# A New Hybrid Extragradient Iterative Method for Approximating the Common Solutions of a System of Variational Inequalities, a Mixed Equilibrium Problem, and a Hierarchical Fixed Point Problem

Authors: Abdellah Bnouhachem; Abdelouahed Hamdi
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We suggest and analyze an iterative scheme for finding
the approximate element of the common set of solutions of a system of variational
inequalities, a mixed equilibrium problem, and a hierarchical fixed point problem in a
real Hilbert space. Strong convergence of the proposed method is proved under some
conditions. The results presented in this paper extend and improve some well-known
results in the literature.


---

## Body

## 1. Introduction

LetH be a real Hilbert space, whose inner product and norm are denoted by 〈·,·〉 and ∥·∥. Let C be a nonempty closed convex subset of H. We consider the system of variational inequalities of finding (x*,y*)∈C×C such that
(1)〈μ1B1y*+x*-y*,x-x*〉≥0;∀x∈C,μ1>0,〈μ2B2x*+y*-x*,x-y*〉≥0;∀x∈C,μ2>0,
where Bi:C→C is a nonlinear mapping for each i=1,2. The solution set of (1) is denoted by S*.IfB1=B2=B, then the problem (1) reduces finding (x*,y*)∈C×C such that
(2)〈μ1By*+x*-y*,x-x*〉≥0;∀x∈C,μ1>0,〈μ2Bx*+y*-x*,x-y*〉≥0;∀x∈C,μ2>0,
which has been introduced and studied by Verma [1, 2].Ifx*=y* and μ1=μ2, then the problem (2) collapses to the classical variational inequality finding x*∈C, such that
(3)〈Bx*,x-x*〉≥0,∀x∈C
is called the classical variational inequality problem, which was introduced by Stampacchia [3] in 1964. For the recent applications, numerical techniques, and physical formulation, see [1–33]. We now have a variety of techniques to suggest and analyze various iterative algorithms for solving the system of variational inequalities (1); see [1, 2, 7, 8, 12, 14, 24, 28, 30].We introduce the following definitions which are useful in the following analysis.Definition 1.
The mappingT:C→H is said to be (a)
monotone, if(4)〈Tx-Ty,x-y〉≥0,∀x,y∈C;(b)
strongly monotone, if there exists anα>0 such that
(5)〈Tx-Ty,x-y〉≥α∥x-y∥2,∀x,y∈C;(c)
α-inverse strongly monotone, if there exists an α>0 such that
(6)〈Tx-Ty,x-y〉≥α∥Tx-Ty∥2,∀x,y∈C;(d)
nonexpansive, if(7)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C;(e)
k-Lipschitz continuous, if there exists a constant k>0 such that
(8)∥Tx-Ty∥≤k∥x-y∥,∀x,y∈C;(f)
contraction onC, if there exists a constant 0≤k<1 such that
(9)∥Tx-Ty∥≤k∥x-y∥,∀x,y∈C.It is easy to observe that every α-inverse strongly monotone T is monotone and Lipschitz continuous. A mapping T:C→H is called k-strict pseudocontraction, if there exists a constant 0≤k<1 such that
(10)∥Tx-Ty∥2≤∥x-y∥2+k∥(I-T)x-(I-T)y∥2,+k∥(I-T)x-(I-T)y∥2,∀x,y∈C.
The fixed-point problem for the mapping T is to find x∈C such that
(11)Tx=x.
We denote by F(T) the set of solutions of (11). It is well-known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings; then F(T) is closed and convex and PF(T) is well defined (see [33]).
The mixed equilibrium problem, denoted byMEP, is to find x∈C such that
(12)F(x,y)+〈Dx,y-x〉≥0,∀y∈C,
where F:C×C→ℝ is bifunction and D:C→H is a nonlinear mapping. This problem was introduced and studied by Moudafi and Théra [21] and Moudafi [22]. The set of solutions of (12) is denoted by
(13)MEP(F):={x∈C:F(x,y)+〈Dx,y-x〉≥0,∀y∈C}.If D=0, then it is reduced to the equilibrium problem is to find x∈C such that(14)F(x,y)≥0,∀y∈C.
The solution set of (14) is denoted by EP(F). Numerous problems in physics, optimization, and economics reduce to find a solution of (14); see [9, 13, 25, 26]. In 1997, Flåm and Antipin [10] introduced an iterative scheme of finding the best approximation to the initial data when EP(F) is nonempty. Recently, Plubtieng and Punpaeng [25] introduced an iterative method for finding the common element of the set F(T)∩Ω*∩EP(F).
LetS:C→H be a nonexpansive mapping. The following problem is called a hierarchical fixed point problem: Find x∈F(T) such that
(15)〈x-Sx,y-x〉≥0,∀y∈F(T).
It is known that the hierarchical fixed-point problem (15) links with some monotone variational inequalities and convex programming problems; see [11, 31]. Various methods have been proposed to solve the hierarchical fixed point problem; see Moudafi [23], Maingé and Moudafi in [17], Marino and Xu in [19], and Cianciaruso et al. [6]. Very recently, Yao et al. [31] introduced the following strong convergence iterative algorithm to solve the problem (15):
(16)yn=βnSxn+(1-βn)xnxn+1=PC[αnf(xn)+(1-αn)Tyn],∀n≥0,
where f:C→H is a contraction mapping and {αn} and {βn} are two sequences in (0,1). Under some certain restrictions on parameters, Yao et al. proved that the sequence {xn} generated by (16) converges strongly to z∈F(T), which is the unique solution of the following variational inequality:
(17)〈(I-f)z,y-z〉≥0,∀y∈F(T).
By changing the restrictions on parameters, the authors obtained another result on the iterative scheme (16); the sequence {xn} generated by (16) converges strongly to a point z∈F(T), which is the unique solution of the following variational inequality:
(18)〈1τ(I-f)z+(I-S)z,y-z〉≥0,∀y∈F(T).
LetS:C→H be a nonexpansive mapping and {Ti}i=1∞:C→C a countable family of nonexpansive mappings. Very recently, Gu et al. [11] introduced the following iterative algorithm:
(19)yn=PC[βnSxn+(1-βn)xn]xn+1=PC[αnf(xn)+∑i=1n(αi-1-αi)Tiyn],∀n≥1,
where α0=1, {αn} is a strictly decreasing sequence in (0,1), and {βn} is a sequence in (0,1). Under some certain conditions on parameters, Gu et al. proved that the sequence {xn} generated by (19) converges strongly to z∈⋂i=1∞‍F(Ti), which is unique solution of one of the variational inequalities (17) and (18).
In this paper, motivated by the work of Yao et al. [31] and Gu et al. [11] and by the recent work going in this direction, we give an iterative method for finding the approximate element of the common set of solutions of (1), (12), and (15) for a strictly pseudocontraction mapping in real Hilbert space. We establish a strong convergence theorem based on this method. The presented method improves and generalizes many known results for solving system of variational inequality problems, mixed equilibrium problems, and hierarchical fixed point problems; see, for example [6, 11, 17, 31] and relevant references cited therein.


## 2. Preliminaries

In this section, we list some fundamental lemmas that are useful in the consequent analysis. The first lemma provides some basic properties of projection ontoC.Lemma 2.
LetPC denote the projection of H onto C. Then, one has the following inequalities:
(20)〈z-PC[z],PC[z]-v〉≥0,∀z∈H,v∈C;(21)〈u-v,PC[u]-PC[v]〉≥∥PC[u]-PC[v]∥2,∀u,v∈H;(22)∥PC[u]-PC[v]∥≤∥u-v∥,∀u,v∈H;∥u-PC[z]∥2≤∥z-u∥2-∥z-PC[z]∥2,∀z∈H,u∈C.Lemma 3 (see [7]).
For any(x*,y*)∈C×C,(x*,y*) is a solution of (1) if and only if x* is a fixed point of the mapping Q:C→C de fined by
(23)Q(x)=PC[PC[x-μ2B2x]-μ1B1PC[x-μ2B2x]],∀x∈C,
where y*=PC[x*-μ2B2x*],μi∈(0,2θi), and Bi:C→C is θi-inverse strongly monotone mappings for each i=1,2.Lemma 4 (see [5]).
LetF:C×C→ℝ be a bifunction satisfying the following assumptions. (i)
F(x,x)=0,∀x∈C.(ii)
F is monotone; that is, F(x,y)+F(y,x)≤0,∀x,y∈C.(iii)
For eachx,y,z∈C,limt→0F(tz+(1-t)x,y)≤F(x,y).(iv)
For eachx∈C,y→F(x,y) is convex and lower semicontinuous. Let r>0 and x∈H. Then, there exists z∈C such that
(24)F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.Lemma 5 (see [10]).
Assume thatF:C×C→ℝ satisfies assumptions (i)–(iv) of Lemma 4. For r>0 and forallx∈H, define a mapping Tr:H→C as follows:
(25)Tr(x)={z∈C:F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}.
Then, the following hold. (i)
Tr is single valued.(ii)
Tr is firmly nonexpansive; that is,
(26)∥Trx-Try∥2≤〈Trx-Try,x-y〉,∀x,y∈H.(iii)
F(Tr)=EP(F).(iv)
EP(F) is closed and convex.Lemma 6 (see [32]).
LetC be a nonempty closed convex subset of a real Hilbert space H. If T:C→C is a k-strict pseudocontraction, then (i)
the mappingI-T is demiclosed at 0; that is, if {xn} is a sequence in C weakly converging to x and if {(I-T)xn} converges strongly to 0, then (I-T)x=0;(ii)
the setF(T) of T is closed and convex so that the projection PF(T) is well defined.Lemma 7 (see [29]).
Assume that{an} is a sequence of nonnegative real numbers such that
(27)an+1≤(1-γn)an+δn,
where {γn} is a sequence in (0,1) and δn is a sequence such that(1)
∑n=1∞γn=∞;(2)
limsupn→∞δn/γn≤0 or ∑n=1∞|δn|<∞.Then, limn→∞an=0.Lemma 8 (see [4]).
LetC be a closed convex subset of H. Let {xn} be a bounded sequence in H. Assume that (i)
the weakw-limit set ww(xn)⊂C where ww(xn)={x:xni⇀x},(ii)
for eachz∈C,limn→∞∥xn-z∥ exists. Then, {xn} is weakly convergent to a point in C.Lemma 9 (see [33]).
LetH be a Hilbert space, C a closed and convex subset of H, and T:C→C a k-strict pseudocontraction mapping. Define a mapping V:C→H by Vx=λx+(1-λ)Tx,  forallx∈C. Then, as k≤λ<1,V is a nonexpansive mapping such that F(V)=F(T).Lemma 10 (see [11]).
LetH be a Hilbert space, C a closed and convex subset of H, and T:C→C a nonexpansive mapping such that F(T)≠∅. Then,
(28)∥Tx-x∥2≤2〈x-Tx,x-x′〉,∀x′∈F(T),∀x∈C.


## 3. The Proposed Method and Some Properties

In this section, we suggest and analyze our method for finding the common solutions of the system of variational inequality problem (1), the mixed equilibrium problem (12), and the hierarchical fixed point problem (15).LetC be a nonempty closed convex subset of a real Hilbert space H. Let D,Bi:C→H be η,θi-inverse strongly monotone mappings for each i=1,2, respectively. Let F:C×C→ℝ be a bifunction satisfying the assumptions (i)–(iv) of Lemma 4, S:C→H a nonexpansive mapping, and {Ti}i=1∞:C→C a countable family of ki-strict pseudocontraction mappings such that S*∩MEP(F)∩F(T)≠∅, where F(T):=⋂i=1∞‍F(Ti)=⋂i=1∞‍F(Vi). Let f be a ρ-contraction mapping.Algorithm 11.
For a givenx0∈C arbitrarily, let the iterative sequences {un},{xn},{yn}, and {zn} be generated by
(29)F(un,y)+〈Dxn,y-un〉+1rn〈y-un,un-xn〉≥0,+〈Dxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C;zn=PC[PC[un-μ2B2un]-μ1B1PC[un-μ2B2un]];yn=PC[βnSxn+(1-βn)zn];xn+1=PC[αnf(xn)+∑i=1n(αi-1-αi)Viyn],∀n≥0,
where Vi=kiI+(1-ki)Ti,0≤ki<1,μi∈(0,2θi) for each i=1,2,{rn}⊂(0,2η),α0=1,{αn} is a strictly decreasing sequence in (0,1), and {βn} is a sequence in (0,1) satisfying the following conditions: (a)
limn→∞αn=0 and ∑n=1∞‍αn=∞,(b)
limn→∞(βn/αn)=0,(c)
∑n=1∞‍|αn-1-αn|<∞ and ∑n=1∞‍|βn-1-βn|<∞,(d)
liminfn→∞rn<limsupn→∞rn<2η and ∑n=1∞‍|rn-1-rn|<∞.Lemma 12.
Letx*∈S*∩MEP(F)∩F(T). Then, {xn},{un},{zn}, and {yn} are bounded.Proof.
First, we show that the mapping(I-rnD) is nonexpansive. For any x,y∈C,
(30)∥(I-rnD)x-(I-rnD)y∥2=∥(x-y)-rn(Dx-Dy)∥2=∥x-y∥2-2rn〈x-y,Dx-Dy〉+rn2∥Dx-Dy∥2≤∥x-y∥2-rn(2η-rn)∥Dx-Dy∥2≤∥x-y∥2.
Similarly, we can show that the mapping (I-μiBi) is nonexpansive for each i=1,2. It follows from Lemma 5 that un=Trn(xn-rnDxn). Let x*∈S*∩MEP(F)∩F(T); we have x*=Trn(x*-rnDx*), and it follows that
(31)∥un-x*∥2=∥Trn(xn-rnDxn)-Trn(x*-rnDx*)∥2≤∥(xn-rnDxn)-(x*-rnDx*)∥2≤∥xn-x*∥2-rn(2η-rn)∥Dxn-Dx*∥2≤∥xn-x*∥2.
Let x*∈S*∩MEP(F)∩F(T); we have
(32)x*=PC[y*-μ1B1y*],
where
(33)y*=PC[x*-μ2B2x*].
Setting vn=PC[un-μ2B2un]. Since B2 is θ2-inverse strongly monotone mapping, it follows that
(34)∥vn-y*∥2=∥PC[un-μ2B2un]-PC[x*-μ2B2x*]∥2≤∥un-u*-μ2(B2un-B2x*)∥2≤∥xn-x*∥2-μ2(2θ-μ2)∥B2un-B2x*∥2≤∥xn-x*∥2.
Since Bi is θi-inverse strongly monotone mappings for each i=1,2, we get
(35)∥zn-x*∥2=∥PC[PC[un-μ2B2un]-μ1B1PC[un-μ2B2un]]-PC[PC[x*-μ2B2x*]-μ1B1PC[x*-μ2B2x*]]∥2≤∥PC[un-μ2B2un]-μ1B1PC[un-μ2B2un]-(PC[x*-μ2B2x*]-μ1B1PC[x*-μ2B2x*])∥2=∥PC[un-μ2B2un]-PC[x*-μ2B2x*]-μ1(B1PC[un-μ2B2un]-B1PC[x*-μ2B2x*])∥2≤∥PC[un-μ2B2un]-PC[x*-μ2B2x*]∥2-μ1(2θ1-μ1)∥B1PC[un-μ2B2un]-B1PC[x*-μ2B2x*]∥2≤∥(un-μ2B2un)-(x*-μ2B2x*)∥2-μ1(2θ1-μ1)∥B1PC[un-μ2B2un]-B1PC[x*-μ2B2x*]∥2≤∥un-x*∥2-μ2(2θ2-μ2)∥B2un-B2x*∥2-μ1(2θ1-μ1)∥B1vn-B1y*∥2≤∥un-x*∥2≤∥xn-x*∥2.
Next, we prove that the sequence {xn} is bounded; without loss of generality we can assume that βn≤αn for all n≥1. From (29), we have
(36)∥xn+1-x*∥≤∥αnf(xn)+∑i=1n‍(αi-1-αi)Viyn-αnx*-∑i=1n(αi-1-αi)Vix*∥≤αn∥f(xn)-f(x*)∥+αn∥f(x*)-x*∥+∑i=1n(αi-1-αi)∥Viyn-Vix*∥≤αn∥f(xn)-f(x*)∥+αn∥f(x*)-x*∥+∑i=1n(αi-1-αi)∥yn-x*∥≤αn∥f(xn)-f(x*)∥+αn∥f(x*)-x*∥+(1-αn)∥βnSxn+(1-βn)zn-x*∥≤αn∥f(xn)-f(x*)∥+αn∥f(x*)-x*∥+(1-αn)(βn∥Sxn-Sx*∥+βn∥Sx*-x*∥+(1-βn)∥zn-x*∥)≤αnρ∥xn-x*∥+αn∥f(x*)-x*∥+(1-αn)(βn∥xn-x*∥+βn∥Sx*-x*∥hhhhhhhhhhhh+(1-βn)∥xn-x*∥)=(1-αn(1-ρ))×∥xn-x*∥+αn∥f(x*)-x*∥+(1-αn)βn∥Sx*-x*∥≤(1-αn(1-ρ))×∥xn-x*∥+αn∥f(x*)-x*∥+βn∥Sx*-x*∥≤(1-αn(1-ρ))∥xn-x*∥+αn(∥f(x*)-x*∥+∥Sx*-x*∥)=(1-αn(1-ρ))∥xn-x*∥+αn(1-ρ)1-ρ×(∥f(x*)-x*∥+∥Sx*-x*∥)≤max{∥xn-x*∥,11-ρ(∥f(x*)-x*∥+∥Sx*-x*∥)}.
By induction on n, we obtain ∥xn-x*∥≤max{∥x0-x*∥,(1/(1-ρ))(∥f(x*)-x*∥+∥Sx*-x*∥)}, for n≥0 and x0∈C. Hence, {xn} is bounded and, consequently, we deduce that {un},{zn},{vn}, and {yn} are bounded.Lemma 13.
Letx*∈S*∩MEP(F)∩F(T) and {xn} be the sequence generated by Algorithm 11. Then one has(a)
limn→∞∥xn+1-xn∥=0.(b)
The weakw-limit set ww(xn)⊂F(T),(ww(xn)={x:xni⇀x}).Proof.
Sinceun=Trn(xn-rnDxn) and un-1=Trn-1(xn-1-rn-1Dxn-1), we have
(37)F(un,y)+〈Dxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,(38)F(un-1,y)+〈Dxn-1,y-un-1〉+1rn-1〈y-un-1,un-1-xn-1〉≥0,∀y∈C.
Take y=un-1 in (37) and y=un in (38), we get
(39)F(un,un-1)+〈Dxn,un-1-un〉+1rn〈un-1-un,un-xn〉≥0,F(un-1,un)+〈Dxn-1,un-un-1〉+1rn-1〈un-un-1,un-1-xn-1〉≥0.
Adding (39) and using the monotonicity of F, we have
(40)〈Dxn-1-Dxn,un-un-1〉+〈un-un-1,un-1-xn-1rn-1-un-xnrn〉≥0,
which implies that
(41)0≤〈un-un-1,rn(Dxn-1-Dxn)+rnrn-1(un-1-xn-1)-(un-xn)rnrn-1〉=〈-(xn-rnDxn)-xn-1+rnrn-1xn-1〉un-1-un,un-un-1+(1-rnrn-1)un-1+(xn-1-rnDxn-1)-(xn-rnDxn)-xn-1+rnrn-1xn-1〉=〈un-1-un,(1-rnrn-1)un-1+(xn-1-rnDxn-1)-(xn-rnDxn)-xn-1+rnrn-1xn-1〉-∥un-un-1∥2=〈un-1-un,(1-rnrn-1)(un-1-xn-1)+(xn-1-rnDxn-1)-(xn-rnDxn)(1-rnrn-1)〉-∥un-un-1∥2≤∥un-1-un∥×{|1-rnrn-1|∥un-1-xn-1∥+∥(xn-1-rnDxn-1)-(xn-rnDxn)∥|1-rnrn-1|}-∥un-un-1∥2≤∥un-1-un∥×{|1-rnrn-1|∥un-1-xn-1∥+∥xn-1-xn∥}-∥un-un-1∥2
and then
(42)∥un-1-un∥≤|1-rnrn-1|∥un-1-xn-1∥+∥xn-1-xn∥.
Without loss of generality, let us assume that there exists a real number μ such that rn>μ>0, for all positive integers n. Then, we get
(43)∥un-1-un∥≤∥xn-1-xn∥+1μ|rn-1-rn|∥un-1-xn-1∥.
Next, we estimate
(44)∥zn-zn-1∥2=∥PC[PC[un-μ2B2un]-μ1B1PC[un-μ2B2un]]-PC[PC[un-1-μ2B2un-1]-μ1B1PC[un-1-μ2B2un-1]]∥2≤∥PC[un-μ2B2un]-μ1B1PC[un-μ2B2un]-(PC[un-1-μ2B2un-1]-μ1B1PC[un-1-μ2B2un-1])∥2=∥PC[un-μ2B2un]-PC[un-1-μ2B2un-1]-μ1(B1PC[un-μ2B2un]-B1PC[un-1-μ2B2un-1])∥2≤∥PC[un-μ2B2un]-PC[un-1-μ2B2un-1]∥2-μ1(2θ1-μ1)×∥B1PC[un-μ2B2un]-B1PC[un-1-μ2B2un-1]∥2≤∥PC[un-μ2B2un]-PC[un-1-μ2B2un-1]∥2≤∥(un-un-1)-μ2(B2un-B2un-1)∥2≤∥un-un-1∥2-μ2(2θ2-μ2)∥B2un-B2un-1∥2≤∥un-un-1∥2.
It follows from (43) and (44) that
(45)∥zn-zn-1∥≤∥xn-1-xn∥+1μ|rn-1-rn|∥un-1-xn-1∥.
From (29) and the previous inequality, we get
(46)∥yn-yn-1∥≤∥βnSxn+(1-βn)zn-(βn-1Sxn-1+(1-βn-1)zn-1)∥=∥βn(Sxn-Sxn-1)+(βn-βn-1)Sxn-1+(1-βn)(zn-zn-1)+(βn-1-βn)zn-1∥≤βn∥xn-xn-1∥+(1-βn)∥zn-zn-1∥+|βn-βn-1|(∥Sxn-1∥+∥zn-1∥)≤βn∥xn-xn-1∥+(1-βn)×{∥xn-1-xn∥+1μ|rn-1-rn|∥un-1-xn-1∥}+|βn-βn-1|(∥Sxn-1∥+∥zn-1∥)≤∥xn-xn-1∥+1μ|rn-1-rn|∥un-1-xn-1∥+|βn-βn-1|(∥Sxn-1∥+∥zn-1∥).
Next, we estimate
(47)∥xn+1-xn∥≤∥αnf(xn)+∑i=1n(αi-1-αi)Viyn-(αn-1f(xn-1)+∑i=1n-1(αi-1-αi)Viyn-1)∥=∥αn(f(xn)-f(xn-1))+(αn-αn-1)f(xn-1)+∑i=1n(αi-1-αi)(Viyn-Viyn-1)+(αn-1-αn)Vnyn-1∥≤αn∥f(xn)-f(xn-1)∥+∑i=1n(αi-1-αi)∥Viyn-Viyn-1∥+|αn-αn-1|(∥f(xn-1)∥+∥Vnyn-1∥)≤αnρ∥xn-xn-1∥+∑i=1n(αi-1-αi)∥yn-yn-1∥+|αn-αn-1|(∥f(xn-1)∥+∥Vnyn-1∥)=αnρ∥xn-xn-1∥+(1-αn)∥yn-yn-1∥+|αn-αn-1|(∥f(xn-1)∥+∥Vnyn-1∥).
From (46) and (47), we have
(48)∥xn+1-xn∥≤αnρ∥xn-xn-1∥+(1-αn)×{∥xn-xn-1∥+1μ|rn-1-rn|∥un-1-xn-1∥+|βn-βn-1|(∥Sxn-1∥+∥zn-1∥)1μ}+|αn-αn-1|(∥f(xn-1)∥+∥Vnyn-1∥)≤(1-(1-ρ)αn)∥xn-xn-1∥+1μ|rn-1-rn|×∥un-1-xn-1∥+|βn-βn-1|(∥Sxn-1∥+∥zn-1∥)+|αn-αn-1|(∥f(xn-1)∥+∥Vnyn-1∥)≤(1-(1-ρ)αn)∥xn-xn-1∥+M×(1μ|rn-1-rn|+|βn-βn-1|+|αn-αn-1|),
where
(49)M=max{supn≥1∥un-1-xn-1∥,supn≥1(∥Sxn-1∥+∥zn-1∥),supn≥1(∥f(xn-1)∥+∥Vnyn-1∥)}.
It follows by conditions (a)–(d) of Algorithm 11 and Lemma 7 that
(50)limn→∞∥xn+1-xn∥=0.
Next, we show that limn→∞∥un-xn∥=0. Since x*∈S*∩MEP(F)∩F(T) and αn+∑i=1n‍(αi-1-αi)=1, by using (31) and (35), we obtain
(51)∥xn+1-x*∥2≤∥αnf(xn)+∑i=1n‍(αi-1-αi)Viyn-αnx*-∑i=1n‍(αi-1-αi)Vix*∥2≤αn∥f(xn)-x*∥2+∑i=1n(αi-1-αi)∥Viyn-Vix*∥2≤αn∥f(xn)-x*∥2+∑i=1n(αi-1-αi)∥yn-x*∥2≤αn∥f(xn)-x*∥2+(1-αn)×(βn∥Sxn-x*∥2+(1-βn)∥zn-x*∥2)≤αn∥f(xn)-x*∥2+(1-αn)βn∥Sxn-x*∥2+(1-αn)(1-βn)×{∥un-x*∥2-μ2(2θ2-μ2)×∥B2un-B2x*∥2-μ1(2θ1-μ1)∥B1vn-B1y*∥2}≤αn∥f(xn)-x*∥2+βn∥Sxn-x*∥2+(1-αn)(1-βn)×{∥xn-x*∥2-rn(2η-rn)∥Dxn-Dx*∥2-μ2(2θ2-μ2)∥B2un-B2x*∥2-μ1(2θ1-μ1)∥B1vn-B1y*∥2}≤αn∥f(xn)-x*∥2+βn∥Sxn-x*∥2+∥xn-x*∥2-(1-αn)(1-βn){rn(2η-rn)∥Dxn-Dx*∥2+μ2(2θ2-μ2)∥B2un-B2x*∥2+μ1(2θ1-μ1)∥B1vn-B1y*∥2}.
Then, from the previous inequality, we get
(52)(1-αn)(1-βn){rn(2η-rn)∥Dxn-Dx*∥2+μ2(2θ2-μ2)∥B2un-B2x*∥2+μ1(2θ1-μ1)∥B1vn-B1y*∥2}≤αn∥f(xn)-x*∥2+βn∥Sxn-x*∥2+∥xn-x*∥2-∥xn+1-x*∥2≤αn∥f(xn)-x*∥2+βn∥Sxn-x*∥2+(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥.
Since liminfn→∞rn≤limsupn→∞rn<2η,limn→∞∥xn+1-xn∥=0,αn→0, and βn→0, we obtain limn→∞∥B2un-B2x*∥=0,limn→∞∥B1vn-B1y*∥=0, and limn→∞∥Dxn-Dx*∥=0.
SinceTrn is firmly nonexpansive, we have
(53)∥un-x*∥2=∥Trn(xn-rnDxn)-Trn(x*-rnDx*)∥2≤〈un-x*,(xn-rnDxn)-(x*-rnDx*)〉=12{∥un-x*∥2+∥(xn-rnDxn)-(x*-rnDx*)∥2-∥un-x*-[(xn-rnDxn)-(x*-rnDx*)]∥2}.
Hence,
(54)∥un-x*∥2≤∥(xn-rnDxn)-(x*-rnDx*)∥2-∥un-xn+rn(Dxn-Dx*)∥2≤∥xn-x*∥2-∥un-xn+rn(Dxn-Dx*)∥2≤∥xn-x*∥2-∥un-xn∥2+2rn∥un-xn∥∥Dxn-Dx*∥.
From (51), (35), and the previous inequality, we have
(55)∥xn+1-x*∥2≤αn∥f(xn)-x*∥2+(1-αn)(βn∥Sxn-x*∥2+(1-βn)∥zn-x*∥2)≤αn∥f(xn)-x*∥2+(1-αn)×(βn∥Sxn-x*∥2+(1-βn)∥un-x*∥2)≤αn∥f(xn)-x*∥2+(1-αn)×{βn∥Sxn-x*∥2+(1-βn)×(∥xn-x*∥2-∥un-xn∥2+2rn∥un-xn∥∥Dxn-Dx*∥)}≤αn∥f(xn)-x*∥2+βn∥Sxn-x*∥2+∥xn-x*∥2-(1-αn)(1-βn)∥un-xn∥2+2rn∥un-xn∥∥Dxn-Dx*∥.
Hence,
(56)(1-αn)(1-βn)∥un-xn∥2≤αn∥f(xn)-x*∥2+βn∥Sxn-x*∥2+∥xn-x*∥2-∥xn+1-x*∥2+2rn∥un-xn∥∥Dxn-Dx*∥≤αn∥f(xn)-x*∥2+βn∥Sxn-x*∥2+(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥+2rn∥un-xn∥∥Dxn-Dx*∥.
Since limn→∞∥xn+1-xn∥=0,αn→0,βn→0, and limn→∞∥Dxn-Dx*∥=0, we obtain
(57)limn→∞∥un-xn∥=0.
From (21), we get
(58)∥vn-y*∥2=∥PC[un-μ2B2un]-PC[x*-μ2B2x*]∥2≤〈vn-y*,(un-μ2B2un)-(x*-μ2B2x*)〉=12{∥vn-y*∥2+∥un-x*-μ2(B2un-B2x*)∥2-∥un-x*-μ2(B2un-B2x*)-(vn-y*)∥2}≤12{∥vn-y*∥2+∥un-x*∥2-μ2(2θ2-μ2)∥B2un-B2x*∥2-∥un-x*-μ2(B2un-B2x*)-(vn-y*)∥2}≤12{∥vn-y*∥2+∥un-x*∥2-∥un-vn-μ2(B2un-B2x*)-(x*-y*)∥2}≤12{∥vn-y*∥2+∥un-x*∥2-∥un-vn-(x*-y*)∥2+2μ2〈un-vn-(x*-y*),B2un-B2x*〉-μ22∥B2un-B2x*∥2}≤12{∥vn-y*∥2+∥un-x*∥2-∥un-vn-(x*-y*)∥2+2μ2∥un-vn-(x*-y*)∥∥B2un-B2x*∥}.
Hence,
(59)∥vn-y*∥2≤∥un-x*∥2-∥un-vn-(x*-y*)∥2+2μ2∥un-vn-(x*-y*)∥∥B2un-B2x*∥≤∥xn-x*∥2-∥un-xn∥2+2rn∥un-xn∥∥Dxn-Dx*∥-∥un-vn-(x*-y*)∥2+2μ2∥un-vn-(x*-y*)∥∥B2un-B2x*∥,
where the last inequality follows from (54). On the other hand, from (29) and (21), we obtain
(60)∥zn-x*∥2=∥PC[vn-μ1B1vn]-PC[y*-μ1B1y*]∥2≤〈zn-x*,(vn-μ1B1vn)-(y*-μ1B1y*)〉=12{∥zn-x*∥2+∥vn-y*-μ1(B1vn-B1y*)∥2-∥vn-y*-μ1(B1vn-B1y*)-(zn-x*)∥2}=12{∥zn-x*∥2+∥vn-y*∥2-2μ1〈vn-y*,B1vn-B1y*〉+μ12∥B1vn-B1y*∥2-∥vn-y*-μ1(B1vn-B1y*)-(zn-x*)∥2}≤12{∥zn-x*∥2+∥vn-y*∥2-μ1(2θ1-μ1)∥B1vn-B1y*∥2-∥vn-y*-μ1(B1vn-B1y*)-(zn-x*)∥2}≤12{∥zn-x*∥2+∥vn-y*∥2-∥vn-zn-μ1(B1vn-B1y*)+(x*-y*)∥2}≤12{∥zn-x*∥2+∥vn-y*∥2-∥vn-zn+(x*-y*)∥2+2μ1〈vn-zn+(x*-y*),B1vn-B1y*〉}≤12{∥zn-x*∥2+∥vn-y*∥2-∥vn-zn+(x*-y*)∥2+2μ1∥vn-zn+(x*-y*)∥×∥B1vn-B1y*∥}.
Hence,
(61)∥zn-x*∥2≤∥vn-y*∥2-∥vn-zn+(x*-y*)∥2+2μ1∥vn-zn+(x*-y*)∥∥B1vn-B1y*∥≤∥xn-x*∥2-∥un-xn∥2+2rn∥un-xn∥∥Dxn-Dx*∥-∥un-vn-(x*-y*)∥2+2μ2∥un-vn-(x*-y*)∥∥B2un-B2x*∥-∥vn-zn+(x*-y*)∥2+2μ1∥vn-zn+(x*-y*)∥∥B1vn-B1y*∥,
where the last inequality follows from (59). From (51) and the previous inequality, we have
(62)∥xn+1-x*∥2≤αn∥f(xn)-x*∥2+(1-αn)×(βn∥Sxn-x*∥2+(1-βn)∥zn-x*∥2)≤αn∥f(xn)-x*∥2+(1-αn)×{βn∥Sxn-x*∥2+(1-βn)(∥xn-x*∥2-∥un-xn∥2+2rn∥un-xn∥∥Dxn-Dx*∥)+(1-βn)(-∥un-vn-(x*-y*)∥2+2μ2∥un-vn-(x*-y*)∥×∥B2un-B2x*∥)+(1-βn)(-∥vn-zn+(x*-y*)∥2+2μ1∥vn-zn+(x*-y*)∥×∥B1vn-B1y*∥-∥vn-zn+(x*-y*)∥2)∥Sxn-x*∥2}≤αn∥f(xn)-x*∥2+βn∥Sxn-x*∥2+∥xn-x*∥2+2rn∥un-xn∥×∥Dxn-Dx*∥+2μ2∥un-vn-(x*-y*)∥×∥B2un-B2x*∥+2μ1∥vn-zn+(x*-y*)∥×∥B1vn-B1y*∥-(1-αn)(1-βn)×{∥un-xn∥2+∥un-vn-(x*-y*)∥2+∥vn-zn+(x*-y*)∥2}.
Hence,
(63)(1-αn)(1-βn)×{∥un-xn∥2+∥un-vn-(x*-y*)∥2+∥vn-zn+(x*-y*)∥2}≤αn∥f(xn)-x*∥2+βn∥Sxn-x*∥2+∥xn-x*∥2-∥xn+1-x*∥2+2rn∥un-xn∥∥Dxn-Dx*∥+2μ2∥un-vn-(x*-y*)∥∥B2un-B2x*∥+2μ1∥vn-zn+(x*-y*)∥∥B1vn-B1y*∥≤αn∥f(xn)-x*∥2+βn∥Sxn-x*∥2+(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥+2rn∥un-xn∥∥Dxn-Dx*∥+2μ2∥un-vn-(x*-y*)∥∥B2un-B2x*∥+2μ1∥vn-zn+(x*-y*)∥∥B1vn-B1y*∥.
Since limn→∞∥xn+1-xn∥=0,αn→0,βn→0, and limn→∞∥Dxn-Dx*∥=0, limn→∞∥B2un-B2x*∥=0, limn→∞∥B1vn-B1y*∥=0, we obtain
(64)limn→∞∥un-vn-(x*-y*)∥=0,limn→∞∥vn-zn+(x*-y*)∥=0.
Since
(65)∥un-zn∥≤∥un-vn-(x*-y*)∥+∥vn-zn+(x*-y*)∥,
we get
(66)limn→∞∥un-zn∥=0.
It follows from (57) and (66) that
(67)limn→∞∥xn-zn∥=0.
Now, let z∈F(T)∩S*∩MEP(F); since for each i≥1,Vixn∈C, and αn+∑i=1n‍(αi-1-αi)=1, we have ∑i=1n‍(αi-1-αi)Vixn+αnz∈C. And
(68)∑i=1n(αi-1-αi)(xn-Vixn)=PC[αnf(xn)+∑i=1n(αi-1-αi)Viyn]+(1-αn)xn-(∑i=1n(αi-1-αi)Vixn+αnz)+αnz-xn+1=PC[αnf(xn)+∑i=1n(αi-1-αi)Viyn]+αn(z-xn+1)-PC[∑i=1n(αi-1-αi)Vixn+αnz]+(1-αn)(xn-xn+1).
It follows that
(69)∑i=1n(αi-1-αi)〈xn-Vixn,xn-x*〉=〈PC[αnf(xn)+∑i=1n(αi-1-αi)Viyn]-PC[∑i=1n(αi-1-αi)Vixn+αnz],xn-x*〉+αn〈z-xn+1,xn-x*〉+(1-αn)〈xn-xn+1,xn-x*〉≤∥αn(f(xn)-z)+∑i=1n(αi-1-αi)(Viyn-Vixn)∥×∥xn-x*∥+αn∥z-xn+1∥∥xn-x*∥+(1-αn)∥xn-xn+1∥∥xn-x*∥≤αn∥f(xn)-z∥∥xn-x*∥+∑i=1n‍(αi-1-αi)∥yn-xn∥∥xn-x*∥+αn∥z-xn+1∥∥xn-x*∥+(1-αn)∥xn-xn+1∥∥xn-x*∥=αn∥f(xn)-z∥∥xn-x*∥+(1-αn)∥yn-xn∥∥xn-x*∥+αn∥z-xn+1∥∥xn-x*∥+(1-αn)∥xn-xn+1∥∥xn-x*∥≤αn∥f(xn)-z∥∥xn-x*∥+(1-αn)∥βnSxn+(1-βn)zn-xn∥×∥xn-x*∥+αn∥z-xn+1∥∥xn-x*∥+(1-αn)∥xn-xn+1∥∥xn-x*∥≤αn∥f(xn)-z∥∥xn-x*∥+(1-αn)βn∥Sxn-xn∥∥xn-x*∥+(1-αn)(1-βn)∥zn-xn∥∥xn-x*∥+αn∥z-xn+1∥∥xn-x*∥+(1-αn)∥xn-xn+1∥∥xn-x*∥.
From Lemma 10 and the previous inequality, we get
(70)12∑i=1n(αi-1-αi)∥xn-Vixn∥2≤∑i=1n(αi-1-αi)〈xn-Vixn,xn-x*〉≤αn∥f(xn)-z∥∥xn-x*∥+(1-αn)βn∥Sxn-xn∥∥xn-x*∥+(1-αn)(1-βn)∥zn-xn∥∥xn-x*∥+αn∥z-xn+1∥∥xn-x*∥+(1-αn)∥xn-xn+1∥∥xn-x*∥.
Since limn→∞∥xn+1-xn∥=0,αn→0,βn→0, and limn→∞∥xn-zn∥=0, we obtain
(71)limn→∞∑i=1n(αi-1-αi)∥xn-Vixn∥2=0.
Since (αi-1-αi)∥xn-Vixn∥2≤∑i=1n‍(αi-1-αi)∥xn-Vixn∥2 and {αn} is strictly decreasing, we have
(72)limn→∞∥xn-Vixn∥=0.
Hence, we obtain
(73)limn→∞∥xn-Tixn∥=limn→∞∥xn-Vixn∥(1-ki)=0,∀i≥1.
Since {xn} is bounded, without loss of generality, we can assume that xn⇀w∈C. It follows from Lemma 6 that w∈F(T). Therefore, ww(xn)⊂F(T).Theorem 14.
The sequence{xn} generated by Algorithm 11 converges strongly to z=PS*∩MEP(F)∩F(T)f(z), which is the unique solution of the variational inequality
(74)〈(I-f)z,x-z〉≥0,∀x∈S*∩MEP(F)∩F(T).Proof.
Since{xn} is bounded xn⇀w and from Lemma 13, we have w∈F(T). Next, we show that w∈MEP(F). Since un=Trn(xn-rnDxn), we have
(75)F(un,y)+〈Dxn,y-un〉+1rn〈y-un,un-xn〉≥0,+〈Dxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C.
It follows from monotonicity of F that
(76)〈Dxn,y-un〉+1rn〈y-un,un-xn〉≥F(y,un),∀y∈C,〈Dxnk,y-unk〉+〈y-unk,unk-xnkrnk〉≥F(y,unk),∀y∈C.
Since limn→∞∥un-xn∥=0 and xn⇀w, it easy to observe that unk→w. For any 0<t≤1 and y∈C, let yt=ty+(1-t)w; we have yt∈C. Then from (76), we obtain
(77)〈Dyt,yt-unk〉≥〈Dyt,yt-unk〉-〈Dxnk,yt-unk〉-〈yt-unk,unk-xnkrnk〉+F(yt,unk)=〈Dyt-Dunk,yt-unk〉+〈Dunk-Dxnk,yt-unk〉-〈yt-unk,unk-xnkrnk〉+F(yt,unk).
Since D is Lipschitz continuous and limn→∞∥un-xn∥=0, we obtain limk→∞∥Dunk-Dxnk∥=0. From the monotonicity of D and unk→w, it follows from (77) that
(78)〈Dyt,yt-w〉≥F(yt,w).
Hence, from assumptions (i)–(iv) of Lemma 4 and (78), we have
(79)0=F(yt,yt)≤tF(yt,y)+(1-t)F(yt,w)≤tF(yt,y)+(1-t)〈Dyt,yt-w〉≤tF(yt,y)+(1-t)t〈Dyt,y-w〉
which implies that F(yt,y)+(1-t)〈Dyt,y-w〉≥0. Letting t→0+, we have
(80)F(w,y)+〈Dw,y-w〉≥0,∀y∈C,
which implies that w∈MEP(F). Next, we show that w∈S*. Since limn→∞∥xn-zn∥=0 and there exists a subsequence {xnk} of {xn} such that xnk→w, it easy to observe that znk→w. For any x,y∈C, using (23), we have
(81)∥Q(x)-Q(y)∥2=∥PC[PC[x-μ2B2x]-μ1B1PC[x-μ2B2x]]-PC[PC[y-μ2B2y]-μ1B1PC[y-μ2B2y]]∥2≤∥(PC[x-μ2B2x]-PC[y-μ2B2y])-μ1(B1PC[x-μ2B2x]-B1PC[y-μ2B2y])∥2≤∥PC[x-μ2B2x]-PC[y-μ2B2y]∥2-μ1(2θ1-μ1)∥PC[x-μ2B2x]-PC[y-μ2B2y]∥2≤∥PC[x-μ2B2x]-PC[y-μ2B2y]∥2≤∥(x-μ2B2x)-(y-μ2B2y)∥2≤∥x-y∥2-μ2(2θ2-μ2)∥B2x-B2y∥2≤∥x-y∥2.
This implies that Q:C→C is nonexpansive. On the other hand,
(82)∥zn-Q(zn)∥2=∥PC[PC[un-μ2B2un]-μ1B1PC[un-μ2B2un]]-Q(zn)∥2=∥Q(un)-Q(zn)∥2≤∥un-zn∥2.
Since limn→∞∥un-zn∥=0 (see (66)), we have limn→∞∥zn-Q(zn)∥=0. It follows from Lemma 6 that w=Q(w), which implies from Lemma 3 that w∈S*.
Thus, we have(83)w∈S*∩MEP(F)∩F(T).
Next, we claim that limsupn→∞〈f(z)-z,xn-z〉≤0 where z=PS*∩MEP(F)∩F(T)f(z).
Since{xn} is bounded, there exists a subsequence {xnk} of {xn} such that
(84)limsupn→∞〈f(z)-z,xn-z〉=limsupk→∞〈f(z)-z,xnk-z〉=〈f(z)-z,w-z〉≤0.
Next, we show that xn→z.
One has(85)∥xn+1-z∥2=〈-∑i=1n(αi-1-αi)Viyn,xn+1-zxn+1-αnf(xn)-∑i=1n(αi-1-αi)Viyn,xn+1-z〉+〈∑i=1n(αi-1-αi)Viyn-z,xn+1-zαnf(xn)+∑i=1n(αi-1-αi)Viyn-z,xn+1-z〉≤〈∑i=1n(αi-1-αi)Viyn-z,xn+1-zαnf(xn)+∑i=1n(αi-1-αi)Viyn-z,xn+1-z〉=αn〈f(xn)-f(z),xn+1-z〉+αn〈f(z)-z,xn+1-z〉+∑i=1n(αi-1-αi)〈Viyn-z,xn+1-z〉≤αn∥f(xn)-f(z)∥∥xn+1-z∥+αn〈f(z)-z,xn+1-z〉+∑i=1n(αi-1-αi)∥Viyn-z∥∥xn+1-z∥≤αnρ∥xn-z∥∥xn+1-z∥+αn〈f(z)-z,xn+1-z〉+∑i=1n(αi-1-αi)∥yn-z∥∥xn+1-z∥≤αnρ∥xn-z∥∥xn+1-z∥+αn〈f(z)-z,xn+1-z〉+(1-αn)×{βn∥Sxn-Sz∥+βn∥Sz-z∥+(1-βn)∥zn-z∥}×∥xn+1-z∥≤αnρ∥xn-z∥∥xn+1-z∥+αn〈f(z)-z,xn+1-z〉×(1-αn){βn∥xn-z∥+βn∥Sz-z∥+(1-βn)∥xn-z∥}∥xn+1-z∥≤(1-αn(1-ρ))∥xn-z∥×∥xn+1-z∥+αn〈f(z)-z,xn+1-z〉+(1-αn)βn∥Sz-z∥∥xn+1-z∥≤1-αn(1-ρ)2(∥xn-z∥2+∥xn+1-z∥2)+αn〈f(z)-z,xn+1-z〉+(1-αn)βn∥Sz-z∥∥xn+1-z∥
which implies that
(86)∥xn+1-z∥2≤(1-2αn(1-ρ)1+αn(1-ρ))∥xn-z∥2+2αn1+αn(1-ρ)〈f(z)-z,xn+1-z〉+2(1-αn)βn1+αn(1-ρ)∥Sz-z∥∥xn+1-z∥≤(1-2αn(1-ρ)1+αn(1-ρ))∥xn-z∥2+2αn(1-ρ)1+αn(1-ρ){11-ρ〈f(z)-z,xn+1-z〉+(1-αn)βnαn(1-ρ)∥Sz-z∥×∥xn+1-z∥11-ρ}.
Let γn=2αn(1-ρ)/(1+αn(1-ρ)) and δn=(2αn(1-ρ)/(1+αn(1-ρ))){(1/(1-ρ))〈f(z)-z,xn+1-z〉+((1-αn)βn/(αn(1-ρ)))∥Sz-z∥∥xn+1-z∥}.
Since(87)∑n=1∞‍αn=∞,1+αn(1-ρ)≤2,limsupn→∞{11-ρ〈f(z)-z,xn+1-z〉+(1-αn)βnαn(1-ρ)∥Sz-z∥∥xn+1-z∥}≤0.
It follows that
(88)∑n=1∞γn=∞limsupn→∞δnγnlim≤0.
Thus, all the conditions of Lemma 7 are satisfied. Hence, we deduce that xn→z.
SincePS*∩MEP(F)∩F(T)f is a contraction, there exists a unique z∈C such that z=PS*∩MEP(F)∩F(T)f(z). From (20), it follows that z is the unique solution of the problem (74). This completes the proof.Theorem 15.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let D,Bi:C→H be η,θi-inverse strongly monotone mappings for each i=1,2, respectively. Let F:C×C→ℝ be a bifunction satisfying the assumptions (i)–(iv) of Lemma 4, S:C→H a nonexpansive mapping, and {Ti}i=1∞:C→C a countable family of ki-strict pseudocontraction mappings such that S*∩MEP(F)∩F(T)≠∅, where F(T):=⋂i=1∞‍F(Ti)=⋂i=1∞‍F(Vi). Let f be a ρ-contraction mapping. For a given x0∈C arbitrarily, let the iterative sequences {un},{xn},{yn}, and {zn} be generated by
(89)F(un,y)+〈Dxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C;zn=PC[PC[un-μ2B2un]-μ1B1PC[un-μ2B2un]];yn=βnSxn+(1-βn)zn;xn+1=PC[αnf(xn)+∑i=1n(αi-1-αi)Viyn],∀n≥0,
where Vi=kiI+(1-ki)Ti,0≤ki<1,μi∈(0,2θi) for each i=1,2,{rn}⊂(0,2η),α0=1, {αn} is a strictly decreasing sequence in (0,1), and {βn} is a sequence in (0,1) satisfying the following conditions: (a)
limn→∞αn=0 and ∑n=1∞‍αn=∞,(b)
limn→∞(βn/αn)=τ∈(0,∞),(c)
∑n=1∞‍(αn-1-αn)<∞ and ∑n=1∞‍|βn-1-βn|<∞,(d)
limn→∞((1/μ)|rn-1-rn|+|αn-1-αn|+|βn-1-βn|/αnβn)=0,(e)
there exists a constantK>0 such that (1/αn)|(1/βn)-(1/βn-1)|≤K,(f)
liminfn→∞rn<limsupn→∞rn<2η and ∑n=1∞‍|rn-1-rn|<∞. Then, sequence {xn} generated by Algorithm (89) converges strongly to x*∈S*∩MEP(F)∩F(T), which is the unique solution of the variational inequality
(90)〈1τ(I-f)x*+(I-S)x*,x-x*〉≥0,∀x∈S*∩MEP(F)∩F(T).Proof.
Fromlimn→∞(βn/αn)=τ∈(0,∞), without loss of generality, we can assume that βn≤(1+τ)αn for all n≥1. Hence βn→0. By similar argument as that lemmas 12 and 13, we can deduce that {xn} is bounded, limn→∞∥xn+1-xn∥=0,limn→∞∥xn-zn∥=0 (see (67)), and ∥(I-Vi)xn∥→0. Then, we have
(91)∥yn-xn∥≤βn∥xn-Sxn∥+(1-βn)∥xn-zn∥⟶0asn⟶∞.
It follows that, for all i≥1,
(92)∥yn-Vixn∥≤∥yn-xn∥+∥xn-Vixn∥⟶0asn⟶∞.
From (91) and (92), we have
(93)∥yn-Viyn∥≤∥yn-Vixn∥+∥Vixn-Viyn∥≤∥yn-Vixn∥+∥yn-xn∥⟶0asn⟶∞.
Set wn=αnf(xn)+∑i=1n‍(αi-1-αi)Viyn. From (47) and (48), we obtain
(94)∥xn+1-xn∥βn≤∥wn-wn-1∥βn≤(1-(1-ρ)αn)∥xn-xn-1∥βn+M((1/μ)|rn-rn-1|βn+|βn-βn-1|βn+|αn-αn-1|βn)=(1-(1-ρ)αn)∥xn-xn-1∥βn-1+(1-(1-ρ)αn)∥xn-xn-1∥(1βn-1βn-1)+M((1/μ)|rn-rn-1|βn+|βn-βn-1|βn+|αn-αn-1|βn)≤(1-(1-ρ)αn)∥xn-xn-1∥βn-1+∥xn-xn-1∥|1βn-1βn-1|+M((1/μ)|rn-rn-1|βn+|βn-βn-1|βn+|αn-αn-1|βn)≤(1-(1-ρ)αn)∥xn-xn-1∥βn-1+αnK∥xn-xn-1∥+M((1/μ)|rn-rn-1|βn+|βn-βn-1|βn+|αn-αn-1|βn)≤(1-(1-ρ)αn)∥wn-1-wn-2∥βn-1+αnK∥xn-xn-1∥+M((1/μ)|rn-rn-1|βn+|βn-βn-1|βn+|αn-αn-1|βn).
Let γn=(1-ρ)αn and δn=αnK∥xn-xn-1∥+M((1/μ)|rn-rn-1|/βn+|βn-βn-1|/βn+|αn-αn-1|/βn). From conditions (a) and (d), we have
(95)∑n=1∞γn=∞,limn→∞δnγn=0.
By Lemma 7, we obtain
(96)limn→∞∥xn+1-xn∥βn=0,limn→∞∥wn+1-wn∥βn=limn→∞∥wn+1-wn∥αn=0.
From (89), we have
(97)xn+1=PC[wn]-wn+αnf(xn)+∑i=1n(αi-1-αi)(Viyn-yn)+(1-αn)yn.
Hence, it follows that
(98)xn-xn+1=(1-αn)xn+αnxn-(∑i=1n(αi-1-αi)(Viyn-yn)PC[wn]-wn+αnf(xn)+∑i=1n(αi-1-αi)(Viyn-yn)+(1-αn)yn)=(1-αn)[βn(xn-Sxn)+(1-βn)(xn-zn)]+(wn-PC[wn])+∑i=1n‍(αi-1-αi)(yn-Viyn)+αn(xn-f(xn))
and hence
(99)xn-xn+1(1-αn)βn=xn-Sxn+(1-βn)βn(xn-zn)+1(1-αn)βn(wn-PC[wn])+1(1-αn)βn∑i=1n(αi-1-αi)(yn-Viyn)+αn(1-αn)βn(xn-f(xn)).
Let vn=(xn-xn+1)/(1-αn)βn. For any z∈S*∩MEP(F)∩F(T), we have
(100)〈vn,xn-z〉=1(1-αn)βn〈wn-PC[wn],PC[wn-1]-z〉+αn(1-αn)βn〈(I-f)xn,xn-z〉+〈xn-Sxn,xn-z〉+(1-βn)βn〈xn-zn,xn-z〉+1(1-αn)βn∑i=1n(αi-1-αi)〈yn-Viyn,xn-z〉.
Since S is nonexpansive mapping, f is ρ-contraction mapping and Vi is ki-strict pseudocontraction mapping. Then, (I-S) and (I-Vi) are monotones, and f is strongly monotone with coefficient (1-ρ). We can deduce that
(101)〈xn-Sxn,xn-z〉=〈(I-S)xn-(I-S)z,xn-z〉+〈(I-S)z,xn-z〉≥〈(I-S)z,xn-z〉,〈(I-f)xn,xn-z〉=〈(I-f)xn-(I-f)z,xn-z〉+〈(I-f)z,xn-z〉≥(1-ρ)∥xn-z∥2+〈(I-f)z,xn-z〉,〈(I-Vi)yn,xn-z〉=〈(I-Vi)yn-(I-Vi)z,xn-yn〉+〈(I-Vi)yn-(I-Vi)z,yn-z〉≥〈(I-Vi)yn-(I-Vi)z,xn-yn〉=〈(I-Vi)yn,xn-yn〉=〈(I-Vi)yn,βn(xn-Sxn)+(1-βn)(xn-zn)〉.
From (20), we get
(102)〈wn-PC[wn],PC[wn-1]-z〉=〈wn-PC[wn],PC[wn-1]-PC[wn]〉+〈wn-PC[wn],PC[wn]-z〉≥〈wn-PC[wn],PC[wn-1]-PC[wn]〉.
Then, from (100)–(102), we have
(103)〈vn,xn-z〉≥1(1-αn)βn×〈wn-PC[wn],PC[wn-1]-PC[wn]〉+αn(1-αn)βn〈(I-f)z,xn-z〉+〈(I-S)z,xn-z〉+(1-βn)βn〈xn-zn,xn-z〉+(1-βn)(1-αn)βn∑i=1n(αi-1-αi)×〈(I-Vi)yn,xn-zn〉+1(1-αn)∑i=1n(αi-1-αi)×〈(I-Vi)yn,xn-Sxn〉+(1-ρ)αn(1-αn)βn∥xn-z∥2.
Then, we obtain
(104)∥xn-z∥2≤1(1-ρ)αn∥wn-PC[wn]∥∥wn-1-wn∥-1(1-ρ)〈(I-f)z,xn-z〉+(1-αn)βn(1-ρ)αn(〈vn,xn-z〉-〈(I-S)z,xn-z〉)-(1-βn)(1-αn)(1-ρ)αn〈xn-zn,xn-z〉-(1-βn)(1-ρ)αn∑i=1n(αi-1-αi)〈(I-Vi)yn,xn-zn〉-βn(1-ρ)αn∑i=1n(αi-1-αi)×〈(I-Vi)yn,xn-Sxn〉≤∥wn-1-wn∥(1-ρ)αn∥wn-PC[wn]∥-1(1-ρ)〈(I-f)z,xn-z〉+(1-αn)βn(1-ρ)αn×(〈vn,xn-z〉-〈(I-S)z,xn-z〉)+1(1-ρ)(1-βn)βnβnαn∥xn-zn∥∥xn-z∥+1(1-ρ)(1-βn)βnβnαn×∑i=1n(αi-1-αi)∥(I-Vi)yn∥∥xn-zn∥-βn(1-ρ)αn×∑i=1n(αi-1-αi)〈(I-Vi)yn,xn-Sxn〉.
By condition (e) of Theorem 15, there exists a constant N>0 such that ((1-βn)/βn)≤N. Since limn→∞∥xn-zn∥=0, vn→0,(I-Vi)yn→0, and ∥wn-1-wn∥/αn→0 as n→∞, then every weak cluster point of {xn} is also a strong cluster point. Since {xn} is bounded, by Lemma 13 there exists a subsequence {xnk} of {xn} converging to a point x*∈F(T); in similar argument as that Theorem 14 we can show that x*∈S*∩MEP(F)∩F(T).
From (100)–(102), it follows that, for any z∈S*∩MEP(F)∩F(T),
(105)〈(I-f)xnk,xnk-z〉=(1-αnk)βnkαnk〈vnk,xnk-z〉-1αnk×〈wnk-PC[wnk],PC[wnk-1]-z〉-(1-αnk)βnkαnk〈xnk-Sxnk,xnk-z〉-(1-αnk)(1-βnk)αnk〈xnk-znk,xnk-z〉-1αnk∑i=1n(αi-1-αi)〈ynk-Viynk,xnk-z〉≤(1-αnk)βnkαnk〈vnk,xnk-z〉+1αnk∥wnk-PC[wnk]∥×∥wnk-1-wnk∥-(1-αnk)βnkαnk×〈xnk-Sxnk,xnk-z〉+(1-βnk)βnkβnkαnk∥xnk-znk∥×∥xnk-z∥+(1-βnk)βnkβnkαnk×∑i=1nk(αi-1-αi)∥(I-Vi)ynk∥∥xnk-znk∥-βnkαnk∑i=1nk(αi-1-αi)〈(I-Vi)ynk,xnk-Sxnk〉.
Since limn→∞∥xn-zn∥=0,  vn→0,   (I-Vi)yn→0, and ∥wn-1-wn∥/αn→0; letting k→∞ in (105), we obtain
(106)〈(I-f)x*,x*-z〉≤-τ〈x*-Sx*,x*-z〉;
that is,
(107)〈1τ(I-f)x*+(I-S)x*,z-x*〉≥0.
In the following, we show that (90) has unique solution. Assume that x′ is another solution. Then, we have
(108)〈(I-f)x′,x′-x*〉≤-τ〈x′-Sx′,x′-x*〉,〈(I-f)x*,x*-x′〉≤-τ〈x*-Sx*,x*-x′〉.
Adding (108), we get
(109)(1-ρ)∥x′-x*∥2≤〈(I-f)x′-(I-f)x*,x′-x*〉≤-τ〈(I-S)x′-(I-S)x*,x′-x*〉≤0.
Then, x′=x*. Since (90) has unique solution, it follows that ww(xn)={x*}. Since every weak cluster point of {xn} is also a strong cluster point, we conclude that {xn}→x*. This completes the proof.


## 4. Applications

In this section, we obtain the following results by using a special case of the proposed method. The first result can be viewed as extension and improvement of the method of Gu et al. [11] for finding the approximate element of the common set of solutions of a generalized equilibrium problem and a hierarchical fixed point problem in a real Hilbert space.Corollary 16.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let D:C→H be η-inverse strongly monotone mappings, respectively. Let F:C×C→ℝ be a bifunction satisfying the assumptions (i)–(iv) of Lemma 4, S:C→H a nonexpansive mapping, and {Ti}i=1∞:C→C a countable family of ki-strict pseudocontraction mappings such that F(T)∩MEP(F)≠∅, where F(T)=∩i=1∞F(Ti). Let f be a ρ-contraction mapping. For a given x0∈C arbitrarily, let the iterative sequences {un},{xn},{yn}, and {zn} be generated by
(110)F(un,y)+〈Dxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C;yn=βnSxn+(1-βn)un;xn+1=PC[αnf(xn)+∑i=1n(αi-1-αi)Tiyn],∀n≥0,
where α0=1, {αn} is a strictly decreasing sequence in (0,1), and {βn} is a sequence in (0,1) satisfying the following conditions: (a)
limn→∞αn=0 and ∑n=1∞‍αn=∞,(b)
limn→∞(βn/αn)=τ∈(0,∞),(c)
∑n=1∞‍(αn-1-αn)<∞ and ∑n=1∞‍|βn-1-βn|<∞,(d)
limn→∞((1/μ)(|rn-rn-1|+|αn-1-αn|+|βn-1-βn|)/(αnβn))=0,(e)
there exists a constantK>0 such that (1/αn)|(1/βn)-(1/βn-1)|≤K,(f)
liminfn→∞rn>0 and ∑n=1∞‍|rn-1-rn|<∞. Then, sequence {xn} generated by Algorithm (110) converges strongly to x*∈MEP(F)∩F(T), which is the unique solution of the variational inequality
(111)〈1τ(I-f)x*+(I-S)x*,x-x*〉≥0,∀x∈MEP(F)∩F(T).Proof.
PuttingB1=B2=0 and ki=0,foralli≥1 in Theorem 15, then conclusion of Corollary 16 is obtained.The following result can be viewed as extension and improvement of the method of Yao et al. [31] for finding the approximate element of the common set of solutions of a generalized equilibrium problem and a hierarchical fixed point problem in a real Hilbert space.Corollary 17.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let D:C→H be η-inverse strongly monotone mappings, respectively. Let F:C×C→ℝ be a bifunction satisfying the assumptions (i)–(iv) of Lemma 4, S:C→H a nonexpansive mapping, and T:C→C a countable family of k-strict pseudocontraction mappings such that F(T)∩MEP(F)≠∅. Let f be a ρ-contraction mapping. For a given x0∈C arbitrarily, let the iterative sequences {un},{xn},{yn}, and {zn} be generated by
(112)F(un,y)+〈Dxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C;yn=βnSxn+(1-βn)un;xn+1=PC[αnf(xn)+(1-αn)Tyn],∀n≥0,
where α0=1, {αn} is a strictly decreasing sequence in (0,1) and {βn} is a sequence in (0,1) satisfying the following conditions: (a)
limn→∞αn=0 and ∑n=1∞‍αn=∞,(b)
limn→∞(βn/αn)=τ∈(0,∞),(c)
∑n=1∞‍(αn-1-αn)<∞ and ∑n=1∞‍|βn-1-βn|<∞,(d)
limn→∞((1/μ)|rn-rn-1|+|αn-1-αn|+|βn-1-βn|)/αnβn=0,(e)
there exists a constantK>0 such that (1/αn)|(1/βn)-(1/βn-1)|≤K,(f)
liminfn→∞rn>0 and ∑n=1∞‍|rn-1-rn|<∞. Then, sequence {xn} generated by Algorithm (112) converges strongly to x*∈MEP(F)∩F(T), which is the unique solution of the variational inequality
(113)〈1τ(I-f)x*+(I-S)x*,x-x*〉≥0,∀x∈MEP(F)∩F(T).Proof.
PuttingB1=B2=0,  ki=0, and Ti=Tforalli≥1 in Theorem 15, then conclusion of Corollary 17 is obtained.


## 5. Conclusions

In this paper, we suggest and analyze an iterative method for finding the approximate element of the common set of solutions of (1), (12), and (15) for a strictly pseudocontraction mapping in real Hilbert space, which can be viewed as a refinement and improvement of some existing methods for solving a system of variational inequality problem, a mixed equilibrium problem, and a hierarchical fixed point problem. It is easy to verify that Algorithm 11 includes some existing methods (e.g., [6, 11, 17, 31]) as special cases. Therefore, the new algorithm is expected to be widely applicable.



---

*Source: 737590-2013-11-27.xml*