# Hybrid Viscosity Approaches to General Systems of Variational Inequalities with Hierarchical Fixed Point Problem Constraints in Banach Spaces

Authors: Lu-Chuan Ceng; Saleh A. Al-Mezel; Abdul Latif
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The purpose of this paper is to introduce and analyze hybrid viscosity methods for a general system of variational inequalities (GSVI) with hierarchical fixed point problem constraint in the setting of real uniformly convex and 2-uniformly smooth Banach spaces. Here, the hybrid viscosity methods are based on Korpelevich’s extragradient method, viscosity approximation method, and hybrid steepest-descent method. We propose and consider hybrid implicit and explicit viscosity iterative algorithms for solving the GSVI with hierarchical fixed point problem constraint not only for a nonexpansive mapping but also for a countable family of nonexpansive mappings inX, respectively. We derive some strong convergence theorems under appropriate conditions. Our results extend, improve, supplement, and develop the recent results announced by many authors.


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## Body

## 1. Introduction

LetX be a real Banach space whose dual space is denoted by X*. Let U={x∈X:∥x∥=1} denote the unit 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,
(1)∥x-y∥≥ϵ⟹∥x+y∥2≤1-δ.
It is known that a uniformly convex Banach space is reflexive and strictly convex. The normalized duality mapping J:X→2X* is defined by
(2)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.LetC be a nonempty closed convex subset of a real Banach space X. A mapping T:C→C is said to be L-Lipschitzian if there exists a constant L>0 such that ∥Tx-Ty∥≤L∥x-y∥ for all x,y∈C. In particular, if L=1, then T is said to be nonexpansive. 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(i)  accretive if, for eachx,y∈C, there exists j(x-y)∈J(x-y) such that
(3)〈Ax-Ay,j(x-y)〉≥0,
where J is the normalized duality mapping of X,(ii)α-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)〉≥α∥x-y∥2,
for some α∈(0,1),(iii)  pseudocontractive if, for eachx,y∈C, there exists j(x-y)∈J(x-y) such that
(5)〈Ax-Ay,j(x-y)〉≤∥x-y∥2,(iv)β-strongly pseudocontractive if, for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(6)〈Ax-Ay,j(x-y)〉≤β∥x-y∥2
for some β∈(0,1),(v)λ-strictly pseudocontractive if, for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(7)〈Ax-Ay,j(x-y)〉≤∥x-y∥2-λ∥x-y-(Ax-Ay)∥2
for some λ∈(0,1).It is worth emphasizing that the definition of the inverse-strongly accretive mapping is based on that of the inverse-strongly monotone mapping, which was studied by so many authors; see, for example, [1–7].A Banach spaceX is said to be smooth if the limit
(8)lim⁡t→0⁡∥x+ty∥-∥x∥t
exists for all x,y∈X; in this case, X is also said to have a Gateaux differentiable norm. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for x,y∈U; in this case, X is also said to have a uniformly Frechet differentiable norm. The norm of X is said to be the Frechet differential if, for each x∈U, this limit is attained uniformly for y∈U. In the meantime, we define a function ρ:[0,∞)→[0,∞) called the modulus of smoothness of X as follows:
(9)ρ(τ)=sup⁡⁡{12(∥x+y∥+∥x-y∥)-1:x,y∈X,hhhhh∥x∥=1,∥y∥=τ{12(∥x+y∥+∥x-y∥)-1:x,y∈X,}.
It is known that X is uniformly smooth if and only if lim⁡τ→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 [8], 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.In a real smooth Banach spaceX, we say that an operator A is strongly positive (see [9]), if there exists a constant γ->0 with the property
(10)〈Ax,J(x)〉≥γ-∥x∥2,∥aI-bA∥=sup⁡∥x∥≤1⁡|〈(aI-bA)x,J(x)〉|,a∈[0,1],b∈[-1,1],
where I is the identity mapping.Proposition CB [see [9, Lemma 2.5]]
LetC be a nonempty closed convex subset of a uniformly smooth Banach space X. Let T:C→C be a continuous pseudocontractive mapping with Fix⁡(T)≠∅ and let f:C→C be a fixed Lipschitzian strongly pseudocontractive mapping with pseudocontractive coefficient β∈(0,1) and Lipschitzian constant L>0. Let A:C→C be a strongly positive linear bounded operator with coefficient γ->0. Assume that C±C⊂C and 0<β<γ-. Let {xt} be defined by
(11)xt=tf(xt)+(I-tA)Txt.
Then, as t→0,{xt} converges strongly to some fixed point p of T such that p is the unique solution in Fix⁡(T) to the VIP:
(12)〈(A-f)p,J(p-u)〉≤0,∀u∈Fix⁡(T).On the other hand, Cai and Bu [10] 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
(13)〈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,B2:C→X are two nonlinear mappings, and μ1 and μ2 are two positive constants. Here the set of solutions of GSVI (13) is denoted by GSVI(C,B1,B2). Very recently, Cai and Bu [10] constructed an iterative algorithm for solving GSVI (13) and a common fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a 2-uniformly smooth Banach space X.Lemma 1 (see [11]).
LetX be a 2-uniformly smooth Banach space. Then, there exists a best smooth constant κ>0 such that
(14)∥x+y∥2≤∥x∥2+2〈y,J(x)〉+2∥κy∥,∀x,y∈X,
where J is the normalized duality mapping from X into X*.The authors [10] have used the following inequality in a real smooth and uniform convex Banach space X.Proposition 2 (see [12]).
LetX be a real smooth and uniform convex Banach space and let r>0. Then, there exists a strictly increasing, continuous, and convex function g:[0,2r]→R, g(0)=0 such that
(15)g(∥x-y∥)≤∥x∥2-2〈x,J(y)〉+∥y∥2,∀x,y∈Br,
where Br={x∈X:∥x∥≤r}.


## 2. Preliminaries

We list some lemmas that will be used in the sequel. Lemma3 can be found in [13]. Lemma 4 is an immediate consequence of the subdifferential inequality of the function (1/2)∥·∥2.Lemma 3.
Let{an} be a sequence of nonnegative real numbers such that
(16)an+1≤(1-bn)an+bncn,∀n≥0,
where {bn} and {cn} are sequences of real numbers satisfying the following conditions:(i)
{bn}⊂[0,1]  and  ∑n=0∞‍bn=∞;(ii)
eitherlimsup⁡n→∞⁡cn⁡≤0 or ∑n=0∞‍|bncn|<∞.Then, lim⁡n→∞⁡an=0.Lemma 4.
In a smooth Banach spaceX, there holds the inequality
(17)∥x∥2+2〈y,J(x)〉≤∥x+y∥2≤∥x∥2+2〈y,J(x+y)〉,∀x,y∈X,
where J is the normalized duality mapping of X.Letμ be a mean if μ is a continuous linear functional on l∞ satisfying ∥μ∥=1=μ(1). Then, we know that μ is a mean on N if and only if
(18)inf⁡⁡{an:n∈N}≤μ(a)≤sup⁡⁡{an:n∈N}
for every a=(a1,a2,…)∈l∞. According to time and circumstances, we use μn(an) instead of μ(a). A mean μ on N is called a Banach limit if and only if
(19)μn(an)=μn(an+1)
for every a=(a1,a2,…)∈l∞. We know that, if μ is a Banach limit, then
(20)liminf⁡n→∞⁡an⁡≤μn(an)≤limsup⁡n→∞⁡an⁡
for every a=(a1,a2,…)∈l∞. So, if a=(a1,a2,…), b=(b1,b2,…)∈l∞, and an→c (resp., an-bn→0), as n→∞, we have
(21)μn(an)=μ(a)=c(resp.,μn(an)=μn(bn)).Further, it is well known that there holds the following result.Lemma 5 (see [14]).
LetC be a nonempty closed convex subset of a uniformly smooth Banach space X. Let {xn} be a bounded sequence of X; let μ be a mean on N and let z∈C. Then,
(22)μn∥xn-z∥2=min⁡y∈C⁡μn∥xn-y∥2
if and only if
(23)μn〈y-z,J(xn-z)〉≤0,∀y∈C,
where J is the normalized duality mapping of X.Lemma 6 (see [9, Lemma 2.6]).
LetC be a nonempty closed convex subset of a real Banach space X which has uniformly Gateaux differentiable norm. Let T:C→C be a continuous pseudocontractive mapping with Fix⁡(T)≠∅ and let f:C→C be a fixed Lipschitzian strongly pseudocontractive mapping with pseudocontractive coefficient β∈(0,1) and Lipschitzian constant L>0. Let A:C→C be a γ--strongly positive linear bounded operator with coefficient γ->0. Assume that C±C⊂C and that {xt} converges strongly to p∈Fix⁡(T) as t→0, where xt is defined by xt=tf(xt)+(I-tA)Txt. Suppose that {xn}⊂C is bounded and that lim⁡n→∞∥xn-Txn∥=0. Then, limsup⁡n→∞⁡〈(f-A)p,J(xn-p)〉⁡≤0.Lemma 7.
LetC be a nonempty closed convex subset of a real smooth Banach space X. Let F:C→X be an α-strongly accretive and λ-strictly pseudocontractive with α+λ≥1. Then, I-F is nonexpansive and F is Lipschitz continuous with constant (1+(1-α)/λ). Further, for any fixed τ∈(0,1),I-τF is contractive with coefficient 1-τ(1-(1-α)/λ).Proof.
From theλ-strictly pseudocontractivity and α-strongly accretivity of F, we have, for all x,y∈C,
(24)λ∥(I-F)x-(I-F)y∥2≤∥x-y∥2-〈F(x)-F(y),J(x-y)〉≤(1-α)∥x-y∥2,
which implies that(25)∥(I-F)x-(I-F)y∥≤1-αλ∥x-y∥.
Because α+λ≥1⇔(1-α)/λ≤1, we know that I-F is nonexpansive. Also note that
(26)∥F(x)-F(y)∥≤∥(I-F)x-(I-F)y∥+∥x-y∥≤(1+1-αλ)∥x-y∥.
Now, take a fixed τ∈(0,1) arbitrarily. Observe that, for all x,y∈C,
(27)∥(I-τF)x-(I-τF)y∥=∥(1-τ)(x-y)+τ[(I-F)x-(I-F)y]∥≤(1-τ)∥x-y∥+τ∥(I-F)x-(I-F)y∥≤(1-τ)∥x-y∥+τ(1-αλ)∥x-y∥=(1-τ(1-1-αλ))∥x-y∥.
Because α+λ>1⇔(1-α)/λ<1, we know that I-τF is contractive with coefficient 1-τ(1-(1-α)/λ).LetD be a subset of C and let Π be a mapping of C into D. Then, Π is said to be sunny if
(28)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 8 (see [15]).
LetC be a nonempty closed 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))〉, for all x,y∈C;(iii)
〈x-Π(x),J(y-Π(x))〉≤0, for all x∈C, y∈D.It is well known that, ifX=H is 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 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 9.
LetC be a nonempty closed convex subset of a smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C and let B1,B2:C→X be nonlinear mappings. For given x*,y*∈C, (x*,y*) is a solution of GSVI (13) if and only if x*=ΠC(y*-μ1B1y*), where y*=ΠC(x*-μ2B2x*).Proof.
We can rewrite GSVI (13) as
(29)〈x*-(y*-μ1B1y*),J(x-x*)〉≥0,∀x∈C,〈y*-(x*-μ2B2x*),J(x-y*)〉≥0,∀x∈C,
which is obviously equivalent to
(30)x*=ΠC(y*-μ1B1y*),y*=ΠC(x*-μ2B2x*),
because of Lemma 8. This completes the proof.In terms of Lemma9, define the mapping G:C→C as follows:
(31)G(x):=ΠC(I-μ1B1)ΠC(I-μ2B2)x,∀x∈C.
Then, we observe that
(32)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.Lemma 10 (see [16]).
LetC be a nonempty closed convex subset of a strictly convex Banach space X. Let {Tn}n=0∞ be a sequence of nonexpansive mappings on C. Suppose ⋂n=0∞‍Fix⁡(Tn) is nonempty. Let {λn} be a sequence of positive numbers with ∑n=0∞‍λn=1. Then, a mapping T on C defined by Tx=∑n=0∞‍λnTnx for x∈C is well-defined, nonexpansive and Fix⁡(T)=⋂n=0∞‍Fix⁡(Tn) holds.Lemma 11 (see [17]).
LetC be a nonempty closed convex subset of a Banach space X. Let S1,S2,… be a sequence of mappings of C into itself. Suppose that ∑n=1∞‍sup⁡{∥Sn+1x-Snx∥: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=lim⁡n→∞Sny, for all y∈C. Then, lim⁡n→∞sup⁡{∥Sx-Snx∥:x∈C}=0.


## 3. GSVI with Hierarchical Fixed Point Problem Constraint for a Nonexpansive Mapping

In this section, we introduce our hybrid implicit viscosity scheme for solving the GSVI (13) with hierarchical fixed point problem constraint for a nonexpansive mapping and show the strong convergence theorem. First, we list several useful and helpful lemmas.Lemma 12 (see [10, Lemma 2.8]).
LetC be a nonempty closed 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
(33)∥(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 (where κ is the best constant of X as in Lemma 1), then I-μiBi is nonexpansive for i=1,2.Lemma 13 (see [10, Lemma 2.9]).
LetC be a nonempty closed 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
(34)Gx=ΠC[ΠC(x-μ2B2x)-μ1B1ΠC(x-μ2B2x)],hhhhhhhhhhhhhhhhhhhhhhhhihhhh∀x∈C.
If 0<μi≤αi/κ2, for i=1,2, then G:C→C is nonexpansive.Lemma 14 (see [18]).
LetX be a Banach space, C a nonempty closed and convex subset of X, and T:C→C a continuous and strong pseudocontraction. Then, T has a unique fixed point in C.Lemma 15 (see [19]).
Assume thatA is a strongly positive linear bounded operator on a smooth Banach space X with coefficient γ->0 and 0<ρ≤∥A∥-1. Then, ∥I-ρA∥2≤1-ργ-.We now state and prove our first result.Theorem 16.
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X such that C±C⊂C. 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 T:C→C be a nonexpansive mapping such that Λ=Fix⁡(T)∩Ω≠∅ where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2) with 0<μi<αi/κ2 for i=1,2. Let f:C→C be a fixed Lipschitzian strongly pseudocontractive mapping with pseudocontractive coefficient β∈(0,1) and Lipschitzian constant L>0, let F:C→C be α-strongly accretive and λ-strictly pseudocontractive with α+λ>1, and let A:C→C be a γ--strongly positive linear bounded operator with 0<γ--β≤1. Let {xt} be defined by
(35)xt=tf(xt)+(I-tA)[G(Txt)-θtFG(Txt)],
where {θt:t∈(0,1)}⊂[0,1) with lim⁡t→0θt/t=0. Then, as t→0,{xt} converges strongly to a point p∈Λ, which is the unique solution in Λ to the VIP,
(36)〈(A-f)p,J(p-u)〉≤0,∀u∈Λ.Proof.
First, let us show that the net{xt} is defined well. As a matter of fact, define the mapping St:C→C as follows:
(37)Stx=tf(x)+(I-tA)[G(Tx)-θtFG(Tx)],∀x∈C.
We may assume, without loss of generality, that t≤∥A∥-1. Utilizing Lemmas 7, 13, and 15, we have
(38)〈Stx-Sty,J(x-y)〉=t〈f(x)-f(y),J(x-y)〉+〈(I-tA)[(I-θtF)G(Tx)-(I-θtF)G(Ty)],hhhhhhJ(x-y)〉≤tβ∥x-y∥2+(1-tγ-)×∥(I-θtF)G(Tx)-(I-θtF)G(Ty)∥∥x-y∥≤tβ∥x-y∥2+(1-tγ-)(1-θt(1-1-αλ))×∥G(Tx)-G(Ty)∥∥x-y∥≤tβ∥x-y∥2+(1-tγ-)∥Tx-Ty∥∥x-y∥≤tβ∥x-y∥2+(1-tγ-)∥x-y∥2=(1-t(γ--β))∥x-y∥2.
Hence, it is known that St:C→C is a continuous and strongly pseudocontractive mapping with pseudocontractive coefficient 1-t(γ--β)∈(0,1) Thus, by Lemma 14, we deduce that there exists a unique fixed point in C, denoted by xt, which uniquely solves the fixed point equation
(39)xt=tf(xt)+(I-tA)[G(Txt)-θtFG(Txt)].
Let us show the uniqueness of the solution of VIP (36). Suppose that both p1∈Λ and p2∈Λ are solutions to VIP (36). Then, we have
(40)〈(A-f)p1,J(p1-p2)〉≤0,〈(A-f)p2,J(p2-p1)〉≤0.
Adding up the above two inequalities, we obtain
(41)〈(A-f)p1-(A-f)p2,J(p1-p2)〉≤0.
Note that
(42)〈(A-f)p1-(A-f)p2,J(p1-p2)〉=〈A(p1-p2),J(p1-p2)〉-〈f(p1)-f(p2),J(p1-p2)〉≥γ-∥p1-p2∥2-β∥p1-p2∥2=(γ--β)∥p1-p2∥2≥0.
Consequently, we have p1=p2, and the uniqueness is proved.
Next, let us show that, for somea∈(0,1), {xt:t∈(0,a]} is bounded. Indeed, since {θt:t∈(0,1)}⊂[0,1) with lim⁡t→0(θt/t)=0, there exists some a∈(0,1) such that 0≤θt/t<1 for all t∈(0,a]. Take a fixed p∈Fix⁡(Λ) arbitrarily. Utilizing Lemma 7, we have
(43)∥xt-p∥2=〈t(f(xt)-f(p))+(I-tA)[G(Txt)-θtFG(Txt)-p]-t(Ap-f(p)),J(xt-p)〉=t〈f(xt)-f(p),J(xt-p)〉+〈(I-tA)[G(Txt)-θtFG(Txt)-p],J(xt-p)〉-t〈(A-f)p,J(xt-p)〉≤tβ∥xt-p∥2+(1-tγ-)∥G(Txt)-θtFG(Txt)-p∥×∥xt-p∥+t∥(A-f)p∥∥xt-p∥≤tβ∥xt-p∥2+(1-tγ-)×[∥(I-θtF)G(Txt)-(I-θtF)G(Tp)∥+∥(I-θtF)G(Tp)-p∥]∥xt-p∥+t∥(A-f)p∥∥xt-p∥≤tβ∥xt-p∥2+(1-tγ-)(1-θt(1-1-αλ))×∥G(Txt)-G(Tp)∥∥xt-p∥+(1-tγ-)θt∥Fp∥∥xt-p∥+t∥(A-f)p∥∥xt-p∥≤tβ∥xt-p∥2+(1-tγ-)(1-θt(1-1-αλ))×∥Txt-Tp∥∥xt-p∥+θt∥Fp∥∥xt-p∥+t∥(A-f)p∥∥xt-p∥≤tβ∥xt-p∥2+(1-tγ-)∥xt-p∥2+θt∥Fp∥∥xt-p∥+t∥(A-f)p∥∥xt-p∥=(1-t(γ--β))∥xt-p∥2+θt∥Fp∥∥xt-p∥+t∥(A-f)p∥∥xt-p∥,
and, hence, for all t∈(0,a],
(44)∥xt-p∥≤1γ--β(∥(A-f)p∥+θtt∥Fp∥)≤1γ--β(∥(A-f)p∥+∥Fp∥).
Thus, this implies that {xt:t∈(0,a]} is bounded and so are {f(xt):t∈(0,a]}, {Txt:t∈(0,a]}, and {G(Txt):t∈(0,a]}.
Let us show that∥Txt-G(Txt)∥→0 as t→0.
Indeed, for simplicity, we putq=ΠC(I-μ2B2)p, x^t=Txt, ut=ΠC(I-μ2B2)x^t, and vt=ΠC(I-μ1B1)ut. Then, it is clear that p=ΠC(I-μ1B1)q and vt=G(x^t)=G(Txt). Hence, from (43), it follows that
(45)∥xt-p∥2≤tβ∥xt-p∥2+(1-tγ-)(1-θt(1-1-αλ))×∥G(Txt)-G(Tp)∥∥xt-p∥+(1-tγ-)θt∥Fp∥∥xt-p∥+t∥(A-f)p∥∥xt-p∥≤tβ∥xt-p∥2+(1-tγ-)∥vt-p∥∥xt-p∥+θt∥Fp∥∥xt-p∥+t∥(A-f)p∥∥xt-p∥.
From Lemma 12, we have
(46)∥ut-q∥2=∥ΠC(x^t-μ2B2x^t)-ΠC(p-μ2B2p)∥2≤∥x^t-p-μ2(B2x^t-B2p)∥2≤∥x^t-p∥2-2μ2(α2-κ2μ2)∥B2x^t-B2p∥2,∥vt-p∥2=∥ΠC(ut-μ1B1ut)-ΠC(q-μ1B1q)∥2≤∥ut-q-μ1(B1ut-B1q)∥2≤∥ut-q∥2-2μ1(α1-κ2μ1)∥B1ut-B1q∥2.
From the last two inequalities, we obtain
(47)∥vt-p∥2≤∥x^t-p∥2-2μ2(α2-κ2μ2)×∥B2x^t-B2p∥2-2μ1(α1-κ2μ1)×∥B1ut-B1q∥2≤∥xt-p∥2-2μ2(α2-κ2μ2)∥B2x^t-B2p∥2-2μ1(α1-κ2μ1)∥B1ut-B1q∥2,
which together with (45) implies that
(48)∥xt-p∥2≤tβ∥xt-p∥2+(1-tγ-)∥vt-p∥∥xt-p∥+θt∥Fp∥×∥xt-p∥+t∥(A-f)p∥∥xt-p∥≤tβ∥xt-p∥2+t∥(A-f)p∥∥xt-p∥+(1-tγ-)×12(∥xt-p∥2+∥vt-p∥2)+θt∥Fp∥∥xt-p∥≤tβ∥xt-p∥2+t∥(A-f)p∥∥xt-p∥+(1-tγ-)12{∥xt-p∥2+∥xt-p∥2hhhhhhhhhhhhh-2μ2(α2-κ2μ2)∥B2x^t-B2p∥2hhhhhhhhhhhhh-2μ1(α1-κ2μ1)hhhhhhhhhhhhh×∥B1ut-B1q∥2}+θt∥Fp∥∥xt-p∥=(1-t(γ--β))∥xt-p∥2+t∥(A-f)p∥∥xt-p∥-(1-tγ-)[μ2(α2-κ2μ2)∥B2x^t-B2p∥2hhhhhhhhhhhh+μ1(α1-κ2μ1)∥B1ut-B1q∥2]+θt∥Fp∥∥xt-p∥≤∥xt-p∥2+t∥(A-f)p∥∥xt-p∥-(1-tγ-)×[μ2(α2-κ2μ2)∥B2x^t-B2p∥2hhhhhh+μ1(α1-κ2μ1)∥B1ut-B1q∥2]+θt∥Fp∥∥xt-p∥.
So, it immediately follows that
(49)(1-tγ-)[μ2(α2-κ2μ2)∥B2x^t-B2p∥2+μ1(α1-κ2μ1)hhhhhhh×∥B1ut-B1q∥2]≤t∥(A-f)p∥∥xt-p∥+θt∥Fp∥∥xt-p∥.
Since 0<μi<αi/κ2, for i=1,2, we have
(50)lim⁡t→0⁡∥B2x^t-B2p∥=0,lim⁡t→0⁡∥B1ut-B1q∥=0.
Utilizing Proposition 2 and Lemma 8, we have that there exists g1 such that
(51)∥ut-q∥2=∥ΠC(x^t-μ2B2x^t)-ΠC(p-μ2B2p)∥2≤〈x^t-μ2B2x^t-(p-μ2B2p),J(ut-q)〉=〈x^t-p,J(ut-q)〉+μ2〈B2p-B2x^t,J(ut-q)〉≤12[∥x^t-p∥2+∥ut-q∥2hihh-g1(∥x^t-ut-(p-q)∥)∥x^t-p∥2]+μ2∥B2p-B2x^t∥∥ut-q∥,
which implies that
(52)∥ut-q∥2≤∥x^t-p∥2-g1(∥x^t-ut-(p-q)∥)+2μ2∥B2p-B2x^t∥∥ut-q∥.
In the same way, we derive that there exists g2:
(53)∥vt-p∥2=∥ΠC(ut-μ1B1ut)-ΠC(q-μ1B1q)∥2≤〈ut-μ1B1ut-(q-μ1B1q),J(vt-p)〉=〈ut-q,J(vt-p)〉+μ1〈B1q-B1ut,J(vt-p)〉≤12[∥ut-q∥2+∥vt-p∥2hhhh-g2(∥ut-vt+(p-q)∥)∥vt-p∥2]+μ1∥B1q-B1ut∥∥vt-p∥,
which implies that
(54)∥vt-p∥2≤∥ut-q∥2-g2(∥ut-vt+(p-q)∥)+2μ1∥B1q-B1ut∥∥vt-p∥.
Substituting (52) for (54), we get
(55)∥vt-p∥2≤∥x^t-p∥2-g1(∥x^t-ut-(p-q)∥)-g2(∥ut-vt+(p-q)∥)+2μ2∥B2p-B2x^t∥∥ut-q∥+2μ1∥B1q-B1ut∥∥vt-p∥≤∥xt-p∥2-g1(∥x^t-ut-(p-q)∥)-g2(∥ut-vt+(p-q)∥)+2μ2∥B2p-B2x^t∥∥ut-q∥+2μ1∥B1q-B1ut∥∥vt-p∥,
which together with (45) implies that
(56)∥xt-p∥2≤tβ∥xt-p∥2+(1-tγ-)∥vt-p∥∥xt-p∥+θt∥Fp∥×∥xt-p∥+t∥(A-f)p∥∥xt-p∥≤tβ∥xt-p∥2+t∥(A-f)p∥∥xt-p∥+(1-tγ-)×12(∥xt-p∥2+∥vt-p∥2)+θt∥Fp∥∥xt-p∥≤tβ∥xt-p∥2+t∥(A-f)p∥∥xt-p∥+(1-tγ-)×12{∥xt-p∥2+∥xt-p∥2hhhhhhh-g1(∥x^t-ut-(p-q)∥)hhhhhhh-g2(∥ut-vt+(p-q)∥)hhhhhhh+2μ2∥B2p-B2x^t∥∥ut-q∥hhhhhhh+2μ1∥B1q-B1ut∥∥vt-p∥∥xt-p∥2}+θt∥Fp∥∥xt-p∥=(1-t(γ--β))∥xt-p∥2+t∥(A-f)p∥∥xt-p∥-(1-tγ-)×12[g1(∥x^t-ut-(p-q)∥)hhhhhh+g2(∥ut-vt+(p-q)∥)]+(1-tγ-)×[μ2∥B2p-B2x^t∥∥ut-q∥hhhhhh+μ1∥B1q-B1ut∥∥vt-p∥]+θt∥Fp∥∥xt-p∥≤∥xt-p∥2+t∥(A-f)p∥∥xt-p∥-(1-tγ-)×12[g1(∥x^t-ut-(p-q)∥)hhhhhhh+g2(∥ut-vt+(p-q)∥)]+μ2∥B2p-B2x^t∥∥ut-q∥+μ1∥B1q-B1ut∥∥vt-p∥+θt∥Fp∥∥xt-p∥.
So, it immediately follows that
(57)(1-tγ-)12[g1(∥x^t-ut-(p-q)∥)hhhhhhhh+g2(∥ut-vt+(p-q)∥)]≤t∥(A-f)p∥∥xt-p∥+μ2∥B2p-B2x^t∥∥ut-q∥+μ1∥B1q-B1ut∥∥vt-p∥+θt∥Fp∥∥xt-p∥.
Hence, from (50), we conclude that
(58)lim⁡t→0⁡g1(∥x^t-ut-(p-q)∥)=0,lim⁡t→0⁡g2(∥ut-vt+(p-q)∥)=0.
Utilizing the properties of g1 and g2, we get
(59)lim⁡t→0⁡∥x^t-ut-(p-q)∥=0,lim⁡t→0⁡∥ut-vt+(p-q)∥=0,
which leads to
(60)∥x^t-vt∥≤∥x^t-ut-(p-q)∥+∥ut-vt+(p-q)∥⟶0ast⟶0.
That is,
(61)lim⁡t→0⁡∥Txt-G(Txt)∥=lim⁡t→0⁡∥x^t-vt∥=0.
Note that{xt:t∈(0,a]} is bounded and so are {f(xt):t∈(0,a]}, {Txt:t∈(0,a]}, and {G(Txt):t∈(0,a]}. Hence, we have
(62)∥xt-G(Txt)∥=t∥f(xt)-AG(Txt)-θtt(I-tA)FG(Txt)∥⟶0,
as t→0. Also, observe that
(63)∥xt-Txt∥≤∥xt-G(Txt)∥+∥G(Txt)-Txt∥.
This together with (61) and (62) implies that
(64)lim⁡t→0⁡∥xt-Txt∥=0.
Utilizing the nonexpansivity of G, we obtain
(65)∥xt-Gxt∥≤∥xt-G(Txt)∥+∥G(Txt)-Gxt∥≤∥xt-G(Txt)∥+∥Txt-xt∥,
which together with (62) and (64) implies that
(66)lim⁡t→0⁡∥xt-Gxt∥=0.
Now, let {tk} be a sequence in (0,a] that converges to 0 as k→∞, and define a function g on C by
(67)g(x)=μk12∥xtk-x∥2,∀x∈C,
where μ is a Banach limit. Define the set
(68)K:={w∈C:g(w)=min⁡⁡{g(y):y∈C}}
and the mapping
(69)Wx=(1-θ)Tx+θGx,∀x∈C,
where θ is a constant in (0,1). Then, by Lemma 10, we know that Fix⁡(W)=Fix⁡(T)∩Fix⁡(G)=Λ. We observe that
(70)∥xt-Wxt∥=∥(1-θ)(xt-Txt)+θ(xt-Gxt)∥≤(1-θ)∥xt-Txt∥+θ∥xt-Gxt∥.
So, from (64) and (66), we obtain
(71)lim⁡n→∞⁡∥xt-Wxt∥=0.
Since X is a uniformly smooth Banach space, K is a nonempty bounded closed convex subset of C; for more details, see [14]. We claim that K is also invariant under the nonexpansive mapping W. Indeed, noticing (71), we have, for w∈K,
(72)g(Ww)=μk12∥xtk-Ww∥2=μk12∥Wxtk-Ww∥2≤μk12∥xtk-w∥2=g(w).
Since every nonempty closed bounded convex subset of a uniformly smooth Banach space X has the fixed point property for nonexpansive mappings and W is a nonexpansive mapping of K, W has a fixed point in K, say p. Utilizing Lemma 5, we get
(73)μk〈x-p,J(xtk-p)〉≤0,∀x∈C.
Putting x=(f-A)p+p∈C, we have
(74)μk〈(f-A)p,J(xtk-p)〉≤0,∀x∈C.
Since xtk-p=tk(f(xtk)-f(p))+(I-tkA)[G(Txtk)-θtkFG(Txtk)-p]-tk(A-f)p, we get
(75)∥xtk-p∥2=tk〈f(xtk)-f(p),J(xtk-p)〉+〈(I-tkA)(G(Txtk)-p),J(xtk-p)〉-θtk〈(I-tkA)FG(Txtk),J(xtk-p)〉-tk〈(A-f)p,J(xtk-p)〉≤tkβ∥xtk-p∥2+tk〈(f-A)p,J(xtk-p)〉+(1-tkγ-)∥G(Txtk)-p∥∥xtk-p∥+(1-tkγ-)θtk∥FG(Txtk)∥∥xtk-p∥≤tkβ∥xtk-p∥2+tk〈(f-A)p,J(xtk-p)〉+(1-tkγ-)∥xtk-p∥2+θtk∥FG(Txtk)∥∥xtk-p∥.
It follows that
(76)∥xtk-p∥2≤1γ--β[+θtktk∥FG(Txtk)∥∥xtk-p∥]〈(f-A)p,J(xtk-p)〉hhhhhhh+θtktk∥FG(Txtk)∥∥xtk-p∥].
Since lim⁡k→∞(θtk/tk)=0, from (74) and the boundedness of sequences {FG(Txtk)}, {xtk}, it follows that
(77)μk∥xtk-p∥2≤1γ--βμk[+θtktk∥FG(Txtk)∥∥xtk-p∥]〈(f-A)p,J(xtk-p)〉hhhhhhhhh+θtktk∥FG(Txtk)∥∥xtk-p∥]=1γ--β[(θtktk∥FG(Txtk)∥∥xtk-p∥)μk〈(f-A)p,J(xtk-p)〉hhhhhhh+μk(θtktk∥FG(Txtk)∥∥xtk-p∥)]≤0.
Therefore, for the sequence {xtk} in {xt:t∈(0,a]}, there exists a subsequence which is still denoted by {xtk} that converges strongly to some fixed point p of W.
Now, we claim that such ap is the unique solution in Λ to the VIP (36).
Indeed, from (35), it follows that for all u∈Λ=Fix⁡(T)∩Ω(78)∥xt-u∥2=t〈f(xt)-f(u),J(xt-u)〉+〈(I-tA)[G(Txt)-θtFG(Txt)-u],J(xt-u)〉-t〈(A-f)u,J(xt-u)〉=〈(I-tA)[(I-θtF)G(Txt)-(I-θtF)uhhhhhihhhh+(I-θtF)u-u],J(xt-u)〉+t〈f(xt)-f(u),J(xt-u)〉-t〈(A-f)u,J(xt-u)〉≤(1-tγ-)[∥(I-θtF)G(Txt)-(I-θtF)u∥hhhihhhhh+∥(I-θtF)u-u∥]×∥xt-u∥+tβ∥xt-u∥2-t〈(A-f)u,J(xt-u)〉≤(1-tγ-)[(1-θt(1-1-αλ))∥xt-u∥+θt∥Fu∥]×∥xt-u∥+tβ∥xt-u∥2-t〈(A-f)u,J(xt-u)〉≤(1-tγ-)[∥xt-u∥+θt∥Fu∥]∥xt-u∥+tβ∥xt-u∥2-t〈(A-f)u,J(xt-u)〉≤(1-t(γ--β))∥xt-u∥2+θt∥Fu∥×∥xt-u∥-t〈(A-f)u,J(xt-u)〉≤∥xt-u∥2+θt∥Fu∥∥xt-u∥-t〈(A-f)u,J(xt-u)〉,
which hence implies that
(79)〈(A-f)u,J(xt-u)〉≤θtt∥Fu∥∥xt-u∥,∀u∈Λ.
Since xtk→p as tk→0 and lim⁡t→0(θt/t)=0, we obtain from the last inequality that
(80)〈(A-f)u,J(p-u)〉≤0,∀u∈Λ.
Utilizing the well-known Minty-type Lemma, we get
(81)〈(A-f)p,J(p-u)〉≤0,∀u∈Λ.
So, p is a solution in Λ to the VIP (36).
In order to prove that the net{xt:t∈(0,a]} converges strongly to p as t→0, suppose that there exists another subsequence {xsk}⊂{xt} such that xsk→q as sk→0; then we also have q∈Fix⁡(W)=Fix⁡(T)∩Ω=:Λ due to (71). Repeating the same argument as above, we know that q is another solution in Λ to the VIP (36). In terms of the uniqueness of solutions in Λ to the VIP (36), we immediately get p=q. This completes the proof.Remark 17.
It is worth emphasizing that, in the assertion of Theorem16, “as t→0,{xt} converges strongly to a point p∈Λ,” this p depends on no one of the mappings f, A, and F. Indeed, although {xt} is defined by
(82)xt=tf(xt)+(I-tA)[G(Txt)-θtFG(Txt)],∀t∈(0,1),
in the proof of Theorem 16, it can be readily seen that p is first found out as a fixed point of the nonexpansive self-mapping W of K. This shows that p depends on no one of the mappings f, A, and F.Remark 18.
Theorem16 improves, extends, supplements, and develops Cai and Bu [9, Lemma 2.5] in the following aspects.(i)  The GSVI (13) with hierarchical fixed point problem constraint for a nonexpansive mapping is more general and more subtle than the problem in Cai and Bu [9, Lemma 2.5] because our problem is to find a point p∈Λ=Fix⁡(T)∩Ω, which is the unique solution in Λ to the VIP:
(83)〈(A-f)p,J(p-u)〉≤0,∀u∈Λ.(ii)  The iterative scheme in [9, Lemma 2.5] is extended to develop the iterative scheme in Theorem 16 by virtue of hybrid steepest-descent method. The iterative scheme in Theorem 16 is more advantageous and more flexible than the iterative scheme of [9, Lemma 2.5] because our iterative scheme involves solving two problems: the GSVI (13) and the fixed point problem of a nonexpansive mapping T.(iii)  The iterative scheme in Theorem16 is very different from the iterative scheme in [9, Lemma 2.5] because our iterative scheme involves hybrid steepest-descent method (namely, we add a strongly accretive and strictly pseudocontractive mapping F in our iterative scheme) and because the mapping T in [9, Lemma 2.5] is replaced by the composite mapping G∘T in the iterative scheme of Theorem 16.(iv)  The argument techniques of Theorem16 are very different from Cai and Bu's ones of [9, Lemma 2.5]. Because the composite mapping G∘T appears in the iterative scheme of Theorem 16, the proof of Theorem 16 depends on the argument techniques in [18], the inequality in 2-uniformly smooth Banach spaces (see Lemma 1), the inequality in smooth and uniform convex Banach spaces (see Proposition 2), and the properties of the strongly positive linear bounded operator (see Lemmas 15), the Banach limit (see Lemma 5), and the strongly accretive and strictly pseudocontractive mapping (see Lemma 7).


## 4. GSVI with Hierarchical Fixed Point Problem Constraint for a Countable Family of Nonexpansive Mappings

In this section, we propose our hybrid explicit viscosity scheme for solving the GSVI (13) with hierarchical fixed point problem constraint for a countable family of nonexpansive mappings and show the strong convergence theorem.Theorem 19.
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X such that C±C⊂C. 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 {Sn}n=0∞ be an infinite family of nonexpansive mappings of C into itself such that Δ=⋂i=0∞‍Fix⁡(Si)∩Ω≠∅, where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2) with 0<μi<αi/κ2 for i=1,2. Let f:C→C be a fixed contractive map with coefficient β∈(0,1), let F:C→C be α-strongly accretive and λ-strictly pseudocontractive with α+λ>1, and let A:C→C be a γ--strongly positive linear bounded operator with 0<γ--β≤1. Given sequences {λn}n=0∞, {μn}n=0∞ in [0,1] and {αn}n=0∞,{βn}n=0∞ in (0,1], suppose that there hold the following conditions:(i)
lim⁡n→∞βn=0 and ∑n=0∞‍βn=∞;(ii)
lim⁡n→∞(λnμn)/βn=0;(iii)
{αn}⊂[a,b] for some a,b∈(0,1);(iv)
∑n=0∞‍(|αn+1-αn|+|βn+1-βn|+|λn+1-λn|+|μn+1-μn|)<∞.Assume that ∑n=0∞‍sup⁡x∈D∥Sn+1x-Snx∥<∞ for any bounded subset D of C and let S be a mapping of C into itself defined by Sx=lim⁡n→∞Snx for all x∈C and suppose that Fix⁡(S)=⋂n=0∞‍Fix⁡(Sn). Then, for any given point x0∈C, the sequence {xn} generated by
(84)yn=αnxn+(1-αn)G(Snxn),xn+1=βnf(xn)+(I-βnA)×[G(Snyn)-λnμnFG(Snyn)],hhhhhhhhhhhhhhihhhhh∀n≥0,
converges strongly to p∈Δ, which is the unique solution in Δ to the VIP:
(85)〈(A-f)p,J(p-u)〉≤0,∀u∈Δ.Proof.
First, let us show that{xn} is bounded. Indeed, taking a fixed u∈Δ arbitrarily, we have
(86)∥yn-u∥=∥αnxn+(1-αn)G(Snxn)-u∥≤αn∥xn-u∥+(1-αn)∥G(Snxn)-u∥≤αn∥xn-u∥+(1-αn)∥Snxn-u∥≤αn∥xn-u∥+(1-αn)∥xn-u∥=∥xn-u∥.
So, ∥yn-u∥≤∥xn-u∥ for all n≥0. Taking into account lim⁡n→∞(λnμn)/βn=0, we may assume, without loss of generality, that λnμn≤βn≤∥A∥-1 for all n≥0. Thus, by Lemma 7 (ii), we have
(87)∥xn+1-u∥=∥βn(f(xn)-f(u))+(I-βnA)hhhh×[G(Snyn)-λnμnFG(Snyn)-u]-βn(A-f)u∥≤βn∥f(xn)-f(u)∥+∥(I-βnA)[G(Snyn)-λnμnFG(Snyn)-u]∥+βn∥(A-f)u∥≤βnβ∥xn-u∥+(1-βnγ-)×[∥(I-λnμnF)G(Snyn)-(I-λnμnF)G(Snu)∥hhhhh+∥(I-λnμnF)G(Snu)-u∥]+βn∥(A-f)u∥≤βnβ∥xn-u∥+(1-βnγ-)×(1-λnμn(1-1-αλ))∥G(Snyn)-G(Snu)∥+(1-βnγ-)λnμn∥Fu∥+βn∥(A-f)u∥≤βnβ∥xn-u∥+(1-βnγ-)(1-λnμn(1-1-αλ))×∥Snyn-Snu∥+λnμn∥Fu∥+βn∥(A-f)u∥≤βnβ∥xn-u∥+(1-βnγ-)∥yn-u∥+λnμn∥Fu∥+βn∥(A-f)u∥≤βnβ∥xn-u∥+(1-βnγ-)∥xn-u∥+βn∥Fu∥+βn∥(A-f)u∥=(1-βn(γ--β))∥xn-u∥+βn(γ--β)∥(A-f)u∥+∥Fu∥γ--β≤max⁡⁡{∥xn-u∥,∥(A-f)u∥+∥Fu∥γ--β},
By induction,
(88)∥xn-u∥≤max⁡⁡{∥x0-u∥,∥(A-f)u∥+∥Fu∥γ--β},∀n≥0.
Thus, {xn} is bounded and so is {yn}. Because G and Sn are nonexpansive for all n≥0,f is contractive, and F is Lipschitzian, {Snxn}, {Snyn}, {G(Snxn)}, {G(Snyn)}, {f(xn)}, and {FG(Snyn)} are bounded. From conditions (i) and (ii) we have
(89)∥xn+1-G(Snyn)∥=βn∥(f(xn)-AG(Snyn))+(I-βnA)hhhhhh×(G(Snyn)-λnμnFG(Snyn)-G(Snyn))∥≤βn∥f(xn)-AG(Snyn)∥+(1-βnγ-)λnμn∥FG(Snyn)∥≤βn∥f(xn)-AG(Snyn)∥+λnμn∥FG(Snyn)∥⟶02asn⟶∞.
Now, we claim that(90)∥xn+1-xn∥⟶0asn⟶∞.
In order to prove (90), we estimate ∥xn+1-xn∥ first. From (84), we have
(91)yn=αnxn+(1-αn)G(Snxn),yn-1=αn-1xn-1+(1-αn-1)G(Sn-1xn-1).
Simple calculations show that
(92)yn-yn-1=(1-αn)(G(Snxn)-G(Sn-1xn-1))+αn(xn-xn-1)+(xn-1-G(Sn-1xn-1))×(αn-αn-1).
It follows that
(93)∥yn-yn-1∥≤(1-αn)∥G(Snxn)-G(Sn-1xn-1)∥+αn∥xn-xn-1∥+∥xn-1-G(Sn-1xn-1)∥|αn-αn-1|≤(1-αn)∥Snxn-Sn-1xn-1∥+αn∥xn-xn-1∥+∥xn-1-G(Sn-1xn-1)∥|αn-αn-1|≤(1-αn)(∥Snxn-Snxn-1∥+∥Snxn-1-Sn-1xn-1∥)+αn∥xn-xn-1∥+∥xn-1-G(Sn-1xn-1)∥|αn-αn-1|≤(1-αn)(∥xn-xn-1∥+∥Snxn-1-Sn-1xn-1∥)+αn∥xn-xn-1∥+∥xn-1-G(Sn-1xn-1)∥|αn-αn-1|≤∥xn-xn-1∥+∥xn-1-G(Sn-1xn-1)∥|αn-αn-1|+∥Snxn-1-Sn-1xn-1∥.
In the meantime, it follows from (84) that
(94)xn+1=βnf(xn)+(I-βnA)[G(Snyn)-λnμnFG(Snyn)],xn=βn-1f(xn-1)+(I-βn-1A)×[G(Sn-1yn-1)-λn-1μn-1FG(Sn-1yn-1)].
Simple calculations show that
(95)xn+1-xn=(βn-βn-1)f(xn-1)+βn(f(xn)-f(xn-1))+(βn-1-βn)A(I-λn-1μn-1F)G(Sn-1yn-1)+(I-βnA)[(I-λnμnF)G(Snyn)hhhhhhhhhhhhh-(I-λn-1μn-1F)G(Sn-1yn-1)]=(βn-βn-1)f(xn-1)+βn(f(xn)-f(xn-1))+(βn-1-βn)A(I-λn-1μn-1F)G(Sn-1yn-1)+(I-βnA)[(I-λnμnF)G(Snyn)hhhhhhhhhhhh-(I-λnμnF)G(Sn-1yn-1)hhhhhhhhhhhh+(λn-1μn-1-λnμn)FG(Sn-1yn-1)].
It follows from Lemma 7 (ii) and (93) that
(96)∥xn+1-xn∥≤|βn-βn-1|∥f(xn-1)∥+βn∥f(xn)-f(xn-1)∥+|βn-1-βn|∥A(I-λn-1μn-1F)G(Sn-1yn-1)∥+(1-βnγ-)[∥(I-λnμnF)G(Snyn)hhhhhhhhhhhhh-(I-λnμnF)G(Sn-1yn-1)∥hhhhhhhhhhhhh+|λn-1μn-1-λnμn|∥FG(Sn-1yn-1)∥]≤|βn-βn-1|∥f(xn-1)∥+βnβ∥xn-xn-1∥+|βn-1-βn|∥A(I-λn-1μn-1F)G(Sn-1yn-1)∥+(1-βnγ-)[(1-λnμn(1-1-αλ))hhhhhhhhhhhhh×∥G(Snyn)-G(Sn-1yn-1)∥hhhhhhhhhhhhh+|λnμn-λn-1μn-1|∥FG(Sn-1yn-1)∥[(1-λnμn(1-1-αλ))]≤|βn-βn-1|∥f(xn-1)∥+βnβ∥xn-xn-1∥+|βn-1-βn|∥A(I-λn-1μn-1F)G(Sn-1yn-1)∥+(1-βnγ-)[∥Snyn-Sn-1yn-1∥+|λnμn-λn-1μn-1|∥FG(Sn-1yn-1)∥]≤|βn-βn-1|∥f(xn-1)∥+βnβ∥xn-xn-1∥+|βn-1-βn|∥A(I-λn-1μn-1F)G(Sn-1yn-1)∥+(1-βnγ-)[∥Snyn-Snyn-1∥+∥Snyn-1-Sn-1yn-1∥+|λnμn-λn-1μn-1|∥FG(Sn-1yn-1)∥]≤|βn-βn-1|∥f(xn-1)∥+βnβ∥xn-xn-1∥+|βn-1-βn|∥A(I-λn-1μn-1F)G(Sn-1yn-1)∥+(1-βnγ-)[∥yn-yn-1∥+∥Snyn-1-Sn-1yn-1∥hhhhhhhhhhhhh+|λnμn-λn-1μn-1|∥FG(Sn-1yn-1)∥]≤|βn-βn-1|∥f(xn-1)∥+βnβ∥xn-xn-1∥+|βn-1-βn|∥A(I-λn-1μn-1F)G(Sn-1yn-1)∥+(1-βnγ-)[∥xn-xn-1∥+∥xn-1-G(Sn-1xn-1)∥×|αn-αn-1|+∥Snxn-1-Sn-1xn-1∥+∥Snyn-1-Sn-1yn-1∥+|λnμn-λn-1μn-1|∥FG(Sn-1yn-1)∥]≤(1-βn(γ--β))∥xn-xn-1∥+|βn-βn-1|×(∥f(xn-1)∥+∥A(I-λn-1μn-1F)G(Sn-1yn-1)∥)+∥xn-1-G(Sn-1xn-1)∥|αn-αn-1|+|λnμn-λn-1μn-1|×∥FG(Sn-1yn-1)∥+∥Snxn-1-Sn-1xn-1∥+∥Snyn-1-Sn-1yn-1∥≤(1-βn(γ--β))∥xn-xn-1∥+|βn-βn-1|M0+M0|αn-αn-1|+|λnμn-λn-1μn-1|M0+∥Snxn-1-Sn-1xn-1∥+∥Snyn-1-Sn-1yn-1∥=(1-βn(γ--β))∥xn-xn-1∥+M0(|αn-αn-1|+|βn-βn-1|+|λnμn-λn-1μn-1|)+∥Snxn-1-Sn-1xn-1∥+∥Snyn-1-Sn-1yn-1∥,≤(1-βn(γ--β))∥xn-xn-1∥+M0(|αn-αn-1|+|βn-βn-1|+|λn-λn-1|+|μn-μn-1|)+∥Snxn-1-Sn-1xn-1∥+∥Snyn-1-Sn-1yn-1∥,
where sup⁡n≥0{∥f(xn)∥+∥A(I-λnμnF)G(Snyn)∥+∥FG(Snyn)∥+∥xn-G(Snxn)∥}≤M0 for some M0>0. Since it follows from conditions (i) and (iv) that ∑n=0∞‍βn(γ--β)=∞ and
(97)∑n=0∞M0(|αn-αn-1|+|βn-βn-1|+|λn-λn-1|+|μn-μn-1|)<∞,
applying Lemma 3 to (96), we obtain from the assumption on {Sn} that
(98)lim⁡n→∞⁡∥xn+1-xn∥=0.
By condition (iii) and (84), we have
(99)∥yn-xn∥=(1-αn)∥G(Snxn)-xn∥≤(1-a)(∥G(Snxn)-G(Snyn)∥hhhhhhhh+∥G(Snyn)-xn+1∥+∥xn+1-xn∥)≤(1-a)(∥xn-yn∥+∥G(Snyn)-xn+1∥hhhhhhhhh+∥xn+1-xn∥),
which implies that
(100)∥yn-xn∥≤1-aa(∥G(Snyn)-xn+1∥+∥xn+1-xn∥).
This together with (89)-(90) implies that
(101)lim⁡n→∞⁡∥xn-yn∥=0.
So, we obtain
(102)∥xn-G(Snxn)∥≤∥xn-yn∥+∥yn-G(Snxn)∥≤∥xn-yn∥+αn∥xn-G(Snxn)∥≤∥xn-yn∥+b∥xn-G(Snxn)∥,
which implies that
(103)∥xn-G(Snxn)∥≤11-b∥xn-yn∥,
and hence
(104)lim⁡n→∞⁡∥xn-G(Snxn)∥=0.
Letu∈Δ. Now, we show that lim⁡n→∞∥xn-Sxn∥=0 and lim⁡n→∞∥xn-Gxn∥=0.
Indeed, for simplicity, putv=ΠC(u-μ2B2u), x^n=Snxn, un=ΠC(x^n-μ2B2x^n), and vn=ΠC(un-μ1B1un). Then, u=ΠC(v-μ1B1v) and vn=Gx^n=G(Snxn) for all n≥0. It is clear from (84) that
(105)∥yn-u∥2≤αn∥xn-u∥2+(1-αn)∥G(Snxn)-u∥2=αn∥xn-u∥2+(1-αn)∥vn-u∥2.
Utilizing Lemma 12, we have
(106)∥un-v∥2=∥ΠC(x^n-μ2B2x^n)-ΠC(u-μ2B2u)∥2≤∥x^n-u-μ2(B2x^n-B2u)∥2≤∥x^n-u∥2-2μ2(α2-κ2μ2)∥B2x^n-B2u∥2,(107)∥vn-u∥2=∥ΠC(un-μ1B1un)-ΠC(v-μ1B1v)∥2≤∥un-v-μ1(B1un-B1v)∥2≤∥un-v∥2-2μ1(α1-κ2μ1)∥B1un-B1v∥2.
Substituting (106) for (107), we obtain
(108)∥vn-u∥2≤∥x^n-u∥2-2μ2(α2-κ2μ2)×∥B2x^n-B2u∥2-2μ1(α1-κ2μ1)×∥B1un-B1v∥2≤∥xn-u∥2-2μ2(α2-κ2μ2)×∥B2x^n-B2u∥2-2μ1(α1-κ2μ1)×∥B1un-B1v∥2,
which together with (105) implies that
(109)∥yn-u∥2≤αn∥xn-u∥2+(1-αn)∥vn-u∥2≤αn∥xn-u∥2+(1-αn)×[∥xn-u∥2-2μ2(α2-κ2μ2)hhhhh×∥B2x^n-B2u∥2hhhhh-2μ1(α1-κ2μ1)∥B1un-B1v∥2]=∥xn-u∥2-2(1-αn)×[μ2(α2-κ2μ2)∥B2x^n-B2u∥2hhhhhh+μ1(α1-κ2μ1)∥B1un-B1v∥2].
It immediately follows that
(110)2(1-αn)[μ2(α2-κ2μ2)∥B2x^n-B2u∥2hhhhhhhh+μ1(α1-κ2μ1)∥B1un-B1v∥2]≤∥xn-u∥2-∥yn-u∥2≤(∥xn-u∥+∥yn-u∥)∥xn-yn∥.
Since {xn} and {yn} are bounded and 0<μi<αi/κ2 for i=1,2, we deduce from (101) and condition (iii) that
(111)lim⁡n→∞⁡∥B2x^n-B2u∥=0,lim⁡n→∞⁡∥B1un-B1v∥=0.
Utilizing Proposition 2 and Lemma 8, we have that there exists g1 such that
(112)∥un-v∥2=∥ΠC(x^n-μ2B2x^n)-ΠC(u-μ2B2u)∥2≤〈x^n-μ2B2x^n-(u-μ2B2u),J(un-v)〉=〈x^n-u,J(un-v)〉+μ2〈B2u-B2x^n,J(un-v)〉≤12[∥x^n-u∥2+∥un-v∥2hhhh-g1(∥x^n-un-(u-v)∥)∥un-v∥2]+μ2∥B2u-B2x^n∥∥un-v∥,
which implies that
(113)∥un-v∥2≤∥x^n-u∥2-g1(∥x^n-un-(u-v)∥)+2μ2∥B2u-B2x^n∥∥un-v∥.
In the same way, we derive that there exists g2 such that
(114)∥vn-u∥2=∥ΠC(un-μ1B1un)-ΠC(v-μ1B1v)∥2≤〈un-μ1B1un-(v-μ1B1v),J(vn-u)〉=〈un-v,J(vn-u)〉+μ1〈B1v-B1un,J(vn-u)〉≤12[∥un-v∥2+∥vn-u∥2-g2(∥un-vn+(u-v)∥)∥un-v∥2]+μ1∥B1v-B1un∥∥vn-u∥,
which implies that
(115)∥vn-u∥2≤∥un-v∥2-g2(∥un-vn+(u-v)∥)+2μ1∥B1v-B1un∥∥vn-u∥.
Substituting (113) for (115), we get
(116)∥vn-u∥2≤∥x^n-u∥2-g1(∥x^n-un-(u-v)∥)-g2(∥un-vn+(u-v)∥)+2μ2∥B2u-B2x^n∥∥un-v∥+2μ1∥B1v-B1un∥∥vn-u∥≤∥xn-u∥2-g1(∥x^n-un-(u-v)∥)-g2(∥un-vn+(u-v)∥)+2μ2∥B2u-B2x^n∥∥un-v∥+2μ1∥B1v-B1un∥∥vn-u∥,
which together with (105) implies that
(117)∥yn-u∥2≤αn∥xn-u∥2+(1-αn)∥vn-u∥2≤αn∥xn-u∥2+(1-αn)×[∥xn-u∥2-g1(∥x^n-un-(u-v)∥)hhhh-g2(∥un-vn+(u-v)∥)hhhh+2μ2∥B2u-B2x^n∥∥un-v∥hhhh+2μ1∥B1v-B1un∥∥vn-u∥∥un-v∥2]=∥xn-u∥2-(1-αn)×[g1(∥x^n-un-(u-v)∥)hhhh+g2(∥un-vn+(u-v)∥)]+2(1-αn)×(μ2∥B2u-B2x^n∥∥un-v∥hhhh+μ1∥B1v-B1un∥∥vn-u∥).
It immediately follows that
(118)(1-αn)[g1(∥x^n-un-(u-v)∥)hhhhhhh+g2(∥un-vn+(u-v)∥)]≤∥xn-u∥2-∥yn-u∥2+2(1-αn)×(μ2∥B2u-B2x^n∥∥un-v∥hhhhhh+μ1∥B1v-B1un∥∥vn-u∥)≤(∥xn-u∥+∥yn-u∥)∥xn-yn∥+2μ2∥B2u-B2x^n∥∥un-v∥+2μ1∥B1v-B1un∥∥vn-u∥.
Since {xn}, {yn}, {un}, and {vn} are bounded, we deduce from (101), (111), and condition (iii) that
(119)lim⁡n→∞g1(∥x^n-un-(u-v)∥)=0,lim⁡n→∞⁡g2(∥un-vn+(u-v)∥)=0.
Utilizing the properties of g1 and g2, we get
(120)lim⁡n→∞⁡∥x^n-un-(u-v)∥=0,lim⁡n→∞⁡∥un-vn+(u-v)∥=0,
which hence yields
(121)∥x^n-vn∥≤∥x^n-un-(u-v)∥+∥un-vn+(u-v)∥⟶0hhhhhhhhhhhhhhasn⟶∞.
That is,
(122)lim⁡n→∞⁡∥Snxn-G(Snxn)∥=lim⁡n→∞⁡∥x^n-vn∥=0.
Note that
(123)∥xn-Snxn∥≤∥xn-G(Snxn)∥+∥G(Snxn)-Snxn∥.
So, from (104) and (122), we have
(124)lim⁡n→∞⁡∥xn-Snxn∥=0,
which together with (104) and the assumption on {Sn} implies that
(125)∥xn-Sxn∥≤∥xn-Snxn∥+∥Snxn-Sxn∥⟶0asn⟶∞,∥xn-Gxn∥≤∥xn-G(Snxn)∥+∥G(Snxn)-Gxn∥≤∥xn-G(Snxn)∥+∥Snxn-xn∥⟶0hhhhhhhhhhhhhhhhhhhhhasn⟶∞.
That is,
(126)lim⁡n→∞⁡∥xn-Sxn∥=0,lim⁡n→∞⁡∥xn-Gxn∥=0.
Define a mapping(127)Wx=(1-θ)Sx+θGx,∀x∈C,
where θ is a constant in (0,1). Then, by Lemma 10, we know that Fix⁡(W)=Fix⁡(S)∩Fix⁡(G)=Δ. We observe that
(128)∥xn-Wxn∥=∥(1-θ)(xn-Sxn)+θ(xn-Gxn)∥≤(1-θ)∥xn-Sxn∥+∥θxn-Gxn∥.
So, from (126), we get
(129)lim⁡n→∞⁡∥xn-Wxn∥=0,
where p is defined below. Now, we claim that
(130)lim⁡sup⁡n→∞⁡〈(f-A)p,J(xn-p)〉≤0.
Indeed, let{xt} be defined by
(131)xt=tf(xt)+(I-tA)Wxt.
Then, as t→0,{xt} converges strongly to p∈Fix⁡(W)=Δ, which by Proposition CB is the unique solution in Δ to the VIP:
(132)〈(A-f)p,J(p-u)〉≤0,∀u∈Δ.
In terms of Lemma 6, we conclude from (129) that (130) holds. It is clear that
(133)lim⁡sup⁡n→∞⁡〈(f-A)p,J(xn+1-p)〉≤0.
Finally, let us show thatxn→p as n→∞. We observe that
(134)∥yn-p∥2≤αn∥xn-p∥2+(1-αn)∥G(Snxn)-p∥2≤αn∥xn-p∥2+(1-αn)∥xn-p∥2=∥xn-p∥2,
and hence
(135)∥xn+1-p∥2=∥βn(f(xn)-f(p))+(I-βnA)×[G(Snyn)-λnμnFG(Snyn)-p]+βn(f-A)p∥2≤∥βn(f(xn)-f(p))+(I-βnA)×[G(Snyn)-λnμnFG(Snyn)-p]∥2+2βn〈(f-A)p,J(xn+1-p)〉≤[βn∥f(xn)-f(p)∥+(1-βnγ-)×∥G(Snyn)-λnμnFG(Snyn)-p∥]2+2βn〈(f-A)p,J(xn+1-p)〉=[βn∥f(xn)-f(p)∥+(1-βnγ-)×∥(I-λnμnF)G(Snyn)-(I-λnμnF)p-λnμnFp∥]2+2βn〈(f-A)p,J(xn+1-p)〉≤[βn∥βxn-p∥+(1-βnγ-)×(∥(I-λnμnF)G(Snyn)-(I-λnμnF)p∥+λnμn∥Fp∥)]2+2βn〈(f-A)p,J(xn+1-p)〉≤[×((1-λnμn(1-1-αλ))∥G(Snyn)-p∥+λnμn∥Fp∥)]2βnβ∥xn-p∥+(1-βnγ-)×((1-λnμn(1-1-αλ))×∥G(Snyn)-p∥+λnμn∥Fp∥((1-λnμn(1-1-αλ)))[×((1-λnμn(1-1-αλ))∥G(Snyn)-p∥+λnμn∥Fp∥)]2βnβ∥xn-p∥+(1-βnγ-)]2+2βn〈(f-A)p,J(xn+1-p)〉≤[βnβ∥xn-p∥+(1-βnγ-)(∥yn-p∥+λnμn∥Fp∥)]2+2βn〈(f-A)p,J(xn+1-p)〉≤[βnβ∥xn-p∥+(1-βnγ-)∥xn-p∥+λnμn∥Fp∥]2+2βn〈(f-A)p,J(xn+1-p)〉=[(1-βn(γ--β))∥xn-p∥+λnμn∥Fp∥]2+2βn〈(f-A)p,J(xn+1-p)〉=(1-βn(γ--β))2∥xn-p∥2+λnμn∥Fp∥×[2(1-βn(γ--β))∥xn-p∥+λnμn∥Fp∥]+2βn〈(f-A)p,J(xn+1-p)〉≤(1-βn(γ--β))∥xn-p∥2+λnμn∥Fp∥(2∥xn-p∥+λnμn∥Fp∥)+2βn〈(f-A)p,J(xn+1-p)〉=(1-βn(γ--β))∥xn-p∥2+βn(γ--β)·1γ--β×[λnμnβn∥Fp∥(2∥xn-p∥+λnμn∥Fp∥)+2〈(f-A)p,J(xn+1-p)〉[λnμnβn∥Fp∥(2∥xn-p∥+λnμn∥Fp∥)].
Taking into account (133) and conditions (i) and (ii), we obtain that ∑n=0∞‍(γ--β)βn=∞ and
(136)lim⁡sup⁡n→∞⁡1γ--β⁡[λnμnβn∥Fp∥(2∥xn-p∥+λnμn∥Fp∥)hhhhhhhhhhhh+2〈(f-A)p,J(xn+1-p)〉[λnμnβn∥Fp∥(2∥xn-p∥+λnμn∥Fp∥)]≤0.
Therefore, applying Lemma 3 to (135), we infer that
(137)lim⁡n→∞⁡∥xn-p∥=0.
This completes the proof.Remark 20.
It is worth pointing out that the sequences{λn}, {μn}, and {βn} can be taken, which satisfy the conditions in Theorem 19. As a matter of fact, put λn=(1+n)-5/6, μn=1, and βn=(1+n)-2/3 for all n≥0. Then, there hold the following statements:(i)
lim⁡n→∞βn=0 and ∑n=0∞‍βn=∞;(ii)
lim⁡n→∞(λnμn)/βn=0;(iii)
∑n=0∞‍|βn+1-βn|<∞, ∑n=0∞‍|λn+1-λn|<∞, and ∑n=0∞‍|μn+1-μn|<∞.By the careful analysis of the proof of Theorem19, we can obtain the following result. Because its proof is much simpler than that of Theorem 19, we omit its proof.Theorem 21.
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X such that C±C⊂C. 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 {Sn}n=0∞ be an infinite family of nonexpansive mappings of C into itself such that Δ=⋂i=0∞‍Fix⁡(Si)∩Ω≠∅, where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2) with 0<μi<αi/κ2 for i=1,2. Let f:C→C be a fixed contractive map with coefficient β∈(0,1), let F:C→C be α-strongly accretive and λ-strictly pseudocontractive with α+λ>1, and let A:C→C be a γ--strongly positive linear bounded operator with 0<γ--β≤1. Given sequences {λn}n=0∞ in [0,1] and {αn}n=0∞,{βn}n=0∞ in (0,1], suppose that there hold the following conditions:(i)
lim⁡n→∞βn=0 and ∑n=0∞‍βn=∞;(ii)
lim⁡n→∞λn/βn=0 and ∑n=0∞‍|λn+1-λn|<∞;(iii)
{αn}⊂[a,b] for some a,b∈(0,1);(iv)
∑n=0∞‍|αn+1-αn|<∞ and ∑n=0∞‍|βn+1-βn|<∞.Assume that ∑n=0∞‍sup⁡x∈D∥Sn+1x-Snx∥<∞ for any bounded subset D of C and let S be a mapping of C into itself defined by Sx=lim⁡n→∞Snx for all x∈C and suppose that Fix⁡(S)=⋂n=0∞‍Fix⁡(Sn). Then, for any given point x0∈C, the sequence {xn} generated by
(138)yn=αnxn+(1-αn)G(Snxn),xn+1=βnf(xn)+(I-βnA)[yn-λnF(yn)],∀n≥0,
converges strongly to p∈Δ, which is the unique solution in Δ to the VIP (85).Remark 22.
Theorems19 and 21 improve, extend, supplement and develop Cai and Bu [10, Theorem 3.1] and Cai and Bu [9, Theorems 3.1] in the following aspects.(i)  The GSVI (13) with hierarchical fixed point problem constraint for a countable family of nonexpansive mappings is more general and more subtle than every problem in Cai and Bu [10, Theorems 3.1] and Cai and Bu [9, Theorem 3.1] because our problem is to find a point p∈Δ=⋂n‍Fix⁡(Sn)∩Ω, which is the unique solution in Δ to the VIP:
(139)〈(A-f)p,J(p-u)〉≤0,∀u∈Δ.(ii)  The iterative scheme in [10, Theorem 3.1] is extended to develop the iterative schemes in Theorems 19 and 21 by virtue of hybrid steepest-descent method. The iterative schemes in Theorems 19 and 21 are more advantageous and more flexible than the iterative scheme of [9, Theorem 3.1] because the iterative scheme of [9, Theorem 3.1] is implicit and our iterative schemes involve solving two problems: the GSVI (13) and the fixed point problem of a countable family of nonexpansive mappings {Sn}.(iii)  The iterative schemes in Theorems19 and 21 are very different from everyone in both [10, Theorem 3.1] and [9, Theorem 3.1] because our iterative schemes involve hybrid steepest-descent method (namely, we add a strongly accretive and strictly pseudocontractive mapping F in our iterative schemes) and because the mappings G and Sn in [10, Theorem 3.1] and the mapping Sn in [9, Theorem 3.1] are replaced by the same composite mapping G∘Sn in the iterative schemes of Theorems 19 and 21.(iv)  Cai and Bu’s proof in [10, Theorem 3.1] depends on the argument techniques in [20], the inequality in 2-uniformly smooth Banach spaces (see Lemma 1), and the inequality in smooth and uniform convex Banach spaces (see Proposition 2). Because the composite mapping G∘Sn appears in the iterative schemes in Theorems 19 and 21, the proof of Theorems 19 and 21 depends on the argument techniques in [20], the inequality in 2-uniformly smooth Banach spaces (see Lemma 1), the inequality in smooth and uniform convex Banach spaces (see Proposition 2), and the properties of the strongly positive linear bounded operator (see Lemmas 15), the Banach limit (see Lemma 5), and the strongly accretive and strictly pseudocontractive mapping (see Lemma 7).Remark 23.
Theorems19 and 21 extend and improve Theorem 16 of Yao et al. [21] to a great extent in the following aspects:(i)
theu is replaced by a fixed contractive mapping;(ii)
one continuous pseudocontractive mapping (including nonexpansive mapping) is replaced by a countable family of nonexpansive mappings;(iii)
we add a strongly positive linear bounded operatorA and a strongly accretive and strictly pseudocontractive mapping F in our iterative algorithms.



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*Source: 945985-2014-02-17.xml*