# The Meir-Keeler Type for Solving Variational Inequalities and Fixed Points of Nonexpansive Semigroups in Banach Spaces

Authors: Phayap Katchang; Poom Kumam
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The aim of this paper is to introduce a new iterative scheme for finding common solutions of the variational inequalities for an inverse strongly accretive mapping and the solutions of fixed point problems for nonexpansive semigroups by using the modified viscosity approximation method associate with Meir-Keeler type mappings and obtain some strong convergence theorem in a Banach spaces under some parameters controlling conditions. Our results extend and improve the recent results of Li and Gu (2010), Wangkeeree and Preechasilp (2012), Yao and Maruster (2011), and many others.


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## Body

## 1. Introduction

The theory of variational inequalities and variational inclusions are among the most interesting and important mathematical problems and have been studied intensively in the past years since they have wide applications in the optimization and control, economics, engineering science, physical sciences, and applied sciences. For these reasons, many existence result and iterative algorithms for various variational inclusion have been studied extensively by many authors (see, e.g., [1–8]). The important generalization of variational inequalities has been extensively studied and generalized in different directions to study a wide class of problems arising in mechanics, optimization, nonlinear programming, finance, and applied sciences (see, e.g., [2, 8]).LetC be a nonempty closed convex subset of a real Banach space E and E* be the dual space of E with norm ∥·∥ and 〈·,·〉 pairing between E and E*. For q>1, the generalized duality mappingJq:E→2E* is defined by
(1.1)Jq(x)={f∈E*:〈x,f〉=∥x∥q,∥f∥=∥x∥q-1}
for all x∈E. In particular, if q=2, the mapping J2 is called the normalized duality mapping and, usually, written J2=J. Further, we have the following properties of the generalized duality mapping Jq: (i) Jq(x)=∥x∥q-2J2(x) for all x∈E with x≠0; (ii) Jq(tx)=tq-1Jq(x) for all x∈E and t∈[0,∞); (iii) Jq(-x)=-Jq(x) for all x∈E.Recall that a mappingA:C→C is said to be (i) Lipschitzian with Lipschitz constant L>0 if ∥Ax-Ay∥≤L∥x-y∥, for all x,y∈C; (ii) contraction if there exists a constant α∈(0,1) such that ∥Ax-Ay∥≤α∥x-y∥, for all x,y∈C; (iii) nonexpansive if ∥Ax-Ay∥≤∥x-y∥, for all x,y∈C. An operator A:C→E is said to be(i)
accretive if there exists j(x-y)∈J(x-y) such that
(1.2)〈Ax-Ay,j(x-y)〉≥0,∀x,y∈C,(ii)
β-strongly accretive if there exists a constant β>0 such that
(1.3)〈Ax-Ay,j(x-y)〉≥β∥x-y∥2,∀x,y∈C,(iii)
β-inverse strongly accretive if, for any β>0,
(1.4)〈Ax-Ay,j(x-y)〉≥β∥Ax-Ay∥2,∀x,y∈C.LetD be a subset of C and Q:C→D. Then Q is said to sunny if Q(Qx+t(x-Qx))=Qx, whenever Qx+t(x-Qx)∈C for x∈C and t≥0. A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction Q of C onto D. A mapping Q:C→C is called a retraction if Q2=Q. If a mapping Q:C→C is a retraction, then Qz=z for all z is in the range of Q.A family𝒮={T(t):0≤t<∞} of mappings of C into itself is called a nonexpansive semigroup on C if it satisfies the following conditions: (i)
T(0)x=x for all x∈C;(ii)
T(s+t)=T(s)T(t) for all s,t≥0;(iii)
∥T(t)x-T(t)y∥≤∥x-y∥ for all x,y∈C and t≥0;(iv)
for allx∈C,t↦T(t)x is continuous.We denote byF(𝒮) the set of all common fixed points of𝒮={T(t):0≤t<∞},thatis,F(𝒮)=∩t≥0F(T(t)). It is known that F(𝒮) is closed and convex (see also [9, 10]).A mappingψ:ℝ+→ℝ+ is said to be an L function if ψ(0)=0,ψ(t)>0 for each t>0, and for every s>0 there exists u>s such that ψ(t)≤s for all t∈[s,u]. As a consequence, every L-function ψ satisfies ψ(t)<t for each t>0.Definition 1.1.
Let(X,d) be a matric space. A mapping f:X→X is said to be:(i)
(ψ,L)-contraction if ψ:ℝ+→ℝ+ is an L-function and d(f(x),f(y))<ψ(d(x,y)) for all x,y∈X with x≠y;(ii)
Meir-Keeler type mapping if for each ϵ>0 there exists δ=δ(ϵ)>0 such that for each x,y∈X with d(x,y)<ϵ+δ we have d(f(x),f(y))<ϵ.Remark 1.2.
From Definition1.1, if ψ(t)=αt,α∈[0,1), t∈ℝ+, then we get the usual contraction mapping with coefficient α.At the same time, we are also interesting in the variational inequality problems for an inverse strongly accretive mappings in Banach spaces. In 2006, Aoyama et al. [11] introduced the following iteration scheme for an inverse strongly accretive operator A in Banach spaces E:
(1.5)x1=x∈C,xn+1=αnxn+(1-αn)QC(xn-λnAxn),
for all n≥1, where C⊂E and QC is a sunny nonexpansive retraction from E onto C. They proved a weak convergence theorem in a Banach spaces. Moreover, the sequence {xn} in (1.5) solved the generalized variational inequality problem for finding a point x∈C such that
(1.6)〈Ax,j(y-x)〉≥0
for all y∈C. The set of solutions of (1.6) is denoted by VI(C,A).An interesting is the proof by using a nonexpansive semigroup and Meir-Keeler type mapping, in 2010, Li and Gu [12] defined the following sequence:
(1.7)x1=x∈E,yn-αnxn+(1-αn)T(tn)xn,xn+1=βnf(xn)+(1-βn)yn,n≥1.
Wangkeeree and Preechasilp [13] introduced the following iterative scheme:
(1.8)x0∈C,zn-γnxn+(1-γn)T(tn)xn,yn-αnxn+(1-αn)T(tn)zn,xn+1=βnf(xn)+(1-βn)yn,n≥0.In 2011, Yao and Maruster [8] proved some strong convergence theorems for finding a solution of variational inequality problem (1.6) in Banach spaces. They defined a sequence {xn} iteratively by given arbitrarily x0∈C and
(1.9)xn+1=βnxn+(1-βn)QC[(1-αn)(xn-λAxn)],∀n≥0,
where QC is a sunny nonexpansive retraction from a uniformly convex and 2-uniformly smooth Banach space E, and A is an α-inverse strongly accretive operator of C into E.Motivated and inspired by the idea of Li and Gu [12], Wangkeeree and Preechasilp [13], and Yao and Maruster [8], in this paper, we introduce a new iterative scheme for finding common solutions of the variational inequalities for an inverse strongly accretive mapping and the solutions of fixed point problems for a nonexpansive semigroup by using the modified viscosity approximation method associated with Meir-Keeler type mapping. We will prove the strong convergence theorem under some parameters controlling conditions. Our results extend and improve the recent results of Li and Gu [12], Wangkeeree and Preechasilp [13], Yao and Maruster [8], and many others.


## 2. Preliminaries

LetU={x∈E:∥x∥=1}. A Banach space E is said to uniformly convex if, for any ϵ∈(0,2], there exists δ>0 such that, for any x,y∈U, ∥x-y∥≥ϵ implies ∥(x+y)/2∥≤1-δ. It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space E is said to be smooth if the limit limt→0((∥x+ty∥-∥x∥)/t) exists for all x,y∈U. It is also said to be uniformly smooth if the limit is attained uniformly for x,y∈U. The modulus of smoothness of E is defined by
(2.1)ρ(τ)=sup{12(∥x+y∥+∥x-y∥)-1:x,y∈E,∥x∥=1,∥y∥=τ},
where ρ:[0,∞)→[0,∞) is a function. It is known that E is uniformly smooth if and only if limτ→0(ρ(τ)/τ)=0. Let q be a fixed real number with 1<q≤2. A Banach space E is said to be q-uniformly smooth if there exists a constant c>0 such that ρ(τ)≤cτq for all τ>0.A Banach spaceE is said to satisfy Opial’s condition if for any sequence {xn} in E, xn⇀x(n→∞) implies that
(2.2)limsupn→∞∥xn-x∥<limsupn→∞∥xn-y∥,∀y∈Ewithx≠y.
By [14, Theorem 1], it is well known that if E admits a weakly sequentially continuous duality mapping, then E satisfies Opial’s condition, and E is smooth.The following result describes a characterization of sunny nonexpansive retractions on a smooth Banach space.Proposition 2.1 (see [15]).
LetE be a smooth Banach space and let C be a nonempty subset of E. Let Q:E→C be a retraction, and let J be the normalized duality mapping on E. Then the following are equivalent:(i)
Q is sunny and nonexpansive;(ii)
∥Qx-Qy∥2≤〈x-y,J(Qx-Qy)〉,∀x,y∈E;(iii)
〈x-Qx,J(y-Qx)〉≤0,∀x∈E,y∈C.Proposition 2.2 (see [16]).
LetC be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, and let T be a nonexpansive mapping of C into itself with F(T)≠∅. Then the set F(T) is a sunny nonexpansive retract of C.Lemma 2.3 (see [17]).
LetC be a nonempty bounded closed convex subset of a uniformly convex Banach space E and T:C→C be a nonexpansive mapping. If {xn} is a sequence of C such that xn⇀x and xn-Txn→0, then x is a fixed point of T.We need the following lemmas for proving our main results.Lemma 2.4 (see [18]).
Letr>0, and let E be a uniformly convex Banach space. Then, there exists a continuous, strictly increasing, and convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(2.3)∥λx+(1-λ)y∥2≤λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)g(∥x-y∥)
for all x,y∈Br:={z∈E:∥z∥≤r} and 0≤λ≤1.Lemma 2.5 (see [19]).
LetE be a real smooth and uniformly convex Banach space, and let r>0. Then there exists a strictly increasing, continuous, and convex function g:[0,2r]→ℝ such that g(0)=0 and
(2.4)g(∥x-y∥)≤∥x∥2-2〈x,jy〉+∥y∥2,∀x,y∈Br.Lemma 2.6 (see [18]).
LetE be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inquality holds:
(2.5)∥x+y∥2≤∥x∥2+2〈y,Jx〉+2∥Ky∥2,∀x,y∈E.Lemma 2.7 (see [20]).
LetE be a real Banach space and J:E→2E* be the normalized duality mapping. Then, for any x,y∈E, one has
(2.6)∥x+y∥2≤∥x∥2+2〈y,J(x+y)〉
for all j(x+y)∈J(x+y) with x≠y.Lemma 2.8 (see [21]).
Let{xn} and {yn} be bounded sequences in a Banach space X, and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose that xn+1=(1-βn)yn+βnxn for all integers n≥0 and limsupn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then, limn→∞∥yn-xn∥=0.Lemma 2.9 (see [22]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.7)an+1≤(1-αn)an+δn,n≥0,
where {αn} is a sequence in (0,1), and {δn} is a sequence in ℝ such that (1)
∑n=1∞‍αn=∞(2)
limsupn→∞(δn/αn)≤0 or ∑n=1∞‍|δn|<∞. Then limn→∞an=0.Theorem 2.10 (see [23]).
Let(X,d) be a complete metric space and f:X→X a Meir-Keeler type mapping. Then f has a unique fixed point.Theorem 2.11 (see [24]).
Let(X,d) be a metric space and f:X→X a mapping. Then the following assertions are equivalent:(i)
f is a Meir-Keeler type mapping;(ii)
there exists an L  functionψ:ℝ+→ℝ+ such that f is a (ψ,L) contraction.Proposition 2.12 (see [21]).
LetC be a convex subset of a Banach space E. Let f:C→C be a Meir-Keeler type mapping. Then for each ϵ>0 there exists r∈(0,1) such that for each x,y∈C with ∥x-y∥≥ϵ, one has
(2.8)∥f(x)-f(y)∥≤r∥x-y∥.Proposition 2.13 (see [21]).
LetC be a convex subset of a Banach space E. Let T be a nonexpansive mapping on C, and let f:C→C be a Meir-Keeler-type mapping. Then the following holds:(i)
T∘f is a Meir-Keeler type mapping on C;(ii)
for eachα∈(0,1) the mapping x↦αf(x)+(1-α)T(x) is a Meir-Keeler-type mapping on C.The following lemma is characterized by the set of solutions of variational inequality by using sunny nonexpansive retractions.Lemma 2.14 (see [11]).
LetC be a nonempty closed convex subset of a smooth Banach space E. Let QC be a sunny nonexpansive retraction from E onto C, and let A be an accretive operator of C into E. Then, for all λ>0,
(2.9)VI(C,A)=F(Q(I-λA)),
where VI(C,A)={x*∈C:〈Ax*,J(x-x*)〉≥0,∀x∈C}.


## 3. Strong Convergence Theorems

In this section, we suppose that the functionψ from the definition of the (ψ,L) contraction is continuous and strictly increasing and limt→∞η(t)=∞, where η(t)=t-ψ(t), t∈ℝ+. In consequence, we have that η is a bijection on ℝ+ and the function ψ satisfies the assumption in Remark 1.2.Suppose that{αn},{βn}⊂(0,1), and {μn},{λn}⊂(0,∞) satisfy the following conditions: (C1)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(C2)
limn→∞|λn+1-λn|=0 and 0<a≤λn≤b<β/K2;(C3)
0<liminfn→∞βn≤limsupn→∞βn<1;(C4)
limn→∞μn=0;(C5)
limn→∞supx∈C~∥T(μn+1)x-T(μn)x∥=0, C~ bounded subset of C.Next, we stat the main result.Theorem 3.1.
LetE be a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant K and C a nonempty closed convex subset of E. Let QC be a sunny nonexpansive retraction from E onto C and A:C→E be an β-inverse-strongly accretive operator. Let 𝒮={T(t):t≥0} be a nonexpansive semigroup from C into itself and f be a Meir-Keeler contraction of C into itself. Suppose that F:=F(𝒮)∩VI(C,A)≠∅ and the conditions (C1)–(C5). For arbitrary given x1∈C, the sequences {xn} are generated by
(3.1)un=QC(xn-λnAxn),yn=QC[αnf(xn)+(1-αn)un],xn+1=βnxn+(1-βn)T(μn)yn.
Then {xn} converges strongly to x*=QFfx*∈F which also solves the following variational inequality:
(3.2)〈(f-I)x*,J(z-x*)〉≤0,∀z∈F.Proof.
First we prove that{xn} bounded. Let p∈F, we have
(3.3)∥un-p∥2=∥QC(xn-λnAxn)-QC(p-λnAp)∥2≤∥(xn-λnAxn)-(p-λnAp)∥2=∥(I-λnA)xn-(I-λnA)p∥2=∥(xn-p)-λn(Axn-Ap)∥2≤∥xn-p∥2-2λn〈Axn-Ap,J(xn-p)〉+2K2λn2∥Axn-Ap∥2≤∥xn-p∥2-2λnβ∥Axn-Ap∥2+2K2λn2∥Axn-Ap∥2=∥xn-p∥2+2λn(λnK2-β)∥Ax-Ay∥2≤∥xn-p∥2.
So, we get ∥un-p∥≤∥xn-p∥, for all n≥1. It follows that
(3.4)∥xn+1-p∥=∥βn(xn-p)+(1-βn)(T(μn)yn-p)∥≤βn∥xn-p∥+(1-βn)∥yn-p∥=βn∥xn-p∥+(1-βn)∥QC[αnf(xn)+(1-αn)un]-QCp∥≤βn∥xn-p∥+(1-βn)[αn∥f(xn)-p∥+(1-αn)∥un-p∥]≤βn∥xn-p∥+(1-βn)[αn∥f(xn)-f(p)∥+αn∥f(p)-p∥+(1-αn)∥xn-p∥]≤βn∥xn-p∥+(1-βn)[αnψ(∥xn-p∥)+αn∥f(p)-p∥+(1-αn)∥xn-p∥]=∥xn-p∥+αn(1-βn)[η(∥xn-p∥)]+αn(1-βn)η[η-1(∥f(p)-p∥)]≤max{∥xn-p∥,η-1(∥f(p)-p∥)}.
By induction, we conclude that(3.5)∥xn-p∥≤max{∥xn-p∥,η-1(∥f(p)-p∥)},∀n≥1.
This implies that {xn} bounded, so are {f(xn)}, {yn}, {un}, {Axn}, and {T(μn)yn}.
Next, we show thatlimn→∞∥xn+1-xn∥=0, we observe that
(3.6)∥un+1-un∥=∥QC(xn+1-λn+1Axn+1)-QC(xn-λnAxn)∥≤∥(xn+1-λn+1Axn+1)-(xn-λnAxn)∥=∥(xn+1-λn+1Axn+1)-(xn-λn+1Axn)+(λn+1-λn)Axn∥≤∥(I-λn+1A)xn+1-(I-λn+1A)xn∥+|λn+1-λn|∥Axn∥≤∥xn+1-xn∥+|λn+1-λn|∥Axn∥,∥yn+1-yn∥=∥QC[αn+1f(xn+1)+(1-αn+1)un+1]-QC[αnf(xn)+(1-αn)un]∥≤∥[αn+1f(xn+1)+(1-αn+1)un+1]-[αnf(xn)+(1-αn)un]∥≤∥un+1-un∥+αn+1(∥un+1∥+∥f(xn+1)∥)+αn(∥un∥+∥f(xn)∥)≤∥xn+1-xn∥+|λn+1-λn|∥Axn∥+αn+1(∥un+1∥+∥f(xn+1)∥)+αn(∥un∥+∥f(xn)∥).
It follows that
(3.7)∥T(μn+1)yn+1-T(μn)yn∥≤∥T(μn+1)yn+1-T(μn+1)yn∥+∥T(μn+1)yn-T(μn)yn∥≤∥yn+1-yn∥+∥T(μn+1)yn-T(μn)yn∥≤∥xn+1-xn∥+|λn+1-λn|∥Axn∥+αn+1(∥un+1∥+∥f(xn+1)∥)+αn(∥un∥+∥f(xn)∥)+∥T(μn+1)yn-T(μn)yn∥≤∥xn+1-xn∥+|λn+1-λn|∥Axn∥+αn+1(∥un+1∥+∥f(xn+1)∥)+αn(∥un∥+∥f(xn)∥)+supy∈{yn}∥T(μn+1)y-T(μn)y∥.By (C1), (C2), and (C4), they imply that
(3.8)limsupn→∞(∥T(μn+1)yn+1-T(μn)yn∥-∥xn+1-xn∥)≤0.
Applying Lemma 2.8, we obtain
(3.9)limn→∞∥T(μn)yn-xn∥=0.
Therefore, we have
(3.10)limn→∞∥xn+1-xn∥=0.
On the other hand, we consider
(3.11)∥xn+1-p∥2≤βn∥xn-p∥2+(1-βn)∥yn-p∥2≤βn∥xn-p∥2+(1-βn)[αn∥f(xn)-p∥2+(1-αn)∥un-p∥2]=βn∥xn-p∥2+(1-βn)×[αn∥f(xn)-p∥2+(1-αn)∥QC(xn-λnAxn)-QC(p-λnAp)∥2]≤βn∥xn-p∥2+(1-βn)×[αn∥f(xn)-p∥2+(1-αn)(∥xn-p∥2+2λn(λnK2-β)∥Ax-Ay∥2)]=[βn+(1-βn)(1-αn)]∥xn-p∥2+αn(1-βn)∥f(xn)-p∥2+2λn(λnK2-β)(1-βn)(1-αn)∥Ax-Ay∥2≤∥xn-p∥2+αn∥f(xn)-p∥2+2λn(λnK2-β)∥Ax-Ay∥2.
Then, we obtain that
(3.12)2λn(β-λnK2)∥Ax-Ay∥2≤∥xn-p∥2-∥xn+1-p∥2+αn∥f(xn)-p∥2≤∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥)+αn∥f(xn)-p∥2.
By (C1), (C2), (C3), and (3.10), we get
(3.13)limn→∞∥Axn-Ap∥=0.
From Proposition 2.1 (ii) and Lemma 2.5, we also have
(3.14)∥un-p∥2=∥QC(xn-λnAxn)-QC(p-λnAp)∥2≤〈(xn-λnAxn)-(p-λnAp),J(un-p)〉=〈(xn-p)-λn(Axn-Ap),J(un-p)〉=〈xn-p,J(un-p)〉-λn〈Axn-Ap,J(un-p)〉≤12[∥xn-p∥2+∥un-p∥2-g∥xn-un∥]+λn∥Axn-Ap∥∥un-p∥.
So, we get,
(3.15)∥un-p∥2≤∥xn-p∥2-g∥xn-un∥+2λn∥Axn-Ap∥∥un-p∥.
Therefore, using (3.11), we obtain
(3.16)∥xn+1-p∥2≤βn∥xn-p∥2+(1-βn)[αn∥f(xn)-p∥2+(1-αn)∥un-p∥2]=βn∥xn-p∥2+αn(1-βn)∥f(xn)-p∥2+(1-αn)(1-βn)∥un-p∥2≤βn∥xn-p∥2+αn(1-βn)∥f(xn)-p∥2+(1-αn)(1-βn)[∥xn-p∥2-g(∥xn-un∥)+2λn∥Axn-Ap∥∥un-p∥]≤∥xn-p∥2+αn∥f(xn)-p∥2-g(∥xn-un∥)+2λn∥Axn-Ap∥∥un-p∥.
Then we get
(3.17)g(∥xn-un∥)≤∥xn-p∥2-∥xn+1-p∥2+αn∥f(xn)-p∥2+2λn∥Axn-Ap∥∥un-p∥≤∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥)+αn∥f(xn)-p∥2+2λn∥Axn-Ap∥∥un-p∥.
By (C1), (3.10), and (3.13), we have
(3.18)limn→∞g(∥xn-un∥)=0.
It follows from the property of g that
(3.19)limn→∞∥xn-un∥=0.
Again, we consider
(3.20)∥yn-p∥2=∥QC[αnf(xn)+(1-αn)un]-QCp∥2≤〈αn(f(xn)-p)+(1-αn)(un-p),J(yn-p)〉=αn〈f(xn)-p,J(yn-p)〉+(1-αn)〈un-p,J(yn-p)〉≤αn∥f(xn)-p∥∥yn-p∥+12[∥un-p∥2+∥yn-p∥2-g(∥un-yn∥)]≤αn∥f(xn)-p∥∥yn-p∥+12[∥xn-p∥2+∥yn-p∥2-g(∥un-yn∥)].
It follows that
(3.21)∥yn-p∥2≤2αn∥f(xn)-p∥∥yn-p∥+∥xn-p∥2-g(∥un-yn∥).
By using (3.11), we obtain
(3.22)∥xn+1-p∥2≤βn∥xn-p∥2+(1-βn)[αn∥f(xn)-p∥2+(1-αn)∥yn-p∥2]=βn∥xn-p∥2+αn(1-βn)∥f(xn)-p∥2+(1-αn)(1-βn)∥yn-p∥2≤βn∥xn-p∥2+αn(1-βn)∥f(xn)-p∥2+(1-αn)(1-βn)[2αn∥f(xn)-p∥∥yn-p∥+∥xn-p∥2-g(∥un-yn∥)]≤∥xn-p∥2+αn∥f(xn)-p∥2+2αn∥f(xn)-p∥∥yn-p∥-g(∥un-yn∥).
Therefore, we have
(3.23)g(∥un-yn∥)≤∥xn-p∥2-∥xn+1-p∥2+αn∥f(xn)-p∥2+2αn∥f(xn)-p∥∥yn-p∥≤∥xn-xn+1∥(∥xn-p∥+∥xn+1-p∥)+2αn∥f(xn)-p∥2+2αn∥f(xn)-un∥∥yn-un∥.
By (C1) and (3.10), we have
(3.24)limn→∞g(∥un-yn∥)=0.
From the property of g, we get
(3.25)limn→∞∥un-yn∥=0.
According (3.19) and (3.25), we also have
(3.26)limn→∞∥xn-yn∥=0.
Since
(3.27)∥xn-T(μn)xn∥≤∥xn-T(μn)yn∥+∥T(μn)yn-T(μn)xn∥≤∥xn-T(μn)yn∥+∥yn-xn∥,
from (3.9) and (3.26), we get
(3.28)limn→∞∥T(μn)xn-xn∥=0.
Now, we show thatz∈F:=F(𝒮)∩VI(C,A). We can choose a sequence {xnk} of {xn} such that {xnk} is bounded, and there exists a subsequence {xnkj} of {xnk} which converges weakly to z. Without loss of generality, we can assume that xnk⇀z.
(I)We show thatz∈VI(C,A). From the assumption, we see that control sequence {λnk} is bounded. So, there exists a subsequence {λnkj} that converges to λ0. We may assume, without loss of generality, that λnk⇀λ0. Observe that
(3.29)∥QC(xnk-λ0Axnk)-xnk∥≤∥QC(xnk-λ0Axnk)-ynk∥+∥ynk-xnk∥≤∥(xnk-λ0Axnk)-(xnk-λnkAxnk)∥+∥ynk-xnk∥≤M∥λnk-λ0∥+∥ynk-xnk∥,
where M is as appropriate constant such that M≥supn≥1{∥Axn∥}. It follows from (3.26) and λnk⇀λ0 that
(3.30)limk→∞∥QC(xnk-λ0Axnk)-xnk∥=0.
We know that QC(I-λ0A) is nonexpansive, and it follows from Lemma 2.3 that z∈F(QC(I-λ0A)). By using Lemma 2.14, we can obtain that z∈F(QC(I-λ0A))=VI(C,A).
(II)Next, we show thatz∈F(𝒮). Let μnk≥0 such that
(3.31)μnk→0,∥T(μnk)xnk-xnk∥μnk→0,k→∞.
Fix t>0. Notice that
(3.32)∥xnk-T(t)z∥≤∑i=0[t/μnk]-1‍∥T((i+1)μnk)xnk-T(iμnk)xnk∥+∥T([tμnk]μk)xnk-T([tμnk]μnk)z∥+∥T([tμnk]μnk)z-T(t)z∥≤[tμnk]∥T(μnk)xnk-xnk∥+∥xnk-z∥+∥T(t-[tμnk]μnk)z-z∥≤t∥T(μnk)xnk-xnk∥μnk+∥xnk-p∥+∥T(t-[tμnk]μnk)z-z∥≤t∥T(μnk)xnk-xnk∥μnk+∥xnk-p∥+max{∥T(s)z-z∥:0≤s≤μnk}.
For all k∈ℕ, we have
(3.33)limsupk→∞∥xnk-T(t)z∥≤limsupk→∞∥xnk-z∥.
Since a Banach space E with a weakly sequentially continuous duality mapping satisfies the Opial’s condition, this implies that T(t)z=z. Therefore z∈F(𝒮), so z∈F.
Next, we show thatlimsupn→∞〈(f-I)x*,J(yn-x*)〉≤0, where x*=QFfx*, QF is a sunny nonexpansive retraction of C onto F. Since we have (3.26) and xnk⇀z, then we have ynk⇀z such that
(3.34)limsupn→∞〈(f-I)x*,J(yn-x*)〉=limk→∞〈(f-I)x*,J(ynk-x*)〉=limk→∞〈(f-I)x*,J(z-x*)〉≤0.
Finally, we show that{xn} converges strongly to x*. Suppose that {xn} does not converge strongly to x*, and then there exist ϵ>0 and a subsequence {xni} of {xn} such that ∥xni-x*∥>ϵ for all i∈ℕ. By Proposition 2.12, for this ϵ there exists r∈(0,1) such that
(3.35)∥f(xni)-f(p)∥≤r∥xni-p∥.
So, by Lemma 2.7, we have
(3.36)∥yni-x*∥2=∥QC[αnf(xni)+(1-αni)uni]-QCx*∥2≤〈αni(f(xni)-x*)+(1-αni)(uni-x*),J(yni-x*)〉=αni〈f(xni)-x*,J(yni-x*)〉+(1-αni)〈uni-x*,J(yni-x*)〉≤αni〈f(xni)-f(x*),J(yni-x*)〉+αni〈f(x*)-x*,J(yni-x*)〉+(1-αni)∥uni-x*∥∥yni-x*∥≤rαni∥xni-x*∥∥yni-x*∥+αni〈f(x*)-x*,J(yni-x*)〉+(1-αni)∥uni-x*∥∥yni-x*∥≤rαni∥xni-x*∥2+αni〈f(x*)-x*,J(yni-x*)〉+(1-αni)∥xni-x*∥2=(1-(1-r)αni)∥xni-x*∥2+αni〈f(x*)-x*,J(yni-x*)〉.
It follows from (3.11) that
(3.37)∥xni+1-p∥2≤βni∥xni-p∥2+(1-βni)∥yni-p∥2≤βni∥xni-p∥2+(1-βni)×[(1-(1-r)αni)∥xni-x*∥2+αni〈f(x*)-x*,J(yni-x*)〉]=[1-(1-r)(1-βni)αni]∥xni-x*∥2+(1-βni)αni〈f(x*)-x*,J(yni-x*)〉.
Now, from (C1), (3.34) and applying Lemma 2.9 to (3.37), we get ∥xn-x*∥→0 as n→∞. This is a contradiction, and hence the sequence {xn} converges strongly to x*∈F. The proof is completed.Corollary 3.2.
LetE be a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant K and C a nonempty closed convex subset of E. Let QC be a sunny nonexpansive retraction from E onto C and A:C→E be an β-inverse strongly accretive operator. Let 𝒮={T(t):t≥0} be a nonexpansive semigroup from C into it self and f be a Meir-Keeler contraction of C into itself. Suppose that F:=F(𝒮)∩VI(C,A)≠∅. For arbitrary given x1∈C, the sequences {xn} are generated by
(3.38)un=QC(xn-λnAxn),yn=QC[αnf(xn)+(1-αn)un],xn+1=βnxn+(1-βn)1tn∫0tn‍T(s)ynds,∀n≥1,
where {αn},{βn}⊂(0,1), and {λn}⊂(0,∞) satisfy the conditions (C1)–(C3) in Theorem 3.1 and assume that limn→∞supx∈C~∥(1/tn+1)∫0tn+1‍T(s)xds-(1/tn)∫0tn‍T(s)xds∥=0, C~ bounded subset of C, limn→∞μn=∞ and limn→∞(μn/μn+1)=1. Then {xn} converges strongly to x*=QFfx*∈F, which also solves the following variational inequality:
(3.39)〈(f-I)x*,J(z-x*)〉≤0,∀z∈F.Corollary 3.3.
LetE be a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant K and C a nonempty closed convex subset of E. Let QC be a sunny nonexpansive retraction from E onto C. Let 𝒮={T(t):t≥0} be a nonexpansive semigroup from C into itself and f be a Meir-Keeler contraction of C into itself. Suppose that F(𝒮)≠∅, {αn},{βn}⊂(0,1), and {μn}⊂(0,∞) satisfy the conditions (C1), (C2), (C4), and (C5) in Theorem 3.1. For arbitrary given x1∈C, the sequences {xn} are generated by
(3.40)yn=QC[αnf(xn)+(1-αn)xn],xn+1=βnxn+(1-βn)T(μn)yn,∀n≥1.
Then {xn} converges strongly to x*=QFfx*∈F(𝒮), which also solves the following variational inequality:
(3.41)〈(f-I)x*,J(z-x*)〉≤0,∀z∈F.Proof.
TakingA=0 in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.Corollary 3.4.
LetE be a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant K and C a nonempty closed convex subset of E. Let QC be a sunny nonexpansive retraction from E onto C and A:C→E be an β-inverse strongly accretive operator. Let f be a Meir-Keeler contraction of C into itself. Suppose that F:=VI(C,A)≠∅, {αn},{βn}⊂(0,1) and {λn}⊂(0,∞) satisfy the conditions (C1)–(C3) in Theorem 3.1. For arbitrary given x1∈C, the sequences {xn} are generated by
(3.42)un=QC(xn-λnAxn),yn=QC[αnf(xn)+(1-αn)un],xn+1=βnxn+(1-βn)yn,∀n≥1.
Then {xn} converges strongly to x*=QFfx*∈F, which also solves the following variational inequality:
(3.43)〈(f-I)x*,J(z-x*)〉≤0,∀z∈F.Proof.
Takingμn=0 for all n≥1 in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.Corollary 3.5.
LetE be a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant K and C a nonempty closed convex subset of E. Let QC be a sunny nonexpansive retraction from E onto C and A:C→E be an β strongly accretive and L-Lipschitz continuous operator. Let 𝒮={T(t):t≥0} be a nonexpansive semigroup from C into it self and f be a Meir-Keeler contraction of C into itself. Suppose that F:=F(𝒮)∩VI(C,A)≠∅, {αn},{βn}⊂(0,1) and {μn},{λn}⊂(0,∞) satisfy the conditions (C1) and (C3)–(C5) in Theorem 3.1. If the sequence {xn} is generated by x1∈C and (3.1) and limn→∞(λn+1-λn)=0 and 0<a≤λn≤b<β/K2L2, then the sequence {xn} converges strongly to x*=QFfx*∈F, which also solves the following variational inequality:
(3.44)〈(f-I)x*,J(z-x*)〉≤0,∀z∈F.Proof.
SinceA be an β strongly accretive and L-Lipschitz continuous operator of C into E, we have
(3.45)〈Ax-Ay,j(x-y)〉≥β∥x-y∥2≥βL2∥Ax-Ay∥2,∀x,y∈C.
Therefore, A is (β/L2)-inverse strongly accretive. Using Theorem 3.1, we can obtain that {xn} converges strongly to x*. This completes the proof.The following corollary is defined in a real Hilbert space. LetC be a closed convex subset of a real Hilbert space H. Let A:C→H be a mapping. The classical variational inequality problems are to find x∈C such that
(3.46)〈Ax,y-x〉≥0,
for all y∈C. For every point x∈H, there exists a unique nearest point in C, denoted by PCx, such that
(3.47)∥x-PCx∥≤∥x-y∥,∀y∈C.PC is called the metric projection of H onto C. It is well known that PC is a nonexpansive mapping of H onto C and satisfies
(3.48)〈x-y,PCx-PCy〉≥∥PCx-PCy∥2,
for every x,y∈H. Moreover, PCx is characterized by the following properties: PCx∈C and
(3.49)〈x-PCx,y-PCx〉≤0,(3.50)∥x-y∥2≥∥x-PCx∥2+∥y-PCx∥2,
for all x∈H,y∈C.It is well known in Hilbert spaces the smooth constantK=2/2 and J=I (identity mapping). From Theorem 3.1, we can obtain the following result immediately.Corollary 3.6.
LetC be a nonempty compact convex subset of a real Hilbert space H. Let PC be a metric projection of H onto C and A:C→E be an β-inverse strongly accretive operator. Let 𝒮={T(t):t≥0} be a nonexpansive semigroup from C into itself and f be a Meir-Keeler contraction of C into itself. Suppose that F:=F(𝒮)∩VI(C,A)≠∅, {αn},{βn}⊂(0,1) and {μn},{λn}⊂(0,∞) satisfy the conditions (C1)–(C5) in Theorem 3.1. For arbitrary given x1∈C, the sequences {xn} are generated by
(3.51)un=PC(xn-λnAxn),yn=PC[αnf(xn)+(1-αn)un],xn+1=βnxn+(1-βn)T(μn)yn,∀n≥1.
Then {xn} converges strongly to x*=PFfx*∈F which also solves the following variational inequality:
(3.52)〈(f-I)x*,z-x*〉≤0,∀z∈F.Remark 3.7.
Question and Open problems. Can we extend Theorem 3.1 to more general variational inequalities in the sense of Noor [1] on Banach spaces?



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