραn)∥xn-p∥+11-αnραn∥f(p)-p∥≤∥xn-p∥+αn1-αnρ∥f(p)-p∥.
Thus, from (27), we have
(31)∥xn+1-p∥=∥βn(xn-p)+γn(yn-p)+δn(SnG(yn)-p)∥≤βn∥xn-p∥+γn∥yn-p∥+δn∥G(yn)-p∥≤βn∥xn-p∥+γn∥yn-p∥+δn∥yn-p∥=βn∥xn-p∥+(1-βn)∥yn-p∥≤βn∥xn-p∥+(1-βn)×{(1-1-ρ1-αnραn)∥xn-p∥+11-αnραn∥f(p)-p∥}=[1-(1-βn)(1-ρ)1-αnραn]∥xn-p∥+(1-βn)(1-ρ)1-αnραn∥f(p)-p∥1-ρ≤max{∥x0-p∥,∥f(p)-p∥1-ρ}.
It immediately follows that {xn} is bounded, and so are the sequences {yn}, {G(xn)}, and {G(yn)} due to (30) and the nonexpansivity of G.
Let us show that∥xn+1-xn∥→0 as n→∞. As a matter of fact, from (27), we have
(32)yn=αnf(yn)+(1-αn)SnG(xn),yn-1=αn-1f(yn-1)+(1-αn-1)Sn-1G(xn-1),∀n≥1.
Simple calculations show that
(33)yn-yn-1=αn(f(yn)-f(yn-1))+(αn-αn-1)(f(yn-1)-Sn-1G(xn-1))+(1-αn)(SnG(xn)-Sn-1G(xn-1)).
It follows that
(34)∥yn-yn-1∥≤αn∥f(yn)-f(yn-1)∥+|αn-αn-1|×∥f(yn-1)-Sn-1G(xn-1)∥+(1-αn)∥SnG(xn)-Sn-1G(xn-1)∥≤αnρ∥yn-yn-1∥+|αn-αn-1|×∥f(yn-1)-Sn-1G(xn-1)∥+(1-αn)×(∥SnG(xn)-SnG(xn-1)∥+∥SnG(xn-1)-Sn-1G(xn-1)∥)≤αnρ∥yn-yn-1∥+|αn-αn-1|×∥f(yn-1)-Sn-1G(xn-1)∥+(1-αn)×(∥xn-xn-1∥+∥SnG(xn-1)-Sn-1G(xn-1)∥),
which hence yields
(35)∥yn-yn-1∥≤1-αn1-αnρ(∥xn-xn-1∥+∥SnG(xn-1)-Sn-1G(xn-1)∥)+|αn-αn-1|1-αnρ∥f(yn-1)-Sn-1G(xn-1)∥≤∥xn-xn-1∥+∥SnG(xn-1)-Sn-1G(xn-1)∥+|αn-αn-1|1-αnρ∥f(yn-1)-Sn-1G(xn-1)∥.
Now, we write xn=βn-1xn-1+(1-βn-1)zn-1, ∀n≥1, where zn-1=(xn-βn-1xn-1)/(1-βn-1). It follows that for all n≥1,
(36)zn-zn-1=xn+1-βnxn1-βn-xn-βn-1xn-11-βn-1=γnyn+δnSnG(yn)1-βn-γn-1yn-1+δn-1Sn-1G(yn-1)1-βn-1=γn(yn-yn-1)+δn(SnG(yn)-Sn-1G(yn-1))1-βn+(γn1-βn-γn-11-βn-1)yn-1+(δn1-βn-δn-11-βn-1)Sn-1G(yn-1).
This together with (35) implies that
(37)∥zn-zn-1∥≤∥γn(yn-yn-1)+δn(SnG(yn)-Sn-1G(yn-1))∥1-βn+|γn1-βn-γn-11-βn-1|∥yn-1∥+|δn1-βn-δn-11-βn-1|∥Sn-1G(yn-1)∥≤(+∥SnG(yn-1)-Sn-1G(yn-1)∥)γn∥yn-yn-1∥+δn(∥SnG(yn)-SnG(yn-1)∥+∥SnG(yn-1)-Sn-1G(yn-1)∥))×(1-βn)-1+|γn1-βn-γn-11-βn-1|∥yn-1∥+|γn1-βn-γn-11-βn-1|∥Sn-1G(yn-1)∥≤γn∥yn-yn-1∥+δn∥yn-yn-1∥γn+δn+δnγn+δn∥SnG(yn-1)-Sn-1G(yn-1)∥+|γn1-βn-γn-11-βn-1|(∥yn-1∥+∥Sn-1G(yn-1)∥)≤∥yn-yn-1∥+∥SnG(yn-1)-Sn-1G(yn-1)∥+|γn1-βn-γn-11-βn-1|(∥yn-1∥+∥Sn-1G(yn-1)∥)≤∥xn-xn-1∥+∥SnG(xn-1)-Sn-1G(xn-1)∥+|αn-αn-1|1-αnρ∥f(yn-1)-Sn-1G(xn-1)∥+∥SnG(yn-1)-Sn-1G(yn-1)∥+|γn1-βn-γn-11-βn-1|(∥yn-1∥+∥Sn-1G(yn-1)∥)≤∥xn-xn-1∥+∥SnG(xn-1)-Sn-1G(xn-1)∥+|αn-αn-1|1-αnρM+∥SnG(yn-1)-Sn-1G(yn-1)∥+|γn1-βn-γn-11-βn-1|M=∥xn-xn-1∥+M(11-αnρ|αn-αn-1|+|γn1-βn-γn-11-βn-1|)+∥SnG(xn-1)-Sn-1G(xn-1)∥+∥SnG(yn-1)-Sn-1G(yn-1)∥,
where supn≥0{∥f(yn)∥+∥SnG(xn)∥+∥yn∥+∥SnG(yn)∥}≤M for some M>0. So, from αn→0, condition (iii), and the assumption on {Sn}, it immediately follows that
(38)limsupn→∞(∥zn-zn-1∥-∥xn-xn-1∥)≤0.
In terms of condition (ii) and Lemma 4, we get
(39)limn→∞∥zn-xn∥=0.
Hence we obtain
(40)limn→∞∥xn+1-xn∥=limn→∞(1-βn)∥zn-xn∥=0.
Next we show that∥xn-G(xn)∥→0 as n→∞.
For simplicity, putq=ΠC(p-μ2B2p), un=ΠC(xn-μ2B2xn), and vn=ΠC(un-μ1B1un). Then vn=G(xn). From Lemma 10, we have
(41)∥un-q∥2=∥ΠC(xn-μ2B2xn)-ΠC(p-μ2B2p)∥2≤∥xn-p-μ2(B2xn-B2p)∥2≤∥xn-p∥2-2μ2(α2-κ2μ2)∥B2xn-B2p∥2,(42)∥vn-p∥2=∥ΠC(un-μ1B1un)-ΠC(q-μ1B1q)∥2≤∥un-q-μ1(B1un-B1q)∥2≤∥un-q∥2-2μ1(α1-κ2μ1)∥B1un-B1q∥2.
Substituting (41) into (42), we obtain
(43)∥vn-p∥2≤∥xn-p∥2-2μ2(α2-κ2μ2)∥B2xn-B2p∥2-2μ1(α1-κ2μ1)∥B1un-B1q∥2.
According to Lemma 3, we have from (27)
(44)∥yn-p∥2=∥αn(f(yn)-f(p))+(1-αn)(Snvn-p)+αn(f(p)-p)(f(yn)-f(p))∥2≤∥αn(f(yn)-f(p))+(1-αn)(Snvn-p)∥2+2αn〈f(p)-p,J(yn-p)〉≤αn∥f(yn)-f(p)∥2+(1-αn)∥Snvn-p∥2+2αn〈f(p)-p,J(yn-p)〉≤αnρ∥yn-p∥2+(1-αn)∥vn-p∥2+2αn∥f(p)-p∥∥yn-p∥,
which hence yields
(45)∥yn-p∥2≤(1-1-ρ1-αnραn)∥vn-p∥2+2αn1-αnρ∥f(p)-p∥∥yn-p∥.
This together with (43) and the convexity of ∥·∥2, we have
(46)∥xn+1-p∥2=∥βn(xn-p)+γn(yn-p)+δn(SnG(yn)-p)∥2≤βn∥xn-p∥2+γn∥yn-p∥2+δn∥SnG(yn)-p∥2≤βn∥xn-p∥2+γn∥yn-p∥2+δn∥yn-p∥2=βn∥xn-p∥2+(1-βn)∥yn-p∥2≤βn∥xn-p∥2+(1-βn)×{(1-1-ρ1-αnραn)∥vn-p∥2+2αn1-αnρ∥f(p)-p∥∥yn-p∥}≤βn∥xn-p∥2+(1-βn)×(1-1-ρ1-αnραn)∥vn-p∥2+αnM1≤βn∥xn-p∥2+(1-βn)(1-1-ρ1-αnραn)×[∥xn-p∥2-2μ2(α2-κ2μ2)∥B2xn-B2p∥2-2μ1(α1-κ2μ1)∥B1un-B1q∥2]+αnM1=(1-(1-βn)(1-ρ)1-αnραn)∥xn-p∥2-2(1-βn)×(1-1-ρ1-αnραn)×[μ2(α2-κ2μ2)∥B2xn-B2p∥2+μ1(α1-κ2μ1)∥B1un-B1q∥2]+αnM1≤∥xn-p∥2-2(1-βn)(1-1-ρ1-αnραn)×[μ2(α2-κ2μ2)∥B2xn-B2p∥2+μ1(α1-κ2μ1)∥B1un-B1q∥2]+αnM1,
where supn≥0{2(1-βn)/(1-αnρ)∥f(p)-p∥∥yn-p∥}≤M1 for some M1>0. So, it follows that
(47)2(1-βn)(1-1-ρ1-αnραn)×[μ2(α2-κ2μ2)∥B2xn-B2p∥2+μ1(α1-κ2μ1)∥B1un-B1q∥2]≤∥xn-p∥2-∥xn+1-p∥2+αnM1≤(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+αnM1.
Since 0<μi<αi/κ2 for i=1,2, from conditions (i), (ii), and (40), we obtain
(48)limn→∞∥B2xn-B2p∥=0,limn→∞∥B1un-B1q∥=0.
Utilizing [14, Proposition 1] and Lemma 5, we have
(49)∥un-q∥2=∥ΠC(xn-μ2B2xn)-ΠC(p-μ2B2p)∥2≤〈xn-μ2B2xn-(p-μ2B2p),J(un-q)〉=〈xn-p,J(un-q)〉+μ2〈B2p-B2xn,J(un-q)〉≤12[∥xn-p∥2+∥un-q∥2-g1(∥xn-un-(p-q)∥)∥un-q∥2]+μ2∥B2p-B2xn∥∥un-q∥,
which implies that
(50)∥un-q∥2≤∥xn-p∥2-g1(∥xn-un-(p-q)∥)+2μ2∥B2p-B2xn∥∥un-q∥.
In the same way, we derive
(51)∥vn-p∥2=∥ΠC(un-μ1B1un)-ΠC(q-μ1B1q)∥2≤〈un-μ1B1un-(q-μ1B1q),J(vn-p)〉=〈un-q,J(vn-p)〉+μ1〈B1q-B1un,J(vn-p)〉≤12[∥un-q∥2+∥vn-p∥2-g2(∥un-vn+(p-q)∥)∥vn-p∥2∥vn-p∥2]+μ1∥B1q-B1un∥∥vn-p∥,
which implies that
(52)∥vn-p∥2≤∥un-q∥2-g2(∥un-vn+(p-q)∥)+2μ1∥B1q-B1un∥∥vn-p∥.
Substituting (50) into (52), we get
(53)∥vn-p∥2≤∥xn-p∥2-g1(∥xn-un-(p-q)∥)-g2(∥un-vn+(p-q)∥)+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥.
From (46) and (53), we have
(54)∥xn+1-p∥2≤αnM1+βn∥xn-p∥2+(1-βn)(1-1-ρ1-αnραn)×[∥xn-p∥2-g1(∥xn-un-(p-q)∥)-g2(∥un-vn+(p-q)∥)+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥∥vn-p∥2]≤αnM1+(1-(1-βn)(1-ρ)1-αnραn)∥xn-p∥2-(1-βn)(1-1-ρ1-αnραn)×[g1(∥xn-un-(p-q)∥)+g2(∥un-vn+(p-q)∥)]+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥≤αnM1+∥xn-p∥2-(1-βn)×(1-1-ρ1-αnραn)×[g1(∥xn-un-(p-q)∥)+g2(∥un-vn+(p-q)∥)]+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥,
which implies that
(55)(1-βn)(1-1-ρ1-αnραn)×[g1(∥xn-un-(p-q)∥)+g2(∥un-vn+(p-q)∥)]≤αnM1+∥xn-p∥2-∥xn+1-p∥2+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥≤αnM1+(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥.
Utilizing conditions (i), (ii), from (40) and (48), we have
(56)limn→∞g1(∥xn-un-(p-q)∥)=0,limn→∞g2(∥un-vn+(p-q)∥)=0.
Utilizing the properties of g1 and g2, we deduce that
(57)limn→∞∥xn-un-(p-q)∥=0,limn→∞∥un-vn+(p-q)∥=0.
From (57), we obtain
(58)∥xn-vn∥≤∥xn-un-(p-q)∥+∥un-vn+(p-q)∥⟶0asn→∞.
That is,
(59)limn→∞∥xn-G(xn)∥=0.
On the other hand, since {yn} and {SnG(yn)} are bounded, by Lemma 6, there exists a continuous strictly increasing function g3:[0,∞)→[0,∞), g3(0)=0 such that for p∈F(60)∥xn+1-p∥2=∥βn(xn-p)+γn(yn-p)+δn(SnG(yn)-p)∥2=∥(γn+δn)[γnγn+δn(yn-p)+δnγn+δn(SnG(yn)-p)]+βn(xn-p)[γnγn+δn]∥2≤(γn+δn)∥γnγn+δn(yn-p)+δnγn+δn(SnG(yn)-p)∥2+βn∥xn-p∥2≤(γn+δn)[γnδn(γn+δn)2γnγn+δn∥yn-p∥2+δnγn+δn∥SnG(yn)-p∥2-γnδn(γn+δn)2g3(∥yn-SnG(yn)∥)]+βn∥xn-p∥2≤γn∥yn-p∥2+δn∥yn-p∥2-γnδnγn+δng3(∥yn-SnG(yn)∥)+βn∥xn-p∥2=(1-βn)∥yn-p∥2-γnδnγn+δn×g3(∥yn-SnG(yn)∥)+βn∥xn-p∥2,
which together with (30) implies that
(61)∥xn+1-p∥2≤(1-βn)(∥xn-p∥+αn1-αnρ∥f(p)-p∥)2-γnδnγn+δng3(∥yn-SnG(yn)∥)+βn∥xn-p∥2≤(∥xn-p∥+αn1-αnρ∥f(p)-p∥)2-γnδnγn+δng3(∥yn-SnG(yn)∥).
It immediately follows that
(62)γnδng3(∥yn-SnG(yn)∥)≤γnδnγn+δng3(∥yn-SnG(yn)∥)≤(∥xn-p∥+αn1-αnρ∥f(p)-p∥)2-∥xn+1-p∥2≤(αn1-αnρ∥xn-p∥+∥xn+1-p∥+αn1-αnρ∥f(p)-p∥)×(∥xn-xn+1∥+αn1-αnρ∥f(p)-p∥).
According to condition (ii), we get
(63)liminfn→∞δn=liminfn→∞(1-βn-γn)=1-limsupn→∞(βn+γn)>0.
Since αn→0, ∥xn+1-xn∥→0, and liminfn→∞γn>0, we conclude that
(64)limn→∞g3(∥yn-SnG(yn)∥)=0.
Utilizing the property of g3, we have
(65)limn→∞∥yn-SnG(yn)∥=0.
We note that
(66)xn+1-xn+xn-yn=xn+1-yn=βn(xn-yn)+δn(SnG(yn)-yn).
So,
(67)(1-βn)∥xn-yn∥=∥δn(SnG(yn)-yn)-(xn+1-xn)∥≤δn∥SnG(yn)-yn∥+∥xn+1-xn∥≤∥SnG(yn)-yn∥+∥xn+1-xn∥⟶0asn⟶∞.
That is,
(68)limn→∞∥xn-yn∥=0.
We observe that
(69)∥G(xn)-SnG(xn)∥≤∥G(xn)-xn∥+∥xn-yn∥+∥yn-SnG(yn)∥+∥SnG(yn)-SnG(xn)∥≤∥G(xn)-xn∥+2∥xn-yn∥+∥yn-SnG(yn)∥.
Thus, from (59)–(68), we obtain that
(70)limn→∞∥G(xn)-SnG(xn)∥=0.
By (70) and Lemma 7, we have
(71)∥SG(xn)-G(xn)∥≤∥SG(xn)-SnG(xn)∥+∥SnG(xn)-G(xn)∥⟶0asn⟶∞.
In terms of (59) and (71), we have
(72)∥xn-Sxn∥≤∥xn-G(xn)∥+∥G(xn)-SG(xn)∥+∥SG(xn)-Sxn∥≤2∥xn-G(xn)∥+∥G(xn)-SG(xn)∥⟶0asn⟶∞.
Define a mapping Wx=(1-θ)Sx+θG(x), where θ∈(0,1) is a constant. Then by Lemma 9, we have that Fix(W)=Fix(S)∩Fix(G)=F. We observe that
(73)∥xn-Wxn∥=∥(1-θ)(xn-Sxn)+θ(xn-G(xn))∥≤(1-θ)∥xn-Sxn∥+θ∥xn-G(xn)∥.
From (59) and (72), we obtain
(74)limn→∞∥xn-Wxn∥=0.
Now, we claim that(75)limsupn→∞〈f(q)-q,J(xn-q)〉≤0,
where q=s-limt→0xt with xt being the fixed point of the contraction
(76)x⟼tf(x)+(1-t)Wx.
Then xt solves the fixed point equation xt=tf(xt)+(1-t)Wxt. Thus we have
(77)∥xt-xn∥=∥(1-t)(Wxt-xn)+t(f(xt)-xn)∥.
By Lemma 3, we conclude that
(78)∥xt-xn∥2=∥(1-t)(Wxt-xn)+t(f(xt)-xn)∥2≤(1-t)2∥Wxt-xn∥2+2t〈f(xt)-xn,J(xt-xn)〉≤(1-t)2(∥Wxt-Wxn∥+∥Wxn-xn∥)2+2t〈f(xt)-xn,J(xt-xn)〉≤(1-t)2(∥xt-xn∥+∥Wxn-xn∥)2+2t〈f(xt)-xn,J(xt-xn)〉=(1-t)2[∥xt-xn∥2+2∥xt-xn∥×∥Wxn-xn∥+∥Wxn-xn∥2]+2t〈f(xt)-xt,J(xt-xn)〉+2t〈xt-xn,J(xt-xn)〉=(1-2t+t2)∥xt-xn∥2+fn(t)+2t〈f(xt)-xt,J(xt-xn)〉+2t∥xt-xn∥2,
where
(79)fn(t)=(1-t)2(2∥xt-xn∥+∥xn-Wxn∥)×∥xn-Wxn∥⟶0,asn⟶∞.
It follows from (78) that
(80)〈xt-f(xt),J(xt-xn)〉≤t2∥xt-xn∥2+12tfn(t).
Letting n→∞ in (80) and noticing (79), we derive
(81)limsupn→∞〈xt-f(xt),J(xt-xn)〉≤t2M2,
where M2>0 is a constant such that ∥xt-xn∥2≤M2 for all t∈(0,1) and n≥0. Taking t→0 in (81), we have
(82)limsupt→0limsupn→∞〈xt-f(xt),J(xt-xn)〉≤0.
On the other hand, we have
(83)〈f(q)-q,J(xn-q)〉=〈f(q)-q,J(xn-q)〉-〈f(q)-q,J(xn-xt)〉+〈f(q)-q,J(xn-xt)〉-〈f(q)-xt,J(xn-xt)〉+〈f(q)-xt,J(xn-xt)〉-〈f(xt)-xt,J(xn-xt)〉+〈f(xt)-xt,J(xn-xt)〉=〈f(q)-q,J(xn-q)-J(xn-xt)〉+〈xt-q,J(xn-xt)〉+〈f(q)-f(xt),J(xn-xt)〉+〈f(xt)-xt,J(xn-xt)〉.
It follows that
(84)limsupn→∞〈f(q)-q,J(xn-q)〉≤limsupn→∞〈f(q)-q,J(xn-q)-J(xn-xt)〉+∥xt-q∥limsupn→∞∥xn-xt∥+ρ∥q-xt∥limsupn→∞∥xn-xt∥+limsupn→∞〈f(xt)-xt,J(xn-xt)〉.
Taking into account that xt→q as t→0, we have from (82)
(85)limsupn→∞〈f(q)-q,J(xn-q)〉=limsupt→0limsupn→∞〈f(q)-q,J(xn-q)〉≤limsupt→0limsupn→∞〈f(q)-q,J(xn-q)-J(xn-xt)〉.
Since X has a uniformly Fréchet differentiable norm, the duality mapping J is norm-to-norm uniformly continuous on bounded subsets of X. Consequently, the two limits are interchangeable, and hence (75) holds. From (68), we get (yn-q)-(xn-q)→0. Noticing that J is norm-to-norm uniformly continuous on bounded subsets of X, we deduce from (75) that
(86)limsupn→∞〈f(q)-q,J(yn-q)〉=limsupn→∞(〈f(q)-q,J(xn-q)〉+〈f(q)-q,J(yn-q)-J(xn-q)〉)=limsupn→∞〈f(q)-q,J(xn-q)〉≤0.
Finally, let us show thatxn→q as n→∞. We observe that
(87)∥yn-q∥2=∥αn(f(yn)-f(q))+(1-αn)×(SnG(xn)-q)+αn(f(q)-q)∥2≤∥αn(f(yn)-f(q))+(1-αn)(SnG(xn)-q)(f(yn)-f(q))∥2+2αn〈f(q)-q,J(yn-q)〉≤αn∥f(yn)-f(q)∥2+(1-αn)∥SnG(xn)-q∥2+2αn〈f(q)-q,J(yn-q)〉≤αnρ∥yn-q∥2+(1-αn)∥xn-q∥2+2αn〈f(q)-q,J(yn-q)〉,
which implies that
(88)∥yn-q∥2≤(1-1-ρ1-αnραn)∥xn-q∥2+αn(1-ρ)1-αnρ·2〈f(q)-q,J(yn-q)〉1-ρ.
By (27) and the convexity of ∥·∥2, we get
(89)∥xn+1-q∥2≤βn∥xn-q∥2+γn∥yn-q∥2+δn∥SnG(yn)-q∥2≤βn∥xn-q∥2+γn∥yn-q∥2+δn∥yn-q∥2=βn∥xn-q∥2+(1-βn)∥yn-q∥2,
which together with (88) leads to
(90)∥xn+1-q∥2≤βn∥xn-q∥2+(1-βn)×{2〈f(q)-q,J(yn-q)〉1-ρ(1-1-ρ1-αnραn)∥xn-q∥2+αn(1-ρ)1-αnρ·2〈f(q)-q,J(yn-q)〉1-ρ}=[1-(1-βn)(1-ρ)1-αnραn]∥xn-q∥2+(1-βn)(1-ρ)1-αnραn·2〈f(q)-q,J(yn-q)〉1-ρ.
Applying Lemma 2 to (88), we obtain that xn→q as n→∞. This completes the proof.Corollary 15.
LetC be a nonempty, closed, and convex subset of a uniformly convex and 2-uniformly smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C. Let the mapping Bi:C→X be αi-inverse-strongly accretive for i=1,2. Let f:C→C be a contraction with coefficient ρ∈(0,1). Let S be a nonexpansive mapping of C into itself such that F=Fix(S)∩Ω≠∅, where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2). For arbitrarily given x0∈C, let {xn} be the sequence generated by
(91)yn=αnf(yn)+(1-αn)SG(xn),xn+1=βnxn+γnyn+δnSG(yn),∀n≥0,
where 0<μi<αi/κ2 for i=1,2 and {αn}, {βn}, {γn}, and {δn} are the sequences in (0,1) such that βn+γn+δn=1, ∀n≥0. Suppose that the following conditions hold:(i)
limn→∞αn=0 and ∑n=0∞‍αn=∞,(ii)
0<liminfn→∞βn≤limsupn→∞(βn+γn)<1 and liminfn→∞γn>0,(iii)
limn→∞|(γn/(1-βn))-(γn-1/(1-βn-1))|=0.Then {xn} converges strongly to q∈F, which solves the following VIP:
(92)〈q-f(q),J(q-p)〉≤0,∀p∈F.Further, we illustrate Theorem14 by virtue of an example, that is, the following corollary.Corollary 16.
LetC be a nonempty, closed, and convex subset of a uniformly convex and 2-uniformly smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C. Let f:C→C be a contraction with coefficient ρ∈(0,1). Let T be an η-strictly pseudocontractive mapping of C into itself, and let S be a nonexpansive mapping of C into itself such that F=Fix(S)∩Fix(T)≠∅. For arbitrarily given x0∈C, let {xn} be the sequence generated by
(93)yn=αnf(yn)+(1-αn)S(I-λ(I-T))xn,xn+1=βnxn+γnyn+δnS(I-λ(I-T))yn,∀n≥0,
where 0<λ<max{1,η/κ2} and {αn}, {βn}, {γn}, and {δn} are the sequences in (0,1) such that βn+γn+δn=1, ∀n≥0. Suppose that the following conditions hold:(i)
limn→∞αn=0 and ∑n=0∞‍αn=∞,(ii)
0<liminfn→∞βn≤limsupn→∞(βn+γn)<1 and liminfn→∞γn>0,(iii)
limn→∞|(γn/(1-βn))-(γn-1/(1-βn-1))|=0.Then {xn} converges strongly to q∈F, which solves the following VIP:
(94)〈q-f(q),J(q-p)〉≤0,∀p∈F.Proof.
In Corollary15, put B1=I-T, B2=0, μ1=λ, and α1=η. Since T is an η-strictly pseudocontractive mapping, it is clear that B1=I-T is an η-inverse strongly accretive mapping. Hence, the GSVI (9) is equivalent to the following VIP of finding x*∈C such that
(95)(B1x*,J(x-x*))≥0,∀x∈C,
which leads to Ω=VI(C,B1). In the meantime, we have
(96)Gxn=ΠC(I-μ1B1)ΠC(I-μ2B2)xn=ΠC(I-μ1B1)xn=ΠC[(1-λ)xn+λTxn]=xn-λ(I-T)xn.
In the same way, we get Gyn=yn-λ(I-T)yn. In this case, it is easy to see that (91) reduces to (93). We claim that Fix(T)=VI(C,B1). As a matter of fact, we have, for λ>0,
(97)u∈VI(C,B1)⟺〈B1u,J(y-u)〉≥0∀y∈C⟺〈u-λB1u-u,J(u-y)〉≥0∀y∈C⟺u=ΠC(u-λB1u)⟺u=ΠC(u-λu+λTu)⟺〈u-λu+λTu-u,J(u-y)〉≥0∀y∈C⟺〈u-Tu,J(u-y)〉≤0∀y∈C⟺u=Tu⟺u∈Fix(T).
So, we conclude that F=Fix(S)∩Ω=Fix(S)∩Fix(T). Therefore, the desired result follows from Corollary 15.Remark 17.
Theorem14 improves, extends, supplements, and develops Cai and Bu [2, Theorem  3.1 and Corollary  3.2] and Jung [5, Theorem  3.1] in the following aspects.(i)
The problem of finding a pointq∈⋂n‍Fix(Sn)∩Ω in Theorem 14 is more general and more subtle than the problem of finding a point q∈Fix(S)∩VI(C,A) in Jung [5, Theorem  3.1].(ii)
The iterative scheme in [2, Theorem  3.1] is extended to develop the iterative scheme (27) of Theorem 14 by virtue of the iterative scheme of [5, Theorem  3.1]. The iterative scheme (27) of Theorem 14 is more advantageous and more flexible than the iterative scheme of [2, Theorem  3.1] because it involves several parameter sequences {αn}, {βn}, {γn}, and {δn}.(iii)
The iterative scheme (27) in Theorem 14 is very different from everyone in both [2, Theorem  3.1] and [5, Theorem  3.1] because the mappings Sn and G in the iterative scheme of [2, Theorem  3.1] and the mapping SPC(I-λnA) in the iterative scheme of [5, Theorem  3.1] are replaced by the same composite mapping SnG in the iterative scheme (27) of Theorem 14.(iv)
The proof in [2, Theorem  3.1] depends on the argument techniques in [3], the inequality in 2-uniformly smooth Banach spaces ([9]), and the inequality in smooth and uniform convex Banach spaces ([14, Proposition 1]). Because the composite mapping SnG appears in the iterative scheme (27) of Theorem 14, the proof of Theorem 14 depends on the argument techniques in [3], the inequality in 2-uniformly smooth Banach spaces, the inequality in smooth and uniform convex Banach spaces, and the inequality in uniform convex Banach spaces (Lemma 6).(v)
The iterative scheme in [2, Corollary 3.2] is extended to develop the new iterative scheme in Corollary 15 because the mappings S and G are replaced by the same composite mapping SG in Corollary 15.


## 4. Explicit Iterative Schemes

In this section, we introduce our explicit iterative schemes and show the strong convergence theorems. First, we give several useful lemmas.Lemma 18.
LetC be a nonempty, closed, and convex subset of a smooth Banach space X, and let the mapping Bi:C→X be λi-strictly pseudocontractive and αi-strongly accretive with αi+λi≥1 for i=1,2. Then, for μi∈(0,1], we have
(98)∥(I-μiBi)x-(I-μiBi)y∥≤{1-αiλi+(1-μi)(1+1λi)}∥x-y∥,∀x,y∈C,
for i=1,2. In particular, if 1-(λi/(1+λi))(1-(1-αi)/λi)≤μi≤1, then I-μiBi is nonexpansive for i=1,2.Proof.
Taking into account theλi-strict pseudocontractivity of Bi, we derive for every x,y∈C(99)λi∥(I-Bi)x-(I-Bi)y∥2≤〈(I-Bi)x-(I-Bi)y,J(x-y)〉≤∥(I-Bi)x-(I-Bi)y∥∥x-y∥,
which implies that
(100)∥(I-Bi)x-(I-Bi)y∥≤1λi∥x-y∥.
Hence,
(101)∥Bix-Biy∥≤∥(I-Bi)x-(I-Bi)y∥+∥x-y∥≤(1+1λi)∥x-y∥.
Utilizing the αi-strong accretivity and λi-strict pseudocontractivity of Bi, we get
(102)λi∥(I-Bi)x-(I-Bi)y∥2≤∥x-y∥2-〈Bix-Biy,J(x-y)〉≤(1-αi)∥x-y∥2.
So, we have
(103)∥(I-Bi)x-(I-Bi)y∥≤1-αiλi∥x-y∥.
Therefore, for μi∈(0,1], we have
(104)∥(I-μiBi)x-(I-μiBi)y∥≤∥(I-Bi)x-(I-Bi)y∥+(1-μi)∥Bix-Biy∥≤1-αiλi∥x-y∥+(1-μi)(1+1λi)∥x-y∥={1-αiλi+(1-μi)(1+1λi)}∥x-y∥.
Since 1-(λi/(1+λi))(1-(1-αi)/λi)≤μi≤1, it follows immediately that
(105)1-αiλi+(1-μi)(1+1λi)≤1.
This implies that I-μiBi is nonexpansive for i=1,2.Lemma 19.
LetC be a nonempty, closed, and convex subset of a smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C, and let the mapping Bi:C→X be λi-strictly pseudocontractive and αi-strongly accretive with αi+λi≥1 for i=1,2. Let G:C→C be the mapping defined by
(106)G(x)=ΠC[ΠC(x-μ2B2x)-μ1B1ΠC(x-μ2B2x)],∀x∈C.
If 1-(λi/(1+λi))(1-(1-αi)/λi)≤μi≤1, then G:C→C is nonexpansive.Proof.
According to Lemma10, we know that I-μiBi is nonexpansive for i=1,2. Hence, for all x,y∈C, we have
(107)∥G(x)-G(y)∥=∥ΠC[ΠC(x-μ2B2x)-μ1B1ΠC(x-μ2B2x)]-ΠC[ΠC(y-μ2B2y)-μ1B1ΠC(y-μ2B2y)]∥=∥ΠC(I-μ1B1)ΠC(I-μ2B2)x-ΠC(I-μ1B1)ΠC(I-μ2B2)y∥≤∥(I-μ1B1)ΠC(I-μ2B2)x-(I-μ1B1)ΠC(I-μ2B2)y∥≤∥ΠC(I-μ2B2)x-ΠC(I-μ2B2)y∥≤∥(I-μ2B2)x-(I-μ2B2)y∥≤∥x-y∥.
This shows that G:C→C is nonexpansive. This completes the proof.Lemma 20.
LetC be a nonempty, closed, and convex subset of a smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C, and let the mapping Bi:C→X be λi-strictly pseudocontractive and αi-strongly accretive for i=1,2. For given x*,y*∈C, (x*,y*) is a solution of GSVI (9) if and only if x*=ΠC(y*-μ1B1y*) where y*=ΠC(x*-μ2B2x*).Proof.
We can rewrite GSVI (9) as
(108)〈x*-(y*-μ1B1y*),J(x-x*)〉≥0,∀x∈C,〈y*-(x*-μ2B2x*),J(x-y*)〉≥0,∀x∈C,
which is obviously equivalent to
(109)x*=ΠC(y*-μ1B1y*),y*=ΠC(x*-μ2B2x*),
because of Lemma 5. This completes the proof.Remark 21.
By Lemma20, we observe that
(110)x*=ΠC[ΠC(x*-μ2B2x*)-μ1B1ΠC(x*-μ2B2x*)],
which implies that x* is a fixed point of the mapping G. Throughout this paper, the set of fixed points of the mapping G is denoted by Ω.We are now in a position to state and prove our result on the explicit iterative scheme.Theorem 22.
LetC be a nonempty, closed, and convex subset of a uniformly convex Banach space X which has a uniformly Gâteaux differentiable norm. Let ΠC be a sunny nonexpansive retraction from X onto C. Let the mapping Bi:C→X be λi-strictly pseudocontractive and αi-strongly accretive with αi+λi≥1 for i=1,2. Let f:C→C be a contraction with coefficient ρ∈(0,1). Let {Sn}n=0∞ be an infinite family of nonexpansive mappings of C into itself such that F=⋂n=0∞‍Fix(Sn)∩Ω≠∅, where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2). For arbitrarily given x0∈C, let {xn} be the sequence generated by
(111)yn=αnG(xn)+(1-αn)SnG(xn),xn+1=βnf(xn)+γnyn+δnSnG(yn),∀n≥0,
where 1-(λi/(1+λi))(1-(1-αi)/λi)≤μi≤1 for i=1,2 and {αn}, {βn}, {γn}, and {δn} are the sequences in (0,1) such that βn+γn+δn=1, ∀n≥0. Suppose that the following conditions hold:(i)
0<liminfn→∞αn≤limsupn→∞αn<1,(ii)
limn→∞βn=0 and ∑n=0∞‍βn=∞,(iii)
∑n=1∞‍|αn-αn-1|<∞ or limn→∞|αn-αn-1|/βn=0,(iv)
∑n=1∞‍|βn-βn-1|<∞ or limn→∞βn-1/βn=1,(v)
∑n=1∞‍|(γn/(1-βn))-(γn-1/(1-βn-1))|<∞ or limn→∞(1/βn)|(γn/(1-βn))-(γn-1/(1-βn-1))|=0,(vi)
0<liminfn→∞γn≤limsupn→∞γn<1.Assume that ∑n=1∞‍supx∈D∥Snx-Sn-1x∥<∞ for any bounded subset D of C, and let S be a mapping of C into itself defined by Sx=limn→∞Snx for all x∈C. Suppose that Fix(S)=⋂n=0∞‍Fix(Sn). Then {xn} converges strongly to q∈F, which solves the following VIP:
(112)〈q-f(q),J(q-p)〉≤0,∀p∈F.Proof.
Take a fixedp∈F arbitrarily. Then by Lemma 20, we know that p=G(p) and p=Snp for all n≥0. Moreover, by Lemma 19, we have
(113)∥yn-p∥≤αn∥G(xn)-p∥+(1-αn)∥SnG(xn)-p∥≤αn∥xn-p∥+(1-αn)∥xn-p∥=∥xn-p∥.
From (113) we obtain
(114)∥xn+1-p∥≤βn∥f(xn)-p∥+γn∥yn-p∥+δn∥SnG(yn)-p∥≤βn(∥f(xn)-f(p)∥+∥f(p)-p∥)+γn∥yn-p∥+δn∥yn-p∥≤βnρ∥xn-p∥+βn∥f(p)-p∥+(1-βn)∥yn-p∥≤βnρ∥xn-p∥+βn∥f(p)-p∥+(1-βn)∥xn-p∥=(1-βn(1-ρ))∥xn-p∥+βn∥f(p)-p∥=(1-βn(1-ρ))∥xn-p∥+βn(1-ρ)·∥f(p)-p∥1-ρ≤max{∥x0-p∥,∥f(p)-p∥1-ρ},
which implies that {xn} is bounded. By Lemma 19 we know from (113) that {yn}, {G(xn)}, and {G(yn)} are bounded.
Let us show that∥xn+1-xn∥→0 and ∥xn-yn∥→0 as n→∞. As a matter of fact, from (113), we have
(115)yn=αnG(xn)+(1-αn)SnG(xn),yn-1=αn-1G(xn-1)+(1-αn-1)Sn-1G(xn-1),∀n≥1.
Simple calculations show that
(116)yn-yn-1=αn(G(xn)-G(xn-1))+(αn-αn-1)(G(xn-1)-Sn-1G(xn-1))+(1-αn)(SnG(xn)-Sn-1G(xn-1)).
It follows that
(117)∥yn-yn-1∥≤αn∥G(xn)-G(xn-1)∥+|αn-αn-1|∥G(xn-1)-Sn-1G(xn-1)∥+(1-αn)∥SnG(xn)-Sn-1G(xn-1)∥≤αn∥G(xn)-G(xn-1)∥+|αn-αn-1|∥G(xn-1)-Sn-1G(xn-1)∥+(1-αn)(∥SnG(xn)-SnG(xn-1)∥+∥SnG(xn-1)-Sn-1G(xn-1)∥)≤αn∥G(xn)-G(xn-1)∥+|αn-αn-1|∥G(xn-1)-Sn-1G(xn-1)∥+(1-αn)(∥G(xn)-G(xn-1)∥+∥SnG(xn-1)-Sn-1G(xn-1)∥)≤∥G(xn)-G(xn-1)∥+|αn-αn-1|∥G(xn-1)-Sn-1G(xn-1)∥+∥SnG(xn-1)-Sn-1G(xn-1)∥≤∥xn-xn-1∥+|αn-αn-1|×∥G(xn-1)-Sn-1G(xn-1)∥+∥SnG(xn-1)-Sn-1G(xn-1)∥.
Now, we write xn=βn-1f(xn-1)+(1-βn-1)vn-1, ∀n≥1, where vn-1=(xn-βn-1f(xn-1))/(1-βn-1). It follows that, for all n≥1,
(118)vn-vn-1=xn+1-βnf(xn)1-βn-xn-βn-1f(xn-1)1-βn-1=γnyn+δnSnG(yn)1-βn-γn-1yn-1+δn-1Sn-1G(yn-1)1-βn-1=γn(yn-yn-1)+δn(SnG(yn)-Sn-1G(yn-1))1-βn+(γn1-βn-γn-11-βn-1)yn-1+(δn1-βn-δn-11-βn-1)Sn-1G(yn-1).
This together with (117) implies that
(119)∥vn-vn-1∥≤∥γn(yn-yn-1)+δn(SnG(yn)-Sn-1G(yn-1))∥1-βn+|γn1-βn-γn-11-βn-1|∥yn-1∥+|δn1-βn-δn-11-βn-1|∥Sn-1G(yn-1)∥≤(∥SnG(yn-1)-Sn-1G(yn-1)∥)γn∥yn-yn-1∥+δn(∥SnG(yn-1)-Sn-1G(yn-1)∥∥SnG(yn)-SnG(yn-1)∥+∥SnG(yn-1)-Sn-1G(yn-1)∥))×(1-βn)-1+|γn1-βn-γn-11-βn-1|∥yn-1∥+|γn1-βn-γn-11-βn-1|∥Sn-1G(yn-1)∥≤γn∥yn-yn-1∥+δn∥yn-yn-1∥γn+δn+δnγn+δn∥SnG(yn-1)-Sn-1G(yn-1)∥+|γn1-βn-γn-11-βn-1|×(∥yn-1∥+∥Sn-1G(yn-1)∥)≤∥yn-yn-1∥+∥SnG(yn-1)-Sn-1G(yn-1)∥+|γn1-βn-γn-11-βn-1|(∥yn-1∥+∥Sn-1G(yn-1)∥)≤∥xn-xn-1∥+|αn-αn-1|∥G(xn-1)-Sn-1G(xn-1)∥+∥SnG(xn-1)-Sn-1G(xn-1)∥+∥SnG(yn-1)-Sn-1G(yn-1)∥+|γn1-βn-γn-11-βn-1|(∥yn-1∥+∥Sn-1G(yn-1)∥).
Furthermore, we note that
(120)xn+1=βnf(xn)+(1-βn)vn,xn=βn-1f(xn-1)+(1-βn-1)vn-1,∀n≥1.
Also, simple calculations show that
(121)xn+1-xn=βn(f(xn)-f(xn-1))+(βn-βn-1)×(f(xn-1)-vn-1)+(1-βn)(vn-vn-1).
This together with (119) implies that
(122)∥xn+1-xn∥≤βn∥f(xn)-f(xn-1)∥+|βn-βn-1|×∥f(xn-1)-vn-1∥+(1-βn)∥vn-vn-1∥≤βnρ∥xn-xn-1∥+|βn-βn-1|∥f(xn-1)-vn-1∥+(1-βn)×[|γn1-βn-γn-11-βn-1|∥xn-xn-1∥+|αn-αn-1|∥G(xn-1)-Sn-1G(xn-1)∥+∥SnG(xn-1)-Sn-1G(xn-1)∥+∥SnG(yn-1)-Sn-1G(yn-1)∥+|γn1-βn-γn-11-βn-1|×(∥yn-1∥+∥Sn-1G(yn-1)∥)|γn1-βn-γn-11-βn-1|]≤(1-βn(1-ρ))∥xn-xn-1∥+|βn-βn-1|∥f(xn-1)-vn-1∥+|αn-αn-1|∥G(xn-1)-Sn-1G(xn-1)∥+∥SnG(xn-1)-Sn-1G(xn-1)∥+∥SnG(yn-1)-Sn-1G(yn-1)∥+|γn1-βn-γn-11-βn-1|(∥yn-1∥+∥Sn-1G(yn-1)∥)≤(1-βn(1-ρ))∥xn-xn-1∥+|βn-βn-1|M+|αn-αn-1|M+∥SnG(xn-1)-Sn-1G(xn-1)∥+∥SnG(yn-1)-Sn-1G(yn-1)∥+|γn1-βn-γn-11-βn-1|M=(1-βn(1-ρ))∥xn-xn-1∥+M(|γn1-βn-γn-11-βn-1||αn-αn-1|+|βn-βn-1|+|γn1-βn-γn-11-βn-1|)+∥SnG(xn-1)-Sn-1G(xn-1)∥+∥SnG(yn-1)-Sn-1G(yn-1)∥,
where supn≥0{∥f(xn)∥+∥vn∥+∥G(xn)∥+∥SnG(xn)∥+∥yn∥+∥SnG(yn)∥}≤M for some M>0. Utilizing Lemma 2, from conditions (ii)–(v) and the assumption on {Sn}, we deduce that
(123)limn→∞∥xn+1-xn∥=0.
Taking into account the boundedness of {G(xn)} and {SnG(xn)}, by Lemma 6, we know that there exists a continuous strictly increasing function g1:[0,∞)→[0,∞), g1(0)=0 such that for p∈F(124)∥yn-p∥2≤αn∥G(xn)-p∥2+(1-αn)∥SnG(xn)-p∥2-αn(1-αn)g1(∥G(xn)-SnG(xn)∥)≤αn∥xn-p∥2+(1-αn)∥xn-p∥2-αn(1-αn)g1(∥G(xn)-SnG(xn)∥)=∥xn-p∥2-αn(1-αn)g1(∥G(xn)-SnG(xn)∥).
Since {yn} and {SnG(yn)} are bounded, by Lemma 6, there exists a continuous strictly increasing function g2:[0,∞)→[0,∞), g2(0)=0 such that for p∈F(125)∥xn+1-p∥2=∥(SnG(yn)-p)βn(f(xn)-p)+γn(yn-p)+δn(SnG(yn)-p)∥2=∥[γnγn+δn(yn-p)(γn+δn)×[γnγn+δn(yn-p)+δnγn+δn(SnG(yn)-p)]+βn(f(xn)-p)[γnγn+δn(yn-p)∥2≤(γn+δn)2∥γnγn+δn(yn-p)+δnγn+δn(SnG(yn)-p)∥2+2βn〈f(xn)-p,J(xn+1-p)〉≤(γn+δn)2[γnδn(γn+δn)2γnγn+δn∥yn-p∥2+δnγn+δn∥SnG(yn)-p∥2-γnδn(γn+δn)2g2(∥yn-SnG(yn)∥)]+2βn∥f(xn)-p∥∥xn+1-p∥≤γn∥yn-p∥2+δn∥yn-p∥2-γnδng2(∥yn-SnG(yn)∥)+2βn∥f(xn)-p∥∥xn+1-p∥≤∥yn-p∥2-γnδng2(∥yn-SnG(yn)∥)+2βn∥f(xn)-p∥∥xn+1-p∥,
which together with (124) implies that
(126)∥xn+1-p∥2≤∥xn-p∥2-αn(1-αn)g1(∥G(xn)-SnG(xn)∥)-γnδng2(∥yn-SnG(yn)∥)+2βn∥f(xn)-p∥∥xn+1-p∥.
It immediately follows that
(127)αn(1-αn)g1(∥G(xn)-SnG(xn)∥)+γnδng2(∥yn-SnG(yn)∥)≤∥xn-p∥2-∥xn+1-p∥2+2βn∥f(xn)-p∥∥xn+1-p∥≤(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+2βn∥f(xn)-p∥∥xn+1-p∥.
According to condition (vi), we get
(128)liminfn→∞δn=liminfn→∞(1-βn-γn)=1-limsupn→∞(βn+γn)>0.
Since βn→0 and ∥xn+1-xn∥→0, we conclude from conditions (i) and (vi) that
(129)limn→∞g1(G(xn)-SnG(xn))=0,limn→∞g2(∥yn-SnG(yn)∥)=0.
Utilizing the properties of g1 and g2, we have
(130)limn→∞∥G(xn)-SnG(xn)∥=0,limn→∞∥yn-SnG(yn)∥=0.
Note that
(131)∥yn-xn∥=∥xn+1-xn-βn(f(xn)-yn)-δn(SnG(yn)-yn)∥≤∥xn+1-xn∥+βn∥f(xn)-yn∥+δn∥SnG(yn)-yn∥.
Thus, from (123), (130), and βn→0, it follows that
(132)limn→∞∥yn-xn∥=0.
On the other hand, from (130), we get
(133)limn→∞∥yn-G(xn)∥=limn→∞(1-αn)×∥SnG(xn)-G(xn)∥=0.
This together with (132) implies that
(134)limn→∞∥xn-G(xn)∥=0.
By (130) and Lemma 7, we have
(135)∥SG(xn)-G(xn)∥≤∥SG(xn)-SnG(xn)∥+∥SnG(xn)-G(xn)∥⟶0asn⟶∞.
In terms of (134) and (135), we have
(136)∥xn-Sxn∥≤∥xn-G(xn)∥+∥G(xn)-SG(xn)∥+∥SG(xn)-Sxn∥≤2∥xn-G(xn)∥+∥G(xn)-SG(xn)∥⟶0asn⟶∞.
Define a mapping Wx=(1-θ)Sx+θG(x), where θ∈(0,1) is a constant. Then by Lemma 9, we have that Fix(W)=Fix(S)∩Fix(G)=F. We observe that
(137)∥xn-Wxn∥=∥(1-θ)(xn-Sxn)+θ(xn-G(xn))∥≤(1-θ)∥xn-Sxn∥+θ∥xn-G(xn)∥.
From (134) and (136), we obtain
(138)limn→∞∥xn-Wxn∥=0.
Now, we claim that(139)limsupn→∞〈f(q)-q,J(xn-q)〉≤0,
where q=s-limt→0xt with xt being the fixed point of the contraction
(140)x⟼tf(x)+(1-t)Wx.
Then xt solves the fixed point equation xt=tf(xt)+(1-t)Wxt. Thus we have
(141)∥xt-xn∥=∥(1-t)(Wxt-xn)+t(f(xt)-xn)∥.
By Lemma 3, we conclude that
(142)∥xt-xn∥2≤(1-t)2∥Wxt-xn∥2+2t〈f(xt)-xn,J(xt-xn)〉≤(1-t)2(∥Wxt-Wxn∥+∥Wxn-xn∥)2+2t〈f(xt)-xn,J(xt-xn)〉≤(1-t)2[∥xt-xn∥2+2∥xt-xn∥×∥Wxn-xn∥+∥Wxn-xn∥2]+2t〈f(xt)-xt,J(xt-xn)〉+2t〈xt-xn,J(xt-xn)〉=(1-2t+t2)∥xt-xn∥2+fn(t)+2t〈f(xt)-xt,J(xt-xn)〉+2t∥xt-xn∥2,
where
(143)fn(t)=(1-t)2(2∥xt-xn∥+∥xn-Wxn∥)×∥xn-Wxn∥⟶0asn⟶∞.
It follows from (142) that
(144)〈xt-f(xt),J(xt-xn)〉≤t2∥xt-xn∥2+12tfn(t).
Letting n→∞ in (144) and noticing (143), we derive
(145)limsupn→∞〈xt-f(xt),J(xt-xn)〉≤t2M2,
where M2>0 is a constant such that ∥xt-xn∥2≤M2 for all t∈(0,1) and n≥0. Taking t→0 in (145), we have
(146)limsupt→0limsupn→∞〈xt-f(xt),J(xt-xn)〉≤0.
On the other hand, we have
(147)〈f(q)-q,J(xn-q)〉=〈f(q)-q,J(xn-q)-J(xn-xt)〉+〈xt-q,J(xn-xt)〉+〈f(q)-f(xt),J(xn-xt)〉+〈f(xt)-xt,J(xn-xt)〉.
Hence it follows that
(148)limsupn→∞〈f(q)-q,J(xn-q)〉≤limsupn→∞〈f(q)-q,J(xn-q)-J(xn-xt)〉+∥xt-q∥limsupn→∞∥xn-xt∥+ρ∥q-xt∥limsupn→∞∥xn-xt∥+limsupn→∞〈f(xt)-xt,J(xn-xt)〉.
Taking into account that xt→q as t→0, we have from (146)
(149)limsupn→∞〈f(q)-q,J(xn-q)〉=limsupt→0limsupn→∞〈f(q)-q,J(xn-q)〉≤limsupt→0limsupn→∞〈f(q)-q,J(xn-q)-J(xn-xt)〉.
Since X has a uniformly Gâteaux differentiable norm, the duality mapping J is norm-to-weak* uniformly continuous on bounded subsets of X. Consequently, the two limits are interchangeable, and hence (139) holds. From (123), we get (xn+1-q)-(xn-q)→0. Noticing the norm-to-weak* uniform continuity of J on bounded subsets of X, we deduce from (139) that
(150)limsupn→∞〈f(q)-q,J(xn+1-q)〉=limsupn→∞(〈f(q)-q,J(xn+1-q)-J(xn-q)〉+〈f(q)-q,J(xn+1-q)〉)=limsupn→∞〈f(q)-q,J(xn-q)〉≤0.
Finally, let us show thatxn→q as n→∞. We observe that
(151)∥yn-q∥=∥αn(G(xn)-q)+(1-αn)(SnG(xn)-q)∥≤αn∥xn-q∥+(1-αn)∥xn-q∥=∥xn-q∥,(152)∥xn+1-q∥2=βn〈f(xn)-f(q)+f(q)-q,J(xn+1-q)〉+〈γn(yn-q)+δn(SnG(yn)-q),J(xn+1-q)〉≤βn∥f(xn)-f(q)∥∥xn+1-q∥+βn〈f(q)-q,J(xn+1-q)〉+∥γn(yn-q)+δn(SnG(yn)-q)∥∥xn+1-q∥≤βnρ∥xn-q∥∥xn+1-q∥+βn〈f(q)-q,J(xn+1-q)〉+(γn∥yn-q∥+δn∥yn-q∥)∥xn+1-q∥=βnρ∥xn-q∥∥xn+1-q∥+βn〈f(q)-q,J(xn+1-q)〉+(γn+δn)∥yn-q∥∥xn+1-q∥≤βnρ∥xn-q∥∥xn+1-q∥+βn〈f(q)-q,J(xn+1-q)〉+(1-βn)∥xn-q∥∥xn+1-q∥=(1-βn(1-ρ))∥xn-q∥∥xn+1-q∥+βn〈f(q)-q,J(xn+1-q)〉≤1-βn(1-ρ)2(∥xn-q∥2+∥xn+1-q∥2)+βn〈f(q)-q,J(xn+1-q)〉≤1-βn(1-ρ)2∥xn-q∥2+12∥xn+1-q∥2+βn〈f(q)-q,J(xn+1-q)〉.
So, we have
(153)∥xn+1-q∥2≤(1-βn(1-ρ))∥xn-q∥2+2βn〈f(q)-q,J(xn+1-q)〉=(1-βn(1-ρ))∥xn-q∥2+βn(1-ρ)2〈f(q)-q,J(xn+1-q)〉1-ρ.
Since ∑n=0∞‍βn=∞ and limsupn→∞〈f(q)-q, J(xn+1-q)〉≤0, by Lemma 2, we conclude from (153) that xn→q as n→∞. This completes the proof.Corollary 23.
LetC be a nonempty, closed, and convex subset of a uniformly convex Banach space X which has a uniformly Gâteaux differentiable norm. Let ΠC be a sunny nonexpansive retraction from X onto C. Let the mapping Bi:C→X be λi-strictly pseudocontractive and αi-strongly accretive with αi+λi≥1 for i=1,2. Let f:C→C be a contraction with coefficient ρ∈(0,1). Let S be a nonexpansive mapping of C into itself such that F=Fix(S)∩Ω≠∅, where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2). For arbitrarily given x0∈C, let {xn} be the sequence generated by
(154)yn=αnG(xn)+(1-αn)SG(xn),xn+1=βnf(xn)+γnyn+δnSG(yn),∀n≥0,
where 1-(λi/(1+λi))(1-(1-αi)/λi)≤μi≤1 for i=1,2. Suppose that {αn}, {βn}, {γn}, and {δn} are the sequences in (0,1) satisfying the following conditions:(i)
0<liminfn→∞αn≤limsupn→∞αn<1,(ii)
limn→∞βn=0 and ∑n=0∞‍βn=∞,(iii)
∑n=1∞‍|αn-αn-1|<∞ or limn→∞|αn-αn-1|/βn=0,(iv)
∑n=1∞‍|βn-βn-1|<∞ or limn→∞βn-1/βn=1,(v)
∑n=1∞‍|(γn/(1-βn))-(γn-1/(1-βn-1))|<∞ or limn→∞(1/βn)|(γn/(1-βn))-(γn-1/(1-βn-1))|=0,(vi)
0<liminfn→∞γn≤limsupn→∞γn<1.Then {xn} converges strongly to q∈F, which solves the following VIP:
(155)〈q-f(q),J(q-p)〉≤0,∀p∈F.Further, we illustrate Theorem22 by virtue of an example, that is, the following corollary.Corollary 24.
LetC be a nonempty, closed, and convex subset of a uniformly convex Banach space X which has a uniformly Gâteaux differentiable norm. Let ΠC be a sunny nonexpansive retraction from X onto C. Let f:C→C be a contraction with coefficient ρ∈(0,1). Let T:C→C be a self-mapping on C such that I-T is ζ-strictly pseudocontractive and θ-strongly accretive with ζ+θ≥1, and let S be a nonexpansive mapping of C into itself such that F=Fix(S)∩Fix(T)≠∅. For arbitrarily given x0∈C, let {xn} be the sequence generated by
(156)yn=αnG(xn)+(1-αn)S(I-λ(I-T))xn,xn+1=βnf(xn)+γnyn+δnS(I-λ(I-T))yn,∀n≥0,
where 1-(ζ/(1+ζ))(1-(1-θ)/ζ)≤λ≤1. Suppose that {αn}, {βn}, {γn}, and {δn} are the sequences in (0,1) satisfying the following conditions:(i)
0<liminfn→∞αn≤limsupn→∞αn<1,(ii)
limn→∞βn=0 and ∑n=0∞‍βn=∞,(iii)
∑n=1∞‍|αn-αn-1|<∞ or limn→∞|αn-αn-1|/βn=0,(iv)
∑n=1∞‍|βn-βn-1|<∞ or limn→∞βn-1/βn=1,(v)
∑n=1∞‍|(γn/(1-βn))-(γn-1/(1-βn-1))|<∞ or limn→∞(1/βn)|(γn/(1-βn))-(γn-1/(1-βn-1))|=0,(vi)
0<liminfn→∞γn≤limsupn→∞γn<1.Then {xn} converges strongly to q∈F, which solves the following VIP:
(157)〈q-f(q),J(q-p)〉≤0,∀p∈F.Proof.
Utilizing the arguments similar to those in the proof of Corollary16, we can obtain the desired result.Remark 25.
As previous, we emphasize that our composite iterative algorithms (i.e., the iterative schemes (27) and (111)) are based on Korpelevich’s extragradient method and viscosity approximation method. It is well known that the so-called viscosity approximation method must contain a contraction f on C. In the meantime, it is worth pointing out that our proof of Theorems 14 and 22 must make use of Lemma 8 for implicit viscosity approximation method; that is, Lemma 8 plays a key role in our proof of Theorems 14 and 22. Therefore, there is no doubt that the contraction f in Theorems 14 and 22 cannot be replaced by a general k-Lipschitzian mapping with constant k≥0.Remark 26.
Theorem22 improves, extends, supplements, and develops [2, Theorem  3.1 and Corollary  3.2] and [5, Theorems  3.1] in the following aspects.(i)
The problem of finding a pointq∈⋂n‍Fix(Sn)∩Ω in Theorem 22 is more general and more subtle than the problem of finding a point q∈Fix(S)∩VI(C,A) in Jung [5, Theorem  3.1].(ii)
The iterative scheme in [2, Theorem  3.1] is extended to develop the iterative scheme (111) of Theorem 22 by virtue of the iterative scheme of [5, Theorem  3.1]. The iterative scheme (111) in Theorem 22 is more advantageous and more flexible than the iterative scheme in [2, Theorem  3.1] because it involves several parameter sequences {αn}, {βn}, {γn}, and {δn}.(iii)
The iterative scheme (111) in Theorem 22 is very different from everyone in both [2, Theorem  3.1] and [5, Theorem  3.1] because the mappings Sn and G in the iterative scheme of [2, Theorem  3.1] and the mapping SPC(I-λnA) in the iterative scheme of  [5, Theorem  3.1] are replaced by the same composite mapping SnG in the iterative scheme (111) of Theorem 22.(iv)
The proof in [2, Theorem  3.1] depends on the argument techniques in [3], the inequality in 2-uniformly smooth Banach spaces, and the inequality in smooth and uniform convex Banach spaces. However, the proof of Theorem 22 does not depend on the argument techniques in [3], the inequality in 2-uniformly smooth Banach spaces, and the inequality in smooth and uniform convex Banach spaces. It depends on only the inequality in uniform convex Banach spaces.(v)
The assumption of the uniformly convex and 2-uniformly smooth Banach spaceX in [2, Theorem  3.1] is weakened to the one of the uniformly convex Banach space X having a uniformly Gateaux differentiable norm in Theorem 22.(vi)
The iterative scheme in [2, Corollary  3.2] is extended to develop the new iterative scheme in Corollary 15 because the mappings S and G are replaced by the same composite mapping SG in Corollary 23.
Finally, we observe that related results can be found in recent papers, for example, [15–24] and the references therein.



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