# Hybrid Projection Algorithm for a New General System of Variational Inequalities in Hilbert Spaces

Authors: S. Imnang
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

A new general system of variational inequalities in a real Hilbert space is introduced and studied. The solution of this system is shown to be a fixed point of a nonexpansive mapping. We also introduce a hybrid projection algorithm for finding a common element of the set of solutions of a new general system of variational inequalities, the set of solutions of a mixed equilibrium problem, and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Several strong convergence theorems of the proposed hybrid projection algorithm are established by using the demiclosedness principle. Our results extend and improve recent results announced by many others.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and C be a nonempty closed convex subset of H. Recall that T:C→C is nonexpansive if ∥Tx-Ty∥≤∥x-y∥, for all x,y∈C. The fixed point set of T is denoted by F(T):={x∈C:Tx=x}.LetA:C→H be a nonlinear mapping. Then A is called (i)
α-strongly monotone, if there exists a positive real number α>0 such that
(1.1)〈Ax-Ay,x-y〉≥α∥x-y∥2,∀x,y∈C;(ii)
L-Lipschitz continuous (or Lipschitzian), if there exists a constant L≥0 such that
(1.2)∥Ax-Ay∥≤L∥x-y∥,∀x,y∈C;(iii)
relaxed(c,d)-cocoercive, if there exist two constants c,d>0 such that
(1.3)〈Ax-Ay,x-y〉≥(-c)∥Ax-Ay∥2+d∥x-y∥2,∀x,y∈C,
for c=0, A is d-strongly monotone. This class of mappings is more general than the class of strongly monotone mappings.Next, we consider the following variational inequality problem of findingx*∈C such that
(1.4)〈Ax*,x-x*〉≥0,∀x∈C.
The set of solutions of the variational inequality (1.4) is denoted by VI(C,A). Variational inequality theory has emerged as an important tool in studying a wide class of obstacle, unilateral, free, moving, equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework. The variational inequality problem has been extensively studied in the literature, see, Piri [1], Qin et al. [2], Shehu [3], Wangkeeree and Preechasilp [4], Yao et al. [5], Yao et al. [6], and the references therein.For solving the variational inequality problem in the finite-dimensional Euclidean spaceℝn under the assumption that a set C⊂ℝn is closed and convex, a mapping A of C into ℝn is monotone and k-Lipschitz-continuous and VI(C,A) is nonempty, Korpelevič [7] introduced the following called extragradient method:
(1.5)x0=x∈C,yn=PC(xn-λAxn),xn+1=PC(xn-λAyn),
for every n=0,1,2,…, where λ∈(0,1/k) and PC is the projection of ℝn onto C. He showed that the sequences {xn} and {yn} generated by this iterative process converge to the same point z∈VI(C,A). Later on, the idea of Korpelevich was generalized and extended by many authors, see for example, [1–5, 8, 9] for finding a common element of the set of fixed points and the set of solutions of the variational inequality.We recall the following well-known result which is called the best approximation result or the projection lemma.Lemma 1.1.
For a givenz∈H, u∈C satisfies the inequality
(1.6)〈u-z,v-u〉≥0,∀v∈C,iffu=PCz,
where PC is the projection of H onto a closed convex set C.Lemma 1.2.
x
*
∈
C is a solution of the variational inequality if and only if x*∈C satisfies the relation
(1.7)x*=PC(x*-λAx*),
where PC is the projection of H onto a closed convex set C and λ>0 is a constant.LetC be a nonempty closed convex subset of a real Hilbert space H. Let Ai:C→H for all i=1,2,3 be three mappings. In this paper, we focus on the problem of finding (x*,y*,z*)∈C×C×C such that
(1.8)〈λ1A1y*+x*-y*,x-x*〉≥0,∀x∈C,〈λ2A2z*+y*-z*,x-y*〉≥0,∀x∈C,〈λ3A3x*+z*-x*,x-z*〉≥0,∀x∈C,
which is called a new general system of variational inequalities, where λi>0 for all i=1,2,3. In particular, if A3=0 and z*=x*, then problem (1.8) reduces to find (x*,y*)∈C×C such that
(1.9)〈λ1A1y*+x*-y*,x-x*〉≥0,∀x∈C,〈λ2A2x*+y*-x*,x-y*〉≥0,∀x∈C,
which is called a general system of variational inequalities and defined by Ceng et al. [10]. If we add up the requirement that A1=A2:=A, then problem (1.9) reduces to find (x*,y*)∈C×C such that
(1.10)〈λ1A1y*+x*-y*,x-x*〉≥0,∀x∈C,〈λ2A2x*+y*-x*,x-y*〉≥0,∀x∈C,
which is defined by Verma [11], and is called the new system of variational inequalities. Further, if we add up the requirement that x*=y* and λ1=λ2=1, then problem (1.10) reduces to the classical variational inequality VI(C,A). Ceng et al. [10] introduced and studied a relaxed extragradient method for finding a common element of the set of solutions of problem (1.9) for the α and β-inverse-strongly monotone mappings and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Some related works, we refer to see [9, 12–16].Recently, in 2012, Ceng et al. [12] considered an iterative method for the system of problem (1.9) and obtained a strong convergence theorem for the two different systems of problem (1.9) and the set of fixed points of a strict pseudocontraction mapping in a real Hilbert space.Letφ:C→ℝ⋃‍{+∞} be a proper extended real-valued function and F be a bifunction from C×C to ℝ, where ℝ is the set of real numbers. Ceng and Yao [17] considered the following mixed equilibrium problem:
(1.11)Findx∈CsuchthatF(x,y)+φ(y)≥φ(x),∀y∈C.
The set of solutions of problem (1.11) is denoted by MEP(F,φ). It is easy to see that x is a solution of problem (1.11) implies that x∈domφ={x∈C∣φ(x)<+∞}.Ifφ=0, then the problem (1.11) becomes the following equilibrium problem:
(1.12)findx∈CsuchthatF(x,y)≥0,∀y∈C.
The set of solution of (1.12) is denoted by EP(F).IfF=0, then the problem (1.11) reduces to the convex minimization problem:
(1.13)Findx∈Csuchthatφ(y)≥φ(x),∀y∈C.Ifφ=0 and F(x,y)=〈Ax,y-x〉 for all x,y∈C, where A is a mapping from C into H, then problem (1.11) reduces to the classical variational inequality and EP(F)=VI(C,A). For solving problem (1.11), Ceng and Yao [17] introduced a hybrid iterative scheme for finding a common element of the set MEP(F,φ) and the set of common fixed points of finite many nonexpansive mappings in a Hilbert space. Some related works, we refer to see [3, 5, 9, 15].Recently, in 2012, Kumam and Katchang [14] introduced an iterative algorithm for finding a common element of the set of solutions of a system of mixed equilibrium problems, the set of solutions of a general system of variational inequalities for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in Hilbert spaces.In this paper, motivated and inspired by the idea of Kumam and Katchang [14], we introduce a hybrid projection algorithm for finding a common element of the set of solutions of a new general system of variational inequalities, the set of solutions of a mixed equilibrium problem and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Starting with an arbitrary v∈C and let x1∈C, {xn},{yn} and {zn} be the sequences generated by
(1.14)F(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,zn=PC(un-λ3A3un),yn=PC(zn-λ2A2zn),xn+1=anv+bnxn+(1-an-bn)TPC(yn-λ1A1yn),n≥1,
where λi>0 for all i=1,2,3, {rn}⊂(0,∞) and {an},{bn}⊂[0,1]. Using the demiclosedness principle for nonexpansive mappings, we show that the sequence {xn} converges strongly to a common element of those three sets under some control conditions. Our results extend and improve recent results announced by many others.


## 2. Preliminaries

In this section, we recall the well-known results and give some useful lemmas that are used in the next section.LetC be a nonempty closed convex subset of a real Hilbert space H. For every point x∈H, there exists a unique nearest point in C, denoted by PCx, such that
(2.1)∥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
(2.2)〈x-y,PCx-PCy〉≥∥PCx-PCy∥2,∀x,y∈H.
Obviously, this immediately implies that
(2.3)∥(x-y)-(PCx-PCy)∥2≤∥x-y∥2-∥PCx-PCy∥2,∀x,y∈H.
Recall that, PCx is characterized by the following properties: PCx∈C and
(2.4)〈x-PCx,y-PCx〉≤0,∥x-y∥2≥∥x-PCx∥2+∥PCx-y∥2,
for all x∈H and y∈C; see Goebel and Kirk [18] for more details.For solving the mixed equilibrium problem, let us give the following assumptions for the bifunctionF,φ and the set C:(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 eachy∈C, x↦F(x,y) is weakly upper semicontinuous;(A4)
For eachx∈C, y↦F(x,y) is convex;(A5)
For eachx∈C, y↦F(x,y) is lower 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,
(2.5)F(z,yx)+φ(yx)+1r〈yx-z,z-x〉<φ(z).(B2)
C is a bounded set.In the sequel we will need to use the following lemma.Lemma 2.1 (see [19]).
LetC be a nonempty closed convex subset of H. Let F be a bifunction from C×C to ℝ satisfying (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:
(2.6)Tr(x)={z∈C:F(z,y)+φ(y)+1r〈y-z,z-x〉≥φ(z),∀y∈C}
for all x∈H. Then the following conclusions hold:(1)
for eachx∈H, Tr(x)≠Ø;(2)
Tr is single-valued;(3)
Tr is firmly nonexpansive, that is, for any x,y∈H,
(2.7)∥Tr(x)-Tr(y)∥2≤〈Trx-Try,x-y〉;(4)
F(Tr)=MEP(F,φ);(5)
MEP(F,φ) is closed and convex.We also need the following lemmas.Lemma 2.2 (see [20]).
Let(H,〈·,·〉) be an inner product space. Then, for all x,y,z∈H and α,β,γ∈[0,1] with α+β+γ=1, one has
(2.8)∥αx+βy+γz∥2=α∥x∥2+β∥y∥2+γ∥z∥2-αβ∥x-y∥2-αγ∥x-y∥2-βγ∥y-z∥2.Lemma 2.3.
In a real Hilbert spaceH, there holds the inequality
(2.9)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H.Lemma 2.4 (see [21]).
Assume{an} is a sequence of nonnegative real numbers such that
(2.10)an+1≤(1-γn)an+δn,
where {γn} is a sequence in (0,1) and {δn} is a sequence such that
(i)∑n=1∞‍γn=∞; (ii)limsupn→∞δn/γn≤0 or ∑n=1∞‍|δn|<∞.
Thenlimn→∞an=0.Lemma 2.5 (see [22]).
Let{xn} and {yn} be bounded sequences in a Banach space X and let {bn} be a sequence in [0,1] with 0<liminfn→∞bn≤limsupn→∞bn<1. Suppose xn+1=(1-bn)yn+bnxn for all integers n≥1 and limsupn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then, limn→∞∥yn-xn∥=0.Lemma 2.6 (see [18]).
Demiclosedness principle. Assume thatT is a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space H. If T has a fixed point, then I-T is demiclosed: that is, whenever {xn} is a sequence in C converging weakly to some x∈C (for short, xn⇀x∈C), and the sequence {(I-T)xn} converges strongly to some y (for short, (I-T)xn→y), it follows that (I-T)x=y. Here I is the identity operator of H.In 2009, Kangtunyakarn and Suantai [23] introduced a new mapping called the S-mapping. Let {Ti}i=1N be a finite family of nonexpansive mappings of C into itself. For each j=1,2,…,N, let αj=(α1j,α2j,α3j), where α1j,α2j,α3j∈[0,1] and α1j+α2j+α3j=1. They defined the new mapping S:C→C as follows:
(2.11)U0=I,U1=α11T1U0+α21U0+α31I,U2=α12T2U1+α22U1+α32I,U3=α13T3U2+α23U2+α33I,⋮UN-1=α1N-1TN-1UN-2+α2N-1UN-2+α3N-1I,S=UN=α1NTNUN-1+α2NUN-1+α3NI.
This mapping is called S-mapping generated by T1,T2,…,TN and α1,α2,…,αN. Nonexpansivity of each Ti ensures the nonexpansivity of S.Lemma 2.7 (see [23]).
LetC be a nonempty closed convex subset of a strictly convex Banach space X. Let {Ti}i=1N be a finite family of nonexpansive mappings of C into itself with ⋂i=1N‍F(Ti)≠Ø and let αj=(α1j,α2j,α3j), j=1,2,…,N, where α1j,α2j,α3j∈[0,1], α1j+α2j+α3j=1, α1j∈(0,1) for all j=1,2,…,N-1, α1N∈(0,1] and α2j,α3j∈[0,1) for all j=1,2,…,N. Let S be the S-mapping generated by T1,T2,…,TN and α1,α2,…,αN. Then F(S)=⋂i=1N‍F(Ti).


## 3. Main Results

In this section, we prove strong convergence theorems of the iterative scheme (1.14) to a common element of the set of solutions of a new general system of variational inequalities for relaxed (c,d)-cocoercive mappings, the set of fixed points of a nonexpansive mapping, and the set of solutions of a mixed equilibrium problem in a real Hilbert space.The next lemmas are crucial for proving the main theorems.Lemma 3.1.
LetC be a nonempty closed and convex subset of a real Hilbert space H and let Ai:C→H be Li-Lipschitzian and relaxed (ci,di)-cocoercive mappings for i=1,2,3. Let G:C→C defined by
(3.1)G(x)=PC[PC(PC(x-λ3A3x)-λ2A2PC(x-λ3A3x))-λ1A1PC(PC(x-λ3A3x)-λ2A2PC(x-λ3A3x))],∀x∈C.
If 0<λi≤2(di-ciLi2)/Li2, for i=1,2,3, then G:C→C is a nonexpansive mapping.Proof.
For allx,y∈C, we have
(3.2)∥G(x)-G(y)∥=∥PC[PC(PC(x-λ3A3x)-λ2A2PC(x-λ3A3x))-λ1A1PC(PC(x-λ3A3x)-λ2A2PC(x-λ3A3x))]-PC[PC(PC(y-λ3A3y)-λ2A2PC(y-λ3A3y))-λ1A1PC(PC(y-λ3A3y)-λ2A2PC(y-λ3A3y))]∥=∥PC[PC(PC(I-λ3A3)x-λ2A2PC(I-λ3A3)x)-λ1A1PC(PC(I-λ3A3)x-λ2A2PC(I-λ3A3)x)]-PC[PC(PC(I-λ3A3)y-λ2A2PC(I-λ3A3)y)-λ1A1PC(PC(I-λ3A3)y-λ2A2PC(I-λ3A3)y)]∥≤∥PC(PC(I-λ3A3)x-λ2A2PC(I-λ3A3)x)-λ1A1PC(PC(I-λ3A3)x-λ2A2PC(I-λ3A3)x)-[PC(PC(I-λ3A3)y-λ2A2PC(I-λ3A3)y)-λ1A1PC(PC(I-λ3A3)y-λ2A2PC(I-λ3A3)y)]∥=∥(I-λ1A1)PC(I-λ2A2)PC(I-λ3A3)x-(I-λ1A1)PC(I-λ2A2)PC(I-λ3A3)y∥.
It is well known that if A:C→H is L-Lipschitzian and relaxed (c,d)-cocoercive, then I-λA is nonexpansive for all 0<λ≤2(d-cL2)/L2. By our assumption, we obtain I-λiAi is nonexpansive for all i=1,2,3. It follows that (I-λ1A1)PC(I-λ2A2)PC(I-λ3A3) is nonexpansive. Therefore, from (3.2), we obtain immediately that the mapping G is nonexpansive.Lemma 3.2.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let Ai:C→H be three possibly nonlinear mappings. For given x*,y*,z*∈C, (x*,y*,z*) is a solution of problem (1.8) if and only if x*∈F(G), y*=PC(z*-λ2A2z*) and z*=PC(x*-λ3A3x*), where G is the mapping defined as in Lemma 3.1.Proof.
Note that we can rewrite (1.8) as
(3.3)〈x*-(y*-λ1A1y*),x-x*〉≥0,∀x∈C,〈y*-(z*-λ2A2z*),x-y*〉≥0,∀x∈C,〈z*-(x*-λ3A3x*),x-z*〉≥0,∀x∈C.
From Lemma 1.1, we can deduce that (3.3) is equivalent to
(3.4)x*=PC(y*-λ1A1y*),y*=PC(z*-λ2A2z*),z*=PC(x*-λ3A3x*).
It is easy to see that (3.4) is equivalent to x*∈F(G), y*=PC(z*-λ2A2z*), and z*=PC(x*-λ3A3x*).Throughout this paper, the set of fixed points of the mappingG is denoted by GVI(C,A1,A2,A3).Now we prove the strong convergence theorems of the algorithm (1.14) for solving problem (1.8), fixed point problem of nonexpansive mapping and mixed equilibrium problem.Theorem 3.3.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let F be a function from C×C to ℝ satisfying (A1)–(A5) and φ:C→ℝ⋃‍{+∞} be a proper lower semicontinuous and convex function. Let the mappings Ai:C→H be Li-Lipschitzian and relaxed (ci,di)-cocoercive for i=1,2,3 and T be a nonexpansive self-mapping of C such that Ω=F(T)⋂‍GVI(C,A1,A2,A3)⋂‍MEP(F,φ)≠Ø. Assume that either (B1) or (B2) holds and that v is an arbitrary point in C. Let x1∈C and {xn},{yn},{zn} be the sequences generated by
(3.5)F(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,zn=PC(un-λ3A3un),yn=PC(zn-λ2A2zn),xn+1=anv+bnxn+(1-an-bn)TPC(yn-λ1A1yn),n≥1,
where {rn}⊂(0,∞) and 0<λi<2(di-ciLi2)/Li2, for i=1,2,3 and {an}, {bn} are two sequences in [0,1] such that(C1)
limn→∞an=0 and ∑n=1∞‍an=∞;(C2)
0<liminfn→∞bn≤limsupn→∞bn<1;(C3)
liminfn→∞rn>0 and limn→∞|rn+1-rn|=0.
Then{xn} converges strongly to x¯=PΩv and (x¯,y¯,z¯) is a solution of problem (1.8), where y¯=PC(z¯-λ2A2z¯) and z¯=PC(x¯-λ3A3x¯).Proof.
Letx*∈Ω and {Trn} be a sequence of mappings defined as in Lemma 2.1. It follows from Lemma 3.2 that
(3.6)x*=PC[PC(PC(x*-λ3A3x*)-λ2A2PC(x*-λ3A3x*))-λ1A1PC(PC(x*-λ3A3x*)-λ2A2PC(x*-λ3A3x*))].
Put y*=PC(z*-λ2A2z*), z*=PC(x*-λ3A3x*) and tn=PC(yn-λ1A1yn). Then x*=PC(y*-λ1A1y*), and
(3.7)xn+1=anv+bnxn+(1-an-bn)Ttn.
By nonexpansiveness of I-λiAi(i=1,2,3), we have
(3.8)∥tn-x*∥=∥PC(I-λ1A1)yn-PC(I-λ1A1)y*∥≤∥yn-y*∥=∥PC(I-λ2A2)zn-PC(I-λ2A2)z*∥≤∥zn-z*∥=∥PC(I-λ3A3)un-PC(I-λ3A3)x*∥≤∥un-x*∥=∥Trnxn-Trnx*∥≤∥xn-x*∥,
which implies that
(3.9)∥xn+1-x*∥=∥anv+bnxn+(1-an-bn)Ttn-x*∥≤an∥v-x*∥+bn∥xn-x*∥+(1-an-bn)∥tn-x*∥≤an∥v-x*∥+bn∥xn-x*∥+(1-an-bn)∥xn-x*∥≤max{∥v-x*∥,∥x1-x*∥}.
Thus, {xn} is bounded. Consequently, the sequences {yn}, {zn}, {tn}, {A1yn}, {A2zn}, {A3un}, and {Ttn} are also bounded. Also, observe that
(3.10)∥tn+1-tn∥=∥PC(yn+1-λ1A1yn+1)-PC(yn-λ1A1yn)∥≤∥yn+1-yn∥=∥PC(zn+1-λ2A2zn+1)-PC(zn-λ2A2zn)∥≤∥zn+1-zn∥=∥PC(un+1-λ3A3un+1)-PC(un-λ3A3un)∥≤∥un+1-un∥.
On the other hand, from un=Trnxn∈domφ and un+1=Trn+1xn+1∈domφ, we have
(3.11)F(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,(3.12)F(un+1,y)+φ(y)-φ(un+1)+1rn+1〈y-un+1,un+1-xn+1〉≥0,∀y∈C.
Putting y=un+1 in (3.11) and y=un in (3.12), we have
(3.13)F(un,un+1)+φ(un+1)-φ(un)+1rn〈un+1-un,un-xn〉≥0,F(un+1,un)+φ(un)-φ(un+1)+1rn+1〈un-un+1,un+1-xn+1〉≥0.
From the monotonicity of F, we obtain that
(3.14)〈un+1-un,un-xnrn-un+1-xn+1rn+1〉≥0,
and hence
(3.15)〈un+1-un,un-un+1+un+1-xn-rnrn+1(un+1-xn+1)〉≥0.
Then, we have
(3.16)∥un+1-un∥2≤〈un+1-un,xn+1-xn+(1-rnrn+1)(un+1-xn+1)〉≤∥un+1-un∥{∥xn+1-xn∥+|1-rnrn+1|∥un+1-xn+1∥},
and hence
(3.17)∥un+1-un∥≤∥xn+1-xn∥+1rn+1|rn+1-rn|∥un+1-xn+1∥.
It follows from (3.10) and (3.17) that
(3.18)∥tn+1-tn∥≤∥xn+1-xn∥+1rn+1|rn+1-rn|∥un+1-xn+1∥.
Let xn+1=bnxn+(1-bn)wn for all n≥1. Then, we obtain
(3.19)wn+1-wn=xn+2-bn+1xn+11-bn+1-xn+1-bnxn1-bn=an+1v+(1-an+1-bn+1)Ttn+11-bn+1-anv+(1-an-bn)Ttn1-bn=an+11-bn+1(v-Ttn+1)+an1-bn(Ttn-v)+Ttn+1-Ttn.
By (3.18) and (3.19), we have
(3.20)∥wn+1-wn∥-∥xn+1-xn∥≤an+11-bn+1∥v-Ttn+1∥+an1-bn∥Ttn-v∥+∥tn+1-tn∥-∥xn+1-xn∥≤an+11-bn+1∥v-Ttn+1∥+an1-bn∥Ttn-v∥+1rn+1|rn+1-rn|∥un+1-xn+1∥.
This together with (C1)–(C3), we obtain that
(3.21)limsupn→∞∥wn+1-wn∥-∥xn+1-xn∥≤0.
Hence, by Lemma 2.5, we get ∥xn-wn∥→0 as n→∞. Consequently,
(3.22)limn→∞∥xn+1-xn∥=limn→∞(1-bn)∥wn-xn∥=0.
Since
(3.23)xn+1-xn=an(v-xn)+(1-an-bn)(Ttn-xn),
we have that
(3.24)∥Ttn-xn∥→0asn→∞.
Next, we prove that limn→∞∥xn-un∥=0. From Lemma 2.1 (3), we have
(3.25)∥un-x*∥2=∥Trnxn-Trnx*∥2≤〈Trnxn-Trnx*,xn-x*〉=〈un-x*,xn-x*〉=12{∥un-x*∥2+∥xn-x*∥2-∥xn-un∥2}.
Hence
(3.26)∥un-x*∥2≤∥xn-x*∥2-∥xn-un∥2.
From Lemma 2.2, (3.8) and (3.26), we have
(3.27)∥xn+1-x*∥2≤an∥v-x*∥2+bn∥xn-x*∥2+(1+an-bn)∥tn-x*∥2≤an∥v-x*∥2+bn∥xn-x*∥2+(1+an-bn)∥un-x*∥2≤an∥v-x*∥2bn∥xn-x*∥2+(1+an-bn)[∥xn-x*∥2-∥xn-un∥2]≤an∥v-x*∥2∥xn-x*∥2-(1+an-bn)∥xn-un∥2.
It follows that
(3.28)(1-an-bn)∥xn-un∥2≤an∥v-x*∥2+∥xn-x*∥2-∥xn+1-x*∥2≤an∥v-x*∥2+(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥.
From the conditions (C1), (C2) and (3.22), we obtain
(3.29)limn→∞∥xn-un∥=0.
By (3.24) and (3.29), we have
(3.30)∥Ttn-un∥≤∥Ttn-xn∥+∥xn-un∥→0,asn→∞.
Next, we show that ∥A1yn-A1y*∥→0,∥A2zn-A2z*∥→0, and ∥A3un-A3u*∥→0, as n→∞.
From (3.8) and the fact that A1 is L1-Lipschitzian and relaxed (c1,d1)-cocoercive, we have
(3.31)∥xn+1-x*∥2≤an∥v-x*∥2+bn∥xn-x*∥2+(1-an-bn)∥tn-x*∥2=an∥v-x*∥2+bn∥xn-x*∥2+(1-an-bn)∥PC(yn-λ1A1yn)-PC(y*-λ1A1y*)∥2≤an∥v-x*∥2+bn∥xn-x*∥2+(1-an-bn)∥(yn-λ1A1yn)-(y*-λ1A1y*)∥2=an∥v-x*∥2+bn∥xn-x*∥2+(1-an-bn)[∥yn-y*∥2+λ12∥A1yn-A1y*∥2-2λ1〈yn-y*,A1yn-A1y*〉{∥yn-y*∥2}]≤an∥v-x*∥2+bn∥xn-x*∥2+(1-an-bn)[∥yn-y*∥2+λ12∥A1yn-A1y*∥2-2λ1(-c1∥A1yn-A1y*∥2+d1∥yn-y*∥2)]≤an∥v-x*∥2+bn∥xn-x*∥2+(1-an-bn)[∥xn-x*∥2+(2λ1c1+λ12-2λ1d1L12)∥A1yn-A1y*∥2]≤an∥v-x*∥2+∥xn-x*∥2+(1-an-bn)(2λ1c1+λ12-2λ1d1L12)∥A1yn-A1y*∥2.
Similarly, since Ai is Li-Lipschitzian and relaxed (ci,di)-cocoercive mappings for i=2,3, ∥tn-x*∥≤∥yn-y*∥ and ∥yn-y*∥≤∥zn-z*∥, we can show that
(3.32)∥xn+1-x*∥2≤an∥v-x*∥2+∥xn-x*∥2+(1-an-bn)(2λ2c2+λ22-2λ2d2L22)∥A2zn-A2z*∥2,∥xn+1-x*∥2≤an∥v-x*∥2+∥xn-x*∥2+(1-an-bn)(2λ3c3+λ32-2λ3d3L32)∥A3un-A3x*∥2.
From (3.31) and (3.32), we have
(3.33)-(1-an-bn)(2λ1c1+λ12-2λ1d1L12)∥A1yn-A1y*∥2≤an∥v-x*∥2+(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥-(1-an-bn)(2λ2c2+λ22-2λ2d2L22)∥A2zn-A2z*∥2≤an∥v-x*∥2+(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥-(1-an-bn)(2λ3c3+λ32-2λ3d3L32)∥A3un-A3x*∥2≤an∥v-x*∥2+(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥.
This together with (C1), (C2) and (3.2), we obtain that
(3.34)limn→∞∥A1yn-A1y*∥=limn→∞∥A2zn-A2z*∥=limn→∞∥A3un-A3x*∥=0.
Next, we prove that ∥Ttn-tn∥→0 as n→∞. From (2.2), (3.8) and nonexpansiveness of I-λ2A2 and I-λ3A3, we get
(3.35)∥yn-y*∥2=∥PC(zn-λ2A2zn)-PC(z*-λ2A2z*)∥2≤〈(zn-λ2A2zn)-(z*-λ2A2z*),yn-y*〉=12[∥(zn-λ2A2zn)-(z*-λ2A2z*)∥2+∥yn-y*∥2-∥(zn-λ2A2zn)-(z*-λ2A2z*)-(yn-y*)∥2]≤12[∥zn-z*∥2+∥yn-y*∥2-∥(zn-yn)-(z*-y*)-λ2(A2zn-A2z*)∥2]≤12[∥xn-x*∥2+∥yn-y*∥2-∥(zn-yn)-(z*-y*)∥2+2λ2〈(zn-yn)-(z*-y*),A2zn-A2z*〉-λ22∥A2zn-A2z*∥2],∥zn-z*∥2=∥PC(un-λ3A3un)-PC(x*-λ3A3x*)∥2≤〈(un-λ3A3un)-(x*-λ3A3x*),zn-z*〉=12[∥(un-λ3A3un)-(x*-λ3A3x*)∥2+∥zn-z*∥2-∥(un-λ3A3un)-(x*-λ3A3x*)-(zn-z*)∥2]≤12[∥un-x*∥2+∥zn-z*∥2-∥(un-zn)-(x*-z*)-λ3(A3un-A3x*)∥2]≤12[∥xn-x*∥2+∥zn-z*∥2-∥(un-zn)-(x*-z*)∥2+2λ3〈(un-zn)-(x*-z*),A3un-A3x*〉-λ32∥A3un-A3x*∥2].
Therefore
(3.36)∥yn-y*∥2≤∥xn-x*∥2-∥(zn-yn)-(z*-y*)∥2+2λ2〈(zn-yn)-(z*-y*),A2zn-A2z*〉,∥zn-z*∥2≤∥xn-x*∥2-∥(un-zn)-(x*-z*)∥2+2λ3〈(un-zn)-(x*-z*),A3un-A3x*〉.
From (3.36), we have
(3.37)∥xn+1-x*∥2≤an∥v-x*∥2+bn∥xn-x*∥2+(1-an-bn)∥yn-y*∥2≤an∥v-x*∥2+bn∥xn-x*∥2+(1-an-bn)[∥xn-x*∥2-∥(zn-yn)-(z*-y*)∥2+2λ2〈(zn-yn)-(z*-y*),A2zn-A2z*〉{∥xn-x*∥2}]≤an∥v-x*∥2+∥xn-x*∥2-(1-an-bn)∥(zn-yn)-(z*-y*)∥2+(1-an-bn)2λ2∥(zn-yn)-(z*-y*)∥∥A2zn-A2z*∥,(3.38)∥xn+1-x*∥2≤an∥v-x*∥2+bn∥xn-x*∥2+(1-an-bn)∥zn-z*∥2≤an∥v-x*∥2+bn∥xn-x*∥2+(1-an-bn)[∥xn-x*∥2-∥(un-zn)-(x*-z*)∥2+2λ3〈(un-zn)-(x*-z*),A3un-A3x*〉{∥xn-x*∥2}]≤an∥v-x*∥2+∥xn-x*∥2-(1-an-bn)∥(un-zn)-(x*-z*)∥2+(1-an-bn)2λ3∥(un-zn)-(x*-z*)∥∥A3un-A3x*∥.
Hence
(3.39)(1-an-bn)∥(zn-yn)-(z*-y*)∥2≤an∥v-x*∥2+(1-an-bn)2λ2∥(zn-yn)-(z*-y*)∥∥A2zn-A2z*∥+(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥,(1-an-bn)∥(un-zn)-(x*-z*)∥2≤an∥v-x*∥2+(1-an-bn)2λ3∥(un-zn)-(x*-z*)∥∥A3un-A3x*∥+(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥.
This together with (C1), (C2), (21), and (3.34), we obtain
(3.40)limn→∞∥(zn-yn)-(z*-y*)∥=limn→∞∥(un-zn)-(x*-z*)∥=0.
Therefore
(3.41)∥(un-yn)-(x*-y*)∥≤∥(zn-yn)-(z*-y*)∥+∥(un-zn)-(x*-z*)∥→0asn→∞.
From Lemma 2.3 and (2.3), it follows that
(3.42)∥(yn-tn)+(x*-y*)∥2=∥(yn-λ1A1yn)-(y*-λ1A1y*)-[PC(yn-λ1A1yn)-PC(y*-λ1A1y*)]+λ1(A1yn-A1y*)∥2≤∥(yn-λ1A1yn)-(y*-λ1A1y*)-[PC(yn-λ1A1yn)-PC(y*-λ1A1y*)]∥2+2λ1〈A1yn-A1y*,(yn-tn)+(x*-y*)〉≤∥(yn-λ1A1yn)-(y*-λ1A1y*)∥2-∥PC(yn-λ1A1yn)-PC(y*-λ1A1y*)∥2+2λ1∥A1yn-A1y*∥∥(yn-tn)+(x*-y*)∥≤∥(yn-λ1A1yn)-(y*-λ1A1y*)∥2-∥TPC(yn-λ1A1yn)-TPC(y*-λ1A1y*)∥2+2λ1∥A1yn-A1y*∥∥(yn-tn)+(x*-y*)∥≤∥(yn-λ1A1yn)-(y*-λ1A1y*)-(Ttn-x*)∥×[∥(yn-λ1A1yn)-(y*-λ1A1y*)∥+∥Ttn-x*∥]+2λ1∥A1yn-A1y*∥∥(yn-tn)+(x*-y*)∥=∥un-Ttn+x*-y*-(un-yn)-λ1(A1yn-A1y*)∥×[∥(yn-λ1A1yn)-(y*-λ1A1y*)∥+∥Ttn-x*∥]+2λ1∥A1yn-A1y*∥∥(yn-tn)+(x*-y*)∥.
This together with (3.30), (3.34), and (3.41), we obtain ∥(yn-tn)+(x*-y*)∥→0 as n→∞. This together with (3.30) and (3.40), we obtain that
(3.43)∥Ttn-tn∥≤∥Ttn-un∥+∥(un-zn)-(x*-z*)∥+∥(zn-yn)-(z*-y*)∥+∥(yn-tn)+(x*-y*)∥→0,asn→∞.
Next, we show that
(3.44)limsupn→∞〈v-x¯,xn-x¯〉≤0,
where x¯=PΩv.
Indeed, since{tn} and {Ttn} are two bounded sequences in C, we can choose a subsequence {tni} of {tn} such that tni⇀z∈C and
(3.45)limsupn→∞〈v-x¯,Ttn-x¯〉=limi→∞〈v-x¯,Ttni-x¯〉.
Sincelimn→∞∥Ttn-tn∥=0, we obtain that Ttni⇀z as i→∞.
Next, we show thatz∈Ω.
Sincetni⇀z and ∥Ttn-tn∥→0, we obtain by Lemma 2.6 that z∈F(T).
From (3.43) and (3.24), we obtain
(3.46)∥tn-xn∥≤∥Ttn-tn∥+∥Ttn-xn∥→0,asn→∞.
Furthermore, by Lemma 3.1, we have G:C→C is nonexpansive. Then, we have
(3.47)∥tn-G(tn)∥=∥PC(yn-λ1A1yn)-G(tn)∥=∥PC[PC(zn-λ2A2zn)-λ1A1PC(zn-λ2A2zn)]-G(tn)∥=∥PC[PC(PC(un-λ3A3un)-λ2A2PC(un-λ3A3un))-λ1A1PC(PC(un-λ3A3un)-λ2A2PC(un-λ3A3un))]-G(tn)∥=∥G(un)-G(tn)∥≤∥un-tn∥≤∥un-xn∥+∥xn-tn∥,
hence limn→∞∥tn-G(tn)∥=0. Again by Lemma 2.6, we have z∈GVI(C,A1,A2,A3).
Sincetni⇀z and ∥xn-tn∥→0, we obtain that xni⇀z. From ∥un-xn∥→0, we also obtain that uni⇀z. By using the same argument as that in the proof of [19, Theorem 3.1, pp. 1825], we can show that z∈MEP(F,φ). Therefore z∈Ω.
On the other hand, it follows from (2.4), (3.24), and Ttni⇀z as i→∞ that
(3.48)limsupn→∞〈v-x¯,xn-x¯〉=limsupn→∞〈v-x¯,Ttn-x¯〉=limi→∞〈v-x¯,Ttni-x¯〉=〈v-x¯,z-x¯〉≤0.
Hence, we have
(3.49)∥xn+1-x¯∥2=〈anv+bnxn+(1-an-bn)Ttn-x¯,xn+1-x¯〉=an〈v-x¯,xn+1-x¯〉+bn〈xn-x¯,xn+1-x¯〉+(1-an-bn)〈Ttn-x¯,xn+1-x¯〉≤an〈v-x¯,xn+1-x¯〉+12bn(∥xn-x¯∥2+∥xn+1-x¯∥2)+12(1-an-bn)(∥tn-x¯∥2+∥xn+1-x¯∥2)≤an〈v-x¯,xn+1-x¯〉+12bn(∥xn-x¯∥2+∥xn+1-x¯∥2)+12(1-an-bn)(∥xn-x¯∥2+∥xn+1-x¯∥2)=an〈v-x¯,xn+1-x¯〉+12(1-an)(∥xn-x¯∥2+∥xn+1-x¯∥2),
which implies that
(3.50)∥xn+1-x¯∥2≤(1-an)∥xn-x¯∥2+2an〈v-x¯,xn+1-x¯〉.
This together with (C1) and (3.48), we have by Lemma 2.4 that {xn} converges strongly to x¯. This completes the proof.Theorem 3.4.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let F be a function from C×C to ℝ satisfying (A1)–(A5) and φ:C→ℝ⋃‍{+∞} be a proper lower semicontinuous and convex function. Let the mappings Ai:C→H be Li-Lipschitzian and relaxed (ci,di)-cocoercive for i=1,2,3 and {Ti}i=1N be a finite family of nonexpansive self-mappings of C such that Ω=⋂i=1N‍F(Ti)⋂‍GVI(C,A1,A2,A3)⋂‍MEP(F,φ)≠Ø. Let αj=(α1j,α2j,α3j), j=1,2,…,N, where α1j,α2j,α3j∈[0,1], α1j+α2j+α3j=1, α1j∈(0,1) for all j=1,2,…,N-1, α1N∈(0,1] and α2j,α3j∈[0,1) for all j=1,2,…,N. Let S be the S-mapping generated by T1,T2,…,TN and α1,α2,…,αN. Assume that either (B1) or (B2) holds and that v is an arbitrary point in C. Let x1∈C and {xn},{yn},{zn} be the sequences generated by
(3.51)F(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,zn=PC(un-λ3A3un),yn=PC(zn-λ2A2zn),xn+1=anv+bnxn+(1-an-bn)SPC(yn-λ1A1yn),n≥1.
If 0<λi<2(di-ciLi2)/Li2, for i=1,2,3 and {rn}, {an}, {bn} are as in Theorem 3.3. Then {xn} converges strongly to x¯=PΩv and (x¯,y¯,z¯) is a solution of problem (1.8), where y¯=PC(z¯-λ2A2z¯) and z¯=PC(x¯-λ3A3x¯).Proof.
By Lemma2.7, we obtain that S is nonexpansive and F(S)=⋂i=1N‍F(Ti). Hence, the result is obtained directly from Theorem 3.3.Ifφ=0 in Theorem 3.3, then, we obtain the following result.Corollary 3.5.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let F be a function from C×C to ℝ satisfying (A1)–(A5) and φ:C→ℝ⋃‍{+∞} be a proper lower semicontinuous and convex function. Let the mappings Ai:C→H be Li-Lipschitzian and relaxed (ci,di)-cocoercive for i=1,2,3 and T be a nonexpansive self-mapping of C such that Ω=F(T)⋂‍GVI(C,A1,A2,A3)⋂‍EP(F)≠Ø.
Letv,x1∈C and {xn},{yn},{zn} be the sequences generated by
(3.52)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,zn=PC(xn-λ3A3xn),yn=PC(zn-λ2A2zn),xn+1=anv+bnxn+(1-an-bn)TPC(yn-λ1A1yn),n≥1.
If 0<λi<2(di-ciLi2)/Li2, for i=1,2,3 and the sequences {an}, {bn}, and {rn} are as in Theorem 3.3, then {xn} converges strongly to x¯=PΩv and (x¯,y¯,z¯) is a solution of problem (1.8), where y¯=PC(z¯-λ2A2z¯) and z¯=PC(x¯-λ3A3x¯).



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*Source: 482869-2012-12-31.xml*