# Iterative Algorithms for Mixed Equilibrium Problems, System of Quasi-Variational Inclusion, and Fixed Point Problem in Hilbert Spaces

Authors: Poom Kumam; Thanyarat Jitpeera
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We introduce a new iterative algorithm for approximating a common element of the set
of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusion,
and the set of fixed points of an infinite family of nonexpansive mappings in a real Hilbert space. Strong
convergence of the proposed iterative algorithm is obtained. Our results generalize, extend, and improve
the results of Peng and Yao, 2009, Qin et al. 2010 and many authors.


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## Body

## 1. Introduction

Throughout this paper, we assume thatH is a real Hilbert space with inner product and norm denoted by 〈·,·〉 and ∥·∥, respectively. Let C be a nonempty closed convex subset of H. A mapping T:C→C is callednonexpansive if ∥Tx-Ty∥≤∥x-y∥, ∀x,y∈C. They use F(T) to denote the set offixed points of T; that is, F(T)={x∈C:Tx=x}. It is assumed throughout the paper that T is a nonexpansive mapping such that F(T)≠∅. Recall that a self-mapping f:C→C is a contraction on C if there exists a constant α∈[0,1), and x,y∈C such that ∥f(x)-f(y)∥≤α∥x-y∥.Letφ:C→R∪{+∞} be a proper extended real-valued function and let F be a bifunction of C×C into R, where R is the set of real numbers. Ceng and Yao [1] considered the followingmixed equilibrium problem for finding x∈C such that
(1)F(x,y)+φ(y)≥φ(x),∀y∈C.
The set of solutions of (1) is denoted by MEP⁡(F,φ). We see that x is a solution of problem (1) which implies that x∈dom⁡φ={x∈C∣φ(x)<+∞}. If φ≡0, then the mixed equilibrium problem (1) becomes the followingequilibrium problem for finding x∈C such that
(2)F(x,y)≥0,∀y∈C.
The set of solutions of (2) is denoted by EP⁡(F). The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of (2). Some methods have been proposed to solve the equilibrium problem (see [2–14]).LetB:C→H be a mapping. Thevariational inequality problem, denoted by VI⁡(C,B), is for finding x∈C such that
(3)〈Bx,y-x〉≥0,
for all y∈C. The variational inequality problem has been extensively studied in the literature. See, for example, [15, 16] and the references therein. A mapping B of C into H is calledmonotone if
(4)〈Bx-By,x-y〉≥0,
for all x,y∈C. B is called β-inverse-strongly monotone if there exists a positive real number β>0 such that for all x,y∈C(5)〈Bx-By,x-y〉≥β∥Bx-By∥2.
We consider asystem of quasi-variational inclusion for finding (x*,y*)∈H×H such that
(6)θ∈x*-y*+ρ1(B1y*+M1x*),θ∈y*-x*+ρ2(B2x*+M2y*),
where Bi:H→H and Mi:H→2H are nonlinear mappings for each i=1,2. The set of solutions of problem (6) is denoted by SQVI(B1,M1,B2,M2). As special cases of problem (6), we have the following.(1)
IfB1=B2=B and M1=M2=M, then problem (6) is reduced to (7) for finding (x*,y*)∈H×H such that
(7)θ∈x*-y*+ρ1(By*+Mx*),θ∈y*-x*+ρ2(Bx*+My*).(2)
Further, ifx*=y*, then problem (7) is reduced to (8) for finding x*∈H such that
(8)θ∈Bx*+Mx*,whereθ is the zero vector in H. The set of solutions of problem (8) is denoted by I(B,M). A set-valued mapping M:H→2H is calledmonotone if for all x,y∈H,f∈M(x) and g∈M(y) imply 〈x-y,f-g〉≥0. A monotone mapping M ismaximal if its graph G(M):={(f,x)∈H×H:f∈M(x)} of M is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for (x,f)∈H×H,〈x-y,f-g〉≥0 for all (y,g)∈G(M) imply f∈M(x). Let B be a monotone mapping of C into H and let NCy¯ be thenormal cone to C at y¯∈C; that is, NCy¯={w∈H:〈u-y¯,w〉≤0,∀u∈C}, and define
(9)My¯={By¯+NCy¯,y¯∈C;∅,y¯∉C.Then,M is themaximal monotone and θ∈My¯ if and only if y¯∈VI⁡(C,B); see [17].LetM:H→2H be a set-valued maximal monotone mapping; then, the single-valued mapping JM,λ:H→H defined by
(10)JM,λx*=(I+λM)-1x*,x*∈H
is called theresolvent operator associated with M, where λ is any positive number and I is the identity mapping. The following characterizes the resolvent operator.(R1)
The resolvent operatorJM,λ is single-valued and nonexpansive for all λ>0; that is,
(11)∥JM,λ(x)-JM,λ(y)∥≤∥x-y∥,∀x,y∈H,∀λ>0.(R2)
The resolvent operatorJM,λ is 1-inverse-strongly monotone; see [18]; that is,
(12)∥JM,λ(x)-JM,λ(y)∥2≤〈x-y,JM,λ(x)-JM,λ(y)〉,∀x,y∈H.(R3)
The solution of problem (8) is a fixed point of the operator JM,λ(I-λB) for all λ>0; see also [19]; that is,
(13)I(B,M)=F(JM,λ(I-λB)),∀λ>0.(R4)
If0<λ≤2β, then the mapping JM,λ(I-λB):H→H is nonexpansive.(R5)
I(B,M) is closed and convex.LetA be a strongly positive linear bounded operator on H; that is, there exists a constant γ¯>0 with property
(14)〈Ax,x〉≥γ¯∥x∥2,∀x∈H.
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:
(15)min⁡x∈F(T)12〈Ax,x〉-h(x),
where A is a strongly positive linear bounded operator and h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H).In 2007, Plubtieng and Punpaeng [20] proposed the following iterative algorithm:
(16)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈H,xn+1=ϵnγf(xn)+(I-ϵnA)Tun.
They proved that if the sequences {ϵn} and {rn} of parameters satisfy appropriate conditions, then the sequences {xn} and {un} both converge to the unique solution z of the variational inequality
(17)〈(A-γf)z,x-z〉≥0,∀x∈F(T)∩EP⁡(F),
which is the optimality condition for the minimization problem
(18)min⁡x∈F(T)∩EP⁡(F)12〈Ax,x〉-h(x),
where h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H).In 2009, Peng and Yao [21] introduced an iterative algorithm based on extragradient method which solves the problem for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings, and the set of the variational inequality for a monotone, Lipschitz continuous mapping in a real Hilbert space. The sequences generated by v∈C are
(19)x1=x∈C,F(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,1111111111111111111111111111111111111∀y∈C,yn=PC(un-γnBun),xn+1=αnv+βnxn+(1-αn-βn)WnPC(un-λnByn),
for all n≥1, where Wn is W-mapping. They proved the strong convergence theorems under some mild conditions.In 2010, Qin et al. [22] introduced an iterative method for finding solutions of a generalized equilibrium problem, the set of fixed points of a family of nonexpansive mappings, and the common variational inclusions. The sequences generated by x1∈C and {xn} are a sequence generated by
(20)F(un,y)+〈A3xn,y-un〉+1rn〈y-un,un-xn〉≥0,11111111111111111111111111111111111111∀y∈C,zn=PC(un-λnA2un),yn=PC(zn-ηnA1zn),xn+1=αnf(xn)+βnxn+γnWnyn,∀n≥1,
where f is a contraction and Ai is inverse-strongly monotone mappings for i=1,2,3 and Wn is called a W-mapping generated by Sn,Sn1,…,S1 and γn,γn-1,…,γ1. They proved the strong convergence theorems under some mild conditions. Liou [23] introduced an algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of variational inclusion in a real Hilbert space. The sequences generated by x0∈C are
(21)F(un,y)+φ(y)-φ(un)+1r〈y-un,un-(xn-rQxn)〉≥0,∀y∈C,xn+1=PC[(1-αnA)JM,λ(un-λBun)],
for all n≥1, where A is a strongly positive bounded linear operator and B,Q are inverse-strongly monotone. They proved the strong convergence theorems under some suitable conditions.Next, Petrot et al. [24] introduced the new following iterative process for finding the set of solutions of quasi-variational inclusion problem and the set of fixed point of a nonexpansive mapping. The sequence is generated by
(22)x0∈H,chosenarbitrary,xn+1=αnf(xn)+βnxn+γnSzn,zn=JM,λ(yn-λAyn),yn=JM,ρ(xn-ρAxn),
for all n∈N∪{0}, where {αn},{βn},{γn} are three sequences in [0,1] and λ∈(0,2α]. They proved that {xn} generated by (22) converges strongly to z0 which is the unique solution in F(S)∩I(A,M).In 2011, Jitpeera and Kumam [25] introduced a shrinking projection method for finding the common element of the common fixed points of nonexpansive semigroups, the set of common fixed point for an infinite family, the set of solutions of a system of mixed equilibrium problems, and the set of solution of the variational inclusion problem. Let {xn}, {yn}, {vn}, {zn}, and {un} be sequences generated by x0∈C, C1=C, x1=PC1x0, un∈C, and
(23)x0=x∈Cchosenarbitrary,un=KrN,nFNKrN-1,nFN-1KrN-2,nFN-2⋯Kr2,nF2Kr1,nF1xn,yn=JM2,δn(un-δnBun),vn=JM1,λn(yn-λnAyn),zn=αnvn+(1-αn)1tn∫0tn‍S(s)Wnvnds,Cn+1={∥vn-1tn∫0tn‍S(s)Wnvnds∥2z∈Cn:∥zn-z∥2≤∥xn-z∥2-αn(1-αn)×∥vn-1tn∫0tn‍S(s)Wnvnds∥2},xn+1=PCn+1x0,n∈N,
where KrkFk:C→C, k=1,2,…,N. We proved the strong convergence theorem under certain appropriate conditions.In this paper, motivated by the above results, we introduce a new iterative method for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusions, and the set of fixed points of an infinite family of nonexpansive mappings in a real Hilbert space. Then, we prove strong convergence theorems which are connected with [5, 26–29]. Our results extend and improve the corresponding results of Jitpeera and Kumam [25], Liou [23], Plubtieng and Punpaeng [20], Petrot et al. [24], Peng and Yao [21], Qin et al. [22], and some authors.


## 2. Preliminaries

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥ and let C be a nonempty closed convex subset of H. Then,
(24)∥x-y∥2=∥x∥2-∥y∥2-2〈x-y,y〉,∥λx+(1-λ)y∥2=λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)×∥x-y∥2,∀x,y∈H,λ∈[0,1].
For every point x∈H, there exists a uniquenearest point in C, denoted by PCx, such that
(25)∥x-PCx∥≤∥x-y∥,∀y∈C.PC is called themetric projection of H onto C. It is well known that PC is a nonexpansive mapping of H onto C and satisfies
(26)〈x-y,PCx-PCy〉≥∥PCx-PCy∥2,∀x,y∈H.
Moreover, PCx is characterized by the following properties: PCx∈C and
(27)〈x-PCx,y-PCx〉≤0,(28)∥x-y∥2≥∥x-PCx∥2+∥y-PCx∥2,∀x∈H,y∈C.
Let B be a monotone mapping of C into H. In the context of the variational inequality problem, the characterization of projection (27) implies the following:
(29)u∈VI⁡(C,B)⟺u=PC(u-λBu),λ>0.
It is also known that H satisfies the Opial condition [30]; that is, for any sequence {xn}⊂H with xn⇀x, the inequality
(30)liminf⁡n→∞∥xn-x∥<liminf⁡n→∞∥xn-y∥
holds for every y∈H with x≠y.For the infinite family of nonexpansive mappings ofT1,T2,…, and sequence {λi}i=1∞ in [0,1), see [31]; we define the mapping Wn of C into itself as follows:
(31)Un,0=I,Un,1=λ1T1Un,0+(1-λ1)Un,0,Un,2=λ2T2Un,1+(1-λ2)Un,1,⋮Un,N-1=λN-1TN-1Un,N-2+(1-λN-1)Un,N-2,Wn=Un,N=λNTNUn,N-1+(1-λN)Un,N-1.Lemma 1 (Shimoji and Takahashi [32]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T={Ti}i=1N be a family of infinitely nonexpanxive mappings with F(T)=⋂i=1∞‍F(Ti)≠∅ and let {λi} be a real sequence such that 0<λi≤b<1 for every i≥1. Then (1)
Wn is nonexpansive and F(Wn)=⋂i=1n‍F(Ti) for each n≥1;(2)
for eachx∈C and for each positive integer k, the limit lim⁡n→∞Un,kx exists;(3)
the mappingW:C→C defined by Wx=lim⁡n→∞Wnx=lim⁡n→∞Un,1x is a nonexpansive mapping satisfying F(W)=F(T) and it is called the W-mapping generated by T1,T2,…, and λ1,λ2,…;(4)
ifK is any bounded subset of C, then lim⁡n→∞sup⁡x∈K∥Wx-Wnx∥=0.For solving the mixed equilibrium problem, let us give the following assumptions for a bifunctionF:C×C→R and a proper extended real-valued function φ:C→R∪{+∞} satisfies the following conditions:(A1)
F(x,x)=0 for all x∈C;(A2)
F is monotone; that is, F(x,y)+F(y,x)≤0 for all x,y∈C;(A3)
for eachx,y,z∈C, lim⁡t→0F(tz+(1-t)x,y)≤F(x,y);(A4)
for eachx∈C, y↦F(x,y) is convex and lower semicontinuous;(A5)
for eachy∈C, x↦F(x,y) is weakly upper semicontinuous;(B1)
for eachx∈H and r>0, there exist a bounded subset Dx⊆C and yx∈C such that for any z∈C∖Dx,
(32)F(z,yx)+φ(yx)+1r〈yx-z,z-x〉<φ(z);(B2)
C is a bounded set.We need the following lemmas for proving our main results.Lemma 2 (Peng and Yao [21]).
LetC be a nonempty closed convex subset of H. Let F:C×C→R be a bifunction that satisfies (A1)–(A5) and let φ:C→R∪{+∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r>0 and x∈H, define a mapping Tr:H→C as follows:
(33)Tr(x)={1rz∈C:F(z,y)+φ(y)+1r〈y-z,z-x〉≥φ(z),∀y∈C},
for all x∈H. Then, the following hold: (1)
for eachx∈H,Tr(x)≠∅;(2)
Tr is single-valued;(3)
Tr is firmly nonexpansive; that is, for any x,y∈H, ∥Trx-Try∥2≤〈Trx-Try,x-y〉;(4)
F(Tr)=MEP(F,φ);(5)
MEP(F,φ) is closed and convex.Lemma 3 (Xu [33]).
Assume{an} is a sequence of nonnegative real numbers such that
(34)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)
limsup⁡n→∞(δn/αn)≤0 or ∑n=1∞|δn|<∞.Then, lim⁡n→∞an=0.Lemma 4 (Suzuki [34]).
Let{xn} and {yn} be bounded sequences in a Banach space X and let {βn} be a sequence in [0,1] with 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1. Suppose xn+1=(1-βn)yn+βnxn for all integers n≥0 and limsup⁡n→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then, lim⁡n→∞∥yn-xn∥=0.Lemma 5 (Marino and Xu [35]).
AssumeA is a strongly positive linear bounded operator on a Hilbert space H with coefficient γ¯>0 and 0<ρ≤∥A∥-1. Then, ∥I-ρA∥≤1-ργ¯.Lemma 6.
For givenx*,y*∈C×C,(x*,y*) is a solution of problem (6) if and only if x* is a fixed point of the mapping G:C→C defined by
(35)G(x)=JM1,λ[JM2,μ(x-μE2x)-λE1JM2,μ(x-μE2x)],1111111111111111111111111111111111111111∀x∈C,
where y*=JM2,μ(x-μE2x), λ,μ are positive constants, and E1,E2:C→H are two mappings.Proof.
(36)
θ
∈
x
*
-
y
*
+
λ
(
E
1
y
*
+
M
1
x
*
)
,
θ
∈
y
*
-
x
*
+
μ
(
E
2
x
*
+
M
2
y
*
)
⇔
(37)
x
*
=
J
M
1
,
λ
(
y
*
-
λ
E
1
y
*
)
,
y
*
=
J
M
2
,
μ
(
x
*
-
μ
E
2
x
*
)
⇔
(38)
G
(
x
*
)
=
J
M
1
,
λ
[
J
M
2
,
μ
(
x
*
-
μ
E
2
x
*
)
J
M
1
,
λ
-
λ
E
1
J
M
2
,
μ
(
x
*
-
μ
E
2
x
*
)
]
=
x
*
.
This completes the proof.Now, we prove the following lemmas which will be applied in the main theorem.Lemma 7.
LetG:C→C be defined as in Lemma 6. If E1,E2:C→H is η1,η2-inverse-strongly monotone and λ∈(0,2η1), and μ∈(0,2η2), respectively, then G is nonexpansive.Proof.
For anyx,y∈C and λ∈(0,2η1),  μ∈(0,2η2), we have
(39)∥G(x)-G(y)∥2=∥JM1,λ[JM2,μ(x-μE2x)-λE1JM2,μ(x-μE2x)]-JM1,λ[JM2,μ(y-μE2y)-λE1JM2,μ(y-μE2y)]∥2≤∥[JM2,μ(x-μE2x)-λE1JM2,μ(x-μE2x)]-[JM2,μ(y-μE2y)-λE1JM2,μ(y-μE2y)]∥2=∥[JM2,μ(x-μE2x)-JM2,μ(y-μE2y)]-λ[E1JM2,μ(x-μE2x)-E1JM2,μ(y-μE2y)]∥2=∥JM2,μ(x-μE2x)-JM2,μ(y-μE2y)∥2-2λ〈JM2,μ(x-μE2x)-JM2,μ(y-μE2y),E1JM2,μ(x-μE2x)-E1JM2,μ(y-μE2y)〉+λ2∥E1JM2,μ(x-μE2x)-E1JM2,μ(y-μE2y)∥2≤∥JM2,μ(x-μE2x)-JM2,μ(y-μE2y)∥2-2λη1∥E1JM2,μ(x-μE2x)-E1JM2,μ(y-μE2y)∥2+λ2∥E1JM2,μ(x-μE2x)-E1JM2,μ(y-μE2y)∥2=∥JM2,μ(x-μE2x)-JM2,μ(y-μE2y)∥2+λ(λ-2η1)∥E1JM2,μ(x-μE2x)-E1JM2,μ(y-μE2y)∥2≤∥JM2,μ(x-μE2x)-JM2,μ(y-μE2y)∥2≤∥(x-μE2x)-(y-μE2y)∥2=∥(x-y)-μ(E2x-E2y)∥2=∥x-y∥2-2μ〈x-y,E2x-E2y〉+μ2∥E2x-E2y∥2≤∥x-y∥2-2η2μ∥E2x-E2y∥2+μ2∥E2x-E2y∥2=∥x-y∥2+μ(μ-2η2)∥E2x-E2y∥2≤∥x-y∥2.
This shows that G is nonexpansive on C.


## 3. Main Results

In this section, we show a strong convergence theorem for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusion, and the set of fixed points of a infinite family of nonexpansive mappings in a real Hilbert space.Theorem 8.
LetC be a nonempty closed convex subset of a real Hilbert Space H. Let F be a bifunction of C×C into real numbers R satisfying (A1)–(A5) and let φ:C→R∪{+∞} be a proper lower semicontinuos and convex function. Let Ti:C→C be nonexpansive mappings for all i=1,2,3,…, such that Θ∶=∩i=1∞F(Ti)∩SQVI(B1,M1,B2,M2)∩MEP(F,φ)≠∅. Let f be a contraction of C into itself with coefficient α∈(0,1) and let Q,E1,E2 be δ,η1,η2-inverse-strongly monotone mapping of C into H. Let A be a strongly positive bounded linear self-adjoint on H with coefficient γ¯>0 and 0<γ<γ¯/α, let M1,M2:H→2H be a maximal monotone mapping. Assume that either B1 or B2 holds and let Wn be the W-mapping defined by (31). Let {xn}, {yn}, {zn}, and {un} be sequences generated by x0∈C, un∈C, and
(40)F(un,y)+φ(y)-φ(un)+1r〈y-un,un-(xn-rQxn)〉≥0,∀y∈C,zn=JM2,μ(un-μE2un),yn=JM1,λ(zn-λE1zn),xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)Wnyn,1111111111111111111111111111111111111∀n≥0,
where {αn} and {βn}⊂(0,1), λ∈(0,2η1), μ∈(0,2η2), and r∈(0,2δ) satisfy the following conditions:(C1)
∑n=0∞αn=∞ and lim⁡n→∞αn=0,(C2)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1,(C3)
lim⁡n→∞|λn,i-λn-1,i|=0, ∀i=1,2,…,N.Then, {xn} converges strongly to x*∈Θ, where x*=PΘ(γf+I-A)(x*), PΘ is the metric projection of H onto Θ and (x*,y*), where y*=JM2,μ(x*-μE2x*) is solution to the problem (6).Proof.
Letx*∈Θ; that is Tr(x*-rQx*)=JM1,λ[JM2,μ(x*-μB2x*)-λB1JM2,μ(x*-μB2x*)]=Ti(x*)=x*, i≥1. Putting y*=JM2,μ(x*-μE2x*), one can see that x*=JM1,λ(y*-λB1y*).
We divide our proofs into the following steps:(1)
sequences{xn}, {yn}, {zn}, and {un} are bounded;(2)
lim⁡n→∞∥xn+1-xn∥=0;(3)
lim⁡n→∞∥Qxn-Qx*∥=0, lim⁡n→∞∥E1zn-E1x*∥=0 and lim⁡n→∞∥E2un-E2x*∥=0;(4)
lim⁡n→∞∥xn-Wxn∥=0;(5)
limsup⁡n→∞〈γf(x*)-Ax*,xn-x*〉≤0, where x*=PΘ(γf+I-A)x*;(6)
lim⁡n→∞∥xn-x*∥=0.
Step
1. From conditions (C1) and (C2), we may assume that αn≤(1-βn)∥A∥-1. By the same argument as that in [9], we can deduce that (1-βn)I-αnA is positive and ∥(1-βn)I-αnA∥≤1-βn-αnγ¯. For all x,y∈C and r∈(0,2δ). since Q is a δ-inverse-strongly monotone and B1,B2 are η1,η2-inverse-strongly monotone, we have
(41)∥(I-rQ)x-(I-rQ)y∥2=∥(x-y)-r(Qx-Qy)∥2=∥x-y∥2-2r〈x-y,Qx-Qy〉+r2∥Qx-Qy∥2≤∥x-y∥2-2rδ∥Qx-Qy∥2+r2∥Qx-Qy∥2=∥x-y∥2+r(r-2δ)∥Qx-Qy∥2≤∥x-y∥2.
It follows that ∥(I-rQ)x-(I-rQ)y∥≤∥x-y∥; hence I-rQ is nonexpansive.
In the same way, we conclude that∥(I-λE1)x-(I-λE1)y∥≤∥x-y∥ and ∥(I-μE2)x-(I-μE2)y∥≤∥x-y∥; hence I-λE1,I-μE2 are nonexpansive. Let yn=JM1,λ(zn-λE1zn),n≥0. It follows that
(42)∥yn-x*∥=∥JM1,λ(zn-λE1zn)-JM1,λ(y*-λE1y*)∥≤∥(zn-λE1zn)-(y*-λE1y*)∥≤∥zn-y*∥,∥zn-y*∥=∥JM2,μ(un-μE2un)-JM2,μ(x*-μE2x*)∥≤∥(un-μE2un)-(x*-μE2x*)∥≤∥un-x*∥.
By Lemma 2, we have un=Tr(xn-rQxn) for all n≥0, ∀x,y∈C. Then, for r∈(0,2δ), we obtain
(43)∥un-x*∥2=∥Tr(xn-rQxn)-Tr(x*-rQx*)∥2≤∥(xn-rQxn)-(x*-rQx*)∥2≤∥xn-x*∥2+r(r-2δ)∥Qxn-Qx*∥2≤∥xn-x*∥2.
Hence, we have
(44)∥yn-x*∥≤∥xn-x*∥.
From (40) and (44), we deduce that
(45)∥xn+1-x*∥=∥αn(γf(xn)-Ax*)+βn(xn-x*)+((1-βn)I-αnA)(Wnyn-x*)∥≤αn∥γf(xn)-Ax*∥+βn∥xn-x*∥+(1-βn-αnγ¯)∥yn-x*∥≤αn∥γf(xn)-Ax*∥+βn∥xn-x*∥+(1-βn-αnγ¯)∥xn-x*∥≤αnγ∥f(xn)-f(x*)∥+αn∥γf(x*)-Ax*∥+(1-αnγ¯)∥xn-x*∥≤αnγα∥xn-x*∥+αn∥γf(x*)-Ax*∥+(1-αnγ¯)∥xn-x*∥=(1-αn(γ¯-γα))∥xn-x*∥+αn(γ¯-γα)∥γf(x*)-Ax*∥(γ¯-γα)≤max⁡⁡{∥xn-x*∥,∥γf(x*)-Ax*∥(γ¯-γα)}.
It follows by mathematical induction that
(46)∥xn+1-x*∥≤max⁡{∥x0-x*∥,∥γf(x*)-Ax*∥(γ¯-γα)},11111111111111111111111111111111111111n≥0.
Hence, {xn} is bounded and also {un}, {zn}, {yn}, {Wnyn}, {AWnyn}, and {fxn} are all bounded.
Step
2. We show that lim⁡n→∞∥xn+1-xn∥=0.
Puttingtn=(xn+1-βnxn)/(1-βn)=(αnγf(xn)+((1-βn)I-αnA)Wnyn)/(1-βn), we get xn+1=(1-βn)tn+βnxn, n≥1. We note that
(47)tn+1-tn=αn+1γf(xn+1)+((1-βn+1)I-αn+1A)Wn+1yn+11-βn+1-αnγf(xn)+((1-βn)I-αnA)Wnyn1-βn=αn+11-βn+1γf(xn+1)-αn1-βnγf(xn)+Wn+1yn+1-Wnyn-αn+11-βn+1AWn+1yn+1+αn1-βnAWnyn=αn+11-βn+1(γf(xn+1)-AWn+1yn+1)+αn1-βn(AWnyn-γf(xn))+Wn+1yn+1-Wn+1yn+Wn+1yn-Wnyn.
It follows that
(48)∥tn+1-tn∥-∥xn+1-xn∥≤αn+11-βn+1(∥γf(xn+1)∥+∥AWn+1yn+1∥)+αn1-βn(∥AWnyn∥+∥γf(xn)∥)+∥Wn+1yn+1-Wn+1yn∥+∥Wn+1yn-Wnyn∥-∥xn+1-xn∥≤αn+11-βn+1(∥γf(xn+1)∥+∥AWn+1yn+1∥)+αn1-βn(∥AWnyn∥+∥γf(xn)∥)+∥yn+1-yn∥+∥Wn+1yn-Wnyn∥-∥xn+1-xn∥.
By the definition of Wn,
(49)∥Wn+1yn-Wnyn∥=∥λn+1,NTNUn+1,N-1yn+(1-λn+1,N)yn-λn,NTNUn,N-1yn-(1-λn,N)yn∥≤|λn+1,N-λn,N|∥yn∥+∥λn+1,NTNUn+1,N-1yn-λn,NTNUn,N-1yn∥≤|λn+1,N-λn,N|∥yn∥+∥λn+1,N(TNUn+1,N-1yn-TNUn,N-1yn)∥+|λn+1,N-λn,N|∥TNUn,N-1yn∥≤2M|λn+1,N-λn,N|+λn+1,N∥Un+1,N-1yn-Un,N-1yn∥,
where M is an approximate constant such that M≥max⁡{sup⁡n≥1{∥yn∥},sup⁡n≥1{∥TmUn,m-1yn∥}∣m=1,2,…,N}. Since 0<λni≤1 for all n≥1 and i=1,2,…,N, we compute
(50)∥Un+1,N-1yn-Un,N-1yn∥=∥λn+1,N-1TN-1Un+1,N-2yn+(1-λn+1,N-1)yn-λn,N-1TN-1Un,N-2yn-(1-λn,N-1)yn∥≤|λn+1,N-1-λn,N-1|∥yn∥+∥λn+1,N-1TN-1Un+1,N-2yn-λn,N-1TN-1Un,N-2yn∥≤|λn+1,N-1-λn,N-1|∥yn∥+∥λn+1,N-1(TN-1Un+1,N-2yn-TN-1Un,N-2yn)∥+|λn+1,N-1-λn,N-1|∥TN-1Un,N-2yn∥≤2M|λn+1,N-1-λn,N-1|+∥Un+1,N-2yn-Un,N-2yn∥.
It follows that
(51)∥Un+1,N-1yn-Un,N-1yn∥≤2M|λn+1,N-1-λn,N-1|+2M|λn+1,N-2-λn,N-2|+∥Un+1,N-3yn-Un,N-3yn∥≤2M∑i=2N-1‍|λn+1,i-λn,i|+∥Un+1,1yn-Un,1yn∥=2M∑i=2N-1‍|λn+1,i-λn,i|2M∑i=2N-1‍+∥λn+1,1T1yn+(1-λn+1,1)yn2M∑i=2N-1‍-λn,1T1yn-(1-λn,1)yn∥≤2M∑i=1N-1‍|λn+1,i-λn,i|.
Substituting (51) into (49),
(52)∥Wn+1yn-Wnyn∥≤2M|λn+1,N-λn,N|+2λn+1,NM∑i=1N-1‍|λn+1,i-λn,i|≤2M∑i=1N‍|λn+1,i-λn,i|.
We note that
(53)∥yn+1-yn∥=∥JM1,λ(zn+1-λE1zn+1)-JM1,λ(zn-λE1zn)∥≤∥(zn+1-λE1zn+1)-(zn-λE1zn)∥≤∥zn+1-zn∥=∥JM2,μ(un+1-μE2un+1)-JM2,μ(un-μE2un)∥≤∥(un+1-μE2un+1)-(un-μE2un)∥≤∥un+1-un∥=∥Tr(xn+1-rDxn+1)-Tr(xn-rDxn)∥≤∥(xn+1-rDxn+1)-(xn-rDxn)∥≤∥xn+1-xn∥.
Applying (52) and (53) in (48), we get
(54)∥tn+1-tn∥-∥xn+1-xn∥≤αn+11-βn+1(∥γf(xn+1)∥+∥AWn+1yn+1∥)+αn1-βn(∥AWnyn∥+∥γf(xn)∥)+∥xn+1-xn∥+2M∑i=1N‍|λn+1,i-λn,i|-∥xn+1-xn∥.
By conditions (C1)–(C3), imply that
(55)limsup⁡n→∞(∥tn+1-tn∥-∥xn+1-xn∥)≤0.
Hence, by Lemma 4, we obtain
(56)lim⁡n→∞∥tn-xn∥=0.
It follows that
(57)lim⁡n→∞∥xn+1-xn∥=lim⁡n→∞(1-βn)∥tn-xn∥=0.
We obtain that
(58)lim⁡n→∞∥xn+1-xn∥=0.
Step
3. We can rewrite (40) as xn+1=αn(γf(xn)-AWnyn)+βn(xn-Wnyn)+Wnyn. We observe that
(59)∥xn-Wnyn∥≤∥xn-xn+1∥+∥xn+1-Wnyn∥≤∥xn-xn+1∥+αn∥γf(xn)-AWnyn∥+βn∥xn-Wnyn∥;
it follows that
(60)∥xn-Wnyn∥≤11-βn∥xn-xn+1∥+αn1-βn∥γf(xn)-AWnyn∥.
By conditions (C1), (C2), and (58), imply that
(61)lim⁡n→∞∥xn-Wnyn∥=0.
From (42) and (43), we get
(62)∥yn-x*∥2=∥JM1,λ(zn-λE1zn)-JM1,λ(x*-λE1x*)∥2≤∥(zn-λE1zn)-(x*-λE1x*)∥2≤∥zn-x*∥2+λ(λ-2η1)∥E1zn-E1x*∥2≤∥JM2,μ(un-μE2un)-JM2,μ(x*-μE2x*)∥2+λ(λ-2η1)∥E1zn-E1x*∥2≤∥(un-μE2un)-(x*-μE2x*)∥2+λ(λ-2η1)∥E1zn-E1x*∥2≤∥un-x*∥2+μ(μ-2η2)∥E2un-E2x*∥2+λ(λ-2η1)∥E1zn-E1x*∥2≤∥xn-x*∥2+r(r-2δ)∥Qxn-Qx*∥2+μ(μ-2η2)∥E2un-E2x*∥2+λ(λ-2η1)∥E1zn-E1x*∥2.
By (40), we obtain
(63)∥xn+1-x*∥2=∥αn(γf(xn)-Ax*)+βn(xn-Wnyn)+(I-αnA)(Wnyn-x*)∥2≤∥(I-αnA)(Wnyn-x*)+βn(xn-Wnyn)∥2+2αn〈γf(xn)-Ax*,xn+1-x*〉≤∥(I-αnA)(yn-x*)+βn(xn-Wnyn)∥2+2αn∥γf(xn)-Ax*∥∥xn+1-x*∥=(1-αnγ¯)2∥yn-x*∥2+βn2∥xn-Wnyn∥2+2(1-αnγ¯)βn∥yn-x*∥∥xn-Wnyn∥+2αn∥γf(xn)-Ax*∥∥xn+1-x*∥.
Substituting (62) into (63), imply that
(64)∥xn+1-x*∥2≤∥xn-x*∥2+r(r-2δ)∥Qxn-Qx*∥2+μ(μ-2η2)∥E2un-E2x*∥2+λ(λ-2η1)∥E1zn-E1x*∥2+βn2∥xn-Wnyn∥2+2(1-αnγ¯)βn∥yn-x*∥∥xn-Wnyn∥+2αn∥γf(xn)-Ax*∥∥xn+1-x*∥.
Thus,
(65)r(2δ-r)∥Qxn-Qx*∥2+μ(2η2-μ)∥E2un-E2x*∥2+λ(2η1-λ)∥E1zn-E1x*∥2≤∥xn-x*∥2-∥xn+1-x*∥2+βn2∥xn-Wnyn∥2+2(1-αnγ¯)βn∥yn-x*∥∥xn-Wnyn∥+2αn∥γf(xn)-Ax*∥∥xn+1-x*∥≤(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥+βn2∥xn-Wnyn∥2+2(1-αnγ¯)βn∥yn-x*∥∥xn-Wnyn∥+2αn∥γf(xn)-Ax*∥∥xn+1-x*∥.
By conditions (C1), (C2), (58), and (61), we deduce immediately that
(66)lim⁡n→∞∥Qxn-Qx*∥=lim⁡n→∞∥E1zn-E1x*∥=lim⁡n→∞∥E2un-E2x*∥=0.
Step
4. We show that lim⁡n→∞∥xn-Wxn∥=0. Since Tr is firmly nonexpansive, we have
(67)∥un-x*∥2=∥Tr(xn-rQxn)-Tr(x*-rQx*)∥2≤〈(xn-rQxn)-(x*-rQx*),un-x*〉=12{∥(xn-rQxn)-(x*-rQx*)∥2+∥un-x*∥2}-12{∥(xn-rQxn)-(x*-rQx*)-(un-x*)∥2}=12{∥xn-x*∥2+∥un-x*∥2-∥(xn-un)-r(Qxn-Qx*)∥2}=12{∥xn-x*∥2+∥un-x*∥2-(∥xn-un∥2+r2∥Qxn-Qx*∥2-2r〈xn-un,Qxn-Qx*〉∥Qxn-Qx*∥2)}≤12{∥xn-x*∥2+∥un-x*∥2-∥xn-un∥2-r2∥Qxn-Qx*∥2+2r∥xn-un∥∥Qxn-Qx*∥},
which implies that
(68)∥un-x*∥2≤∥xn-x*∥2-∥xn-un∥2+2r∥xn-un∥∥Qxn-Qx*∥.
Since JM1,λ is 1-inverse-strongly monotone, we have
(69)∥yn-x*∥2=∥JM1,λ(zn-λE1zn)-JM1,λ(x*-λE1x*)∥2≤〈(zn-λE1zn)-(x*-λE1x*),yn-x*〉=12{∥(zn-λE1zn)-(x*-λE1x*)∥2+∥yn-x*∥2}-12{∥(zn-λE1zn)-(x*-λE1x*)-(yn-x*)∥2}=12{∥zn-x*∥2+∥yn-x*∥2-∥(zn-yn)-λ(E1zn-E1x*)∥2}=12{∥zn-x*∥2+∥yn-x*∥2-(∥zn-yn∥2+λ2∥E1zn-E1x*∥2-2λ〈zn-yn,E1zn-E1x*〉∥E1zn-E1x*∥2)}≤12{∥zn-x*∥2+∥yn-x*∥2-∥zn-yn∥2-λ2∥E1zn-E1x*∥2+2λ∥zn-yn∥∥E1zn-E1x*∥},
which implies that
(70)∥yn-x*∥2≤∥zn-x*∥2-∥zn-yn∥2+2λ∥zn-yn∥∥E1zn-E1x*∥.
In the same way with (70), we can get
(71)∥zn-x*∥2≤∥un-x*∥2-∥un-zn∥2+2μ∥un-zn∥∥E2un-E2x*∥.
Substituting (71) into (70), imply that
(72)∥yn-x*∥2≤∥un-x*∥2-∥un-zn∥2+2μ∥un-zn∥∥E2un-E2x*∥-∥zn-yn∥2+2λ∥zn-yn∥∥E1zn-E1x*∥.
Again, substituting (68) into (72), we get
(73)∥yn-x*∥2≤{∥xn-x*∥2-∥xn-un∥2+2r∥xn-un∥∥Qxn-Qx*∥}-∥un-zn∥2+2μ∥un-zn∥∥E2un-E2x*∥-∥zn-yn∥2+2λ∥zn-yn∥∥E1zn-E1x*∥.
Substituting (73) into (63), imply that
(74)∥xn+1-x*∥2≤(1-αnγ¯)2{∥xn-x*∥2-∥xn-un∥2≤(1-αnγ¯)2+2r∥xn-un∥∥Qxn-Qx*∥-∥un-zn∥2≤(1-αnγ¯)2+2μ∥un-zn∥∥E2un-E2x*∥-∥zn-yn∥2≤(1-αnγ¯)2+2λ∥zn-yn∥∥E1zn-E1x*∥∥xn-x*∥2}+βn2∥xn-Wnyn∥2+2(1-αnγ¯)βn∥yn-x*∥∥xn-Wnyn∥+2αn∥γf(xn)-Ax*∥∥xn+1-x*∥.
Then, we derive
(75)(1-αnγ¯)2(∥xn-un∥2+∥un-zn∥2+∥zn-yn∥2)≤∥xn-x*∥2-∥xn+1-x*∥2+2r∥xn-un∥∥Qxn-Qx*∥+2μ∥un-zn∥∥E2un-E2x*∥+2λ∥zn-yn∥∥E1zn-E1x*∥+βn2∥xn-Wnyn∥2+2(1-αnγ¯)βn∥yn-x*∥∥xn-Wnyn∥+2αn∥γf(xn)-Ax*∥∥xn+1-x*∥≤(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥+2r∥xn-un∥∥Qxn-Qx*∥+2μ∥un-zn∥∥E2un-E2x*∥+2λ∥zn-yn∥∥E1zn-E1x*∥+βn2∥xn-Wnyn∥2+2(1-αnγ¯)βn∥yn-x*∥∥xn-Wnyn∥+2αn∥γf(xn)-Ax*∥∥xn+1-x*∥.
By conditions (C1), (C2), (58), (61), and (66), we obtain
(76)lim⁡n→∞∥xn-un∥=lim⁡n→∞∥un-zn∥=lim⁡n→∞∥zn-yn∥=0.
Observe that
(77)∥Wnyn-yn∥≤∥Wnyn-xn∥+∥xn-un∥+∥un-zn∥+∥zn-yn∥.
By (61) and (76), we have
(78)lim⁡n→∞∥Wnyn-yn∥=0.
Note that
(79)∥Wyn-yn∥≤∥Wyn-Wnyn∥+∥Wnyn-yn∥.
From Lemma 1, we get
(80)lim⁡n→∞∥Wyn-Wnyn∥=0.
By (78) and (80), we have lim⁡n→∞∥Wyn-yn∥=0. It follows that lim⁡n→∞∥Wxn-xn∥=0.
Step
5. We show that limsup⁡n→∞〈(γf-A)z,xn-z〉≤0, where z=PΘ(γf+I-A)z. It is easy to see that PΘ(γf+(I-A)) is a contraction of H into itself. Indeed, since 0<γ<γ¯/α, we have
(81)∥PΘ(γf+(I-A))x-PΘ(γf+(I-A))y∥≤∥(γf+(I-A))x-(γf+(I-A))y∥≤γ∥f(x)-f(y)∥+|I-A|∥x-y∥≤γα∥x-y∥+(1-γ¯)∥x-y∥=(1-γ¯+γα)∥x-y∥.
Since H is complete, there exists a unique fixed point z∈H such that z=PΘ(γf+I-A)(z). Since {xn} is bounded, there exists a subsequence {xni} of {xn}, such that
(82)lim⁡i→∞〈(A-γf)z,z-xni〉=limsup⁡n→∞〈(A-γf)z,z-xn〉.
Also, since {xni} is bounded, there exists a subsequence {xnij} of {xni} which converges weakly to w∈C. Without loss of generality, we can assume that xni⇀w. From ∥Wxn-xn∥→0, we obtain Wxni⇀w. Then, by the demiclosed principle of nonexpansive mappings, we obtain w∈∩i=1∞F(Ti).
Next, we show thatw∈MEP⁡(F,φ). Since un=Tr(xn-rQxn), we obtain
(83)F(un,y)+φ(y)-φ(un)+1r〈y-un,un-(xn-rQxn)〉≥0,∀y∈C.
From (A2), we also have
(84)φ(y)-φ(un)+1r〈y-un,un-(xn-rQxn)〉≥F(y,un),111111111111111111111111111111111111111111∀y∈C,
and hence,
(85)φ(y)-φ(uni)+〈y-uni,uni-(xni-rQxni)r〉≥F(y,uni),∀y∈C.
For t with 0<t≤1 and y∈H, let yt=ty+(1-t)w. From (85) we have
(86)〈yt-uni,Qyt〉≥〈yt-uni,Qyt〉-φ(yt)+φ(uni)-〈yt-uni,uni-(xni-rQxni)r〉+F(yt,uni)=〈yt-uni,Qyt-Quni〉+〈yt-uni,Quni-Qxni〉-φ(yt)+φ(uni)-〈yt-uni,uni-xnir〉+F(yt,uni).
Since ∥uni-xni∥→0, we have ∥Quni-Qxni∥→0. Further, from an inverse-strongly monotonicity of Q, we have 〈yt-uni,Qyt-Quni〉≥0. So, from (A4), (A5), and the weakly lower semicontinuity of φ,〈uni-xni〉/r→0 and uni→w weakly, we have
(87)〈yt-w,Qyt〉≥-φ(yt)+φ(w)+F(yt,w).
From (A1), (A4), and (87), we also have
(88)0=F(yt,yt)+φ(yt)-φ(yt)≤tF(yt,y)+(1-t)F(yt,w)+tφ(y)+(1-t)φ(w)-φ(yt)=t(F(yt,y)+φ(y)-φ(yt))+(1-t)(F(yt,w)+φ(w)-φ(yt))≤t(F(yt,y)+φ(y)-φ(yt))+(1-t)〈yt-w,Qyt〉=t(F(yt,y)+φ(y)-φ(yt))+(1-t)t〈y-w,Qyt〉,
and hence,
(89)0≤F(yt,y)+φ(y)-φ(yt)+(1-t)〈y-w,Qyt〉.
Letting t→0, we have, for each y∈C,
(90)F(w,y)+φ(y)-φ(w)+〈y-w,Qw〉≥0.
This implies that w∈MEP⁡(F,φ).
Lastly, we show thatw∈SQVI(B1,M1,B2,M2). Since ∥un-zn∥→0 and ∥zn-yn∥→0 as n→∞, we get
(91)∥un-yn∥≤∥un-zn∥+∥zn-yn∥,
we conclude that ∥un-yn∥→0 as n→∞. Moreover, by the nonexpansivity of G in Lemma 6, we have
(92)∥yn-G(yn)∥=∥JM1,λ[JM2,μ(un-μE2un)-λE1JM2,μ(un-μE2un)]-G(yn)[JM2,μ(un-μE2un)-λE1JM2,μ(un-μE2un)]∥=∥G(un)-G(yn)∥≤∥un-yn∥.
Thus, lim⁡n→∞∥yn-G(yn)∥=0. According to Lemma 7, we obtain that w∈SQVI(B1,M1,B2,M2). Hence, w∈Θ. Since z=PΘ(I-A+γf)(z), we have
(93)limsup⁡n→∞〈(γf-A)z,xn-z〉=limsup⁡i→∞〈(γf-A)z,xni-z〉=〈(γf-A)z,w-z〉≤0.
Step
6. We show that {xn} converges strongly to z; we compute that
(94)∥xn+1-z∥2=∥αnγf(xn)+βnxn+((1-βn)I-αnA)Wnyn-z∥2=∥((1-βn)I-αnA)αn(γf(xn)-Az)+βn(xn-z)+((1-βn)I-αnA)(Wnyn-z)∥2=αn2∥γf(xn)-Az∥2+∥βn(xn-z)+((1-βn)I-αnA)(Wnyn-z)∥2+2〈βn(xn-z)+((1-βn)I-αnA)×(Wnyn-z),αn(γf(xn)-Az)〉≤αn2∥γf(xn)-Az∥2+{βn∥xn-z∥+(1-βn-αnγ¯)∥yn-z∥}2+2αnβn〈xn-z,γf(xn)-Az〉+2αn(1-βn-αnγ¯)〈Wnyn-z,γf(xn)-Az〉≤αn2∥γf(xn)-Az∥2+{βn∥xn-z∥+(1-βn-αnγ¯)∥xn-z∥}2+2αnβn〈xn-z,γf(xn)-γf(z)〉+2αnβn〈xn-z,γf(z)-Az〉+2αn(1-βn-αnγ¯)〈Wnyn-z,γf(xn)-γf(z)〉+2αn(1-βn-αnγ¯)〈Wnyn-z,γf(z)-Az〉≤αn2∥γf(xn)-Az∥2+(1-αnγ¯)2∥xn-z∥2+2αnβnγ∥xn-z∥∥f(xn)-f(z)∥+2αnβn〈xn-z,γf(z)-Az〉+2αn(1-βn-αnγ¯)γ∥Wnyn-z∥∥f(xn)-f(z)∥+2αn(1-βn-αnγ¯)〈Wnyn-z,γf(z)-Az〉≤αn2∥γf(xn)-Az∥2+(1-αnγ¯)2∥xn-z∥2+2αnβnγα∥xn-z∥2+2αnβn〈xn-z,γf(z)-Az〉+2αn(1-βn-αnγ¯)γα∥xn-z∥2+2αn(1-βn-αnγ¯)〈Wnyn-z,γf(z)-Az〉=αn2∥γf(xn)-Az∥2+(1-2αnγ¯+αn2γ¯2+2αnγα-2αn2γ¯γα)×∥xn-z∥2+2αnβn〈xn-z,γf(z)-Az〉+2αn(1-βn-αnγ¯)〈Wnyn-z,γf(z)-Az〉≤{1-αn(2γ¯-αnγ¯2-2γα+2αnγ¯γα)}∥xn-z∥2+αn2∥γf(xn)-Az∥2+2αnβn〈xn-z,γf(z)-Az〉+2αn(1-βn-αnγ¯)〈Wnyn-z,γf(z)-Az〉≤{1-αn(2γ¯-αnγ¯2-2γα+2αnγ¯γα)}∥xn-z∥2+αnσn,
where σn=αn∥γf(xn)-Az∥2+2βn〈xn-z,γf(z)-Az〉+2(1-βn-αnγ¯)〈Wnyn-z,γf(z)-Az〉. It is easy to see that limsup⁡n→∞σn≤0. Applying Lemma 3 to (94), we conclude that xn→z. This completes the proof.Next, the following example shows that all conditions of Theorem8 are satisfied.Example 9.
For instance, letαn=1/2(n+1), let βn=(2n+2)/2(2n), let λn=n/(n+1). Then, we will show that the sequences {αn} satisfy condition (C1). Indeed, we take αn=1/2(n+1); then, we have
(95)∑n=1∞‍αn=∑n=1∞‍12(n+1)=∞,lim⁡n→∞αn=lim⁡n→∞12(n+1)=0.
We will show that the sequences{βn} satisfy condition (C2). Indeed, we set βn=(2n+2)/2(2n)=(1/2)+(1/2n). It is easy to see that 0<liminf⁡n→∞βn<limsup⁡n→∞βn<1.
Next, we will show the condition (C3) is satisfied. We takeλn=n/(n+1); then we compute
(96)lim⁡n→∞|λn-λn-1|=lim⁡n→∞|nn+1-n-1(n-1)+1|=lim⁡n→∞|n(n)-(n-1)(n+1)(n+1)n|=lim⁡n→∞|n2-n2+1(n+1)n|=lim⁡n→∞|1n(n+1)|.
Then, we have lim⁡n→∞|λn+1-λn|=0. The sequence {λn} satisfies condition (C3).Using Theorem8, we obtain the following corollaries.Corollary 10.
LetC be a nonempty closed convex subset of a real Hilbert Space H. Let F be a bifunction of C×C into real numbers R satisfying (A1)–(A5) and let φ:C→R∪{+∞} be a proper lower semicontinuos and convex function. Let Ti:C→C be nonexpansive mappings for all i=1,2,3,…, such that Θ∶=∩i=1∞F(Ti)∩SQVI(B1,M1,B2,M2)∩MEP(F,φ)≠∅. Let f be a contraction of C into itself with coefficient α∈(0,1) and let Q,E1,E2 be δ,η1,η2-inverse-strongly monotone mapping of C into H. Let M1,M2:H→2H be a maximal monotone mapping. Assume that either B1 or B2 holds and let Wn be the W-mapping defined by (31). Let {xn}, {yn}, {zn}, and {un} be sequences generated by x0∈C, un∈C, and
(97)F(un,y)+φ(y)-φ(un)+1r〈y-un,un-(xn-rQxn)〉≥0,∀y∈C,zn=JM2,μ(un-μE2un),yn=JM1,λ(zn-λE1zn),xn+1=αnf(xn)+βnxn+(1-βn-αn)Wnyn,∀n≥0,
where {αn} and {βn}⊂(0,1), λ∈(0,2η1), μ∈(0,2η2), and r∈(0,2δ) satisfy the following conditions:(C1)
∑n=0∞αn=∞ and lim⁡n→∞αn=0,(C2)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1,(C3)
lim⁡n→∞|λn,i-λn-1,i|=0, ∀i=1,2,…,N.Then, {xn} converges strongly to x*∈Θ, where x*=PΘ(f+I)(x*), PΘ is the metric projection of H onto Θ and (x*,y*), where y*=JM2,μ(x*-μE2x*) is solution to the problem (6).Proof.
Takingγ≡1 and A≡I in Theorem 8, we can conclude the desired conclusion easily.Corollary 11.
LetC be a nonempty closed convex subset of a real Hilbert Space H. Let F be a bifunction of C×C into real numbers R satisfying (A1)–(A5) and let φ:C→R∪{+∞} be a proper lower semicontinuos and convex function. Let Ti:C→C be a nonexpansive mappings for all i=1,2,3,…, such that Θ∶=∩i=1∞F(Ti)∩SQVI(B1,M1,B2,M2)∩MEP(F,φ)≠∅. Let f be a contraction of C into itself with coefficient α∈(0,1) and let E1,E2 be η1,η2-inverse-strongly monotone mapping of C into H. Let A be strongly positive bounded linear self-adjoint on H with coefficient γ¯>0 and 0<γ<γ¯/α, let M1,M2:H→2H be a maximal monotone mapping. Assume that either B1 or B2 holds and let Wn be the W-mapping defined by (31). Let {xn}, {yn}, {zn}, and {un} be sequences generated by x0∈C, un∈C, and
(98)F(un,y)+φ(y)-φ(un)+1r〈y-un,un-xn〉≥0,111111111111111111111111111111111111∀y∈C,zn=JM2,μ(un-μE2un),yn=JM1,λ(zn-λE1zn),xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)Wnyn,1111111111111111111111111111111111111∀n≥0,
where {αn} and {βn}⊂(0,1), λ∈(0,2η1), μ∈(0,2η2), and r∈(0,∞) satisfy the following conditions:(C1)
∑n=0∞αn=∞ and lim⁡n→∞αn=0,(C2)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1,(C3)
lim⁡n→∞|λn,i-λn-1,i|=0, ∀i=1,2,…,N.Then, {xn} converges strongly to x*∈Θ, where x*=PΘ(γf+I-A)(x*), PΘ is the metric projection of H onto Θ and (x*,y*), where y*=JM2,μ(x*-μE2x*) is solution to the problem (6).Proof.
TakingQ≡0 in Theorem 8, we can conclude the desired conclusion easily.Corollary 12.
LetC be a nonempty closed convex subset of a real Hilbert Space H. Let F be a bifunction of C×C into real numbers R satisfying (A1)–(A5) and let φ:C→R∪{+∞} be a proper lower semicontinuos and convex function such that Θ:=SQVI(B1,M1,B2,M2)∩MEP(F,φ)≠∅. Let f be a contraction of C into itself with coefficient α∈(0,1) and let Q,E1,E2 be δ,η1,η2-inverse-strongly monotone mapping of C into H. Let A be a strongly positive bounded linear self-adjoint on H with coefficient γ¯>0 and 0<γ<γ¯/α, let M1,M2:H→2H be a maximal monotone mapping. Assume that either B1 or B2 holds, let {xn}, {yn}, {zn}, and {un} be sequences generated by x0∈C, un∈C, and
(99)F(un,y)+φ(y)-φ(un)+1r〈y-un,un-(xn-rQxn)〉≥0,∀y∈C,zn=JM2,μ(un-μE2un),yn=JM1,λ(zn-λE1zn),xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)yn,∀n≥0,
where {αn} and {βn}⊂(0,1), λ∈(0,2η1), μ∈(0,2η2), and r∈(0,2δ) satisfy the following conditions:(C1)
∑n=0∞‍αn=∞ and lim⁡n→∞αn=0,(C2)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1,(C3)
lim⁡n→∞|λn,i-λn-1,i|=0, ∀i=1,2,…,N.Then, {xn} converges strongly to x*∈Θ, where x*=PΘ(γf+I-A)(x*), PΘ is the metric projection of H onto Θ and (x*,y*), where y*=JM2,μ(x*-μE2x*) is solution to the problem (7), which is the unique solution of the variational inequality
(100)〈(γf-A)x*,x-x*〉≥0,∀x∈Θ,Proof.
TakingWn≡I in Theorem 8, we can conclude the desired conclusion easily.Corollary 13.
LetC be a nonempty closed convex subset of a real Hilbert Space H. Let F be a bifunction of C×C into real numbers R satisfying (A1)–(A5). Let Ti:C→C be nonexpansive mappings for all i=1,2,3,…, such that Θ∶=∩i=1∞F(Ti)∩SQVI(B1,M1,B2,M2)∩EP(F)≠∅. Let f be a contraction of C into itself with coefficient α∈(0,1) and let Q,E1,E2 be δ,η1,η2-inverse-strongly monotone mapping of C into H. Let A be a strongly positive bounded linear self-adjoint on H with coefficient γ¯>0 and 0<γ<γ¯/α, let M1,M2:H→2H be a maximal monotone mapping. Assume that either B1 or B2 holds and let Wn be the W-mapping defined by (31). Let {xn}, {yn}, {zn}, and {un} be sequences generated by x0∈C, un∈C, and
(101)F(un,y)+1r〈y-un,un-(xn-rQxn)〉≥0,1111111111111111111111111111111∀y∈C,zn=JM2,μ(un-μE2un),yn=JM1,λ(zn-λE1zn),xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)Wnyn,1111111111111111111111111111111111111∀n≥0,
where {αn} and {βn}⊂(0,1), λ∈(0,2η1), μ∈(0,2η2), and r∈(0,2δ) satisfy the following conditions:(C1)
∑n=0∞αn=∞ and lim⁡n→∞αn=0,(C2)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1,(C3)
lim⁡n→∞|λn,i-λn-1,i|=0, ∀i=1,2,…,N.Then, {xn} converges strongly to x*∈Θ, where x*=PΘ(γf+I-A)(x*), PΘ is the metric projection of H onto Θ and (x*,y*), where y*=JM2,μ(x*-μE2x*) is solution to the problem (6).Proof.
Takingφ≡0 in Theorem 8, we can conclude the desired conclusion easily.Corollary 14.
LetC be a nonempty closed convex subset of a real Hilbert Space H. Let F be a bifunction of C×C into real numbers R satisfying (A1)–(A5) such that Θ∶=SQVI(B1,M1,B2,M2)∩EP(F)≠∅. Let f be a contraction of C into itself with coefficient α∈(0,1) and let Q,E1,E2 be δ,η1,η2-inverse-strongly monotone mapping of C into H. Let A be a strongly positive bounded linear self-adjoint on H with coefficient γ¯>0 and 0<γ<γ¯/α, let M1,M2:H→2H be a maximal monotone mapping. Assume that either B1 or B2 holds, let {xn}, {yn}, {zn}, and {un} be sequences generated by x0∈C, un∈C, and
(102)F(un,y)+1r〈y-un,un-(xn-rQxn)〉≥0,1111111111111111111111111111111∀y∈C,zn=JM2,μ(un-μE2un),yn=JM1,λ(zn-λE1zn),xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)yn,11111111111111111111111111111111111∀n≥0,
where {αn} and {βn}⊂(0,1), λ∈(0,2η1), μ∈(0,2η2), and r∈(0,2δ) satisfy the following conditions:(C1)
∑n=0∞‍αn=∞ and lim⁡n→∞αn=0,(C2)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1,(C3)
lim⁡n→∞|λn,i-λn-1,i|=0, ∀i=1,2,…,N.Then, {xn} converges strongly to x*∈Θ, where x*=PΘ(γf+I-A)(x*), PΘ is the metric projection of H onto Θ and (x*,y*), where y*=JM2,μ(x*-μE2x*) is solution to the problem (6).Proof.
Takingφ≡0 and Wn≡I in Theorem 8, we can conclude the desired conclusion easily.Corollary 15.
LetC be a nonempty closed convex subset of a real Hilbert Space H. Let F be a bifunction of C×C into real numbers R satisfying (A1)–(A5) such that Θ∶=SQVI(B1,M1,B2,M2)∩EP(F)≠∅. Let f be a contraction of C into itself with coefficient α∈(0,1) and let Q,E1,E2 be δ,η1,η2-inverse-strongly monotone mapping of C into H. Let M1,M2:H→2H be a maximal monotone mapping. Let {xn}, {yn}, {zn}, and {un} be sequences generated by x0∈C, un∈C, and
(103)F(un,y)+1r〈y-un,un-(xn-rQxn)〉≥0,1111111111111111111111111111111∀y∈C,zn=JM2,μ(un-μE2un),yn=JM1,λ(zn-λE1zn),xn+1=αnf(xn)+βnxn+(1-βn-αn)yn,111111111111111111111111111111∀n≥0,
where {αn} and {βn}⊂(0,1), λ∈(0,2η1), μ∈(0,2η2), and r∈(0,2δ) satisfy the following conditions:(C1)
∑n=0∞αn=∞ and lim⁡n→∞αn=0,(C2)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1,(C3)
lim⁡n→∞|λn,i-λn-1,i|=0, ∀i=1,2,…,N.Then, {xn} converges strongly to x*∈Θ, where x*=PΘ(f+I)(x*), PΘ is the metric projection of H onto Θ and (x*,y*), where y*=JM2,μ(x*-μE2x*) is solution to the problem (6).Proof.
Takingγ≡1, A≡I, φ≡0, and Wn≡I in Theorem 8, we can conclude the desired conclusion easily.Corollary 16.
LetC be a nonempty closed convex subset of a real Hilbert Space H. Let F be a bifunction of C×C into real numbers R satisfying (A1)–(A5) such that Θ∶=SQVI(B1,M1,B2,M2)∩EP(F)≠∅. Let f be a contraction of C into itself with coefficient α∈(0,1) and let Q,E be δ,η-inverse-strongly monotone mapping of C into H. Let M1,M2:H→2H be a maximal monotone mapping. Let {xn}, {yn}, {zn}, and {un} be sequences generated by x0∈C, un∈C, and
(104)F(un,y)+1r〈y-un,un-(xn-rQxn)〉≥0,1111111111111111111111111111111∀y∈C,zn=JM2,μ(un-μEun),yn=JM1,λ(zn-λEzn),xn+1=αnf(xn)+βnxn+(1-βn-αn)yn,111111111111111111111111111111∀n≥0,
where {αn} and {βn}⊂(0,1), λ∈(0,2η), μ∈(0,2η), and r∈(0,2δ) satisfy the following conditions:(C1)
∑n=0∞αn=∞ and lim⁡n→∞αn=0,(C2)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1,(C3)
lim⁡n→∞|λn,i-λn-1,i|=0, ∀i=1,2,…,N.Then, {xn} converges strongly to x*∈Θ, where x*=PΘ(f+I)(x*), PΘ is the metric projection of H onto Θ and (x*,y*), where y*=JM2,μ(x*-μEx*) is solution to the problem (6).Proof.
TakingE1=E2=E in Corollary 15, we can conclude the desired conclusion easily.



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*Source: 271208-2014-07-24.xml*