# Modified Hybrid Steepest-Descent Methods for General Systems of Variational Inequalities with Solutions to Zeros ofm-Accretive Operators in Banach Spaces

Authors: Lu-Chuan Ceng; Ching-Feng Wen
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

The purpose of this paper is to introduce and analyze modified hybrid steepest-descent methods for a general system of variational inequalities (GSVI), with solutions being
also zeros of anm-accretive operator A in the setting of real uniformly convex and 2-uniformly
smooth Banach space X. Here the modified hybrid steepest-descent methods are based on
Korpelevich's extragradient method, hybrid steepest-descent method, and viscosity approximation method. We propose and consider modified implicit and explicit hybrid steepest-descent
algorithms for finding a common element of the solution set of the GSVI and the set A-1(0) of zeros of A in X. 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 literature.


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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 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 → 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
(2)〈Ax-Ay,j(x-y)〉≥0;(ii)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);(iii)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 someβ>0;(iv)λ-strictly pseudocontractive [1] (see also [2]) 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).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, [3–5]. Let U={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 strictly 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 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:
(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 [6], 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.LetA be an operator with domain D(A) and range R(A) in X is said to be accretive if for each xi∈D(A) and yi∈Axi(i=1,2) there exists j(x2-x1)∈J(x2-x1) such that
(9)〈y2-y1,j(x2-x1)〉≥0.
An accretive operator A is said to be m-accretive if R(I+λA)=X for all λ>0. Denote by Jr the resolvent of A; that is, for each r>0,
(10)Jr=(I+rA)-1.
It is known that Jr is a nonexpansive mapping from X to C:=D(A)¯ which will be assumed convex. In 2008, Chen and Zhu [7] derived the following strong convergence theorems for viscosity approximation methods for accretive operators in a uniformly smooth Banach space X.Theorem  CZ1.LetXbe a uniformly smooth Banach space. Suppose thatAis anm-accretive operator inXsuch thatC=D(A)¯is convex andf:C→Cis a fixed contractive map. For eacht∈(0,1), {xt,n}is defined by(11)xt,n=tf(xt,n)+(1-t)Jrnxt,n.Then ast→0, {xt,n}converges strongly to a zero ofA.Theorem  CZ2.LetXbe a uniformly smooth Banach space. Suppose thatAis anm-accretive operator inXsuch thatC=D(A)¯is convex andf:C→Cis a fixed contractive map. The sequence{xn}is defined by(12)xn+1=αnf(xn)+(1-αn)Jrnxn,∀n≥0,where{αn}and{rn}satisfy the following conditions:(i)
αn→0,∑n=0∞‍αn=∞, and∑n=0∞‍|αn+1-αn|<∞,(ii)
rn≥εfor allnand∑n=0∞‍|rn+1-rn|<∞.Then as
n
→
∞
,
{
x
n
}
converges strongly to a zero of
A
.In the meantime, Ceng et al. [8] derived some strong convergence theorems of composite iterative schemes for zeros of m-accretive operators in uniformly smooth Banach spaces. Furthermore, motivated by strong convergence results for hybrid steepest-descent methods in [9, 10], Ceng et al. [11] established some strong convergence theorems for hybrid steepest-descent methods for nonexpansive and m-accretive operators in a uniformly smooth Banach space X. Subsequently, Ceng et al. [12] introduced hybrid viscosity approximation method for finding zeros of m-accretive operators, which combine viscosity approximation method with hybrid steepest-descent method, and obtained the following strong convergence theorems.Theorem  CASY1.LetXbe a uniformly smooth Banach space, letAbe anm-accretive operator inXwithA-1(0)≠∅, and letf:X→C(=D(A)¯)be a contractive map. Assume thatF:X→Xisδ-strongly accretive andλ-strictly pseudocontractive withδ+λ>1. For eacht∈(0,1)and each integern≥0, let{xt,n}be defined by(13)xt,n=tf(xt,n)+(1-t)[Jrnxt,n-θtF(xt,n)],where{rn}⊂[ε,∞)for someε>0and{θt:t∈(0,1)}⊂[0,1)withlimt→0(θt/t)=0. Then ast→0,{xt,n}converges strongly to a zeropofA, which is a unique solution of the variational inequality problem (VIP)(14)〈(I-f)p,J(p-u)〉≤0,∀u∈A-1(0).Theorem  CASY2.LetXbe a uniformly smooth Banach space, letAbe anm-accretive operator inXwithA-1(0)≠∅, and letf:X→C(=D(A)¯)be a contractive map. Assume thatF:X→Xisδ-strongly accretive andλ-strictly pseudocontractive withδ+λ>1. Given sequences{λn}n=0∞,{μn}n=0∞in[0,1],{αn}n=0∞,{βn}n=0∞in(0,1], and{rn}n=0∞in[ε,∞)for someε>0, suppose that there hold the following conditions:(i)
limn→∞βn=0and∑n=0∞‍βn=∞,(ii)
limn→∞(λnμn)/βn=0,(iii)
{αn}⊂[a,b]for somea,b∈(0,1),(iv)
∑n=0∞‍|αn+1-αn|<∞,∑n=0∞‍|βn+1-βn|<∞,∑n=0∞‍|λn+1-λn|<∞,∑n=0∞‍|μn+1-μn|<∞and∑n=0∞‍|rn+1-rn|<∞.Then for any given point
x
0
∈
X
, the sequence
{
x
n
}
generated by
(15)
y
n
=
α
n
x
n
+
(
1
-
α
n
)
J
r
n
x
n
,
x
n
+
1
=
β
n
f
(
x
n
)
+
(
1
-
β
n
)
[
J
r
n
y
n
-
λ
n
μ
n
F
(
J
r
n
y
n
)
]
,
∀
n
≥
0
,
converges strongly to a zero
p
of
A
, which is a unique solution of the VIP as above.On the other hand, Cai and Bu [13] 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
(16)〈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 (16) is denoted by GSVI (C,B1,B2). In particular, if X=H, a real Hilbert space, then GSVI (16) reduces to the following GSVI of finding (x*,y*)∈C×C such that
(17)〈μ1B1y*+x*-y*,x-x*〉≥0,∀x∈C,〈μ2B2x*+y*-x*,x-y*〉≥0,∀x∈C,
where μ1 and μ2 are two positive constants. The set of solutions of problem (17) is still denoted by GSVI (C,B1,B2). In particular, if B1=B2=A, then problem (17) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [14]. Further, if x*=y* additionally, then the NSVI reduces to the classical variational inequality problem (VIP) of finding x*∈C such that
(18)〈Ax*,x-x*〉≥0,∀x∈C.
The solution set of the VIP (18) is denoted by VI (C,A). Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, equilibrium problems. It is now well known that the variational inequalities are equivalent to the fixed point problems, the origin of which can be traced back to Lions and Stampacchia [15]. This alternative formulation has been used to suggest and analyze projection iterative method for solving variational inequalities under the condition that the involved operator must be strongly monotone and Lipschitz continuous.Recently, Ceng et al. [16] transformed problem (17) into a fixed point problem in the following way.Lemma 1 (see [16]).
For givenx-,y-∈C,(x-,y-) is a solution of problem (17) if and only if x- is a fixed point of the mapping G:C→C defined by
(19)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 mappingBi: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.In 1976, Korpelevič [17] proposed an iterative algorithm for solving the VIP (18) in Euclidean space Rn:
(20)yn=PC(xn-τAxn),xn+1=PC(xn-τAyn),∀n≥0,
with τ>0 being a given number, which is known as the extragradient method (see also [18]). The literature on the VIP is vast, and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [3, 13, 19–32] and references therein, to name but a few.In particular, wheneverX is still a real smooth Banach space, B1=B2=A, and x*=y*, then GSVI (16) reduces to the variational inequality problem (VIP) of finding x*∈C such that
(21)〈Ax*,J(x-x*)〉≥0,∀x∈C,
which was considered by Aoyama et al. [33]. Note that VIP (21) is connected with the fixed point problem for nonlinear mapping (see, e.g., [34]), the problem of finding a zero point of a nonlinear operator (see, e.g., [35]), and so on. It is clear that VIP (21) extends VIP (18) from Hilbert spaces to Banach spaces.In order to find a solution of VIP (21), Aoyama et al. [33] introduced the following iterative scheme for an accretive operator A:
(22)xn+1=αnxn+(1-αn)ΠC(xn-λnAxn),∀n≥1,
where ΠC is a sunny nonexpansive retraction from X onto C. Then they proved a weak convergence theorem.Beyond doubt, it is an interesting and valuable problem of constructing some algorithms with strong convergence for solving GSVI (16) which contains VIP (21) as a special case. Very recently, Cai and Bu [13] constructed an iterative algorithm for solving GSVI (16) 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 2 (see [36]).
LetX be a 2-uniformly smooth Banach space. Then
(23)∥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*.Define the mappingG:C→C as follows:
(24)G(x):=ΠC(I-μ1B1)ΠC(I-μ2B2)x,∀x∈C.
The fixed point set of G is denoted by Ω. Then their strong convergence theorem on the proposed method is stated as follows.Theorem  CB (see [13, Theorem 3.1]). LetCbe a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach spaceX. LetΠCbe a sunny nonexpansive retraction fromXontoC. Let the mappingBi:C→Xbeαi-inverse strongly accretive with0<μi<αi/κ2fori=1,2. Letfbe a contraction ofCinto itself with coefficientρ∈(0,1). Let{Sn}n=1∞be an infinite family of nonexpansive mappings ofCinto itself such thatΔ=⋂i=1∞‍Fix(Si)∩Ω≠∅, whereΩis the fixed point set of the mappingGdefined by (24). For arbitrarily givenx1∈C, let{xn}be the sequence generated by(25)xn+1=βnxn+(1-βn)Snyn,yn=αnf(xn)+(1-αn)zn,zn=ΠC(un-μ1B1un),un=ΠC(xn-μ2B2xn),∀n≥1.Suppose that{αn}and{βn}are two sequences in(0,1)satisfying the following conditions:(i)
limn→∞αn=0and∑n=1∞‍αn=∞,(ii)
0<liminfn→∞βn≤limsupn→∞βn<1.Assume that
∑
n
=
1
∞
‍
sup
x
∈
D
∥
S
n
+
1
x
-
S
n
x
∥
<
∞
for any bounded subset
D
of
C
, let
S
be a mapping of
C
into
X
defined by
S
x
=
lim
n
→
∞
S
n
x
for all
x
∈
C
, and suppose that
Fix
(
S
)
=
⋂
n
=
1
∞
‍
Fix
(
S
n
)
. Then
{
x
n
}
converges strongly to
p
∈
Δ
, which solves the VIP:
(26)
〈
(
I
-
f
)
p
,
J
(
p
-
u
)
〉
≤
0
,
∀
u
∈
Δ
.It is easy to see that the iterative scheme in Theorem CB is essentially equivalent to the following two-step iterative scheme:
(27)
y
n
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
G
x
n
,
x
n
+
1
=
β
n
x
n
+
(
1
-
β
n
)
S
n
y
n
,
∀
n
≥
1
.For the convenience of implementing the argument techniques in [16], the authors [13] have used the following inequality in a real smooth and uniform convex Banach space
X.Proposition 3 (see [37]).
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
(28)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 uniformly convex and 2-uniformly 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). Motivated and inspired by the research going on in this area, we introduce and analyze modified hybrid steepest-descent methods for the GSVI (16), with solutions being also zeros of an m-accretive operator A in X. Here the modified hybrid steepest-descent methods are based on Korpelevich’s extragradient method, hybrid steepest-descent method, and viscosity approximation method. We propose and consider modified implicit and explicit hybrid steepest-descent algorithms for finding a common element of the solution set of the GSVI (16) and the set A-1(0) of zeros of A in X. 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 literature [12, 13, 16, 32].


## 2. Preliminaries

We list some lemmas that will be used in the sequel. Lemma4 can be found in [38]. Lemma 5 is an immediate consequence of the subdifferential inequality of the function (1/2)∥·∥2.Lemma 4.
Let{an}be a sequence of nonnegative real numbers such that
(29)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)
eitherlimsupn→∞cn≤0 or ∑n=0∞‍|bncn|<∞.
Then,limn→∞an=0.Lemma 5.
In a real smooth Banach spaceX, there holds the inequality
(30)∥x∥2+2〈y,J(x)〉≤∥x+y∥2≤∥x∥2+2〈y,J(x+y)〉,∀x,y∈X,
where J:X→X* is the normalized duality mapping.LetD be a subset of C and let Π be a mapping of C into D. Then Π is said to be sunny if
(31)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 ofCinto 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 6 (see [39]).
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))〉,∀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 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. The following lemma follows easily from Lemma 6.Lemma 7.
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 (16) if and only if x*=ΠC(y*-μ1B1y*), where y*=ΠC(x*-μ2B2x*).In terms of Lemma7, 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.Lemma8 is the resolvent identity which can be found in [40], and Lemma 9 can be found in [41].Lemma 8.
Forλ,μ>0, there holds the identity
(33)Jλx=Jμ(μλx+(1-μλ)Jλx),∀x∈X.Lemma 9.
Assume thatc2≥c1>0. Then ∥Jc1x-x∥≤2∥Jc2x-x∥ for all x∈X.Lemma 10 (see [42]).
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 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.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
(34)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
(35)μn(an)=μn(an+1)
for every a=(a1,a2,…)∈l∞. We know that if μ is a Banach limit, then
(36)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
(37)μn(an)=μ(a)=c(resp.,μn(an)=μn(bn)).Further, it is well known that there holds the following result.Lemma 11 (see [43]).
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
(38)μn∥xn-z∥2=miny∈Cμn∥xn-y∥2
if and only if
(39)μn〈y-z,J(xn-z)〉≤0,∀y∈C,
where J is the normalized duality mapping of X.LetC be a nonempty closed convex subset of a Banach space X and let T:C→C be a nonexpansive mapping with Fix(T)≠∅. Let ΞC be the set of all contractive self-mappings 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,
(40)xt=tf(xt)+(1-t)Txt.Lemma 12 (see [34]).
LetX be a uniformly smooth Banach space. Let C be a nonempty closed convex subset of X, let T:C→C be a nonexpansive mapping with Fix(T)≠∅, and let f∈ΞC. Then the net {xt} defined by xt=tf(xt)+(1-t)Txt converges strongly to a point in Fix(T). If one defines a mapping Q:ΞC→Fix(T) by Q(f):=s-limt→0xt,for all f∈ΞC, then Q(f) solves the VIP:
(41)〈(I-f)Q(f),J(Q(f)-u)〉≤0,∀f∈ΞC,u∈Fix(T).The following proposition will be used frequently throughout this paper.Proposition 13 (see [11]).
LetX be a real smooth Banach space and let F:X→X be a mapping.(i)
IfFisλ-strictly pseudocontractive, thenFis Lipschitzian with constant1+1/λ.(ii)
IfFisδ-strongly accretive andλ-strictly pseudocontractive withδ+λ>1, thenI-Fis contractive with constant(1-δ)/λ.(iii)
IfFisδ-strongly accretive andλ-strictly pseudocontractive withδ+λ>1, then, for any fixed numberτ∈(0,1),I-τFis contractive with constant1-τ(1-(1-δ)/λ).


## 3. Main Results

In this section, we introduce our modified hybrid steepest-descent schemes and show the strong convergence theorems. We will need the following useful lemmas in the sequel.Lemma 14 (see [13, 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
(42)∥(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 15 (see [13, 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
(43)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.LetX be a real smooth Banach space and let A be an m-accretive operator in Xsuch that C=D(A)¯ is convex. Let ΠC be a sunny nonexpansive retraction from X onto C. Let F:X→X be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1. Let the mapping Bi:C→X be αi-inverse strongly accretive for i=1,2 and let f:X→X be a contractive map with coefficient ρ∈(0,1). In this section, we will consider the problem of finding a point p∈Δ=A-1(0)∩Ω(≠∅), which is a unique solution of the VIP:
(44)〈(I-f)p,J(p-u)〉≤0,∀u∈Δ,
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. For each t∈(0,1) and each integer n≥0, we choose a number θt∈[0,1) arbitrarily and then consider the following mapping Γt,n:X→X defined as
(45)Γt,nx=tf(x)+(1-t)[G(Jrnx)-θtFG(Jrnx)],∀x∈X.
Then, Γt,n:X→X is a contractive map. Indeed, utilizing Proposition 13(iii) and Lemma 15, we have for all x,y∈X(46)∥Γt,nx-Γt,ny∥=∥tf(x)+(1-t)[G(Jrnx)-θtFG(Jrnx)]-tf(y)-(1-t)[G(Jrny)-θtFG(Jrny)]∥≤t∥f(x)-f(y)∥+(1-t)∥(I-θtF)G(Jrnx)-(I-θtF)G(Jrny)∥≤tρ∥x-y∥+(1-t)(1-θt(1-1-δλ))×∥G(Jrnx)-G(Jrny)∥≤tρ∥x-y∥+(1-t)(1-θt(1-1-δλ))×∥Jrnx-Jrny∥≤tρ∥x-y∥+(1-t)(1-θt(1-1-δλ))∥x-y∥≤tρ∥x-y∥+(1-t)∥x-y∥=(1-(1-ρ)t)∥x-y∥,
and hence Γt,n:X→X is contractive due to (1-ρ)t∈(0,1). By Banach’s Contraction Mapping Principle, there exists a unique fixed point xt,n of  Γt,n in X; that is,
(47)xt,n=tf(xt,n)+(1-t)[G(Jrnxt,n)-θtFG(Jrnxt,n)].Theorem 16.
LetX be a uniformly convex and 2-uniformly smooth Banach space and let A be an m-accretive operator in X such that C=D(A)¯ is convex. 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:X→C be a contractive map with coefficient ρ∈(0,1), and let F:X→X be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1. Assume that Δ=A-1(0)∩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. For each t∈(0,1) and each integer n≥0, let {xt,n} be defined by (47), where {rn}⊂[ε,∞) for some ε>0 and {θt:t∈(0,1)}⊂[0,1) with limt→0(θt/t)=0. Then as t→0,{xt,n} converges strongly to p∈Δ, which is a unique solution of the VIP (44).Proof.
First let us show that for somea∈(0,1),{xt,n:t∈(0,a],n≥0} is bounded. Indeed, since {θt:t∈(0,1)}⊂[0,1) with limt→0(θt/t)=0, there exists some a∈(0,1) such that0≤θt/t<1 for all t∈(0,a].
Takep∈Δ. Then utilizing Proposition 13, we have
(48)∥xt,n-p∥≤t∥f(xt,n)-p∥+(1-t)×∥[G(Jrnxt,n)-θtFG(Jrnxt,n)]-p∥≤tρ∥xt,n-p∥+t∥f(p)-p∥+(1-t)×[∥(I-θtF)G(Jrnxt,n)-(I-θtF)G(Jrnp)∥+∥(I-θtF)G(Jrnp)-p∥]≤tρ∥xt,n-p∥+t∥f(p)-p∥+(1-t)×(1-θt(1-1-δλ))∥G(Jrnxt,n)-G(Jrnp)∥+(1-t)θt∥F(p)∥≤tρ∥xt,n-p∥+t∥f(p)-p∥+(1-t)×(1-θt(1-1-δλ))∥Jrnxt,n-Jrnp∥+θt∥F(p)∥≤tρ∥xt,n-p∥+t∥f(p)-p∥+(1-t)×(1-θt(1-1-δλ))∥xt,n-p∥+θt∥F(p)∥≤tρ∥xt,n-p∥+t∥f(p)-p∥+(1-t)∥xt,n-p∥+θt∥F(p)∥,
and, hence, for all t∈(0,a](49)∥xt,n-p∥≤11-ρ(∥f(p)-p∥+θtt∥F(p)∥)≤11-ρ(∥f(p)-p∥+∥F(p)∥).
Thus, this implies that {xt,n:t∈(0,a],n≥0} is bounded and so are {f(xt,n):t∈(0,a],n≥0}, {Jrnxt,n:t∈(0,a],n≥0}, and {G(Jrnxt,n):t∈(0,a],n≥0}.
Let us show that∥Jrnxt,n-G(Jrnxt,n)∥→0 as t→0.
Indeed, for simplicity we putq=ΠC(I-μ2B2)p, x^t,n=Jrnxt,n, ut,n=ΠC(I-μ2B2)x^t,n, and vt,n=ΠC(I-μ1B1)ut,n. Then it is clear that p=ΠC(I-μ1B1)q and vt,n=G(x^t,n)=G(Jrnxt,n). Taking into account xt,n-p=t(f(xt,n)-p)+(1-t)[G(Jrnxt,n)-θtFG(Jrnxt,n)-p], we get
(50)∥xt,n-p∥2=t〈f(xt,n)-p,J(xt,n-p)〉+(1-t)〈G(Jrnxt,n)-p,J(xt,n-p)〉-(1-t)θt〈FG(Jrnxt,n),J(xt,n-p)〉≤t〈f(xt,n)-f(p),J(xt,n-p)〉+t〈f(p)-p,J(xt,n-p)〉+(1-t)∥G(Jrnxt,n)-p∥∥J(xt,n-p)∥+(1-t)θt∥FG(Jrnxt,n)∥∥J(xt,n-p)∥≤ρt∥xt,n-p∥2+t∥f(p)-p∥∥xt,n-p∥+(1-t)∥vt,n-p∥∥xt,n-p∥+θt∥Fvt,n∥∥xt,n-p∥.
From Lemma 14 we have
(51)∥ut,n-q∥2=∥ΠC(x^t,n-μ2B2x^t,n)-ΠC(p-μ2B2p)∥2≤∥x^t,n-p-μ2(B2x^t,n-B2p)∥2≤∥x^t,n-p∥2-2μ2(α2-κ2μ2)∥B2x^t,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
(52)∥vt,n-p∥2≤∥x^t,n-p∥2-2μ2(α2-κ2μ2)∥B2x^t,n-B2p∥2-2μ1(α1-κ2μ1)∥B1ut,n-B1q∥2≤∥xt,n-p∥2-2μ2(α2-κ2μ2)∥B2x^t,n-B2p∥2-2μ1(α1-κ2μ1)∥B1ut,n-B1q∥2,
which, together with (50), implies that
(53)∥xt,n-p∥2≤ρt∥xt,n-p∥2+t∥f(p)-p∥∥xt,n-p∥+(1-t)∥vt,n-p∥∥xt,n-p∥+θt∥Fvt,n∥∥xt,n-p∥≤ρt∥xt,n-p∥2+t∥f(p)-p∥∥xt,n-p∥+(1-t)12(∥xt,n-p∥2+∥vt,n-p∥2)+θt∥Fvt,n∥∥xt,n-p∥≤ρt∥xt,n-p∥2+t∥f(p)-p∥∥xt,n-p∥+(1-t)12{∥xt,n-p∥2+∥xt,n-p∥2-2μ2(α2-κ2μ2)∥B2x^t,n-B2p∥2-2μ1(α1-κ2μ1)∥B1ut,n-B1q∥2}+θt∥Fvt,n∥∥xt,n-p∥=(1-t(1-ρ))∥xt,n-p∥2+t∥f(p)-p∥∥xt,n-p∥-(1-t)[μ2(α2-κ2μ2)∥B2x^t,n-B2p∥2+μ1(α1-κ2μ1)∥B1ut,n-B1q∥2]+θt∥Fvt,n∥∥xt,n-p∥≤∥xt,n-p∥2+t∥f(p)-p∥∥xt,n-p∥-(1-t)[μ2(α2-κ2μ2)∥B2x^t,n-B2p∥2+μ1(α1-κ2μ1)∥B1ut,n-B1q∥2]+θt∥Fvt,n∥∥xt,n-p∥.
So it immediately follows that
(54)(1-t)[μ2(α2-κ2μ2)∥B2x^t,n-B2p∥2+μ1(α1-κ2μ1)∥B1ut,n-B1q∥2]≤t∥f(p)-p∥∥xt,n-p∥+θt∥Fvt,n∥∥xt,n-p∥.
Since 0<μi<αi/κ2 for i=1,2, we have
(55)limt→0∥B2x^t,n-B2p∥=0,limt→0∥B1ut,n-B1q∥=0.
Utilizing Proposition 3 and Lemma 6, we have
(56)∥ut,n-q∥2=∥ΠC(x^t,n-μ2B2x^t,n)-ΠC(p-μ2B2p)∥2≤〈x^t,n-μ2B2x^t,n-(p-μ2B2p),J(ut,n-q)〉=〈x^t,n-p,J(ut,n-q)〉+μ2〈B2p-B2x^t,n,J(ut,n-q)〉≤12[∥x^t,n-p∥2+∥ut,n-q∥2-g1(∥x^t,n-ut,n-(p-q)∥)∥x^t,n-p∥2]+μ2∥B2p-B2x^t,n∥∥ut,n-q∥,
which implies that
(57)∥ut,n-q∥2≤∥x^t,n-p∥2-g1(∥x^t,n-ut,n-(p-q)∥)+2μ2∥B2p-B2x^t,n∥∥ut,n-q∥.
In the same way, we derive
(58)∥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
(59)∥vt,n-p∥2≤∥ut,n-q∥2-g2(∥ut,n-vt,n+(p-q)∥)+2μ1∥B1q-B1ut,n∥∥vt,n-p∥.
Substituting (57) for (59), we get
(60)∥vt,n-p∥2≤∥x^t,n-p∥2-g1(∥x^t,n-ut,n-(p-q)∥)-g2(∥ut,n-vt,n+(p-q)∥)+2μ2∥B2p-B2x^t,n∥∥ut,n-q∥+2μ1∥B1q-B1ut,n∥∥vt,n-p∥≤∥xt,n-p∥2-g1(∥x^t,n-ut,n-(p-q)∥)-g2(∥ut,n-vt,n+(p-q)∥)+2μ2∥B2p-B2x^t,n∥∥ut,n-q∥+2μ1∥B1q-B1ut,n∥∥vt,n-p∥,
which, together with (50), implies that
(61)∥xt,n-p∥2≤ρt∥xt,n-p∥2+t∥f(p)-p∥∥xt,n-p∥+(1-t)12(∥xt,n-p∥2+∥vt,n-p∥2)+θt∥Fvt,n∥∥xt,n-p∥≤ρt∥xt,n-p∥2+t∥f(p)-p∥∥xt,n-p∥+(1-t)12{∥xt,n-p∥2+∥xt,n-p∥2-g1(∥x^t,n-ut,n-(p-q)∥)-g2(∥ut,n-vt,n+(p-q)∥)+2μ2∥B2p-B2x^t,n∥∥ut,n-q∥+2μ1∥B1q-B1ut,n∥∥vt,n-p∥∥xt,n-p∥2}+θt∥Fvt,n∥∥xt,n-p∥=(1-t(1-ρ))∥xt,n-p∥2+t∥f(p)-p∥∥xt,n-p∥-(1-t)12[g1(∥x^t,n-ut,n-(p-q)∥)+g2(∥ut,n-vt,n+(p-q)∥)]+(1-t)[μ2∥B2p-B2x^t,n∥∥ut,n-q∥+μ1∥B1q-B1ut,n∥∥vt,n-p∥]+θt∥Fvt,n∥∥xt,n-p∥≤∥xt,n-p∥2+t∥f(p)-p∥∥xt,n-p∥-(1-t)12[g1(∥x^t,n-ut,n-(p-q)∥)+g2(∥ut,n-vt,n+(p-q)∥)]+μ2∥B2p-B2x^t,n∥∥ut,n-q∥+μ1∥B1q-B1ut,n∥∥vt,n-p∥+θt∥Fvt,n∥∥xt,n-p∥.
So it immediately follows that
(62)(1-t)12[g1(∥x^t,n-ut,n-(p-q)∥)+g2(∥ut,n-vt,n+(p-q)∥)]≤t∥f(p)-p∥∥xt,n-p∥+μ2∥B2p-B2x^t,n∥∥ut,n-q∥+μ1∥B1q-B1ut,n∥∥vt,n-p∥+θt∥Fvt,n∥∥xt,n-p∥.
Hence, from (55) we conclude that
(63)limt→0g1(∥x^t,n-ut,n-(p-q)∥)=0,limt→0g2(∥ut,n-vt,n+(p-q)∥)=0.
Utilizing the properties of g1 and g2, we get
(64)limt→0∥x^t,n-ut,n-(p-q)∥=0,limt→0∥ut,n-vt,n+(p-q)∥=0,
which leads to
(65)∥x^t,n-vt,n∥≤∥x^t,n-ut,n-(p-q)∥+∥ut,n-vt,n+(p-q)∥⟶0ast⟶0.
That is,
(66)limt→0∥Jrnxt,n-G(Jrnxt,n)∥=limt→0∥x^t,n-vt,n∥=0.
Note thatFix(Jrn)=A-1(0) for all n≥0 and that {xt,n:t∈(0,a],n≥0} is bounded and so are {f(xt,n):t∈(0,a],n≥0}, {Jrnxt,n:t∈(0,a],n≥0}, and {G(Jrnxt,n):t∈(0,a],n≥0}. Hence, we have
(67)∥xt,n-G(Jrnxt,n)∥=t∥f(xt,n)-G(Jrnxt,n)-(1-t)θttFG(Jrnxt,n)∥⟶0,
as t→0. Also, observe that
(68)∥xt,n-Jrnxt,n∥≤∥xt,n-G(Jrnxt,n)∥+∥G(Jrnxt,n)-Jrnxt,n∥.
This, together with (66) and (67), implies that
(69)limt→0∥xt,n-Jrnxt,n∥=0.
Utilizing the nonexpansivity of G, we obtain from (67) and (69) that
(70)∥xt,n-Gxt,n∥≤∥xt,n-G(Jrnxt,n)∥+∥G(Jrnxt,n)-Gxt,n∥≤∥xt,n-G(Jrnxt,n)∥+∥Jrnxt,n-xt,n∥⟶0,ast⟶0.
Since rn≥ε for all n, utilizing Lemma 9 we have
(71)∥xt,n-Jεxt,n∥≤2∥xt,n-Jrnxt,n∥⟶0,ast⟶0.
For any integern≥0, for simplicity put wt=xt,n for all t∈(0,a]. Now let {tk} be a sequence in (0,a] that converges to 0 as k→∞ and define a function g on C by
(72)g(w)=μk12∥wtk-w∥2,∀w∈C,
where μ is a Banach limit. Define the set
(73)K:={w∈C:g(w)=min{g(y):y∈C}}
and the mapping
(74)Wx=(1-θ)Jεx+θGx,∀x∈C,
where θ is a constant in (0,1). Then by Lemma 10, we know that 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 (70) 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; for more details, see [43]. We claim that K is also invariant under the nonexpansive mapping W. Indeed, noticing (76), we have for 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 11, we get
(78)μk〈x-p,J(wtk-p)〉≤0,∀x∈C.
Putting x=f(p) we have
(79)μk〈f(p)-p,J(wtk-p)〉≤0,∀x∈C.
Since wtk-p=tk(f(wtk)-p)+(1-tk)[G(Jrnwtk)-θtkFG(Jrnwtk)-p], we get
(80)∥wtk-p∥2=tk〈f(wtk)-p,J(wtk-p)〉+(1-tk)〈G(Jrnwtk)-p,J(wtk-p)〉-(1-tk)θtk〈FG(Jrnwtk),J(wtk-p)〉≤tk〈f(wtk)-f(p),J(wtk-p)〉+tk〈f(p)-p,J(wtk-p)〉+(1-tk)∥G(Jrnwtk)-p∥∥J(wtk-p)∥+(1-tk)θtk∥FG(Jrnwtk)∥∥J(wtk-p)∥≤ρtk∥wtk-p∥2+tk〈f(p)-p,J(wtk-p)〉+(1-tk)∥wtk-p∥2+θtk∥FG(Jrnwtk)∥∥wtk-p∥.
It follows that
(81)∥wtk-p∥2≤11-ρ[θtktk〈f(p)-p,J(wtk-p)〉+θtktk∥FG(Jrnwtk)∥∥wtk-p∥].
Since limk→∞(θtk/tk)=0, from (79) and the boundedness of sequences {FG(Jrnwtk)},{wtk}, it follows that
(82)μk∥wtk-p∥2≤11-ρμk[θtktk〈f(p)-p,J(wtk-p)〉+θtktk∥FG(Jrnwtk)∥∥wtk-p∥]=11-ρ[θtktkμk〈f(p)-p,J(wtk-p)〉+μk(θtktk∥FG(Jrnwtk)∥∥wtk-p∥)]≤0.
Therefore, it is known that for any sequence {wtk} in {wt:t∈(0,a]} there exists a subsequence which is still denoted by {wtk} that converges strongly to some fixed point p of W. To prove that the net {wt:t∈(0,a]} converges strongly to p as t→0, suppose that there exists another subsequence {wsk}⊂{wt} such that wsk→q as sk→0, and then we also have q∈Fix(W)=A-1(0)∩Ω=:Δ due to (76).
Since the sets{wt-u:t∈(0,a]} and {wt-f(wt):t∈(0,a]} are bounded and the duality map J is single-valued and norm-to-norm uniformly continuous on bounded sets of uniformly smooth Banach space X, for any u∈Δ=A-1(0)∩Ω, from wsk→q  (sk→0) we obtain
(83)∥(I-f)wsk-(I-f)q∥⟶0(sk→0),|〈wsk-f(wsk),J(wsk-u)〉-〈(I-f)q,J(q-u)〉|=|〈(I-f)wsk-(I-f)q,J(wsk-u)〉+〈(I-f)q,J(wsk-u)-J(q-u)〉|≤∥(I-f)wsk-(I-f)q∥∥wsk-u∥+|〈(I-f)q,J(wsk-u)-J(q-u)〉|⟶0,
as sk→0. Therefore,
(84)〈(I-f)q,J(q-u)〉=limsk→0〈wsk-f(wsk),J(wsk-u)〉.
Since u∈Δ=A-1(0)∩Ω,
(85)∥xt,n-[(1-t)u+tf(xt,n)]∥2=∥(1-t)G(Jrnxt,n)+tf(xt,n)-(1-t)u-tf(xt,n)-(1-t)θtFG(Jrnxt,n)∥2≤[(1-t)∥G(Jrnxt,n)-u∥+(1-t)θt∥FG(Jrnxt,n)∥]2≤[(1-t)∥xt,n-u∥+(1-t)θt∥FG(Jrnxt,n)∥]2=(1-t)2∥xt,n-u∥2+2(1-t)2θt∥xt,n-u∥∥FG(Jrnxt,n)∥+(1-t)2θt2∥FG(Jrnxt,n)∥2.
Utilizing Lemma 5, we have
(86)∥xt,n-[(1-t)u+tf(xt,n)]∥2=∥(1-t)(xt,n-u)+t(xt,n-f(xt,n))∥2≥(1-t)2∥xt,n-u∥2+2t(1-t)〈xt,n-f(xt,n),J(xt,n-u)〉.
Consequently, from the last two inequalities we deduce that
(87)2t(1-t)〈xt,n-f(xt,n),J(xt,n-u)〉≤2(1-t)2θt∥xt,n-u∥∥FG(Jrnxt,n)∥+(1-t)2θt2∥FG(Jrnxt,n)∥2,
and hence
(88)〈xt,n-f(xt,n),J(xt,n-u)〉≤(1-t)θtt∥xt,n-u∥∥FG(Jrnxt,n)∥+(1-t)θt22t∥FG(Jrnxt,n)∥2.
Noticing (84), from limt→0(θt/t)=0 and the boundedness of sequences {FG(Jrnxt,n)},{xt,n}, we conclude that
(89)〈(I-f)q,J(q-u)〉≤0.
Interchanging p and u leads to
(90)〈(I-f)q,J(q-p)〉≤0.
Interchanging q and p leads to
(91)〈(I-f)p,J(p-q)〉≤0.
This implies that
(92)〈(q-p)-(f(q)-f(p)),J(p-q)〉≤0,
and hence
(93)∥q-p∥2≤ρ∥q-p∥2.
Taking into account ρ∈(0,1), we obtain p=q. Furthermore, by the careful analysis of the above proof, we can readily see that p is also a unique solution of the VIP:
(94)〈(I-f)p,J(p-u)〉≤0,∀u∈Δ.
This completes the proof.Remark 17.
In the assertion of Theorem16, “as t→0, {xt,n} converges strongly to p∈Δ;” this p does not depend on n. Indeed, it is known that there holds the condition that {rn}⊂[ε,∞) for some ε>0. Moreover, 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 neither n nor t.Theorem 18.
LetX be a uniformly convex and 2-uniformly smooth Banach space and let A be an m-accretive operator in X such that C=D(A)¯ is convex. 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, f:X→C be a contractive map with coefficient ρ∈(0,1), and let F:X→X be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1. Assume that Δ=A-1(0)∩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. Given sequences {λn}n=0∞,{μn}n=0∞ in [0,1], {αn}n=0∞,{βn}n=0∞ in (0,1], and {rn}n=0∞ in [ε,∞) for some ε>0, suppose that there hold the following conditions:(i)
limn→∞βn=0 and ∑n=0∞‍βn=∞,(ii)
limn→∞(λnμn)/βn=0,(iii)
{αn}⊂[a,b] for some a,b∈(0,1),(iv)
∑n=0∞‍|αn+1-αn|<∞, ∑n=0∞‍|βn+1-βn|<∞, ∑n=0∞‍|λn+1-λn|<∞, ∑n=0∞‍|μn+1-μn|<∞ and ∑n=0∞‍|rn+1-rn|<∞.
Then for any given pointx0∈X, the sequence {xn} generated by
(95)yn=αnxn+(1-αn)G(Jrnxn),xn+1=βnf(xn)+(1-βn)[G(Jrnyn)-λnμnFG(Jrnyn)],∀n≥0,
converges strongly to p∈Δ, which is a unique solution of the VIP (44).Proof.
First, let us show that{xn} is bounded. Indeed, taking a fixed u∈Δ arbitrarily, we have
(96)∥yn-u∥=∥αnxn+(1-αn)G(Jrnxn)-u∥≤αn∥xn-u∥+(1-αn)∥G(Jrnxn)-u∥≤αn∥xn-u∥+(1-αn)∥Jrnxn-u∥≤αn∥xn-u∥+(1-αn)∥xn-u∥=∥xn-u∥.
So ∥yn-u∥≤∥xn-u∥ for all n≥0. Thus, by Proposition 13(iii), we have(97)∥xn+1-u∥=∥βnf(xn)+(1-βn)[G(Jrnyn)-λnμnFG(Jrnyn)]-u∥≤βn∥f(xn)-u∥+(1-βn)×∥λn(I-μnF)G(Jrnyn)+(1-λn)G(Jrnyn)-u∥≤βnρ∥xn-u∥+βn∥f(u)-u∥+(1-βn)[λn∥(I-μnF)G(Jrnyn)-u∥+(1-λn)∥G(Jrnyn)-u∥]≤βnρ∥xn-u∥+βn∥f(u)-u∥+(1-βn)[λn∥(I-μnF)G(Jrnyn)-(I-μnF)u∥+λn∥(I-μnF)u-u∥+(1-λn)∥G(Jrnyn)-u∥]≤βnρ∥xn-u∥+βn∥f(u)-u∥+(1-βn)[λn(1-μn(1-1-δλ))∥G(Jrnyn)-u∥+λnμn∥Fu∥+(1-λn)∥G(Jrnyn)-u∥(1-μn(1-1-δλ))]=βnρ∥xn-u∥+βn∥f(u)-u∥+(1-βn)[(1-λnμn(1-1-δλ))∥G(Jrnyn)-u∥+λnμn∥Fu∥(1-λnμn(1-1-δλ))]=βnρ∥xn-u∥+βn∥f(u)-u∥+(1-βn)[(1-1-δλ)-1(1-λnμn(1-1-δλ))∥G(Jrnyn)-u∥+λnμn(1-1-δλ)(1-1-δλ)-1∥Fu∥]≤βnρ∥xn-u∥+βn∥f(u)-u∥+(1-βn)max{∥G(Jrnyn)-u∥,(1-1-δλ)-1∥Fu∥}≤βnρ∥xn-u∥+βn∥f(u)-u∥+(1-βn)max{∥yn-u∥,(1-1-δλ)-1∥Fu∥}≤βnρ∥xn-u∥+βn∥f(u)-u∥+(1-βn)max{∥xn-u∥,(1-1-δλ)-1∥Fu∥}≤βn∥f(u)-u∥+(1-(1-ρ)βn)×max{∥xn-u∥,(1-1-δλ)-1∥Fu∥}=(1-ρ)βn∥f(u)-u∥1-ρ+(1-(1-ρ)βn)×max{∥xn-u∥,(1-1-δλ)-1∥Fu∥}≤max{∥xn-u∥,∥f(u)-u∥1-ρ,(1-1-δλ)-1∥Fu∥}.By induction,
(98)∥xn-u∥≤max{∥x0-u∥,∥f(u)-u∥1-ρ,(1-1-δλ)-1∥Fu∥},∀n≥0.
Thus, {xn} is bounded and so is {yn}. Because G and Jrn are nonexpansive for all n≥0, f is contractive, and F is Lipschitzian, {Jrnxn}, {Jrnyn}, {G(Jrnxn)}, {G(Jrnyn)}, {f(xn)}, and {FG(Jrnyn)} are bounded. From conditions (i), (ii) we have
(99)∥xn+1-G(Jrnyn)∥≤βn∥f(xn)-G(Jrnyn)∥+(1-βn)∥G(Jrnyn)-λnμnFG(Jrnyn)-G(Jrnyn)∥=βn∥f(xn)-G(Jrnyn)∥+(1-βn)λnμn∥FG(Jrnyn)∥⟶0asn⟶∞.
Now, we claim that(100)∥xn+1-xn∥⟶0asn⟶∞.
In order to prove (100), we estimate ∥xn+1-xn∥ first. From (95) we have
(101)yn=αnxn+(1-αn)G(Jrnxn),yn-1=αn-1xn-1+(1-αn-1)G(Jrn-1xn-1).
Simple calculations show that
(102)yn-yn-1=(1-αn)(G(Jrnxn)-G(Jrn-1xn-1))+αn(xn-xn-1)+(xn-1-G(Jrn-1xn-1))(αn-αn-1).
It follows that
(103)∥yn-yn-1∥≤(1-αn)∥G(Jrnxn)-G(Jrn-1xn-1)∥+αn∥xn-xn-1∥+∥xn-1-G(Jrn-1xn-1)∥|αn-αn-1|≤(1-αn)∥Jrnxn-Jrn-1xn-1∥+αn∥xn-xn-1∥+∥xn-1-G(Jrn-1xn-1)∥|αn-αn-1|.
On the other hand, ifrn-1≤rn, using the resolvent identity in Lemma 8(104)Jrnxn=Jrn-1(rn-1rnxn+(1-rn-1rn)Jrnxn),
we get
(105)∥Jrnxn-Jrn-1Gn-1∥=∥Jrn-1(rn-1rnxn+(1-rn-1rn)Jrnxn)-Jrn-1xn-1∥≤rn-1rn∥xn-xn-1∥+(1-rn-1rn)∥Jrnxn-xn-1∥≤∥xn-xn-1∥+rn-rn-1rn∥Jrnxn-xn-1∥≤∥xn-xn-1∥+1ε|rn-rn-1|∥Jrnxn-xn-1∥.
If rn≤rn-1, it is easy to see that
(106)∥Jrnxn-Jrn-1xn-1∥≤∥xn-1-xn∥+1ε|rn-1-rn|∥Jrn-1xn-1-xn∥.
So combining the above cases we obtain
(107)∥Jrnxn-Jrn-1xn-1∥≤∥xn-1-xn∥+|rn-1-rn|ε×supn≥1{∥Jrnxn-xn-1∥+∥Jrn-1xn-1-xn∥},∀n≥1.
In the similar way we can derive
(108)∥Jrnyn-Jrn-1yn-1∥≤∥yn-1-yn∥+|rn-1-rn|ε×supn≥1{∥Jrnyn-yn-1∥+∥Jrn-1yn-1-yn∥},∀n≥1.
Therefore, we have
(109)∥Jrnxn-Jrn-1xn-1∥≤∥xn-1-xn∥+|rn-1-rn|M0,∥Jrnyn-Jrn-1yn-1∥≤∥yn-1-yn∥+|rn-1-rn|M0,
for all n≥1, where supn≥1{(1/ε)(∥Jrnxn-xn-1∥+∥Jrn-1xn-1-xn∥)}≤M0 and supn≥1{(1/ε)(∥Jrnyn-yn-1∥+∥Jrn-1yn-1-yn∥)}≤M0 for some M0>0.
Substituting (109) for (103), we obtain
(110)∥yn-yn-1∥≤(1-αn)∥Jrnxn-Jrn-1xn-1∥+αn∥xn-xn-1∥+∥xn-1-G(Jrn-1xn-1)∥|αn-αn-1|≤(1-αn)(∥xn-xn-1∥+|rn-rn-1|M0)+αn∥xn-xn-1∥+∥xn-1-G(Jrn-1xn-1)∥|αn-αn-1|≤∥xn-xn-1∥+M1(|rn-rn-1|+|αn-αn-1|),
where supn≥0{M0,∥xn-G(Jrnxn)∥}≤M1 for some constant M1>0.
In the meantime, it follows from (95) that
(111)xn+1=βnf(xn)+(1-βn)[G(Jrnyn)-λnμnFG(Jrnyn)],xn=βn-1f(xn-1)+(1-βn-1)×[G(Jrn-1yn-1)-λn-1μn-1FG(Jrn-1yn-1)].
Simple calculations show that
(112)xn+1-xn=(βn-βn-1)f(xn-1)+βn(f(xn)-f(xn-1))+(βn-1-βn)(I-λn-1μn-1F)G(Jrn-1yn-1)+(1-βn)[(I-λnμnF)G(Jrnyn)-(I-λn-1μn-1F)G(Jrn-1yn-1)]=(βn-βn-1)f(xn-1)+βn(f(xn)-f(xn-1))+(βn-1-βn)(I-λn-1μn-1F)G(Jrn-1yn-1)+(1-βn)[(I-λnμnF)G(Jrnyn)-(I-λnμnF)G(Jrn-1yn-1)+(λn-1μn-1-λnμn)FG(Jrn-1yn-1)].
It follows from Proposition 13 (iii) and (109) that
(113)∥xn+1-xn∥≤|βn-βn-1|∥f(xn-1)∥+βn∥f(xn)-f(xn-1)∥+|βn-1-βn|∥(I-λn-1μn-1F)G(Jrn-1yn-1)∥+(1-βn)×[∥(I-λnμnF)G(Jrnyn)-(I-λnμnF)G(Jrn-1yn-1)∥+|λn-1μn-1-λnμn|∥FG(Jrn-1yn-1)∥]≤|βn-βn-1|∥f(xn-1)∥+βnρ∥xn-xn-1∥+|βn-1-βn|∥(I-λn-1μn-1F)G(Jrn-1yn-1)∥+(1-βn)[(1-λnμn(1-1-δλ))×∥G(Jrnyn)-G(Jrn-1yn-1)∥+|λnμn-λn-1μn-1|∥FG(Jrn-1yn-1)∥(1-λnμn(1-1-δλ))]≤|βn-βn-1|∥f(xn-1)∥+βnρ∥xn-xn-1∥+|βn-1-βn|∥(I-λn-1μn-1F)G(Jrn-1yn-1)∥+(1-βn)[(1-λnμn(1-1-δλ))×∥Jrnyn-Jrn-1yn-1∥+|λnμn-λn-1μn-1|∥FG(Jrn-1yn-1)∥(1-λnμn(1-1-δλ))]≤|βn-βn-1|∥f(xn-1)∥+βnρ∥xn-xn-1∥+|βn-1-βn|∥(I-λn-1μn-1F)G(Jrn-1yn-1)∥+(1-βn)[(1-λnμn(1-1-δλ))×(∥yn-1-yn∥+|rn-1-rn|M0)+|λnμn-λn-1μn-1|∥FG(Jrn-1yn-1)∥(1-λnμn(1-1-δλ))]≤βnρ∥xn-xn-1∥+|βn-βn-1|×(∥f(xn-1)∥+∥(I-λn-1μn-1F)G(Jrn-1yn-1)∥)+(1-βn)∥yn-1-yn∥+|rn-1-rn|M0+|λnμn-λn-1μn-1|∥FG(Jrn-1yn-1)∥≤βnρ∥xn-xn-1∥+|βn-βn-1|M2+(1-βn)∥yn-1-yn∥+|rn-1-rn|M2+|λnμn-λn-1μn-1|M2=βnρ∥xn-xn-1∥+(1-βn)∥yn-yn-1∥+M2(|βn-βn-1|+|λnμn-λn-1μn-1|+|rn-rn-1|),
where supn≥0{∥f(xn)∥+∥(I-λnμnF)G(Jrnyn)∥, ∥FG(Jrnyn)∥,M0,M1}≤M2 for some M2>0.
Substituting (110) for (113), we get
(114)∥xn+1-xn∥≤βnρ∥xn-xn-1∥+(1-βn)∥yn-yn-1∥+M2(|βn-βn-1|+|λnμn-λn-1μn-1|+|rn-rn-1|)≤βnρ∥xn-xn-1∥+(1-βn)×[∥xn-xn-1∥+M1(|rn-rn-1|+|αn-αn-1|)]+M2(|βn-βn-1|+|λnμn-λn-1μn-1|+|rn-rn-1|)≤(1-(1-ρ)βn)∥xn-xn-1∥+2M2[|αn-αn-1|+|βn-βn-1|+|λn-λn-1|+|μn-μn-1|+|rn-rn-1|].
Since it follows from conditions (i) and (iv) that ∑n=0∞‍(1-ρ)βn=∞ and
(115)∑n=0∞‍2M2[|αn-αn-1|+|βn-βn-1|+|λn-λn-1|+|μn-μn-1|+|rn-rn-1|]<∞,
Lemma 4 is applicable to (114) and we obtain
(116)limn→∞∥xn+1-xn∥=0.
By condition (iii) and (95), we have
(117)∥yn-xn∥=(1-αn)∥G(Jrnxn)-xn∥≤(1-a)(∥G(Jrnxn)-G(Jrnyn)∥+∥G(Jrnyn)-xn+1∥+∥xn+1-xn∥)≤(1-a)(∥xn-yn)∥+∥G(Jrnyn)-xn+1∥+∥xn+1-xn∥∥G(Jrnyn)-xn+1∥),
which implies that
(118)∥yn-xn∥≤1-aa(∥G(Jrnyn)-xn+1∥+∥xn+1-xn∥).
This together with (99)-(100) implies that
(119)limn→∞∥xn-yn∥=0.
So we obtain
(120)∥xn-G(Jrnxn)∥≤∥xn-yn∥+∥yn-G(Jrnxn)∥≤∥xn-yn∥+αn∥xn-G(Jrnxn)∥≤∥xn-yn∥+b∥xn-G(Jrnxn)∥,
which implies that
(121)∥xn-G(Jrnxn)∥≤11-b∥xn-yn∥,
and hence
(122)limn→∞∥xn-G(Jrnxn)∥=0.
Next let us show thatlimn→∞∥xn-Jrnxn∥=0 and limn→∞∥xn-Gxn∥=0.
Indeed, for simplicity, putv=ΠC(u-μ2B2u), x^n=Jrnxn, un=ΠC(x^n-μ2B2x^n) and vn=ΠC(un-μ1B1un). Then u=ΠC(v-μ1B1v), and vn=Gx^n=G(Jrnxn) for all n≥0. It is clear from (95) that
(123)∥yn-u∥2≤αn∥xn-u∥2+(1-αn)∥G(Jrnxn)-u∥2=αn∥xn-u∥2+(1-αn)∥vn-u∥2.
Utilizing Lemma 14, we have
(124)∥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,(125)∥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 (124) for (125), we obtain
(126)∥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 (123), implies that
(127)∥yn-u∥2≤αn∥xn-u∥2+(1-αn)∥vn-u∥2≤αn∥xn-u∥2+(1-αn)×[∥xn-u∥2-2μ2(α2-κ2μ2)∥B2x^n-B2u∥2-2μ1(α1-κ2μ1)∥B1un-B1v∥2]=∥xn-u∥2-2(1-αn)×[μ2(α2-κ2μ2)∥B2x^n-B2u∥2+μ1(α1-κ2μ1)∥B1un-B1v∥2].
It immediately follows that
(128)2(1-αn)[μ2(α2-κ2μ2)∥B2x^n-B2u∥2+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 (119) and condition (iii) that
(129)limn→∞∥B2x^n-B2u∥=0,limn→∞∥B1un-B1v∥=0.
Utilizing Proposition 3 and Lemma 6, we have
(130)∥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〈B2p-B2x^n,J(un-v)〉≤12[∥x^n-u∥2+∥un-v∥2-g1(∥x^n-un-(u-v)∥)]+μ2∥B2u-B2x^n∥∥un-v∥,
which implies that
(131)∥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
(132)∥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)∥)]+μ1∥B1v-B1un∥∥vn-u∥,
which implies that
(133)∥vn-u∥2≤∥un-v∥2-g2(∥un-vn+(u-v)∥)+2μ1∥B1v-B1un∥∥vn-u∥.
Substituting (131) for (133), we get
(134)∥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 (123), implies that
(135)∥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)∥)-g2(∥un-vn+(u-v)∥)+2μ2∥B2u-B2x^n∥∥un-v∥+2μ1∥B1v-B1un∥∥vn-u∥]=∥xn-u∥2-(1-αn)×[g1(∥x^n-un-(u-v)∥)+g2(∥un-vn+(u-v)∥)∥xng1-u∥2]+2(1-αn)(μ2∥B2u-B2x^n∥∥un-v∥+μ1∥B1v-B1un∥∥vn-u∥).
It immediately follows that
(136)(1-αn)[g1(∥x^n-un-(u-v)∥)+g2(∥un-vn+(u-v)∥)]≤∥xn-u∥2-∥yn-u∥2+2(1-αn)(μ2∥B2u-B2x^n∥∥un-v∥+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 (119), (129), and condition (iii) that
(137)limn→∞g1(∥x^n-un-(u-v)∥)=0,limn→∞g2(∥un-vn+(u-v)∥)=0.
Utilizing the properties of g1 and g2, we get
(138)limn→∞∥x^n-un-(u-v)∥=0,limn→∞∥un-vn+(u-v)∥=0,
which hence yields
(139)∥x^n-vn∥≤∥x^n-un-(u-v)∥+∥un-vn+(u-v)∥⟶0asn⟶∞.
That is,
(140)limn→∞∥Jrnxn-G(Jrnxn)∥=limn→∞∥x^n-vn∥=0.
Note that
(141)∥xn-Jrnxn∥≤∥xn-G(Jrnxn)∥+∥G(Jrnxn)-Jrnxn∥.
So from (122) and (140) we have
(142)limn→∞∥xn-Jrnxn∥=0,
which, together with (122), leads to
(143)∥xn-Gxn∥≤∥xn-G(Jrnxn)∥+∥G(Jrnxn)-Gxn∥≤∥xn-G(Jrnxn)∥+∥Jrnxn-xn∥⟶0asn⟶∞.
That is,
(144)limn→∞∥xn-Gxn∥=0.
In addition, utilizing Lemma 9 we obtain from {rn}⊂[ε,∞) that
(145)∥xn-Jεxn∥≤2∥xn-Jrnxn∥,
which, together with (142), implies that
(146)limn→∞∥xn-Jεxn∥=0.
Define a mapping(147)Wx=(1-θ)Jεx+θGx,∀x∈C,
where θ is a constant in (0,1). Then by Lemma 10, we know that Fix(W)=Fix(Jε)∩Fix(G)=Δ. We observe that
(148)∥xn-Wxn∥=∥(1-θ)(xn-Jεxn)+θ(xn-Gxn)∥≤(1-θ)∥xn-Jεxn∥+θ∥xn-Gxn∥.
So from (144) and (146) we obtain
(149)limn→∞∥xn-Wxn∥=0.
Now, we claim that(150)limsupn→∞〈f(p)-p,J(xn-p)〉≤0,
where p=s-limt→0xt, with xt being the fixed point of the contraction x↦tf(x)+(1-t)Wx of C into itself (due to Lemma 12). Then xt solves the fixed point equation xt=tf(xt)+(1-t)Wxt. Thus we have
(151)∥xt-xn∥=∥(1-t)(Wxt-xn)+t(f(xt)-xn)∥.
By Lemma 5 we conclude that
(152)∥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
(153)fn(t)=(1-t)2(2∥xt-xn∥+∥xn-Wxn∥)×∥xn-Wxn∥⟶0,asn⟶∞.
It follows from (152) that
(154)〈xt-f(xt),J(xt-xn)〉≤t2∥xt-xn∥2+12tfn(t).
Letting n→∞ in (154) and noticing (153), we derive
(155)limsupn→∞〈xt-f(xt),J(xt-xn)〉≤t2M3,
where M3>0 is a constant such that ∥xt-xn∥2≤M3 for all t∈(0,1) and n≥0. Taking t→0 in (155), we have
(156)limsupt→0limsupn→∞〈xt-f(xt),J(xt-xn)〉≤0.
On the other hand, we have
(157)〈f(p)-p,J(xn-p)〉=〈f(p)-p,J(xn-p)〉-〈f(p)-p,J(xn-xt)〉+〈f(p)-p,J(xn-xt)〉-〈f(p)-xt,J(xn-xt)〉+〈f(p)-xt,J(xn-xt)〉-〈f(xt)-xt,J(xn-xt)〉+〈f(xt)-xt,J(xn-xt)〉=〈f(p)-p,J(xn-p)-J(xn-xt)〉+〈xt-p,J(xn-xt)〉+〈f(p)-f(xt),J(xn-xt)〉+〈f(xt)-xt,J(xn-xt)〉.
It follows that
(158)limsupn→∞〈f(p)-p,J(xn-p)〉≤limsupn→∞〈f(p)-p,J(xn-p)-J(xn-xt)〉+∥xt-p∥limsupn→∞∥xn-xt∥+ρ∥p-xt∥limsupn→∞∥xn-xt∥+limsupn→∞〈f(xt)-xt,J(xn-xt)〉.
Taking into account that xt→p as t→0, we have from (156)
(159)limsupn→∞〈f(p)-p,J(xn-p)〉=limsupt→0limsupn→∞〈f(p)-p,J(xn-p)〉≤limsupt→0limsupn→∞〈f(p)-p,J(xn-p)-J(xn-xt)〉.
Since X has a uniformly Frechet 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 (150) holds. It is clear from (150) that
(160)limsupn→∞〈f(p)-p,J(xn+1-p)〉≤0.
Finally, let us show thatxn→p as n→∞. We observe that
(161)∥yn-p∥2≤αn∥xn-p∥2+(1-αn)∥G(Jrnxn)-p∥2≤αn∥xn-p∥2+(1-αn)∥xn-p∥2=∥xn-p∥2,
and hence
(162)∥xn+1-p∥2=∥βn(f(xn)-f(p))+(1-βn)[G(Jrnyn)-λnμnFG(Jrnyn)-p]+βn(f(p)-p)βn(f(xn)-f(p))∥2≤∥[G(Jrnyn)-λnμnFG(Jrnyn)-p]βn(f(xn)-f(p))+(1-βn)[G(Jrnyn)-λnμnFG(Jrnyn)-p]∥2+2βn〈f(p)-p,J(xn+1-p)〉≤βn∥f(xn)-f(p)∥2+(1-βn)∥G(Jrnyn)-λnμnFG(Jrnyn)-p∥2+2βn〈f(p)-p,J(xn+1-p)〉=βn∥f(xn)-f(p)∥2+(1-βn)∥(I-λnμnF)G(Jrnyn)-(I-λnμnF)p-λnμnFp(Jrnyn)∥2+2βn〈f(p)-p,J(xn+1-p)〉≤βnρ∥xn-p∥2+(1-βn)[∥(I-λnμnF)G(Jrnyn)-(I-λnμnF)p∥+λnμn∥Fp∥(Jrnyn)]2+2βn〈f(p)-p,J(xn+1-p)〉≤βnρ∥xn-p∥2+(1-βn)×[(1-λnμn(1-1-δλ))∥G(Jrnyn)-p∥+λnμn∥Fp∥(1-λnμn(1-1-δλ))]2+2βn〈f(p)-p,J(xn+1-p)〉≤βnρ∥xn-p∥2+(1-βn)(∥yn-p∥+λnμn∥Fp∥)2+2βn〈f(p)-p,J(xn+1-p)〉≤βnρ∥xn-p∥2+(1-βn)(∥xn-p∥+λnμn∥Fp∥)2+2βn〈f(p)-p,J(xn+1-p)〉=βnρ∥xn-p∥2+(1-βn)[∥xn-p∥2+λnμn∥Fp∥×(2∥xn-p∥+λnμn∥Fp∥)∥xn-p∥2]+2βn〈f(p)-p,J(xn+1-p)〉≤(1-(1-ρ)βn)∥xn-p∥2+λnμn∥Fp∥(2∥xn-p∥+λnμn∥Fp∥)+2βn〈f(p)-p,J(xn+1-p)〉=(1-(1-ρ)βn)∥xn-p∥2+(1-ρ)βn{λnμnβn∥Fp∥(2∥xn-p∥+λnμn∥Fp∥)1-ρ+2〈f(p)-p,J(xn+1-p)〉1-ρ}.
Taking into account (160) and conditions (i), (ii), we obtain that ∑n=0∞‍(1-ρ)βn=∞ and
(163)limsupn→∞{λnμnβn∥Fp∥(2∥xn-p∥+λnμn∥Fp∥)1-ρ+2〈f(p)-p,J(xn+1-p)〉1-ρ∥Fp∥(2∥xn-p∥+λnμn∥Fp∥)1-ρ}≤0.
Therefore, applying Lemma 4 to (162), we infer that
(164)limn→∞∥xn-p∥=0.
This completes the proof.Remark 19.
As pointed out in [12, Remark 3.2], the sequences {λn}, {μn}, and {βn} can be taken, which satisfy the conditions in Theorem 18. 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)
limn→∞βn=0 and ∑n=0∞‍βn=∞,(ii)
limn→∞(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 Theorem18, we can obtain the following result. Because its proof is much simpler than that of Theorem 18, we omit its proof.Theorem 20.
LetX be a uniformly convex and 2-uniformly smooth Banach space and let A be an m-accretive operator in X such that C=D(A)¯ is convex. 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:X→C be a contractive map with coefficient ρ∈(0,1), and let F:X→X be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1. Assume that Δ=A-1(0)∩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. Given sequences {λn}n=0∞ in [0,1], {αn}n=0∞,{βn}n=0∞ in (0,1], and {rn}n=0∞ in [ε,∞) for some ε>0, suppose that there hold the following conditions:(i)
limn→∞βn=0 and ∑n=0∞‍βn=∞,(ii)
limn→∞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|<∞, ∑n=0∞‍|βn+1-βn|<∞, and ∑n=0∞‍|rn+1-rn|<∞.
Then for any given pointx0∈X, the sequence {xn} generated by
(165)yn=αnxn+(1-αn)G(Jrnxn),xn+1=βnf(xn)+(1-βn)[yn-λnF(yn)],∀n≥0,
converges strongly to p∈Δ, which is a unique solution of the VIP (44).Remark 21.
Our Theorems16–20 improve, extend, supplement, and develop Cai and Bu [13, Theorem 3.1] and Ceng et al. [12, Theorems 3.1–3.3] in the following aspects.(i)
The problem of finding a pointp∈A-1(0)∩Ω in our Theorems 16–20 is very different from everyone of both the problem of finding a point p∈⋂n‍Fix(Sn)∩Ω in Cai and Bu [13, Theorem 3.1] and the problem of finding a point p∈A-1(0) in Ceng et al. [12, Theorems 3.1–3.3]. There is no doubt that our problem of finding a point p∈A-1(0)∩Ω is more general than the problem of finding a point p∈A-1(0) in [12, Theorems 3.1–3.3].(ii)
Compared with the choice of iterative parameters in [12, Theorems 3.1–3.3], the choice of iterative parameters in our Theorems 16–20 is the same as that in [12, Theorems 3.1–3.3].(iii)
The iterative schemes in [12, Theorems 3.1–3.3] are extended to develop the iterative schemes in our Theorems 16–20 by virtue of the iterative scheme of [13, Theorems 3.1]. The iterative schemes in our Theorems 16–20 are more advantageous and more flexible than the iterative schemes of [12, Theorems 3.1–3.3] because they involve solving two problems: the GSVI (16) and the problem of finding zeros of an m-accretive operator.(iv)
The iterative schemes in our Theorems16–20 are very different from everyone in both [13, Theorem 3.1] and [12, Theorems 3.1–3.3] because the iterative scheme in our Theorem 16 is implicit and because the mapping G in [13, Theorem 3.1] and the mapping Jrn in [12, Theorems 3.1–3.3] are replaced by the same composite mapping G∘Jrn in the iterative schemes of our Theorems 16–20.(v)
Cai and Bu’s proof in [13, Theorem 3.1] depends on the argument techniques in [16], the inequality in 2-uniformly smooth Banach spaces (see Lemma 2), and the inequality in smooth and uniform convex Banach spaces (see Proposition 3). Because the composite mapping G∘Jrn appears in the iterative schemes in our Theorems 16–20, the proof of our Theorems 16–20 depends on the argument techniques in [16], the inequality in 2-uniformly smooth Banach spaces (see Lemma 2), the inequality in smooth and uniform convex Banach spaces (see Proposition 3), and the properties of the resolvent of anm-accretive operator (see Lemmas 8 and 9), the Banach limit (see Lemma 11) and the strongly accretive and strictly pseudocontractive mapping (see Proposition 13).



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*Source: 852760-2013-09-05.xml*