# Implicit and Explicit Iterative Methods for Systems of Variational Inequalities and Zeros of Accretive Operators

Authors: Lu-Chuan Ceng; Saleh Abdullah Al-Mezel; Qamrul Hasan Ansari
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Based on Korpelevich's extragradient method, hybrid steepest-descent method, and viscosity approximation method, we propose implicit and explicit iterative schemes for computing a common element of the solution set of a system of variational inequalities and the set of zeros of an accretive operator, which is also a unique solution of a variational inequality. Under suitable assumptions, we study the strong convergence of the sequences generated by the proposed algorithms. The results of this paper improve and extend several known results in the literature.


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## Body

## 1. Introduction and Preliminaries

Throughout the paper, we use the notation⇀ to indicate the weak convergence and → to indicate the strong convergence.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.A Banach spaceX is said to be smooth if the limit
(2)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. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for all x,y∈U; in this case, X is also said to have a uniformly Fréchet differentiable norm. The norm of X is said to be Fréchet differentiable if for each x∈U, the limit is attained uniformly for all y∈U. A function ρ:[0,∞)→[0,∞) defined by
(3)ρ(τ)=sup{12(∥x+y∥+∥x-y∥)-1:x,y∈X,∥x∥=1,∥y∥=τ12}
is called the modulus of smoothness of X. It is known that X is uniformly smooth if and only if limτ→0ρ(τ)/τ=0. If q∈(1,2] is a fixed number, 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.Thenormalized duality mapping J:X→2X* is defined by
(4)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 all x∈X. 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. For further details on the geometry of Banach spaces, we refer to [2, 3] and the references therein.Lemma 1 (see [4]).
LetX be a 2-uniformly smooth Banach space. Then
(5)∥x+y∥2≤∥x∥2+2〈y,J(x)〉+2∥κy∥2,∀x,y∈X,
where κ is the 2-uniformly smooth constant of X and J is the normalized duality mapping from X into X*.The following lemma is an immediate consequence of the subdifferential inequality of the function(1/2)∥·∥2.Lemma 2.
LetX be a smooth Banach space. Then,
(6)∥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.Proposition 3 (see [5]).
LetX be a real smooth and uniform convex Banach space and r>0. Then, there exists a strictly increasing, continuous, and convex function g:[0,2r]→R, g(0)=0 such that
(7)g(∥x-y∥)≤∥x∥2-2〈x,J(y)〉+∥y∥2,∀x,y∈Br,
where Br={x∈X:∥x∥≤r}.LetC be a nonempty closed convex subset of a real Banach space X. A mapping T:C→C is said to be L-Lipschitzian (or Lipschtiz continuous) if there exists a constant L>0 such that ∥Tx-Ty∥≤L∥x-y∥ for all x,y∈C. If L∈(0,1), then T is called L-contraction. If L=1, then T is called nonexpansive.The set of fixed points ofT is denoted by Fix(T).Lemma 4 (see [6]).
LetC be a nonempty closed convex subset of a strictly convex Banach space X, and 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 T on C defined by Tx=∑n=0∞‍λnTnx for x∈C is defined well, nonexpansive, and Fix(T)=⋂n=0∞‍Fix(Tn).Lemma 5 (see [7]).
LetC be a nonempty closed and convex subset of a Banach space X and T:C→C be a continuous and strong pseudocontraction mapping. Then, T has a unique fixed point in C.LetC be a nonempty subset of a Banach space X. A mapping A:C→X is said to be(a)
accretive if for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(8)〈Ax-Ay,j(x-y)〉≥0,whereJ is the normalized duality mapping of X;(b)
α-strongly accretive if for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(9)〈Ax-Ay,j(x-y)〉≥α∥x-y∥2,forsomeα∈(0,1);(c)
pseudocontractive if for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(10)〈Ax-Ay,j(x-y)〉≤∥x-y∥2;(d)
β-strongly pseudocontractive if for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(11)〈Ax-Ay,j(x-y)〉≤β∥x-y∥2,forsomeβ∈(0,1);(e)
λ-strictly pseudocontractive if for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(12)〈Ax-Ay,j(x-y)〉≤∥x-y∥2-λ∥x-y-(Ax-Ay)∥2,forsomeλ∈(0,1).It is worth to mention 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, [8, 9].Lemma 6.
LetC be a nonempty closed convex subset of a real smooth Banach space X, and let F:C→X be a mapping.(a)
IfF:C→X is 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/λ.(b)
IfF:C→X is α-strongly accretive and λ-strictly pseudocontractive with α+λ>1, then for any fixed τ∈(0,1),I-τF is contractive with coefficient 1-τ(1-(1-α)/λ).Proof.
(a) From theλ-strictly pseudocontractivity of F, we have for all x,y∈C,
(13)λ∥(I-F)x-(I-F)y∥2≤〈(I-F)x-(I-F)y,J(x-y)〉≤∥(I-F)x-(I-F)y∥∥x-y∥,
which implies that
(14)∥(I-F)x-(I-F)y∥≤1λ∥x-y∥.
Hence,
(15)∥F(x)-F(y)∥≤∥(I-F)x-(I-F)y∥+∥x-y∥≤(1+1λ)∥x-y∥.
From the λ-strictly pseudocontractiveness and α-strongly accretiveness of F, we have for all x,y∈C,
(16)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
(17)∥(I-F)x-(I-F)y∥≤1-αλ∥x-y∥.
Since α+λ≥1⇔(1-α)/λ≤1, I-F is nonexpansive.
(b) Take a fixedτ∈(0,1) arbitrarily. Observe that for all x,y∈C,
(18)∥(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∥.
Since α+λ>1⇔(1-α)/λ<1, I-τF is contractive with coefficient 1-τ(1-(1-α)/λ).LetX be a real smooth Banach space X. An operator V:X→X* is said to bestrongly positive [10] if there exists a constant γ->0 such that
(19)〈Vx,J(x)〉≥γ-∥x∥2,∥aI-bV∥=sup∥x∥≤1|〈(aI-bV)x,J(x)〉|,a∈[0,1],b∈[-1,1],
where I is the identity mapping.Lemma 7 (see [7]).
LetV be a strongly positive linear bounded operator on a smooth Banach space X with coefficient γ->0 and 0<ρ≤∥V∥-1. Then, ∥I-ρA∥≤1-ργ-.Recall that a possibly multivalued operatorA⊂X×X with domain D(A) and range R(A) in X is accretive if for each xi∈D(A) and yi∈Axi (i=1,2), there exists j(x1-x2)∈J(x1-x2) such that 〈y1-y2,j(x1-x2)〉≥0. An accretive operator A is said to satisfy the range condition if D(A)¯⊂R(I+rA) for all r>0. An accretive operator A is m-accretive if R(I+rA)=X for each r>0. If A is an accretive operator which satisfies the range condition, then we can define, for each r>0, a mapping Jr:R(I+rA)→D(A) by Jr=(I+rA)-1, which is called the resolvent of A. It is well known that Jr is nonexpansive and Fix(Jr)=A-10 for all r>0. Hence,
(20)Fix(Jr)=A-10={z∈D(A):0∈Az}.
If A-10≠∅, then the inclusion 0∈Az is solvable.The following resolvent identity is well-known.Lemma 8 (resolvent identity).
Forλ>0, μ>0, and x∈X,
(21)Jλx=Jμ(μλx+(1-μλ)Jλx).Recently, Aoyama et al. [11] studied an iterative scheme in the setting of a uniformly convex Banach space X having a uniformly Gâteaux differentiable norm to compute the approximate zero of an accretive operator A. They proved that the sequence generated by their scheme converges strongly to a zero of A under some appropriate assumptions. Subsequently, Ceng et al. [12] introduced and analyzed a composite iterative scheme in the setting of a uniformly smooth Banach space or a reflexive Banach space having a weakly sequentially continuous duality mapping. Further, Jung [13] proposed and analyzed a composite iterative scheme by the viscosity approximation method for finding a zero of an accretive operator A and established the strong convergence of the sequence generated by the proposed scheme.LetC be a closed and convex subset of a Banach space X, and let D be a nonempty subset of C. A mapping Π:C→D is said to be sunny if
(22)Π[Π(x)+t(x-Π(x))]=Π(x),
whenever Π(x)+t(x-Π(x))∈C for all x∈C and t≥0. A mapping Π:C→C is called a retraction if Π2=Π. If a mapping Π:C→C is a retraction, then Π(z)=z for all 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 9 (see [14]).
LetC be a nonempty closed convex subset of a real smooth Banach space X, let D be a nonempty subset of C, and let Π be a retraction of C onto D. Then, the following statements are equivalent:(a)
Π is sunny and nonexpansive;(b)
∥Π(x)-Π(y)∥2≤〈x-y,J(Π(x)-Π(y))〉,∀x,y∈C;(c)
〈x-Π(x),J(y-Π(x))〉≤0,∀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.LetX be a smooth Banach space, let C⊆X be a nonempty closed convex set, and let B1,B2:C→X be two nonlinear mappings. We consider the following system of variational inequalities (in short, SVI) which is earlier studied by Cai and Bu [15]. Find (x*,y*)∈C×C such that
(23)〈μ1B1y*+x*-y*,J(x-x*)〉≥0,∀x∈C,〈μ2B2x*+y*-x*,J(x-y*)〉≥0,∀x∈C,
where μ1 and μ2 are two positive constants. When X=H is a real Hilbert space, then SVI (23) is considered and studied by Ceng et al. [16]. In addition, if B1=B2=A and x*=y*, then the SVI reduces to the classical variational inequality problem (VIP) of finding x*∈C such that
(24)〈Ax*,x-x*〉≥0,∀x∈C.
It is worth to mention that the system of variational inequalities is used as a tool to study the Nash equilibrium problem; see, for example, [17–19] and the references therein. We believe that the problem (23) could be used to study Nash equilibrium problem for two-player game. The theory of variational inequalities is well established because it has a wide range of applications in science, engineering, management, and social sciences. In 1976, Korpelevič [20] proposed an iterative algorithm for solving VIP (24) in Euclidean space, which is known as the extragradient method or Korpelevich’s extragradient method. For different generalizations of extragradient method for variational inequalities and fixed point problems, we refer to [21–23] and the references therein. For further details on variational inequalities, we refer to [2, 19, 24, 25] and the references therein.WheneverX is a real smooth Banach space, B1=B2=A, and x*=y*, then SVI (23) reduces to the variational inequality problem (VIP) of finding x*∈C such that
(25)〈Ax*,J(x-x*)〉≥0,∀x∈C,
which was considered by Aoyama et al. [26]. Very recently, Cai and Bu [15] constructed an iterative algorithm for solving SVI (23) 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 sequence generated by the proposed algorithm.Lemma 10.
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 SVI (23) if and only if x*=ΠC(y*-μ1B1y*), where y*=ΠC(x*-μ2B2x*).Proof.
We can rewrite SVI (23) as
(26)〈x*-(y*-μ1B1y*),J(x-x*)〉≥0,∀x∈C,〈y*-(x*-μ2B2x*),J(x-y*)〉≥0,∀x∈C,
which is equivalent to
(27)x*=ΠC(y*-μ1B1y*),y*=ΠC(x*-μ2B2x*).
From Lemma 9, we get the conclusion.Remark 11.
In view of Lemma10, we observe that
(28)x*=ΠC[ΠC(x*-μ2B2x*)-μ1B1ΠC(x*-μ2B2x*)],
which implies that x* is a fixed point of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2).Lemma 12 (see [15, Lemma 2.8]).
LetC be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X, and for each i=1,2, let Bi:C→X be an αi-inverse-strongly accretive. Then,
(29)∥(I-μiBi)x-(I-μiBi)y∥2≤∥x-y∥2+2μi(μiκ2-αi)∥Bix-Biy∥2,∀x,y∈C,i=1,2,
where μi>0. In particular, if 0<μi≤αi/κ2, then I-μiBi is nonexpansive for i=1,2.Lemma 13 (see [15, Lemma 2.9]).
LetC be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X, and let ΠC be a sunny nonexpansive retraction from X onto C. For each i=1,2, let Bi:C→X be a αi-inverse-strongly accretive mapping and let G:C→C be the mapping defined by
(30)Gx=ΠC[ΠC(x-μ2B2x)-μ1B1ΠC(x-μ2B2x)],∀x∈C.
If 0<μi≤αi/κ2 for i=1,2, then G:C→C is nonexpansive.Throughout this paper, the set of fixed points of the mappingG=ΠC(I-μ1B1)ΠC(I-μ2B2) is denoted by Ω.LetX be a real uniformly convex and 2-uniformly smooth Banach space, and let C be a nonempty closed convex subset of X such that C±C⊂C. Let V be a γ--strongly positive linear bounded operator with γ-∈(1,2). The purpose of this paper is to introduce and analyze implicit and explicit iterative schemes for computing a common element of the solution set of system of variational inequalities and set of zeros of an accretive operator A with domain D(A) and range R(A) in X such that D(A)¯⊂C⊂⋂r>0‍R(I+rA). Our methods are based on Korpelevich’s extragradient method, hybrid steepest-descent method, and viscosity approximation method. In particular, we propose the iterative methods for computing a point p∈Ω∩A-10, which is also a unique solution of the following variational inequality:
(31)〈(V-I)p,J(p-u)〉≤0,∀u∈Ω∩A-10.
Under suitable assumptions, the strong convergence of the sequences generated by the proposed schemes is studied. Our results extend and improve several results presented in the recent past in the literature; see, for example, [10, 13, 15, 27–29] and the references therein.Now we present some known results and definitions which will be used in the sequel.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
(32)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
(33)μn(an)=μn(an+1)
for every a=(a1,a2,…)∈l∞. We know that if μ is a Banach limit, then
(34)liminfn→∞an≤μn(an)≤limsupn→∞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
(35)μn(an)=μ(a)=c(resp.,μn(an)=μn(bn)).Lemma 14 (see [30]).
LetC be a nonempty closed convex subset of a uniformly smooth Banach space X, and let {xn} be a bounded sequence in X. Let μ be a mean on N and z∈C. Then,
(36)μn∥xn-z∥2=miny∈Cμn∥xn-y∥2
if and only if
(37)μn〈y-z,J(xn-z)〉≤0,∀y∈C,
where J is the normalized duality mapping of X.Recall that a Banach spaceX is said to satisfy Opial’s condition if whenever {xn} is a sequence in X which converges weakly to x as n→∞, then
(38)limsupn→∞∥xn-x∥<limsupn→∞∥xn-y∥,∀y∈X,y≠x.Lemma 15 (demiclosedness principle, see [31, Theorem 10.3]).
LetX be a reflexive Banach space satisfying Opial’s condition, let C be a nonempty closed convex subset of X, and let T:C→C be a nonexpansive mapping. Then the mapping I-T is demiclosed on C, where I is the identity mapping; that is, if {xn} is a sequence of C such that xn⇀x and (I-T)xn→y, then (I-T)x=y.Remark 16.
If the duality mappingJ:x↦{x*∈X*:〈x,x*〉=∥x∥2=∥x*∥2} from X into X* is single-valued and weakly sequentially continuous, then X satisfies Opial’s condition; see [32, Theorem 1]. Each Hilbert space and the sequence spaces lp with 1<p<∞ satisfy Opial’s condition; see [32]. Though an Lp-space with p≠2 does not usually satisfy Opial’s condition, each separable Banach space can be equivalently renormed, so that it satisfies Opial’s condition; see [33].We close this section by mentioning a result on the sequence which will be used in the proof of the main results of this paper.Lemma 17 (see [34]).
Let{sn} be a sequence of nonnegative real numbers satisfying
(39)sn+1≤(1-αn)sn+αnβn+γn,∀n≥0,
where {αn},{βn}, and {γn} satisfy the following conditions:(i)
{αn}⊂[0,1] and ∑n=0∞‍αn=∞;(ii)
limsupn→∞βn≤0;(iii)
γn≥0,∀n≥0, and ∑n=0∞‍γn<∞. Then limn→∞sn=0.


## 2. Implicit Iterative Scheme and Its Convergence Criteria

In this section, we propose an implicit hybrid steepest-descent viscosity scheme for finding the solution of SVI (23) such that such solution is also a zero of an accretive operator. We also study the strong convergence of the sequence generated by the proposed algorithm.Theorem 18.
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, and let A⊂X×X be an accretive operator in X such that D(A)¯⊂C⊂⋂r>0‍R(I+rA). For each i=1,2, let Bi:C→X be an αi-inverse-strongly accretive mapping such that Δ=Ω∩A-10≠∅, 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 α-strongly accretive and λ-strictly pseudocontractive with α+λ>1, and let f:C→C be a fixed Lipschitzian strongly pseudocontractive mapping with pseudocontractive coefficient β∈(0,γ0), γ0=1-(1-α)/λ, and Lipschitzian constant L>0. Let V:C→C be a γ--strongly positive linear bounded operator with γ-∈(1,2). For each integer n≥0, let {xt,n} be defined by
(40)xt,n=(I-θtV)JrnGxt,n+θt[JrnGxt,n-t(F(JrnGxt,n)-f(xt,n))],
where {rn}⊂[ε,∞) for some ε>0 and 0<θt≤∥V∥-1 for all t∈(0,(2-γ-)/(γ0-β)) with limt→0θt=0. Then, as t→0,{xt,n} converges strongly to p∈Δ, which is a unique solution of the following variational inequality:
(41)〈(V-I)p,J(p-u)〉≤0,∀u∈Δ.Proof.
We first show that{xt,n} is defined well. Since 0<θt≤∥V∥-1, for all t∈(0,(2-γ-)/(γ0-β)) and ∥V∥≥γ->1, we have 0<θt≤∥V∥-1<1, for all t∈(0,(2-γ-)/(γ0-β)). Define a mapping Γt,n:C→C by
(42)Γt,nx=(I-θtV)JrnGx+θt[JrnGx-t(F(JrnGx)-f(x))],∀x∈C.
Observe that
(43)〈Γt,nx-Γt,ny,J(x-y)〉=〈(I-θtV)JrnGx-(I-θtV)JrnGy,J(x-y)〉+θt〈(I-tF)JrnGx-(I-tF)JrnGy,J(x-y)〉+θtt〈f(x)-f(y),J(x-y)〉≤∥I-θtV∥∥JrnGx-JrnGy∥∥x-y∥+θt∥(I-tF)JrnGx-(I-tF)JrnGy∥∥x-y∥+tθtβ∥x-y∥2≤(1-θtγ-)∥JrnGx-JrnGy∥∥x-y∥+θt(1-t(1-1-αλ))∥JrnGx-JrnGy∥∥x-y∥+tθtβ∥x-y∥2≤(1-θtγ-)∥Gx-Gy∥∥x-y∥+θt(1-t(1-1-αλ))∥Gx-Gy∥∥x-y∥+tθtβ∥x-y∥2≤(1-θtγ-)∥x-y∥2+θt(1-t(1-1-αλ))∥x-y∥2+tθtβ∥x-y∥2=[1-θt(γ--1+t(γ0-β))]∥x-y∥2.
Whenever n≥0 and t∈(0,(2-γ-)/(γ0-β)), Γt,n:C→C is a continuous and strongly pseudocontractive mapping with pseudocontractive coefficient 1-θt(γ--1+t(γ0-β))∈(0,1). By Lemma 5, we deduce that there exists a unique fixed point in C, denoted by xt,n, which uniquely solves the fixed point equation
(44)xt,n=(I-θtV)JrnGxt,n+θt[JrnGxt,n-t(F(JrnGxt,n)-f(xt,n))].
Let us show the uniqueness of a solution of VIP (41). Suppose that both p1∈Δ and p2∈Δ are solutions to VIP (41). Then, we have
(45)〈(V-I)p1,J(p1-p2)〉≤0,〈(V-I)p2,J(p2-p1)〉≤0.
Adding up the above two inequalities, we obtain
(46)〈(V-I)p1-(V-I)p2,J(p1-p2)〉≤0.
Note that
(47)〈(V-I)p1-(V-I)p2,J(p1-p2)〉=〈V(p1-p2),J(p1-p2)〉-〈p1-p2,J(p1-p2)〉≥γ-∥p1-p2∥2-∥p1-p2∥2=(γ--1)∥p1-p2∥2≥0.
Taking into account 0<γ--1<1, we have p1=p2, and hence the uniqueness is proved.
Next, we show that{xt,n} is bounded. Indeed, we note that 0<θt≤∥V∥-1, for all t∈(0,(2-γ-)/(γ0-β)). Take a fixed p∈Δ arbitrarily. Utilizing Lemmas 6 and 7, we obtain
(48)∥xt,n-p∥2=〈(I-θtV)JrnGxt,n+θt[JrnGxt,n-t(F(JrnGxt,n)-f(xt,n))]-p,J(xt,n-p)Jrn〉=〈(I-θtV)JrnGxt,n-(I-θtV)JrnGp,J(xt,n-p)〉+θt〈JrnGxt,n-t(F(JrnGxt,n)-f(xt,n))-p,J(xt,n-p)〉+θt〈(I-V)p,J(xt,n-p)〉=〈(I-θtV)JrnGxt,n-(I-θtV)JrnGp,J(xt,n-p)〉+θt[〈(I-tF)JrnGxt,n-(I-tF)JrnGp,J(xt,n-p)〉+t(〈f(xt,n)-f(p),J(xt,n-p)〉+〈f(p)-F(p),J(xt,n-p)〉)Jrn]+θt〈(I-V)p,J(xt,n-p)〉≤∥(I-θtV)JrnGxt,n-(I-θtV)JrnGp∥∥xt,n-p∥+θt[∥(I-tF)JrnGxt,n-(I-tF)JrnGp∥∥xt,n-p∥+t(〈f(xt,n)-f(p),J(xt,n-p)〉+∥f(p)-F(p)∥∥xt,n-p∥)Jrn]+θt∥I-V∥∥p∥∥xt,n-p∥≤(1-θtγ-)∥JrnGxt,n-JrnGp∥∥xt,n-p∥+θt[∥xt,n-p∥2(1-tγ0)∥JrnGxt,n-JrnGp∥∥xt,n-p∥+t(β∥xt,n-p∥2+∥f(p)-F(p)∥∥xt,n-p∥)]+θt∥I-V∥∥p∥∥xt,n-p∥≤(1-θtγ-)∥Gxt,n-Gp∥∥xt,n-p∥+θt[(1-tγ0)∥Gxt,n-Gp∥∥xt,n-p∥+t(β∥xt,n-p∥2+∥f(p)-F(p)∥∥xt,n-p∥)]+θt∥I-V∥∥p∥∥xt,n-p∥≤(1-θtγ-)∥xt,n-p∥2+θt[(1-tγ0)∥xt,n-p∥2+t(β∥xt,n-p∥2+∥f(p)-F(p)∥∥xt,n-p∥)]+θt∥I-V∥∥p∥∥xt,n-p∥,
which immediately yields
(49)∥xt,n-p∥≤(1-θtγ-)∥xt,n-p∥+θt[(1-tγ0)∥xt,n-p∥+t(β∥xt,n-p∥+∥f(p)-F(p)∥)]+θt∥I-V∥∥p∥.
It follows that
(50)∥xt,n-p∥≤∥I-V∥∥p∥+t∥f(p)-F(p)∥γ--1+t(γ0-β)≤∥I-V∥∥p∥+∥f(p)-F(p)∥γ--1.
Thus, {xt,n} is bounded and so are {f(xt,n)}, {Gxt,n}, and {JrnGxt,n}.
Let us show that∥xt,n-Gxt,n∥→0 as t→0.
Indeed, for simplicity we putq=ΠC(I-μ2B2)p,ut,n=ΠC(I-μ2B2)xt,n and vt,n=ΠC(I-μ1B1)ut,n. Then, p=ΠC(I-μ1B1)q and vt,n=Gxt,n. Hence from (48), we get
(51)∥xt,n-p∥2≤(1-θtγ-)∥vt,n-p∥∥xt,n-p∥+θt[∥xt,n-p∥2(1-tγ0)∥vt,n-p∥∥xt,n-p∥+t(β∥xt,n-p∥2+∥f(p)-F(p)∥∥xt,n-p∥)]+θt∥I-V∥∥p∥∥xt,n-p∥=[1-θt(γ--1+tγ0)]∥vt,n-p∥∥xt,n-p∥+θtt(β∥xt,n-p∥2+∥f(p)-F(p)∥∥xt,n-p∥)+θt∥I-V∥∥p∥∥xt,n-p∥≤[1-θt(γ--1+tγ0)]12(∥vt,n-p∥2+∥xt,n-p∥2)+θtt(β∥xt,n-p∥2+∥f(p)-F(p)∥∥xt,n-p∥)+θt∥I-V∥∥p∥∥xt,n-p∥.
From Lemma 12, we have
(52)∥ut,n-q∥2=∥ΠC(xt,n-μ2B2xt,n)-ΠC(p-μ2B2p)∥2≤∥xt,n-p-μ2(B2xt,n-B2p)∥2≤∥xt,n-p∥2-2μ2(α2-κ2μ2)∥B2xt,n-B2p∥2,∥vt,n-p∥2=∥ΠC(ut,n-μ1B1ut,n)-ΠC(q-μ1B1q)∥2≤∥ut,n-q-μ1(B1ut,n-B1q)∥2≤∥ut,n-q∥2-2μ1(α1-κ2μ1)∥B1ut,n-B1q∥2.
From the last two inequalities, we obtain
(53)∥vt,n-p∥2≤∥xt,n-p∥2-2μ2(α2-κ2μ2)∥B2xt,n-B2p∥2-2μ1(α1-κ2μ1)∥B1ut,n-B1q∥2,
which together with (51) implies that
(54)∥xt,n-p∥2≤[1-θt(γ--1+tγ0)]12(∥xt,n-p∥2+∥vt,n-p∥2)+θtt(β∥xt,n-p∥2+∥f(p)-F(p)∥∥xt,n-p∥)+θt∥I-V∥∥p∥∥xt,n-p∥≤[1-θt(γ--1+tγ0)]×12{∥xt,n-p∥2+∥xt,n-p∥2-2μ2(α2-κ2μ2)∥B2xt,n-B2p∥2-2μ1(α1-κ2μ1)∥B1ut,n-B1q∥2}+θtt(β∥xt,n-p∥2+∥f(p)-F(p)∥∥xt,n-p∥)+θt∥I-V∥∥p∥∥xt,n-p∥=[1-θt(γ--1+t(γ0-β))]∥xt,n-p∥2+θtt∥f(p)-F(p)∥∥xt,n-p∥+θt∥I-V∥∥p∥∥xt,n-p∥-[1-θt(γ--1+tγ0)]×{μ2(α2-κ2μ2)∥B2xt,n-B2p∥2+μ1(α1-κ2μ1)∥B1ut,n-B1q∥2}≤∥xt,n-p∥2+θtt∥f(p)-F(p)∥∥xt,n-p∥+θt∥I-V∥∥p∥∥xt,n-p∥-[1-θt(γ--1+tγ0)]×{μ2(α2-κ2μ2)∥B2xt,n-B2p∥2+μ1(α1-κ2μ1)∥B1ut,n-B1q∥2}.
Therefore,
(55)[1-θt(γ--1+tγ0)]{μ2(α2-κ2μ2)∥B2xt,n-B2p∥2+μ1(α1-κ2μ1)∥B1ut,n-B1q∥2}≤θtt∥f(p)-F(p)∥∥xt,n-p∥+θt∥I-V∥∥p∥∥xt,n-p∥.
Since 0<μi<αi/κ2 for i=1,2, it follows from limt→0θt=0 that
(56)limt→0∥B2xt,n-B2p∥=0,limt→0∥B1ut,n-B1q∥=0.
Utilizing Proposition 3 and Lemma 9, we have
(57)∥ut,n-q∥2=∥ΠC(xt,n-μ2B2xt,n)-ΠC(p-μ2B2p)∥2≤〈xt,n-μ2B2xt,n-(p-μ2B2p),J(ut,n-q)〉=〈xt,n-p,J(ut,n-q)〉+μ2〈B2p-B2xt,n,J(ut,n-q)〉≤12[∥xt,n-p∥2+∥ut,n-q∥2-g1(∥xt,n-ut,n-(p-q)∥)∥ut,n-q∥2]+μ2∥B2p-B2xt,n∥∥ut,n-q∥,
which implies that
(58)∥ut,n-q∥2≤∥xt,n-p∥2-g1(∥xt,n-ut,n-(p-q)∥)+2μ2∥B2p-B2xt,n∥∥ut,n-q∥.
Similarly, we derive
(59)∥vt,n-p∥2=∥ΠC(ut,n-μ1B1ut,n)-ΠC(q-μ1B1q)∥2≤〈ut,n-μ1B1ut,n-(q-μ1B1q),J(vt,n-p)〉=〈ut,n-q,J(vt,n-p)〉+μ1〈B1q-B1ut,n,J(vt,n-p)〉≤12[∥ut,n-q∥2+∥vt,n-p∥2-g2(∥ut,n-vt,n+(p-q)∥)∥ut,n-q∥2]+μ1∥B1q-B1ut,n∥∥vt,n-p∥,
which implies that
(60)∥vt,n-p∥2≤∥ut,n-q∥2-g2(∥ut,n-vt,n+(p-q)∥)+2μ1∥B1q-B1ut,n∥∥vt,n-p∥.
Combining (58) and (60), we get
(61)∥vt,n-p∥2≤∥xt,n-p∥2-g1(∥xt,n-ut,n-(p-q)∥)-g2(∥ut,n-vt,n+(p-q)∥)+2μ2∥B2p-B2xt,n∥∥ut,n-q∥+2μ1∥B1q-B1ut,n∥∥vt,n-p∥,
which together with (51) implies that
(62)∥xt,n-p∥2≤[1-θt(γ--1+tγ0)]12(∥xt,n-p∥2+∥vt,n-p∥2)+θtt(β∥xt,n-p∥2+∥f(p)-F(p)∥∥xt,n-p∥)+θt∥I-V∥∥p∥∥xt,n-p∥≤[1-θt(γ--1+tγ0)]×12{∥xt,n-p∥2+∥xt,n-p∥2-g1(∥xt,n-ut,n-(p-q)∥)-g2(∥ut,n-vt,n+(p-q)∥)+2μ2∥B2p-B2xt,n∥∥ut,n-q∥+2μ1∥B1q-B1ut,n∥∥vt,n-p∥∥xt,n-p∥2}+θtt(β∥xt,n-p∥2+∥f(p)-F(p)∥∥xt,n-p∥)+θt∥I-V∥∥p∥∥xt,n-p∥=[1-θt(γ--1+t(γ0-β))]∥xt,n-p∥2-12[1-θt(γ--1+tγ0)]×{g1(∥xt,n-ut,n-(p-q)∥)+g2(∥ut,n-vt,n+(p-q)∥)}+[1-θt(γ--1+tγ0)]×(μ2∥B2p-B2xt,n∥∥ut,n-q∥+μ1∥B1q-B1ut,n∥∥vt,n-p∥)+θtt(β∥xt,n-p∥2+∥f(p)-F(p)∥∥xt,n-p∥)+θt∥I-V∥∥p∥∥xt,n-p∥≤∥xt,n-p∥2-12[1-θt(γ--1+tγ0)]×{g1(∥xt,n-ut,n-(p-q)∥)+g2(∥ut,n-vt,n+(p-q)∥)}+[1-θt(γ--1+tγ0)]×(μ2∥B2p-B2xt,n∥∥ut,n-q∥+μ1∥B1q-B1ut,n∥∥vt,n-p∥)+θtt(β∥xt,n-p∥2+∥f(p)-F(p)∥∥xt,n-p∥)+θt∥I-V∥∥p∥∥xt,n-p∥.
Therefore,
(63)12[1-θt(γ--1+tγ0)]{g1(∥xt,n-ut,n-(p-q)∥)+g2(∥ut,n-vt,n+(p-q)∥)}≤[1-θt(γ--1+tγ0)](μ2∥B2p-B2xt,n∥∥ut,n-q∥+μ1∥B1q-B1ut,n∥∥vt,n-p∥)+θtt(β∥xt,n-p∥2+∥f(p)-F(p)∥∥xt,n-p∥)+θt∥I-V∥∥p∥∥xt,n-p∥.
Hence, from (56) and limt→0θt=0, we conclude that
(64)limt→0g1(∥xt,n-ut,n-(p-q)∥)=0,limt→0g2(∥ut,n-vt,n+(p-q)∥)=0.
Utilizing the properties of g1 and g2, we get
(65)limt→0∥xt,n-ut,n-(p-q)∥=0,limt→0∥ut,n-vt,n+(p-q)∥=0,
which leads to
(66)∥xt,n-vt,n∥≤∥xt,n-ut,n-(p-q)∥+∥ut,n-vt,n+(p-q)∥⟶0ast⟶0;
that is,
(67)limt→0∥xt,n-Gxt,n∥=limt→0∥xt,n-vt,n∥=0.
Note that Fix(Jrn)=A-10 for all n≥0 and that {xt,n:t∈(0,b),n≥0} is bounded and so are {f(xt,n):t∈(0,b),n≥0}, {Jrnxt,n:t∈(0,b),n≥0} and {JrnGxt,n:t∈(0,b),n≥0}, where b=(2-γ-)/(γ0-β). Hence, we have
(68)∥xt,n-JrnGxt,n∥=θt∥(A-I)JrnGxt,n-t(f(xt,n)-F(JrnGxt,n))∥⟶0,ast⟶0.
Also, observe that
(69)∥xt,n-Jrnxt,n∥≤∥xt,n-JrnGxt,n∥+∥JrnGxt,n-Jrnxt,n∥≤∥xt,n-JrnGxt,n∥+∥Gxt,n-xt,n∥.
This together with (67) and (68) implies that
(70)limt→0∥xt,n-Jrnxt,n∥=0.
Since rn≥ε for all n≥0, utilizing Lemma 8, we have
(71)∥xt,n-Jεxt,n∥≤∥xt,n-Jrnxt,n∥+∥Jrnxt,n-Jεxt,n∥=∥xt,n-Jrnxt,n∥+∥Jε(εrnxt,n+(1-εrn)Jrnxt,n)-Jεxt,n∥≤2∥xt,n-Jrnxt,n∥⟶0,ast⟶0.
For any integer n≥0, for simplicity we put wt=xt,n for all t∈(0,b). Now let {tk} be a sequence in (0,b) that converges to 0 as k→∞, and define a function g:C→ℝ by
(72)g(w)=μk12∥wtk-w∥2,∀w∈C,
where μ is a Banach limit. Define a set
(73)K:={w∈C:g(w)=min{g(y):y∈C}}
and a mapping
(74)Wx=(1-θ)Jεx+θGx,∀x∈C,
where θ is a constant in (0,1). Then by Lemma 4, we have Fix(W)=Fix(Jε)∩Fix(G)=Δ. We observe that
(75)∥wt-Wwt∥=∥(1-θ)(wt-Jεwt)+θ(wt-Gwt)∥≤(1-θ)∥wt-Jεwt∥+θ∥wt-Gwt∥.
So, from (67) and (71), we obtain
(76)limn→∞∥wt-Wwt∥=0.
Since X is a uniformly smooth Banach space, K is a nonempty bounded closed convex subset of C (see [30]). We claim that K is also invariant under the nonexpansive mapping W. Indeed, noticing (76), we have for all w∈K,
(77)g(Ww)=μk12∥wtk-Ww∥2=μk12∥Wwtk-Ww∥2≤μk12∥wtk-w∥2=g(w).
Since every nonempty closed bounded convex subset of 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 15, we get
(78)μk〈x-p,J(wtk-p)〉≤0,∀x∈C.
Putting x=(2I-V)p, we have
(79)μk〈(I-V)p,J(wtk-p)〉≤0.
Noticing wtk=xtk,n, we conclude from (48) that
(80)∥wtk-p∥2=〈(I-θtkV)JrnGwtk-(I-θtkV)JrnGp,J(wtk-p)〉+θtk[〈(I-tkF)JrnGwtk-(I-tkF)JrnGp,J(wtk-p)〉+tk(〈f(wtk)-f(p),J(wtk-p)〉+〈f(p)-F(p),J(wtk-p)〉)]+θtk〈(I-V)p,J(wtk-p)〉≤(1-θtkγ-)∥JrnGwtk-JrnGp∥∥wtk-p∥+θtk[(β∥wtk-p∥2+∥f(p)-F(p)∥∥wtk-p∥)(1-tkγ0)∥JrnGwtk-JrnGp∥∥wtk-p∥+tk(β∥wtk-p∥2+∥f(p)-F(p)∥∥wtk-p∥)]+θtk〈(I-V)p,J(wtk-p)〉≤(1-θtkγ-)∥wtk-p∥2+θtk[(1-tkγ0)∥wtk-p∥2+tk(β∥wtk-p∥2+∥f(p)-F(p)∥∥wtk-p∥)]+θtk〈(I-V)p,J(wtk-p)〉=[1-θtk(γ--1+tk(γ0-β))]∥wtk-p∥2+θtktk∥f(p)-F(p)∥∥wtk-p∥+θtk〈(I-V)p,J(wtk-p)〉.
It follows that
(81)∥wtk-p∥2≤1γ--1+tk(γ0-β)[〈(I-V)p,J(wtk-p)〉+tk∥f(p)-F(p)∥∥wtk-p∥]≤1γ--1[〈(I-V)p,J(wtk-p)〉+tk∥f(p)-F(p)∥∥wtk-p∥].
Since limk→∞tk=0, from (79) and the boundedness of {wtk}, it follows that
(82)μk∥wtk-p∥2≤1γ--1μk[(I-V)p,J(wtk-p)+tk∥f(p)-F(p)∥∥wtk-p∥]=1γ--1[μk〈(I-V)p,J(wtk-p)〉+μktk∥f(p)-F(p)∥∥wtk-p∥]≤0.
Therefore, it is known that for any sequence {wtk} in {wt:t∈(0,b)}, there exists a subsequence which is still denoted by {wtk} that converges strongly to some fixed point p of W.
Now we claim that such ap∈Δ is a unique solution of variational inequality (41).
Indeed, from (48), it follows that for all u∈Δ=Ω∩A-10(83)∥wt-u∥2=〈(I-θtV)JrnGwt-(I-θtV)JrnGu,J(wt-u)〉+θt[〈(I-tF)JrnGwt-(I-tF)JrnGu,J(wt-u)〉+t(〈f(wt)-f(u),J(wt-u)〉+〈f(u)-F(u),J(wt-u)〉)Jrn]+θt〈(I-V)u,J(wt-u)〉≤(1-θtγ-)∥JrnGwt-JrnGu∥∥wt-u∥+θt[(1-tγ0)∥JrnGwt-JrnGu∥∥wt-u∥+t(β∥wt-u∥2+∥f(u)-F(u)∥∥wt-u∥)Jrn]+θt〈(I-V)u,J(wt-u)〉≤(1-θtγ-)∥wt-u∥2+θt[(1-tγ0)∥wt-u∥2+t(β∥wt-u∥2+∥f(u)-F(u)∥∥wt-u∥)]+θt〈(I-V)u,J(wt-u)〉=[1-θt(γ--1+t(γ0-β))]∥wt-u∥2+θtt∥f(u)-F(u)∥∥wt-u∥+θt〈(I-V)u,J(wt-u)〉≤∥wt-u∥2+θtt∥f(u)-F(u)∥∥wt-u∥+θt〈(I-V)u,J(wt-u)〉,
which implies that
(84)〈(V-I)u,J(wt-u)〉≤t∥f(u)-F(u)∥∥wt-u∥,∀u∈Δ.
Since wtk→p as tk→0, we obtain from the last inequality that
(85)〈(V-I)u,J(p-u)〉≤0,∀u∈Δ.
Utilizing the well-known Minty-type Lemma [25], we get
(86)〈(V-I)p,J(p-u)〉≤0,∀u∈Δ.
So, p∈Δ is a solution of variational inequality (41).
In order to prove that the net{xt:t∈(0,b)} converges strongly to p as t→0, suppose that there exists another subsequence {wsk}⊂{wt:t∈(0,b)} such that wsk→q as sk→0; then we also have q∈Fix(W)=Ω∩A-10=:Δ due to (76). Repeating the same argument as above, we know that q∈Δ is another solution of variational inequality (41). In terms of the uniqueness of solutions in Δ to the variational inequality (41), we get p=q.Remark 19.
In the assertion of Theorem18, “as t→0,{xt,n} converges strongly to p∈Δ”, p does not depend on n. Indeed, it is known that {rn}⊂[ε,∞) for some ε>0. Moreover, in the proof of Theorem 18, it can be readily seen that p is a fixed point of the nonexpansive self-mapping W of K. This shows that p depends on neither n nor t.Remark 20.
Theorem18 improves and extends [15, Theorem 3.1], [10, Lemma 2.5], and [13, Theorem 3.1] in the following aspects.(a)
The problem and iterative scheme in Theorem18 are different from those considered in [15, Theorem 3.1], [10, Lemma 2.5], and [13, Theorem 3.1].(b)
Theorem18 drops the assumption of the asymptotical regularity for the iterative scheme in [13, Theorem 3.1].


## 3. Explicit Iterative Scheme and Its Convergence Criteria

In this section, we propose an explicit iterative scheme to compute a common element of the solution set of SVI (23) and the solution set of zeros of an accretive operator. We study the strong convergence of the sequence generated by the proposed scheme.Theorem 21.
LetX be an uniformly convex and 2-uniformly smooth Banach space which has the weakly sequentially continuous duality mapping J. Let C be a nonempty closed convex subset of X such that C±C⊂C, and let ΠC be a sunny nonexpansive retraction from X onto C. Let A⊂X×X be an accretive operator in X such that D(A)¯⊂C⊂⋂r>0‍R(I+rA). For each i=1,2, let Bi:C→X be an αi-inverse-strongly accretive mapping such that Δ=Ω∩A-10≠∅, 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 α-strongly accretive and λ-strictly pseudocontractive with α+λ>1, and let f:C→C be a contractive mapping with coefficient β∈(0,γ0), where γ0=1-(1-α)/λ. Let V:C→C be a γ--strongly positive linear bounded operator with γ-∈(1,2). For any given point x0∈C, the sequence {xn} is generated by
(87)yn=αnxn+(1-αn)JrnGxn,xn+1=(I-βnV)JrnGyn+βn[JrnGyn-λn(F(JrnGyn)-f(yn))],∀n≥0,
where {rn}n=0∞⊂[ε,∞) for some ε>0 and {αn}n=0∞, {βn}n=0∞, and {λn}n=0∞ are the sequences in (0,1] satisfying the following conditions:(i)
limn→∞λn=0;(ii)
limn→∞βn=0 and ∑n=0∞‍βn=∞;(iii)
{αn}⊂[a,b] for some a,b∈(0,1);(iv)
∑n=0∞‍(|αn+1-αn|+|βn+1-βn|+|rn+1-rn|)<∞. Then, {xn} converges strongly to p∈Δ, which is a unique solution of the following variational inequality:
(88)〈(V-I)p,J(p-u)〉≤0,∀u∈Δ.Proof.
We first show that{xn} is bounded. Indeed, since limn→∞βn=0 and ∥V∥≥γ->1, without loss of generality, we may assume that 0<βn≤∥V∥-1<1, for all n≥0. Take a fixed p∈Δ arbitrarily. Then, p=Gp, p=Jrnp, for all n≥0, and
(89)∥yn-p∥≤αn∥xn-p∥+(1-αn)∥JrnGxn-p∥≤αn∥xn-p∥+(1-αn)∥Gxn-p∥≤αn∥xn-p∥+(1-αn)∥xn-p∥=∥xn-p∥.
Note that
(90)xn+1-p=(I-βnV)JrnGxn-(I-βnV)JrnGp+βn[JrnGyn-λn(F(JrnGyn)-f(yn))-p]+βn(I-V)p=(I-βnV)(JrnGxn-JrnGp)+βn[Jrnλn(f(yn)-F(p))+(I-λnF)JrnGyn-(I-λnF)JrnGp]+βn(I-V)p.
Therefore, we have
(91)∥xn+1-p∥=∥(I-βnV)(JrnGyn-JrnGp)+βn[Jrnλn(f(yn)-F(p))+(I-λnF)JrnGyn-(I-αnF)JrnGp]+βn(I-V)p(JrnGyn-JrnGp)∥≤∥(I-βnV)∥∥JrnGyn-JrnGp∥+βn[∥(I-λnF)JrnGyn-(I-λnF)JrnGp∥λn∥f(yn)-F(p)∥+∥(I-λnF)JrnGyn-(I-λnF)JrnGp∥]+βn∥(I-V)p∥≤(1-βnγ-)∥yn-p∥+βn[∥JrnGyn-JrnGp∥Jrnλn(∥f(yn)-f(p)∥+∥f(p)-F(p)∥)+(1-λnγ0)∥JrnGyn-JrnGp∥]+βn∥I-V∥∥p∥≤(1-βnγ-)∥yn-p∥+βn[λn(β∥yn-p∥+∥f(p)-F(p)∥)+(1-λnγ0)∥yn-p∥]+βn∥I-V∥∥p∥≤(1-βnγ-)∥yn-p∥+βn[(1-λn(γ0-β))∥yn-p∥+αn∥f(p)-F(p)∥(1-λn(γ0-β))]+βn∥I-V∥∥p∥≤(1-βnγ-)∥yn-p∥+βn[∥yn-p∥+∥f(p)-F(p)∥]+βn∥I-V∥∥p∥=(1-βn(γ--1))∥yn-p∥+βn(∥f(p)-F(p)∥+∥I-V∥∥p∥)=(1-βn(γ--1))∥yn-p∥+βn(γ--1)∥f(p)-F(p)∥+∥I-V∥∥p∥γ--1≤max{∥yn-p∥,∥f(p)-F(p)∥+∥I-V∥∥p∥γ--1}≤max{∥xn-p∥,∥f(p)-F(p)∥+∥I-V∥∥p∥γ--1}.
By induction, we obtain
(92)∥xn-p∥≤max{∥x0-p∥,∥f(p)-F(p)∥+∥I-V∥∥p∥γ--1},∀n≥0,
and hence {xn} is bounded and so are {yn}, {JrnGxn}, {JrnGyn}, {f(xn)}, and {F(JrnGyn)}. From conditions (i), (ii), we have
(93)∥xn+1-JrnGyn∥=βn∥(I-V)JrnGyn-λn(F(JrnGyn)-f(yn))∥⟶0,asn⟶∞.
Now we claim that(94)limn→∞∥xn+1-xn∥=0.
Indeed, (88) can be rewritten as
(95)yn=αnxn+(1-αn)JrnGxn,zn=JrnGyn-λn(F(JrnGyn)-f(yn)),xn+1=(I-βnV)JrnGyn+βnzn,∀n≥0.
If rn-1≤rn, using the resolvent identity in Lemma 8,
(96)JrnGxn=Jrn-1(rn-1rnGxn+(1-rn-1rn)JrnGxn).
We get
(97)∥JrnGxn-Jrn-1Gxn-1∥=∥Jrn-1(rn-1rnGxn+(1-rn-1rn)JrnGxn)-Jrn-1Gxn-1∥≤rn-1rn∥Gxn-Gxn-1∥+(1-rn-1rn)∥JrnGxn-Gxn-1∥≤∥xn-xn-1∥+rn-rn-1rn∥JrnGxn-Gxn-1∥≤∥xn-xn-1∥+1ε|rn-rn-1|∥JrnGxn-Gxn-1∥.
If rn≤rn-1, then it can be easily seen that
(98)∥JrnGxn-Jrn-1Gxn-1∥≤∥xn-1-xn∥+1ε|rn-1-rn|∥Jrn-1Gxn-1-Gxn∥.
By combining the above two cases, we obtain
(99)∥JrnGxn-Jrn-1Gxn-1∥≤∥xn-1-xn∥+|rn-1-rn|ε×supn≥1{∥JrnGxn-Gxn-1∥+∥Jrn-1Gxn-1-Gxn∥},∀n≥1.
Similarly, we obtain
(100)∥JrnGyn-Jrn-1Gyn-1∥≤∥yn-1-yn∥+|rn-1-rn|ε×supn≥1{∥JrnGyn-Gyn-1∥+∥Jrn-1Gyn-1-Gyn∥},∀n≥1.
Thus,
(101)∥JrnGxn-Jrn-1Gxn-1∥≤∥xn-1-xn∥+|rn-1-rn|M0,∥JrnGyn-Jrn-1Gyn-1∥≤∥yn-1-yn∥+|rn-1-rn|M0,∀n≥1,
where supn≥1{(1/ε)(∥JrnGxn-Gxn-1∥+∥Jrn-1Gxn-1-Gxn∥)}≤M0 and supn≥1{(1/ε)(∥JrnGyn-Gyn-1∥+∥Jrn-1Gyn-1-Gyn∥)}≤M0 for some M0>0. From (95) and (101), we have
(102)∥yn-yn-1∥=∥(1-αn)(JrnGxn-Jrn-1Gxn-1)+αn(xn-xn-1)+(xn-1-Jrn-1Gxn-1)(αn-αn-1)∥≤(1-αn)∥JrnGxn-Jrn-1Gxn-1∥+αn∥xn-xn-1∥+∥xn-1-Jrn-1Gxn-1∥|αn-αn-1|≤(1-αn)∥JrnGxn-Jrn-1Gxn-1∥+αn∥xn-xn-1∥+∥xn-1-Jrn-1Gxn-1∥|αn-αn-1|≤(1-αn)[∥xn-xn-1∥+|rn-rn-1|M0]+αn∥xn-xn-1∥+∥xn-1-Jrn-1Gxn-1∥|αn-αn-1|≤∥xn-xn-1∥+|rn-rn-1|M0+∥xn-1-Jrn-1Gxn-1∥|αn-αn-1|,∥zn-zn-1∥=∥(I-λnF)JrnGyn+λnf(yn)-(I-λn-1F)Jrn-1Gyn-1-λn-1f(yn-1)∥=∥λn(f(yn)-f(yn-1))+(λn-λn-1)×(f(yn-1)-F(Jrn-1Gyn-1))+(I-λnF)JrnGyn-(I-λnF)Jrn-1Gyn-1∥≤λn∥f(yn)-f(yn-1)∥+|λn-λn-1|×∥f(yn-1)-F(Jrn-1Gyn-1)∥+∥(I-λnF)JrnGyn-(I-λnF)Jrn-1Gyn-1∥≤λnβ∥yn-yn-1∥+|λn-λn-1|×∥f(yn-1)-F(Jrn-1Gyn-1)∥+(1-λnγ0)∥JrnGyn-Jrn-1Gyn-1∥≤λnβ∥yn-yn-1∥+|λn-λn-1|×∥f(yn-1)-F(Jrn-1Gyn-1)∥+(1-λnγ0)[∥yn-yn-1∥+|rn-rn-1|M0]=(1-λn(γ0-β))∥yn-yn-1∥+(1-λnγ0)|rn-rn-1|M0+|λn-λn-1|∥f(yn-1)-F(Jrn-1Gyn-1)∥≤∥yn-yn-1∥+|rn-rn-1|M0+|λn-λn-1|∥f(yn-1)-F(Jrn-1Gyn-1)∥,
and hence,
(103)∥xn+1-xn∥=∥(I-βnV)JrnGyn+βnzn-(I-βn-1V)Jrn-1Gyn-1-βn-1zn-1∥=∥βn(zn-zn-1)+(βn-βn-1)(zn-1-VJrn-1Gyn-1)+(I-βnV)(JrnGyn-Jrn-1Gyn-1)∥≤βn∥zn-zn-1∥+|βn-βn-1|∥zn-1-VJrn-1Gyn-1∥+(1-βnγ-)∥JrnGyn-Jrn-1Gyn-1∥≤βn[(Jrn-1Gyn-1)∥yn-yn-1∥+|rn-rn-1|M0+|λn-λn-1|∥f(yn-1)-F(Jrn-1Gyn-1)∥]+|βn-βn-1|∥zn-1-VJrn-1Gyn-1∥+(1-βnγ-)[∥yn-yn-1∥+|rn-rn-1|M0]=(1-βn(γ--1))[∥yn-yn-1∥+|rn-rn-1|M0]+βn|λn-λn-1|∥f(yn-1)-F(Jrn-1Gyn-1)∥+|βn-βn-1|∥zn-1-VJrn-1Gyn-1∥≤(1-βn(γ--1))×[∥xn-1-Jrn-1Gxn-1∥∥xn-xn-1∥+2|rn-rn-1|M0+∥xn-1-Jrn-1Gxn-1∥|αn-αn-1|]+βn|λn-λn-1|∥f(yn-1)-F(Jrn-1Gyn-1)∥+|βn-βn-1|∥zn-1-VJrn-1Gyn-1∥≤(1-βn(γ--1))∥xn-xn-1∥+βn|λn-λn-1|M1+(|αn-αn-1|+|βn-βn-1|+|rn-rn-1|)M1,
where supn≥0{∥xn-JrnGxn∥+∥zn-VJrnGyn∥+∥f(yn)-F(JrnGyn)∥+2M0}≤M1 for some M1>0 (it is easy to see that {zn} is bounded due to the boundedness of {yn}). Utilizing Lemma 17, we conclude from conditions (i), (ii), and (iv) that
(104)limn→∞∥xn+1-xn∥=0.
By condition (iii) and (95), we have
(105)∥yn-xn∥=(1-αn)∥JrnGxn-xn∥≤(1-a)(∥JrnGxn-JrnGyn∥+∥JrnGyn-xn+1∥+∥xn+1-xn∥)≤(1-a)(∥xn-yn∥+∥JrnGyn-xn+1∥+∥xn+1-xn∥),
which implies that
(106)∥yn-xn∥≤1-aa(∥JrnGyn-xn+1∥+∥xn+1-xn∥).
This together with (93)-(94) implies that
(107)limn→∞∥xn-yn∥=0.
Therefore,
(108)∥xn-JrnGxn∥≤∥xn-yn∥+∥yn-JrnGxn∥≤∥xn-yn∥+αn∥xn-JrnGxn∥≤∥xn-yn∥+b∥xn-JrnGxn∥,
which implies that
(109)∥xn-JrnGxn∥≤11-b∥xn-yn∥,
and hence,
(110)limn→∞∥xn-JrnGxn∥=0.
Next, we show thatlimn→∞∥xn-Jrnxn∥=0 and limn→∞∥xn-Gxn∥=0.
Indeed, for simplicity, putq=ΠC(p-μ2B2p), un=ΠC(xn-μ2B2xn), and vn=ΠC(un-μ1B1un). Then, p=ΠC(q-μ1B1q) and vn=Gxn for all n≥0. From (95), we have
(111)∥yn-p∥2≤αn∥xn-p∥2+(1-αn)∥JrnGxn-p∥2=αn∥xn-p∥2+(1-αn)∥vn-p∥2.
Utilizing Lemma 12, we have
(112)∥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,(113)∥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.
Combining (112) and (113), we obtain
(114)∥vn-p∥2≤∥xn-p∥2-2μ2(α2-κ2μ2)∥B2xn-B2p∥2-2μ1(α1-κ2μ1)∥B1un-B1q∥2,
which together with (111) implies that
(115)∥yn-p∥2≤αn∥xn-p∥2+(1-αn)∥vn-p∥2≤αn∥xn-p∥2+(1-αn)×[∥xn-p∥2-2μ2(α2-κ2μ2)∥B2xn-B2p∥2-2μ1(α1-κ2μ1)∥B1un-B1q∥2]=∥xn-p∥2-2(1-αn)[μ2(α2-κ2μ2)∥B2xn-B2p∥2+μ1(α1-κ2μ1)∥B1un-B1q∥2].
It immediately follows that
(116)2(1-αn)[μ2(α2-κ2μ2)∥B2xn-B2p∥2+μ1(α1-κ2μ1)∥B1un-B1q∥2]≤∥xn-p∥2-∥yn-p∥2≤(∥xn-p∥+∥yn-p∥)∥xn-yn∥.
Since {xn} and {yn} are bounded and 0<μi<αi/κ2 for i=1,2, we deduce from (107) and condition (iii) that
(117)limn→∞∥B2xn-B2p∥=0,limn→∞∥B1un-B1q∥=0.
Utilizing Proposition 3 and Lemma 9, we have
(118)∥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)∥)]+μ2∥B2p-B2xn∥∥un-q∥,
which implies that
(119)∥un-q∥2≤∥xn-p∥2-g1(∥xn-un-(p-q)∥)+2μ2∥B2p-B2xn∥∥un-q∥.
Similarly, we derive
(120)∥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)∥)]+μ1∥B1q-B1un∥∥vn-p∥,
which implies that
(121)∥vn-p∥2≤∥un-q∥2-g2(∥un-vn+(p-q)∥)+2μ1∥B1q-B1un∥∥vn-p∥.
Combining (119) and (121), we get
(122)∥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∥,
which together with (111) implies that
(123)∥yn-p∥2≤αn∥xn-p∥2+(1-αn)∥vn-p∥2≤αn∥xn-p∥2+(1-αn)×[∥xn-p∥2-g1(∥xn-un-(p-q)∥)-g2(∥un-vn+(p-q)∥)+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥∥xn-p∥2]=∥xn-p∥2-(1-αn)×[g1(∥xn-un-(p-q)∥)+g2(∥un-vn+(p-q)∥)]+2(1-αn)(μ2∥B2p-B2xn∥∥un-q∥+μ1∥B1q-B1un∥∥vn-p∥).
It immediately follows that
(124)(1-αn)[g1(∥xn-un-(p-q)∥)+g2(∥un-vn+(p-q)∥)]≤∥xn-p∥2-∥yn-p∥2+2(1-αn)×(μ2∥B2p-B2xn∥∥un-q∥+μ1∥B1q-B1un∥∥vn-p∥)≤(∥xn-p∥+∥yn-p∥)∥xn-yn∥+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥.
Since {xn},{yn},{un}, and {vn} are bounded, we deduce from (107), (117) and condition (iii) that
(125)limn→∞g1(∥xn-un-(p-q)∥)=0,limn→∞g2(∥un-vn+(p-q)∥)=0.
Utilizing the properties of g1 and g2, we get
(126)limn→∞∥xn-un-(p-q)∥=0,limn→∞∥un-vn+(p-q)∥=0,
which yields
(127)∥xn-vn∥≤∥xn-un-(p-q)∥+∥un-vn+(p-q)∥⟶0asn⟶∞;
that is,
(128)limn→∞∥xn-Gxn∥=0.
Note that
(129)∥xn-Jrnxn∥≤∥xn-JrnGxn∥+∥JrnGxn-Jrnxn∥≤∥xn-JrnGxn∥+∥Gxn-xn∥.
From (110) and (128), we have
(130)limn→∞∥xn-Jrnxn∥=0.
By utilizing the resolvent identity in Lemma 8, we obtain from {rn}⊂[ε,∞) that
(131)∥xn-Jεxn∥≤∥xn-Jrnxn∥+∥Jrnxn-Jεxn∥=∥xn-Jrnxn∥+∥Jrnxn-Jrn(rnεxn+(1-rnε)Jrnxn)∥≤2∥xn-Jrnxn∥,
which together with (130) implies that
(132)limn→∞∥xn-Jεxn∥=0.
For everyn≥0 and t∈(0,(2-γ-)/(γ0-β)), let
(133)xt,n=(I-θtV)JrnGxt,n+θt[JrnGxt,n-t(F(JrnGxt,n)-f(xt,n))],
where {rn}⊂[ε,∞) and 0<θt≤∥V∥-1, for all t∈(0,(2-γ-)/(γ0-β)) with limt→0θt=0. Then, by Theorem 18, {xt,n} converges strongly to p∈Δ as t→0, which is the unique solution of the variational inequality (88).
Define a mapping(134)Wx=(1-θ)Jεx+θGx,∀x∈C,
where θ is a constant in (0,1). Then by Lemma 4, we have Fix(W)=Fix(Jε)∩Fix(G)=Δ. Observe that
(135)∥xn-Wxn∥=∥(1-θ)(xn-Jεxn)+θ(xn-Gxn)∥≤(1-θ)∥xn-Jεxn∥+θ∥xn-Gxn∥.
From (128) and (132), we obtain
(136)limn→∞∥xn-Wxn∥=0.
Further, we show that(137)limsupn→∞〈(I-V)p,J(xn-p)〉≤0.
Indeed, take a subsequence {xni} of {xn} such that
(138)limsupn→∞〈(I-V)p,J(xn-p)〉=limi→∞〈(I-V)p,J(xni-p)〉.
Without loss of generality, we may assume that xni⇀x~. Utilizing Lemma 15, we obtain from (136) that x~∈Fix(W)=Fix(Jε)∩Fix(G)=Δ. Hence from (88) and (138), we get
(139)limsupn→∞〈(I-V)p,J(xn-p)〉=〈(I-V)p,J(x~-p)〉≤0.
Finally, we show thatxn→p as n→∞.
Indeed,(140)∥yn-p∥≤αn∥xn-p∥+(1-αn)∥JrnGxn-p∥≤αn∥xn-p∥+(1-αn)∥xn-p∥=∥xn-p∥,∥xn+1-p∥2=〈(I-βnV)JrnGyn-(I-βnV)JrnGp,J(xn+1-p)〉+βn[〈(I-λnF)JrnGyn-(I-λnF)JrnGp,J(xn+1-p)〉+λn(〈f(yn)-f(p),J(xn+1-p)〉+〈f(p)-F(p),J(xn+1-p)〉)Jrn]+βn〈(I-V)p,J(xn+1-p)〉≤(1-βnγ-)∥JrnGyn-JrnGp∥∥xn+1-p∥+βn[(1-λnγ0)∥JrnGyn-JrnGp∥∥xn+1-p∥+λn(β∥yn-p∥∥xn+1-p∥+∥f(p)-F(p)∥∥xn+1-p∥)∥JrnGyn-JrnGp∥]+βn〈(I-V)p,J(xn+1-p)〉≤(1-βnγ-)∥yn-p∥∥xn+1-p∥+βn[(1-λnγ0)∥yn-p∥∥xn+1-p∥+λn(β∥yn-p∥∥xn+1-p∥+∥f(p)-F(p)∥∥xn+1-p∥)]+βn〈(I-V)p,J(xn+1-p)〉=(1-βnγ-)∥yn-p∥∥xn+1-p∥+βn[(1-λn(γ0-β))∥yn-p∥∥xn+1-p∥+λn∥f(p)-F(p)∥∥xn+1-p∥]+βn〈(I-V)p,J(xn+1-p)〉≤(1-βnγ-)∥yn-p∥∥xn+1-p∥+βn[∥yn-p∥∥xn+1-p∥+λn∥f(p)-F(p)∥∥xn+1-p∥]+βn〈(I-V)p,J(xn+1-p)〉=(1-βn(γ--1))∥yn-p∥∥xn+1-p∥+βnλn∥f(p)-F(p)∥∥xn+1-p∥+βn〈(I-V)p,J(xn+1-p)〉≤(1-βn(γ--1))12(∥yn-p∥2+∥xn+1-p∥2)+βnλn∥f(p)-F(p)∥∥xn+1-p∥+βn〈(I-V)p,J(xn+1-p)〉,
which yields
(141)∥xn+1-p∥2≤1-βn(γ--1)1+βn(γ--1)∥yn-p∥2+βn11+βn(γ--1)×[2λn∥f(p)-F(p)∥∥xn+1-p∥+2〈(I-V)p,J(xn+1-p)〉]≤(1-βn(γ--1))∥xn-p∥2+βn11+βn(γ--1)×[2λn∥f(p)-F(p)∥∥xn+1-p∥+2〈(I-V)p,J(xn+1-p)〉]=(1-γn)∥xn-p∥2+γnδn,
where γn=βn(γ--1) and
(142)δn=1(1+βn(γ--1))(γ--1)×[2λn∥f(p)-F(p)∥∥xn+1-p∥+2〈(I-V)p,J(xn+1-p)〉].
It can be easily seen from (137) and conditions (i), (ii) that
(143)∑n=0∞‍γn=∞,limsupn→∞δn≤0.
Applying Lemma 17 to (141), we infer that xn→p as n→∞. This completes the proof.Remark 22.
By a careful analysis of the proof of Theorem21, we reached to the conclusion that the iterative method (87) in Theorem 21 can be replaced by the following iterative method:
(144)yn=αnxn+(1-αn)JrnGxn,xn+1=(I-βnV)yn+βn[yn-λn(F(yn)-f(yn))],∀n≥0.Remark 23.
Theorem21 improves and extends [15, Theorem 3.1], [10, Theorem 3.1], and [13, Theorem 3.1] in the following aspects.(a)
The problem in Theorem21 is different from the one considered in [15, Theorem 3.1], [10, Theorem 3.1], and [13, Theorem 3.1]. Also, the iterative scheme in Theorem 21 is different from the one considered in [15, Theorem 3.1], [10, Theorem 3.1] and [13, Theorem 3.1].(b)
The iterative scheme in Theorem21 is more advantageous and more flexible than the iterative scheme in [13, Theorem 3.1].(c)
Theorem21 drops the assumption of the asymptotical regularity for the iterative scheme in [13, Theorem 3.1].Remark 24.
Theorem21 extends and improves [28, Theorem 3.1] to a great extent in the following aspects.(a)
Theu is replaced by a fixed contractive mapping.(b)
One continuous pseudocontractive mapping (including nonexpansive mapping) is replaced by the mappingJrn∘G.(c)
We add a strongly positive linear bounded operatorA and a strongly accretive and strictly pseudocontractive mapping F in our iterative algorithm.



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*Source: 631382-2013-11-24.xml*