# Hybrid Proximal-Point Methods for Zeros of Maximal Monotone Operators, Variational Inequalities and Mixed Equilibrium Problems

Authors: Kriengsak Wattanawitoon; Poom Kumam
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

We prove strong and weak convergence theorems of modified hybrid proximal-point
algorithms for finding a common element of the zero point of a maximal monotone operator, the set
of solutions of equilibrium problems, and the set of solution of the variational inequality operators of
an inverse strongly monotone in a Banach space under different conditions. Moreover, applications to
complementarity problems are given. Our results modify and improve the recently announced ones by
Li and Song (2008) and many authors.


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## Body

## 1. Introduction

LetE be a Banach space with norm ∥·∥, C a nonempty closed convex subset of E, let E* denote the dual of E and <·,·> is the pairing between E and E*.Consider the problem of finding(1.1)v∈Esuchthat0∈T(v),
where T is an operator from E into E*. Such v∈E is called a zero point of T. When T is a maximal monotone operator, a well-known method for solving (1.1) in a Hilbert space H is the proximal point algorithmx1=x∈H and(1.2)xn+1=Jrnxn,n=1,2,3,…,
where {rn}⊂(0,∞) and Jrn=(I+rnT)-1, then Rockafellar [1] proved that the sequence {xn} converges weakly to an element of T-1(0).In 2000, Kamimura and Takahashi [2] proved the following strong convergence theorem in Hilbert spaces, by the following algorithm:(1.3)xn+1=αnx+(1-αn)Jrnxn,n=1,2,3,…,
where Jr=(I+rT)-1J, then the sequence {xn} converges strongly to PT-10(x), where PT-10 is the projection from H onto T-1(0). These results were extended to more general Banach spaces see [3, 4].In 2004, Kohsaka and Takahashi [4] introduced the following iterative sequence for a maximal monotone operator T in a smooth and uniformly convex Banach space: x1=x∈E and(1.4)xn+1=J-1(αnJx+(1-αn)J(Jrnxn)),n=1,2,3,…,
where J is the duality mapping from E into E* and Jr=(I+rT)-1J.Recently, Li and Song [5] proved a strong convergence theorem in a Banach space, by the following algorithm: x1=x∈E and(1.5)yn=J-1(βnJ(xn)+(1-βn)J(Jrnxn)),xn+1=J-1(αnJx1+(1-αn)J(yn)),
with the coefficient sequences {αn}, {βn}⊂[0,1] and {rn}⊂(0,∞) satisfying limn→∞αn=0, ∑n=1∞αn=∞, limn→∞βn=0, and limn→∞rn=∞. Where J is the duality mapping from E into E* and Jr=(I+rT)-1J. Then, they proved that the sequence {xn} converges strongly to ΠCx, where ΠC is the generalized projection from E onto C.LetC be a nonempty closed convex subset of E, and let A be a monotone operator of C into E*. The variational inequality problem is to find a point x*∈C such that(1.6)〈v-x*,Ax*〉≥0,∀v∈C.
The set of solutions of the variational inequality problem is denoted by VI(C,A). Such a problem is connected with the convex minimization problem, the complementarity problem, the problem of finding a point u∈E satisfying 0=Au, and so on. An operator A of C into E* is said to be inverse-strongly monotone if there exists a positive real number α such that(1.7)〈x-y,Ax-Ay〉≥α‖Ax-Ay‖2,
for all x,y∈C. In such a case, A is said to be α-inverse-strongly monotone. If an operator A of C into E* is α-inverse-strongly monotone, then A is Lipschitz continuous, that is, ∥Ax-Ay∥≤(1/α)∥x-y∥ for all x,y∈C.In a Hilbert spaceH, Iiduka et al. [6] proved that the sequence {xn} defined by: x1=x∈C and(1.8)xn+1=PC(xn-λnAxn),
where PC is the metric projection of H onto C and {λn} is a sequence of positive real numbers, converges weakly to some element of VI(C,A).In 2008, Iiduka and Takahashi [7] introduced the folowing iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly monotone operator A in a Banach space x1=x∈C and(1.9)xn+1=ΠCJ-1(Jxn-λnAxn),
for every n=1,2,3,…, where ΠC is the generalized metric projection from E onto C, J is the duality mapping from E into E* and {λn} is a sequence of positive real numbers. They proved that the sequence {xn} generated by (1.9) converges weakly to some element of VI(C,A).LetΘ be a bifunction of C×C into ℝ and φ:C→ℝ a real-valued function. The mixed equilibrium problem, denoted by MEP(Θ,φ), is to find x∈C such that(1.10)Θ(x,y)+φ(y)-φ(x)≥0,∀y∈C.
If φ≡0, the problem (1.10) reduces into the equilibrium problem forΘ, denoted by EP(Θ), is to find x∈C such that(1.11)Θ(x,y)≥0,∀y∈C.
If Θ≡0, the problem (1.10) reduces into the minimize problem, denoted by Argmin(φ), is to find x∈C such that(1.12)φ(y)-φ(x)≥0,∀y∈C.
The above formulation (1.11) was shown in [8] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem, and optimization problem, which can also be written in the form of an EP(Θ). In other words, the EP(Θ) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, and so forth. In the last two decades, many papers have appeared in the literature on the existence of solutions of EP(Θ); see, for example, [8–11] and references therein. Some solution methods have been proposed to solve the EP(Θ); see, for example, [9, 11–21] and references therein. In 2005, Combettes and Hirstoaga [12] introduced an iterative scheme of finding the best approximation to the initial data when EP(Θ) is nonempty and they also proved a strong convergence theorem.Recall, a mappingS:C→C is said to be nonexpansive if (1.13)‖Sx-Sy‖≤‖x-y‖,
for all x,y∈C. We denote by F(S) the set of fixed points of S. If C is bounded closed convex and S is a nonexpansive mapping of C into itself, then F(S) is nonempty (see [22]). A mapping S is said to be quasi-nonexpansive if F(S)≠∅ and ∥Sx-y∥≤∥x-y∥ for all x∈C and y∈F(S). It is easy to see that if S is nonexpansive with F(S)≠∅, then it is quasi-nonexpansive. We write xn→x(xn⇀x, resp.) if {xn} converges (weakly, resp.) to x. Let E be a real Banach space with norm ∥·∥ and let J be the normalized duality mapping from E into 2E* given by (1.14)Jx={x*∈E*:〈x,x*〉=‖x‖‖x*‖,‖x‖=‖x*‖},
for all x∈E, where E* denotes the dual space of E and 〈·,·〉 the generalized duality pairing between E and E*. It is well known that if E* is uniformly convex, then J is uniformly continuous on bounded subsets of E.LetC be a closed convex subset of E, and let S be a mapping from C into itself. A point p in C is said to be an asymptotic fixed point of S [23] if C contains a sequence {xn} which converges weakly to p such that limn→∞∥xn-Sxn∥=0. The set of asymptotic fixed points of S will be denoted by F(S)̃. A mapping S from C into itself is said to be relatively nonexpansive [24–26] if F(S)̃=F(S) and ϕ(p,Sx)≤ϕ(p,x) for all x∈C and p∈F(S). The asymptotic behavior of a relatively nonexpansive mapping was studied in [27, 28]. S is said to be ϕ-nonexpansive, if ϕ(Sx,Sy)≤ϕ(x,y) for x,y∈C. S is said to be relatively quasi-nonexpansive if F(S)≠∅ and ϕ(p,Sx)≤ϕ(p,x) for x∈C and p∈F(S).In 2009, Takahashi and Zembayashi [29] introduced the following shrinking projection method of closed relatively nonexpansive mappings as follows:(1.15)x0=x∈C,C0=C,yn=J-1(αnJ(xn)+(1-αn)JS(xn)),un∈CsuchthatΘ(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x,
for every n∈ℕ∪{0}, where J is the duality mapping on E, {αn}⊂[0,1] satisfies liminfn→∞αn(1-αn)>0 and {rn}⊂[a,∞) for some a>0. Then, they proved that the sequence {xn} converges strongly to ΠF(S)∩EP(Θ)x.In 2009, Qin et al. [30] modified the Halpern-type iteration algorithm for closed quasi-ϕ-nonexpansive mappings (or relatively quasi-nonexpansive) defined by(1.16)x0∈Echosenarbitrarily,C1=C,x1=ΠC1x0,yn=J-1(αnJ(x1)+(1-αn)JT(xn)),Cn+1={z∈Cn:ϕ(z,yn)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)},xn+1=ΠCn+1x1,∀n≥1.
Then, they proved that under appropriate control conditions the sequence {xn} converges strongly to ΠF(T)x1.Recently, Ceng et al. [31] proved the following strong convergence theorem for finding a common element of the set of solutions for an equilibrium and the set of a zero point for a maximal monotone operator T in a Banach space E(1.17)yn=J-1(αnJ(x0)+(1-αn)(βnJxn+(1-βn)JJrn(xn))),Hn={z∈C:ϕ(z,Trnyn)≤αnϕ(z,x0)+(1-αn)ϕ(z,xn)},Wn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠHn∩Wnx0.
Then, the sequence {xn} converges strongly to ΠT-10∩EP(Θ)x0, where ΠT-10∩EP(Θ) is the generalized projection of E onto T-10∩EP(Θ).In this paper, motivated and inspired by Li and Song [5], Iiduka and Takahashi [7], Takahashi and Zembayashi [29], Ceng et al. [31] and Qin et al. [30], we introduce the following new hybrid proximal-point algorithms defined by x1=x∈C:(1.18)wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJ(xn)+(1-βn)J(Jrnwn)),yn=J-1(αnJ(x1)+(1-αn)J(zn)),un∈CsuchthatΘ(un,y)+φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)},xn+1=ΠCn+1x
and(1.19)un∈CsuchthatΘ(un,y)+φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,zn=ΠCJ-1(Jun-λnAun),yn=J-1(βnJ(xn)+(1-βn)J(Jrnzn)),xn+1=ΠCJ-1(αnJ(x1)+(1-αn)J(yn)).
Under appropriate conditions, we will prove that the sequence {xn} generated by algorithms (1.18) and (1.19) converges strongly to the point ΠVI(C,A)∩T-1(0)∩MEP(Θ,φ)x and converges weakly to the point limn→∞ΠVI(C,A)∩T-1(0)∩MEP(Θ,φ)xn, respectively. The results presented in this paper extend and improve the corresponding ones announced by Li and Song [5] and many authors in the literature.


## 2. Preliminaries

A Banach spaceE is said to be strictly convex if ∥(x+y)/2∥<1 for all x,y∈E with ∥x∥=∥y∥=1 and x≠y. Let U={x∈E:∥x∥=1} be the unit sphere of E. Then, the Banach space E is said to be smooth provided (2.1)lim⁡t→0‖x+ty‖-‖x‖t
exists for each x,y∈U. It is also said to be uniformly smooth if the limit is attained uniformly for x,y∈E. The modulus of convexity of E is the function δ:[0,2]→[0,1] defined by(2.2)δ(ɛ)=inf⁡{1-‖x+y2‖:x,y∈E,‖x‖=‖y‖=1,‖x-y‖≥ɛ}.
A Banach space E is uniformly convex if and only if δ(ɛ)>0 for all ɛ∈(0,2]. Let p be a fixed real number with p≥2. A Banach space E is said to be p-uniformly convex if there exists a constant c>0 such that δ(ɛ)≥cɛp for all ɛ∈[0,2]; see [32, 33] for more details. Observe that every p-uniform convex is uniformly convex. One should note that no Banach space is p-uniform convex for 1<p<2. It is well known that a Hilbert space is 2-uniformly convex and uniformly smooth. For each p>1, the generalized duality mappingJp:E→2E* is defined by(2.3)Jp(x)={x*∈E*:〈x,x*〉=‖x‖p,‖x*‖=‖x‖p-1},
for all x∈E. In particular, J=J2 is called the normalized duality mapping. If E is a Hilbert space, then J=I, where I is the identity mapping. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.We know the following (see [34]):(1)
ifE is smooth, then J is single-valued,(2)
ifE is strictly convex, then J is one-to-one and 〈x-y,x*-y*〉>0 holds for all (x,x*),(y,y*)∈J with x≠y,(3)
ifE is reflexive, then J is surjective,(4)
ifE is uniformly convex, then it is reflexive,(5)
ifE* is uniformly convex, then J is uniformly norm-to-norm continuous on each bounded subset of E.The dualityJ from a smooth Banach space E into E* is said to be weakly sequentially continuous [35] if xn⇀x implies Jxn⇀*Jx, where ⇀* implies the weak* convergence.Lemma 2.1 (see [36, 37]).
IfE be a 2-uniformly convex Banach space. Then, for all x,y∈E one has
(2.4)‖x-y‖≤2c2‖Jx-Jy‖,
where J is the normalized duality mapping of E and 0<c≤1.The best constant1/c in Lemma is called the 2-uniformly convex constant of E; see [32].Lemma 2.2 (see [36, 38]).
IfE a p-uniformly convex Banach space and let p be a given real number with p≥2. Then, for all x,y∈E,Jx∈Jp(x) and Jy∈Jp(y)(2.5)〈x-y,Jx-Jy〉≥cp2p-2p‖x-y‖p,
where Jp is the generalized duality mapping of E and 1/c is the p-uniformly convexity constant of E.Lemma 2.3 (see Xu [37]).
LetE be a uniformly convex Banach space. Then, for each r>0, there exists a strictly increasing, continuous, and convex function K:[0,∞)→[0,∞) such that K(0)=0 and
(2.6)‖λx+(1-λy)‖2≤λ‖x‖2+(1-λ)‖y‖2-λ(1-λ)K(‖x-y‖),
for all x,y∈{z∈E:∥z∥≤r} and λ∈[0,1].LetE be a smooth, strictly convex, and reflexive Banach space and let C be a nonempty closed convex subset of E. Throughout this paper, we denote by ϕ the function defined by(2.7)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2,forx,y∈E.
Following Alber [39], the generalized projectionΠC:E→C is a map that assigns to an arbitrary point x∈E the minimum point of the functional ϕ(x,y), that is, ΠCx=x¯, where x¯ is the solution to the minimization problem(2.8)ϕ(x¯,x)=inf⁡y∈Cϕ(y,x)
existence and uniqueness of the operator ΠC follows from the properties of the functional ϕ(x,y) and strict monotonicity of the mapping J. It is obvious from the definition of function ϕ that (see [39])(2.9)(‖y‖-‖x‖)2≤ϕ(y,x)≤(‖y‖+‖x‖)2,∀x,y∈E.
If E is a Hilbert space, then ϕ(x,y)=∥x-y∥2.IfE is a reflexive, strictly convex and smooth Banach space, then for x,y∈E, ϕ(x,y)=0 if and only if x=y. It is sufficient to show that if ϕ(x,y)=0, then x=y. From (2.9), we have ∥x∥=∥y∥. This implies that 〈x,Jy〉=∥x∥2=∥Jy∥2. From the definition of J, one has Jx=Jy. Therefore, we have x=y; see [34, 40] for more details.Lemma 2.4 (see Kamimura and Takahashi [3]).
LetE be a uniformly convex and smooth real Banach space and let {xn},{yn} be two sequences of E. If ϕ(xn,yn)→0 and either {xn} or {yn} is bounded, then ∥xn-yn∥→0.Lemma 2.5 (see Alber [39]).
LetC be a nonempty, closed, convex subset of a smooth Banach space  E and  x∈E. Then,  x0=ΠCx if and only if
(2.10)〈x0-y,Jx-Jx0〉≥0,∀y∈C.Lemma 2.6 (see Alber [39]).
LetE be a reflexive, strictly convex, and smooth Banach space, let  C be a nonempty closed convex subset of  E and let  x∈E. Then,
(2.11)ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x),∀y∈C.LetE be a strictly convex, smooth, and reflexive Banach space, let J be the duality mapping from E into E*. Then, J-1 is also single-valued, one-to-one, and surjective, and it is the duality mapping from E* into E. Define a function V:E×E*→ℝ as follows (see [4]):(2.12)V(x,x*)=‖x‖2-2〈x,x*〉+‖x*‖2,
for all x∈E,x∈E and x*∈E*. Then, it is obvious that V(x,x*)=ϕ(x,J-1(x*)) and V(x,J(y))=ϕ(x,y).Lemma 2.7 (see Kohsaka and Takahashi [4, Lemma 3.2]).
LetE be a strictly convex, smooth, and reflexive Banach space, and let  V be as in (2.12). Then,
(2.13)V(x,x*)+2〈J-1(x*)-x,y*〉≤V(x,x*+y*),
for all x∈E and x*,y*∈E*.LetE be a reflexive, strictly convex, and smooth Banach space. Let C be a closed convex subset of E. Because ϕ(x,y) is strictly convex and coercive in the first variable, we know that the minimization problem infy∈Cϕ(x,y) has a unique solution. The operator ΠCx:=argminy∈Cϕ(x,y) is said to be the generalized projection of x on C.A set-valued mappingT:E→E* with domain D(T)={x∈E:T(x)≠∅} and range R(T)={x*∈E*:x*∈T(x),x∈D(T)} is said to be monotone if 〈x-y,x*-y*〉≥0 for all x*∈T(x),y*∈T(y). We denote the set {s∈E:0∈Tx} by T-10. T is maximal monotone if its graph G(T) is not properly contained in the graph of any other monotone operator. If T is maximal monotone, then the solution set T-10 is closed and convex.LetE be a reflexive, strictly convex, and smooth Banach space, it is known that T is a maximal monotone if and only if R(J+rT)=E* for all r>0.Define theresolvent of T by Jrx=xr. In other words, Jr=(J+rT)-1J for all r>0. Jr is a single-valued mapping from E to D(T). Also, T-1(0)=F(Jr) for all r>0, where F(Jr) is the set of all fixed points of Jr. Define, for r>0, the Yosida approximation of T by Ar=(J-JJr)/r. We know that Arx∈T(Jrx) for all r>0 and x∈E.Lemma 2.8 (see Kohsaka and Takahashi [4, Lemma 3.1]).
LetE be a smooth, strictly convex, and reflexive Banach space, T⊂E×E* a maximal monotone operator with T-10≠∅, r>0 and Jr=(J+rT)-1J. Then,
(2.14)ϕ(x,Jry)+ϕ(Jry,y)≤ϕ(x,y),
for all x∈T-10 and y∈E.LetA be an inverse-strongly monotone mapping of C into E* which is said to be hemicontinuous if for all x,y∈C, the mapping F of [0,1] into E*, defined by F(t)=A(tx+(1-t)y), is continuous with respect to the weak* topology of E*. We define by NC(v) the normal cone for C at a point v∈C, that is,(2.15)NC(v)={x*∈E*:〈v-y,x*〉≥0,∀y∈C}.Theorem 2.9 (see Rockafellar [1]).
LetC be a nonempty, closed, convex subset of a Banach space E and A a monotone, hemicontinuous operator of C into E*. Let T⊂E×E* be an operator defined as follows:
(2.16)Tv={Av+NC(v),v∈C,∅,otherwise.
Then, T is maximal monotone and T-10=VI(C,A).Lemma 2.10 (see Tan and Xu [41]).
Let{an} and {bn} be two sequence of nonnegative real numbers satisfying the inequality
(2.17)an+1=an+bn,∀n≥0.
If ∑n=1∞bn<∞, then limn→∞an exists.For solving the mixed equilibrium problem, let us assume that the bifunctionΘ:C×C→ℝ and φ:C→ℝ is convex and lower semicontinuous satisfies the following conditions:  (A1)
Θ(x,x)=0 for all x∈C,(A2)
Θ is monotone, that is, Θ(x,y)+Θ(y,x)≤0 for all x,y∈C,(A3)
for eachx,y,z∈C,
(2.18)limsup⁡t↓0Θ(tz+(1-t)x,y)≤Θ(x,y),(A4)
for eachx∈C, y↦Θ(x,y) is convex and lower semicontinuous.Motivated by Blum and Oettli [8], Takahashi and Zembayashi [29, Lemma 2.7] obtained the following lemmas.Lemma 2.11 (see [29, Lemma 2.7]).
LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space  E, let  θ be a bifunction from  C×C to  ℝ satisfying (A1)–(A4), let  r>0, and let  x∈E. Then, there exists  z∈C such that
(2.19)Θ(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Lemma 2.12 (see Takahashi and Zembayashi [29]).
LetC be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space  E and let  Θ be a bifunction from  C×C to  ℝ satisfying (A1)–(A4). For all  r>0 and  x∈E, define a mapping  Tr:E→C as follows:
(2.20)Trx={z∈C:Θ(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},
for all x∈E. Then, the followings hold:  (1)
Tr is single-valued,(2)
Tr is a firmly nonexpansive-type mapping, that is, for all x,y∈E,
(2.21)〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉,(3)
F(Tr)=EP(Θ),(4)
EP(Θ) is closed and convex.Lemma 2.13 (see Takahashi and Zembayashi [29]).
LetC be a closed, convex subset of a smooth, strictly convex, and reflexive Banach space  E, let  Θ a bifunction from  C×C to  ℝ satisfying (A1)–(A4) and let  r>0. Then, for  x∈E and  q∈F(Tr),
(2.22)ϕ(q,Trx)+ϕ(Trx,x)≤ϕ(q,x).Lemma 2.14.
LetC be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Let φ:C→ℝ is convex and lower semicontinuous and Θ be a bifunction from C×C to ℝ satisfying (A1)–(A4). For r>0 and x∈E, then there exists u∈C such that
(2.23)Θ(u,y)+φ(y)-φ(u)+1r〈y-u,Ju-Jx〉.
Define a mapping Kr:E→C as follows:
(2.24)Kr(x)={u∈C:Θ(u,y)+φ(y)-φ(u)+1r〈y-u,Ju-Jx〉≥0,∀y∈C}
for all x∈E. Then, the followings hold:  (1)
Kr is single-valued,(2)
Kr is firmly nonexpansive, that is, for all x,y∈E, 〈Krx-Kry,JKrx-JKry〉≤〈Krx-Kry,Jx-Jy〉,(3)
F(Kr)=MEP(Θ,φ),(4)
MEP(Θ,φ) is closed and convex.Proof.
Define a bifunctionF:C×C→ℝ as follows:
(2.25)F(u,y)=Θ(u,y)+φ(y)-φ(u),∀u,y∈C.
It is easily seen thatF satisfies (A1)–(A4). Therefore, Kr in Lemma 2.14 can be obtained from Lemma 2.12 immediately.


## 3. Strong Convergence Theorem

In this section, we prove a strong convergence theorem for finding a common element of the zero point of a maximal monotone operator, the set of solutions of equilibrium problems, and the set of solution of the variational inequality operators of an inverse strongly monotone in a Banach space by using the shrinking hybrid projection method.Theorem 3.1.
LetE be a 2-uniformly convex and uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let Θ be a bifunction from C×C to ℝ satisfying (A1)–(A4) let φ:C→ℝ be a lower semicontinuous and convex function, and let T:E→E* be a maximal monotone operator. Let Jr=(J+rT)-1J for r>0 and let A be an α-inverse-strongly monotone operator of C into E* with F:=VI(C,A)∩T-1(0)∩MEP(Θ,φ)≠∅ and ∥Ay∥≤∥Ay-Au∥ for all y∈C and u∈F. Let {xn} be a sequence generated by x0∈E with x1=ΠC1x0 and C1=C,
(3.1)wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJ(xn)+(1-βn)J(Jrnwn)),yn=J-1(αnJ(x1)+(1-αn)J(zn)),un∈Csuch thatΘ(un,y)+φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥0,∀yC,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)},xn+1=ΠCn+1x0,
for n∈ℕ, where ΠC is the generalized projection from E onto C, J is the duality mapping on E. The coefficient sequence {αn},{βn}⊂(0,1), {rn}⊂(0,∞) satisfying limn→∞αn=0,limsupn→∞βn<1, liminfn→∞rn>0, and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, 1/c is the 2-uniformly convexity constant of E. Then, the sequence {xn} converges strongly to ΠFx0.Proof.
We first show that{xn} is bounded. Put vn=J-1(Jxn-λnAxn), let p∈F:=VI(C,A)∩T-1(0)∩MEP(Θ,φ), and let {Krn} be a sequence of mapping define as Lemma 2.14 and un=Krnyn. By (3.1) and Lemma 2.7, the convexity of the function V in the second variable, we have
(3.2)ϕ(p,wn)=ϕ(p,ΠCvn)≤ϕ(p,vn)=ϕ(p,J-1(Jxn-λnAxn))≤V(p,Jxn-λnAxn+λnAxn)-2〈J-1(Jxn-λnAxn)-p,λnAxn〉=V(p,Jxn)-2λn〈vn-p,Axn〉=ϕ(p,xn)-2λn〈xn-p,Axn〉+2〈vn-xn,-λnAxn〉.
Since p∈VI(A,C) and A is α-inverse-strongly monotone, we have
(3.3)-2λn〈xn-p,Axn〉=-2λn〈xn-p,Axn-Ap〉-2λn〈xn-p,Ap〉≤-2αλn‖Axn-Ap‖2,
and by Lemma 2.1, we obtain
(3.4)2〈vn-xn,-λnAxn〉=2〈J-1(Jxn-λnAxn)-xn,-λnAxn〉≤2‖J-1(Jxn-λnAxn)-xn‖‖λnAxn‖≤4c2‖Jxn-λnAxn-Jxn‖‖λnAxn‖=4c2λn2‖Axn‖2≤4c2λn2‖Axn-Ap‖2.
Substituting (3.3) and (3.4) into (3.2), we get
(3.5)ϕ(p,wn)≤ϕ(p,xn)-2αλn‖Axn-Ap‖2+4c2λn2‖Axn-Ap‖2≤ϕ(p,xn)+2λn(2c2λn-α)‖Axn-Ap‖2≤ϕ(p,xn).
By Lemmas 2.7, 2.8 and (3.5), we have
(3.6)ϕ(p,zn)=ϕ(p,J-1(βnJ(xn)+(1-βn)J(Jrnwn)))=V(p,βnJ(xn)+(1-βn)J(Jrnwn))≤βnV(p,J(xn))+(1-βn)V(p,J(Jrnwn))=βnϕ(p,xn)+(1-βn)ϕ(p,Jrnwn)≤βnϕ(p,xn)+(1-βn)(ϕ(p,wn)-ϕ(Jrnwn,wn))≤βnϕ(p,xn)+(1-βn)ϕ(p,wn)≤βnϕ(p,xn)+(1-βn)ϕ(p,xn)=ϕ(p,xn).
It follows that
(3.7)ϕ(p,yn)=ϕ(p,J-1(αnJ(x1)+(1-αn)J(zn)))=V(p,αnJ(x1)+(1-αn)J(zn))≤αnV(p,J(x1))+(1-αn)V(p,J(zn))=αnϕ(p,x1)+(1-αn)ϕ(p,zn)≤αnϕ(p,x1)+(1-αn)ϕ(p,xn).
From (3.1) and (3.7), we obtain
(3.8)ϕ(p,un)=ϕ(p,Krnyn)≤ϕ(p,yn)≤ϕ(p,x1)+(1-αn)ϕ(p,xn).
So, we have p∈Cn+1. This implies that F⊂Cn, for all n∈ℕ.
From Lemma2.5 and xn=ΠCnx0, we have
(3.9)〈xn-z,Jx0-Jxn〉≥0,∀z∈Cn,〈xn-p,Jx0-Jxn〉≥0,∀p∈F.
From Lemma 2.6, one has
(3.10)ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(p,x0)-ϕ(p,xn)≤ϕ(p,x0),
for all p∈F⊂Cn and n≥1. Then, the sequence {ϕ(xn,x0)} is bounded. Since xn=ΠCnx0 and xn+1=ΠCn+1x0∈Cn+1⊂Cn, we have
(3.11)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n∈N.
Therefore, {ϕ(xn,x0)} is nondecreasing. Hence, the limit of {ϕ(xn,x0)} exists. By the construction of Cn, one has that Cm⊂Cn and xm=ΠCmx0∈Cn for any positive integer m≥n. It follows that
(3.12)ϕ(xm,xn)=ϕ(xm,ΠCnx0)≤ϕ(xm,x0)-ϕ(ΠCnx0,x0)=ϕ(xm,x0)-ϕ(xn,x0).
Letting m,n→∞ in (3.12), we get ϕ(xm,xn)→0. It follows from Lemma 2.4, that ∥xm-xn∥→0 as m,n→∞, that is, {xn} is a Cauchy sequence. Since E is a Banach space and C is closed and convex, we can assume that xn→u∈C, as n→∞. Since
(3.13)ϕ(xn+1,xn)=ϕ(xn+1,ΠCnx0)≤ϕ(xn+1,x0)-ϕ(ΠCnx0,x0)=ϕ(xn+1,x0)-ϕ(xn,x0),
for all n∈ℕ, we also have limn→∞ϕ(xn+1,xn)=0. From Lemma 2.4, we get limn→∞∥xn+1-xn∥=0. Since xn+1=ΠCn+1x0∈Cn+1 and by definition of Cn+1, we have
(3.14)ϕ(xn+1,un)≤αnϕ(xn+1,x1)+(1-αn)ϕ(xn+1,xn).
Noticing the conditions limn→∞αn=0 and limn→∞ϕ(xn+1,xn)=0, we obtain
(3.15)lim⁡n→∞ϕ(xn+1,un)=0.
From again Lemma 2.4,
(3.16)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞‖xn+1-un‖=0.
So, by the triangle inequality, we get
(3.17)lim⁡n→∞‖xn-un‖=0.
Since J is uniformly norm-to-norm continuous on bounded sets, we have
(3.18)lim⁡n→∞‖Jxn-Jun‖=0.
On the other hand, we observe that
(3.19)ϕ(p,xn)-ϕ(p,un)=‖xn‖2-‖un‖2-2〈p,Jxn-Jun〉≤‖xn-un‖(‖xn‖+‖un‖)+2‖p‖‖Jxn-Jun‖.
It follows that
(3.20)ϕ(p,xn)-ϕ(p,un)⟶0,asn⟶∞.
From (3.1), (3.5), (3.6), (3.7), and (3.8), we have
(3.21)ϕ(p,un)≤ϕ(p,yn)≤αnϕ(p,x1)+(1-αn)ϕ(p,zn)≤αnϕ(p,x1)+(1-αn)[βnϕ(p,xn)+(1-βn)(ϕ(p,wn)-ϕ(Jrnwn,wn))]≤αnϕ(p,x1)+(1-αn)[βnϕ(p,xn)+(1-βn)(ϕ(p,xn)-ϕ(Jrnwn,wn))]≤αnϕ(p,x1)+(1-αn)ϕ(p,xn)-(1-αn)(1-βn)ϕ(Jrnwn,wn)
and then
(3.22)(1-αn)(1-βn)ϕ(Jrnwn,wn)≤αnϕ(p,x1)+(1-αn)ϕ(p,xn)-ϕ(p,un).
From conditions limn→∞αn=0, limsupn→∞βn<1 and (3.20), we obtain
(3.23)lim⁡n→∞ϕ(Jrnwn,wn)=0.
By again Lemma 2.4, we have limn→∞∥Jrnwn-wn∥=0.
SinceJ is uniformly norm-to-norm continuous on bounded sets, we obtain
(3.24)lim⁡n→∞‖J(Jrnwn)-J(wn)‖=0.
Applying (3.5) and (3.6), we observe that
(3.25)ϕ(p,un)≤ϕ(p,yn)≤αnϕ(p,x1)+(1-αn)ϕ(p,zn)≤αnϕ(p,x1)+(1-αn)[βnϕ(p,xn)+(1-βn)ϕ(p,wn)]≤αnϕ(p,x1)+(1-αn)[βnϕ(p,xn)+(1-βn)[ϕ(p,xn)-2λn(α-2c2λn)‖Axn-Ap‖2]]≤αnϕ(p,x1)+(1-αn)ϕ(p,xn)-(1-αn)(1-βn)2λn(α-2c2λn)‖Axn-Ap‖2
and, hence,
(3.26)2λn(α-2c2λn)‖Axn-Ap‖2≤1(1-αn)(1-βn)(αnϕ(p,x1)+(1-αn)ϕ(p,xn)-ϕ(p,un)),
for all n∈ℕ. Since 0<a≤λn≤b<c2α/2, limn→∞αn=0, limsupn→∞βn<1 and (3.20), we have
(3.27)lim⁡n→∞‖Axn-Ap‖=0.
From Lemmas 2.6, 2.7, and (3.4), we get
(3.28)ϕ(xn,wn)=ϕ(xn,ΠCvn)≤ϕ(xn,vn)=ϕ(xn,J-1(Jxn-λnAxn))=V(xn,Jxn-λnAxn)≤V(xn,(Jxn-λnAxn)+λnAxn)-2〈J-1(Jxn-λnAxn)-xn,λnAxn〉=ϕ(xn,xn)+2〈vn-xn,-λnAxn〉=2〈vn-xn,-λnAxn〉≤4λn2c2‖Axn-Ap‖2.
From Lemma 2.4 and (3.27), we have
(3.29)lim⁡n→∞‖xn-wn‖=0.
Since J is uniformly norm-to-norm continuous on bounded sets, we obtain
(3.30)lim⁡n→∞‖J(xn)-J(wn)‖=0.
Since {xn} is bounded, there exists a subsequence {xni} of {xn} such that xni⇀u∈E. Since xn-wn→0, then we get wni⇀u as i→∞.
Now, we claim thatu∈F. First, we show that u∈T-10. Indeed, since liminfn→∞rn>0, it follows from (3.24) that
(3.31)lim⁡n→∞‖Arnwn‖=lim⁡n→∞1rn‖Jwn-J(Jrnwn)‖=0.
If (z,z*)∈T, then it holds from the monotonicity of T that
(3.32)〈z-wni,z*-Arniwni〉≥0,
for all i∈ℕ. Letting i→∞, we get 〈z-u,z*〉≥0. Then, the maximality of T implies u∈T-10.
Next, we show thatu∈VI(C,A). Let B⊂E×E* be an operator as follows:
(3.33)Bv={Av+NC(v),v∈C,∅,otherwise.
By Theorem 2.9, B is maximal monotone and B-10=VI(A,C). Let (v,w)∈G(B). Since w∈Bv=Av+NC(v), we get w-Av∈NC(v). From wn∈C, we have
(3.34)〈v-wn,w-Av〉≥0.
On the other hand, since wn=ΠCJ-1(Jxn-λnAxn), then by Lemma 2.5, we have
(3.35)〈v-wn,Jwn-(Jxn-λnAxn)〉≥0.
Thus,
(3.36)〈v-wn,Jxn-Jwnλn-Axn〉≤0.
It follows from (3.34) and (3.36) that
(3.37)〈v-wn,w〉≥〈v-wn,Av〉≥〈v-wn,Av〉+〈v-wn,Jxn-Jwnλn-Axn〉=〈v-wn,Av-Axn〉+〈v-wn,Jxn-Jwnλn〉=〈v-wn,Av-Awn〉+〈v-wn,Awn-Axn〉+〈v-wn,Jxn-Jwnλn〉≥-‖v-wn‖‖wn-xn‖α-‖v-wn‖‖Jxn-Jwn‖a≥-M(‖wn-xn‖α+‖Jxn-Jwn‖a),
where M=supn≥1{∥v-wn∥}. From (3.29) and (3.30), we obtain 〈v-u,w〉≥0. By the maximality of B, we have u∈B-10 and, hence, u∈VI(C,A).
Next, we show thatu∈MEP(Θ,φ). Since un=Krnyn. From Lemmas 2.13 and 2.14, we have
(3.38)ϕ(un,yn)=ϕ(Krnyn,yn)≤ϕ(u,yn)-ϕ(u,Krnyn)≤ϕ(u,xn)-ϕ(u,un).
Similarly by (3.20),
(3.39)lim⁡n→∞ϕ(un,yn)=0,
and so
(3.40)lim⁡n→∞‖un-yn‖=0.
Since J is uniformly norm-to-norm continuous on bounded sets, we obtain
(3.41)lim⁡n→∞‖Jun-Jyn‖=0.
From (3.1) and (A2), we also have
(3.42)φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥Θ(y,un),∀y∈C.
Hence,
(3.43)φ(y)-φ(uni)+〈y-uni,Juni-Jynirni〉≥Θ(y,uni),∀y∈C.
From ∥xn-un∥→0, ∥xn-wn∥→0, we get uni⇀u. Since (Juni-Jyni/rni)→0, it follows by (A4) and the weak, lower semicontinuous of φ that
(3.44)Θ(y,u)+φ(u)-φ(y)≤0,∀y∈C.
For t with 0<t≤1 and y∈C, let yt=ty+(1-t)u. Since y∈C and u∈C, we have yt∈C and hence Θ(yt,u)+φ(u)-φ(yt)≤0. So, from (A1), (A4), and the convexity of φ, we have
(3.45)0=Θ(yt,yt)+φ(yt)-φ(yt)≤tΘ(yt,y)+(1-t)Θ(yt,u)+tφ(y)+(1-t)φ(y)-φ(yt)≤t(Θ(yt,y)+φ(y)-φ(yt)).
Dividing by t, we get Θ(yt,y)+φ(y)-φ(yt)≥0. From (A3) and the weakly lower semicontinuity of φ, we have Θ(u,y)+φ(y)-φ(u)≥0 for all y∈C implies u∈MEP(Θ,φ). Hence, u∈F:=VI(C,A)∩T-1(0)∩MEP(Θ,φ).
Finally, we show thatu=ΠFx. Indeed, from xn=ΠCnx and Lemma 2.5, we have
(3.46)〈Jx-Jxn,xn-z〉≥0,∀z∈Cn.
Since F⊂Cn, we also have
(3.47)〈Jx-Jxn,xn-p〉≥0,∀p∈F.
Taking limit n→∞, we have
(3.48)〈Jx-Ju,u-p〉≥0,∀p∈F.
By again Lemma 2.5, we can conclude that u=ΠFx0. This completes the proof.Corollary 3.2.
LetE be a 2-uniformly convex and uniformly smooth Banach space, let C be a nonempty, closed, convex subset of E. Let Θ be a bifunction from C×C to ℝ satisfying (A1)–(A4) let φ:C→ℝ be a lower semicontinuous and convex function, and let T:E→E* be a maximal monotone operator. Let Jr=(J+rT)-1J for r>0 with F:=T-1(0)∩MEP(Θ,φ)≠∅. Let {xn} be a sequence generated by x0∈E with x1=ΠC1x0 and C1=C,
(3.49)zn=J-1(βnJ(xn)+(1-βn)J(Jrnxn)),yn=J-1(αnJ(x1)+(1-αn)J(zn)),un∈Csuch thatΘ(un,y)+φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)},xn+1=ΠCn+1x0,
for n∈ℕ, where ΠC is the generalized projection from E onto C, J is the duality mapping on E. The coefficient sequence {αn},{βn}⊂(0,1), {rn}⊂(0,∞) satisfying limn→∞αn=0, limsupn→∞βn<1 and liminfn→∞rn>0. Then, the sequence {xn} converges strongly to ΠFx0.Proof.
In Theorem3.1 if A≡0, then (3.1) reduced to (3.49).


## 4. Weak Convergence Theorem

In this section, we first prove the following strong convergence theorem by using the idea of Plubtieng and Sriprad [42].Theorem 4.1.
LetE be a 2-uniformly convex and uniformly smooth Banach space whose duality mapping J is weak sequentially continuous. Let T:E→E* be a maximal monotone operator and let Jr=(J+rT)-1J for r>0. Let C be a nonempty, closed, convex subset of E such that D(T)⊂C⊂J-1(⋂r>0R(J+rT)), let Θ be a bifunction from C×C to ℝ satisfying (A1)–(A4), let φ:C→ℝ be a lower semicontinuous and convex function, and let A be an α-inverse-strongly monotone operator of C into E* with F:=VI(C,A)∩T-1(0)∩MEP(Θ,φ)≠∅ and ∥Ay∥≤∥Ay-Au∥ for all y∈C and u∈F. Let {xn} be a sequence generated by x1=x∈C and
(4.1)un=Krnxn,zn=ΠCJ-1(Jun-λnAun),yn=J-1(βnJ(xn)+(1-βn)J(Jrnzn)),xn+1=ΠCJ-1(αnJ(x1)+(1-αn)J(yn)),
for n∈ℕ∪{0}, where ΠC is the generalized projection from E onto C, J is the duality mapping on E. The coefficient sequence {αn},{βn}⊂[0,1], {rn}⊂(0,∞) satisfying ∑n=0∞αn<∞, limsupn→∞βn<1liminfn→∞rn>0 and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, 1/c is the 2-uniformly convexity constant of E. Then, the sequence {ΠFxn} converges strongly to an element of F, which is a unique element v∈F such that
(4.2)lim⁡n→∞ϕ(v,xn)=min⁡y∈Flim⁡n→∞ϕ(y,xn),
where ΠF is the generalized projection from C onto F.Proof.
Putvn=J-1(Jun-λnAun). Let p∈F:=VI(C,A)∩T-1(0)∩MEP(Θ,φ), by Lemma 2.14 and nonexpansiveness of Kr, we have
(4.3)ϕ(p,un)=ϕ(p,Krnxn)≤ϕ(p,xn).
By (4.1) and Lemma 2.7, the convexity of the function V in the second variable, we obtain
(4.4)ϕ(p,zn)=ϕ(p,ΠCvn)≤ϕ(p,vn)=ϕ(p,J-1(Jun-λnAun))≤V(p,Jun-λnAun+λnAun)-2〈J-1(Jun-λnAun)-p,λnAun〉=V(p,Jxun)-2λn〈vn-p,Aun〉=ϕ(p,un)-2λn〈un-p,Aun〉+2〈vn-un,-λnAun〉.
Since p∈VI(A,C) and A is α-inverse-strongly monotone, we also have
(4.5)-2λn〈un-p,Aun〉=-2λn〈un-p,Aun-Ap〉-2λn〈un-p,Ap〉≤-2αλn‖Aun-Ap‖2,(4.6)2〈vn-un,-λnAun〉=2〈J-1(Jun-λnAun)-xn,-λnAun〉≤2‖J-1(Jun-λnAun)-xn‖‖λnAun‖≤4c2‖Jun-λnAun-Jun‖‖λnAun‖≤4c2λn2‖Aun-Ap‖2.
Substituting (4.5) and (4.6) into (4.4) and (4.3), we get
(4.7)ϕ(p,zn)≤ϕ(p,un)-2αλn‖Aun-Ap‖2+4c2λn2‖Aun-Ap‖2≤ϕ(p,un)-2λn(α-2c2λn)‖Aun-Ap‖2≤ϕ(p,un)≤ϕ(p,xn).
By Lemmas 2.7, 2.8, (4.7), and using the same argument in Theorem 3.1, (3.6), we obtain
(4.8)ϕ(p,yn)≤ϕ(p,xn),
and hence by Lemma 2.6 and (4.7), we note that
(4.9)ϕ(p,xn+1)=ϕ(p,J-1(αnJ(x1)+(1-αn)J(yn)))=V(p,αnJ(x1)+(1-αn)J(yn))≤αnV(p,J(x1))+(1-αn)V(p,J(yn))=αnϕ(p,x1)+(1-αn)ϕ(p,yn)≤αnϕ(p,x1)+(1-αn)ϕ(p,xn),
for all n≥0. So, from ∑n=0∞αn<∞ and Lemma 2.10, we deduce that limn→∞ϕ(p,xn) exists. This implies that {ϕ(p,xn)} is bounded. It implies that {xn}, {yn}, {zn}, and {Jrnzn} are bounded. Define a function g:F→[0,∞) as follows:
(4.10)g(p)=lim⁡n→∞ϕ(p,xn),∀p∈F.
Then, by the same argument as in proof of [43, Theorem 3.1], we obtain g is a continuous convex function and if ∥zn∥→∞, then g(zn)→∞. Hence, by [34, Theorem 1.3.11], there exists a point v∈F such that
(4.11)g(v)=min⁡y∈Fg(y)(:=l).
Put wn=ΠFxn for all n≥0. We next prove that wn→v as n→∞. Suppose on the contrary that there exists ϵ0>0 such that, for each n∈ℕ, there is n′≥n satisfying ∥wn′-v∥≥ϵ0. Since v∈F, we have
(4.12)ϕ(wn,xn)=ϕ(ΠFxn,xn)≤ϕ(v,ΠFxn)+ϕ(ΠFxn,xn)≤ϕ(v,xn),
for all n≥0. This implies that
(4.13)limsup⁡n→∞ϕ(wn,xn)≤lim⁡n→∞ϕ(v,xn)=l.
Since (∥v∥-∥ΠFxn∥)2≤ϕ(v,wn)≤ϕ(v,xn) for all n≥0 and {xn} is bounded, {wn} is bounded. By Lemma 2.3, there exists a stricly increasing, continuous, and convex function K:[0,∞)→[0,∞) such that K(0)=0 and
(4.14)‖wn+v2‖2≤12‖wn‖2+12‖v‖2-14K(‖wn-v‖),
for all n≥0. Now, choose σ satisfying 0<σ<(1/4)K(ϵ0). Hence, there exists n0∈ℕ such that
(4.15)ϕ(wn,xn)≤l+σ,ϕ(v,xn)≤l+σ,
for all n≥0. Thus, there exists k≥n0 satisfying the following:
(4.16)ϕ(wk,xk)≤l+σ,ϕ(v,xk)≤l+σ,‖wk-v‖≥ϵ0.
From (4.9), (4.14), and (4.16), we obtain
(4.17)ϕ(wk+v2,xn+k)≤ϕ(wk+v2,xk)=‖wk+v2‖2-2〈wk+v2,Jxk〉+‖xk‖2≤12‖wk‖2+12‖v‖2-14K(‖wk-v‖)-〈wk+v,Jxk〉+‖xk‖2=12ϕ(wk,xk)+12ϕ(v,xk)-14K(‖wk-v‖)≤l+σ-14K(ϵ0),
for all n≥0. Hence,
(4.18)l≤lim⁡n→∞ϕ(wk+v2,xn)=lim⁡n→∞ϕ(wk+v2,xn+k)≤l+σ-14K(ϵ0)<l+σ-σ=l.
This is a contradiction. So, {wn} converges strongly to v∈F:=VI(C,A)∩T-1(0)∩MEP(Θ,φ). Consequently, v∈F is the unique element of F such that
(4.19)lim⁡n→∞ϕ(v,xn)=min⁡y∈Flim⁡n→∞ϕ(y,xn).
This completes the proof.Now, we prove a weak convergence theorem for the algorithm (4.20) below under different condition on data.Theorem 4.2.
LetE be a 2-uniformly convex and uniformly smooth Banach space whose duality mapping J is weakly sequentially continuous. Let T:E→E* be a maximal monotone operator and let Jr=(J+rT)-1J for r>0. Let C be a nonempty closed convex subset of E such that D(T)⊂C⊂J-1(⋂r>0R(J+rT)), let Θ be a bifunction from C×C to ℝ satisfying (A1)–(A4), let φ:C→ℝ be a lower semicontinuous and convex function, and let A be an α-inverse-strongly monotone operator of C into E* with F:=VI(C,A)∩T-1(0)∩MEP(Θ,φ)≠∅ and ∥Ay∥≤∥Ay-Au∥ for all y∈C and u∈F. Let {xn} be a sequence generated by x1=x∈C and
(4.20)un=Krnxn,zn=ΠCJ-1(Jun-λnAun),yn=J-1(βnJ(xn)+(1-βn)J(Jrnzn)),xn+1=ΠCJ-1(αnJ(x1)+(1-αn)J(yn)),
for n∈ℕ∪{0}, where ΠC is the generalized projection from E onto C, J is the duality mapping on E. The coefficient sequence {αn},{βn}⊂[0,1], {rn}⊂(0,∞) satisfying ∑n=0∞αn<∞, limsupn→∞βn<1liminfn→∞rn>0 and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, 1/c is the 2-uniformly convexity constant of E. Then, the sequence {xn} converges weakly to an element v of F, where v=limn→∞ΠFxn.Proof.
By Theorem4.1, we have {xn} is bounded and so are {zn}, {Jrnzn}.
From (4.9), we obtain
(4.21)ϕ(p,xn+1)≤αnϕ(p,x1)+(1-αn)ϕ(p,yn)≤αnϕ(p,x1)+(1-αn)[βnϕ(p,xn)+(1-βn)(ϕ(p,zn)-ϕ(Jrnzn,zn))]≤αnϕ(p,x1)+(1-αn)[βnϕ(p,xn)+(1-βn)(ϕ(p,xn)-ϕ(Jrnzn,zn))]≤αnϕ(p,x1)+(1-αnβn)ϕ(p,xn)-(1-αn)(1-βn)ϕ(Jrnzn,zn),
and then
(4.22)(1-αn)(1-βn)ϕ(Jrnzn,zn)≤αnϕ(p,x1)+(1-αnβn)ϕ(p,xn)-ϕ(p,xn+1).
Since limn→∞αn=0, limsupn→∞βn<1 and {ϕ(p,xn)} exists, then we have
(4.23)lim⁡n→∞ϕ(Jrnzn,zn)=0.
By again Lemma 2.4, we have limn→∞∥Jrnzn-zn∥=0. Since J is uniformly norm-to-norm continuous on bounded sets, we obtain
(4.24)lim⁡n→∞‖J(Jrnzn)-J(zn)‖=0.
Apply (4.7), (4.8), and (4.9), we observe that
(4.25)ϕ(p,xn+1)≤αnϕ(p,x1)+(1-αn)ϕ(p,yn)≤αnϕ(p,x1)+(1-αn)[βnϕ(p,xn)+(1-βn)ϕ(p,zn)]≤αnϕ(p,x1)+(1-αn)[βnϕ(p,xn)+(1-βn)[ϕ(p,un)-2λn(α-2c2λn)‖Aun-Ap‖2]]≤αnϕ(p,x1)+(1-αn)[βnϕ(p,xn)+(1-βn)[ϕ(p,xn)-2λn(α-2c2λn)‖Aun-Ap‖2]]≤αnϕ(p,x1)+(1-αn)ϕ(p,xn)-(1-αn)(1-βn)2λn(α-2c2λn)‖Aun-Ap‖2,
and hence
(4.26)2λn(α-2c2λn)‖Aun-Ap‖2≤1(1-αn)(1-βn)(αnϕ(p,x1)+(1-αn)ϕ(p,xn)-ϕ(p,xn+1)),
for all n∈ℕ. Since 0<a≤λn≤b<c2α/2, limn→∞αn=0 and limsupn→∞βn<1, we have
(4.27)lim⁡n→∞‖Aun-Ap‖=0.
From Lemmas2.6, 2.7, and (4.7), we get
(4.28)ϕ(un,zn)=ϕ(un,ΠCvn)≤ϕ(un,vn)=ϕ(un,J-1(Jun-λnAun))=V(un,Jun-λnAun)≤V(un,(Jun-λnAun)+λnAun)-2〈J-1(Jun-λnAun)-xn,λnAun〉=ϕ(un,un)+2〈vn-un,λnAun〉=2〈vn-un,λnAun〉≤4λn2c2‖Aun-Ap‖2.
From Lemma 2.4 and (4.27), we have
(4.29)lim⁡n→∞‖un-zn‖=0.
Since J is uniformly norm-to-norm continuous on bounded sets, we obtain
(4.30)lim⁡n→∞‖J(un)-J(zn)‖=0.
Since{zn} is bounded, there exists a subsequence {zni} of {zn} such that zni⇀u∈C. It follows that Jrnizni⇀u and uni⇀u∈C as i→∞.
Now, we claim thatu∈F. First, we show that u∈T-10. Indeed, since liminfn→∞rn>0, it follows that
(4.31)lim⁡n→∞‖Arnzn‖=lim⁡n→∞1rn‖Jzn-J(Jrnzn)‖=0.
If (z,z*)∈T, then it holds from the monotonicity of T that
(4.32)〈z-Jrnizni,z*-Arnizni〉≥0,
for all i∈ℕ. Letting i→∞, we get 〈z-u,z*〉≥0. Then, the maximality of T implies u∈T-10.
Next, we show thatu∈VI(C,A). Let B⊂E×E* be an operator as follows:
(4.33)Bv={Av+NC(v),v∈C,∅,otherwise.
By Theorem 2.9, B is maximal monotone and B-10=VI(A,C). Let (v,w)∈G(B). Since w∈Bv=Av+NC(v), we get w-Av∈NC(v). From zn∈C, we have
(4.34)〈v-zn,w-Av〉≥0.
On the other hand, since zn=ΠCJ-1(Jun-λnAun). Then, by Lemma 2.5, we have
(4.35)〈v-zn,Jwn-(Jun-λnAun)〉≥0.
Thus,
(4.36)〈v-zn,Jun-Jznλn-Aun〉≤0.
It follows from (4.34) and (4.36) that
(4.37)〈v-zn,w〉≥〈v-zn,Av〉≥〈v-zn,Av〉+〈v-zn,Jun-Jznλn-Axn〉=〈v-zn,Av-Aun〉+〈v-zn,Jun-Jznλn〉=〈v-zn,Av-Azn〉+〈v-zn,Azn-Aun〉+〈v-zn,Jun-Jznλn〉≥-‖v-zn‖‖zn-un‖α-‖v-zn‖‖Jun-Jzn‖a≥-M(‖zn-un‖α+‖Jun-Jzn‖a),
where M=supn≥1{∥v-zn∥}. From (4.29) and (4.30), we obtain 〈v-u,w〉≥0. By the maximality of B, we have u∈B-10 and hence u∈VI(C,A).
Next, we showu∈MEP(f)=F(Kr). From un=Krnxn. It follows from (4.7), (4.8), and (4.9) that
(4.38)ϕ(p,xn+1)≤αnϕ(p,x1)+(1-αn)ϕ(p,yn)≤αnϕ(p,x1)+(1-αn)[βnϕ(p,xn)+(1-βn)ϕ(p,zn)]≤αnϕ(p,x1)+(1-αn)[βnϕ(p,xn)+(1-βn)ϕ(p,un)]≤αnϕ(p,x1)+(1-αn)[βnϕ(p,xn)+(1-βn)ϕ(p,xn)],
or, equivalently,
(4.39)ϕ(p,xn+1)-αnϕ(p,x1)≤(1-αn)[βnϕ(p,xn)+(1-βn)ϕ(p,un)]≤(1-αn)ϕ(p,xn),
with limn→∞αn=0 and limsupn→∞βn<1, yield that limn→∞ϕ(p,un)=limn→∞ϕ(p,xn).
From Lemmas2.13 and 2.14, for p∈F,
(4.40)ϕ(un,xn)≤ϕ(p,xn)-ϕ(p,un).
This implies that limn→∞ϕ(un,xn)=0. Noticing Lemma 2.4, we get
(4.41)‖un-xn‖⟶0,asn⟶∞.
Since J is uniformly norm-to-norm continuous on bounded sets, we obtain
(4.42)lim⁡n→∞‖Jun-Jxn‖=0.
From (4.20) and (A2), we also have
(4.43)φ(y)-φ(un)+1rn〈y-un,Jun-Jxn〉≥Θ(y,un),∀y∈C.
Hence,
(4.44)φ(y)-φ(uni)+〈y-uni,Juni-Jxnirni〉≥Θ(y,uni),∀y∈C.
From ∥un-zn∥→0, we get uni⇀u. Since (Juni-Jxni/rni)→0, it follows by (A4) and the weakly lower semicontinuous of φ that
(4.45)Θ(y,u)+φ(u)-φ(y)≤0,∀y∈C.
For t with 0<t≤1 and y∈C, let yt=ty+(1-t)u. Since y∈C and u∈C, we have yt∈C and hence Θ(yt,u)+φ(u)-φ(yt)≤0. So, from (A1), (A4), and the convexity of φ, we have
(4.46)0=Θ(yt,yt)+φ(yt)-φ(yt)≤tΘ(yt,y)+(1-t)Θ(yt,u)+tφ(y)+(1-t)φ(y)-φ(yt)≤t(Θ(yt,y)+φ(y)-φ(yt)).
Dividing by t, we get Θ(yt,y)+φ(y)-φ(yt)≥0. From (A3) and the weakly lower semicontinuity of φ, we have Θ(u,y)+φ(y)-φ(u)≥0 for all y∈C implies u∈MEP(Θ,φ). Hence, u∈F:=VI(C,A)∩T-1(0)∩MEP(Θ,φ).
By Theorem4.1, the {ΠFxn} converges strongly to a point v∈F which is a unique element of F such that
(4.47)lim⁡n→∞ϕ(v,xn)=min⁡y∈Flim⁡n→∞ϕ(y,xn).
By the uniform smoothness of E, we also have limn→∞∥JΠFxni-Jv∥=0.
Finally, we proveu=v. From Lemma 2.5 and u∈F, we have
(4.48)〈ΠFxni-u,Jxni-JΠFxni〉≥0.
Since J is weakly sequentially continuous, uni⇀u and un-xn→0. Then,
(4.49)〈v-u,Ju-Jv〉≥0.
On the other hand, since J is monotone, we have
(4.50)〈v-u,Ju-Jv〉≤0.
Hence,
(4.51)〈v-u,Ju-Jv〉=0.
Since E is strict convexity, it follows that u=v. Therefore, the sequence {xn} converges weakly to v=limn→∞ΠFxn. This completes the proof.


## 5. Application to Complementarity Problems

LetC be a nonempty, closed convex cone in E and A an operator of C into E*. We define its polar in E* to be the set(5.1)K*={y*∈E*:〈x,y*〉≥0,∀x∈C}.
Then, the element u∈C is called a solution of the complementarity problem if(5.2)Au∈K*,〈u,Au〉=0.
The set of solutions of the complementarity problem is denoted by CP(K,A); see [34], for more detial.Theorem 5.1.
LetE be a 2-uniformly convex and uniformly smooth Banach space and let K be a nonempty closed convex subset of E. Let Θ be a bifunction from K×K to ℝ satisfying (A1)–(A4) let φ:K→ℝ be a lower semicontinuous and convex function, and let T:E→E* be a maximal monotone operator. Let Jr=(J+rT)-1J for r>0 and let A be an α-inverse-strongly monotone operator of K into E* with F:=T-1(0)∩CP(K,A)∩MEP(Θ,φ)≠∅ and ∥Ay∥≤∥Ay-Au∥ for all y∈K and u∈F. For an initial point x0∈E with x1=ΠC1x0 and K1=K,
(5.3)wn=ΠKJ-1(Jxn-λnAxn),zn=J-1(βnJ(xn)+(1-βn)J(Jrnwn)),yn=J-1(αnJ(x1)+(1-αn)J(zn)),un∈Ksuch thatΘ(un,y)+φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈K,Kn+1={z∈Kn:ϕ(z,un)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)},xn+1=ΠKn+1x0,
for n∈ℕ, where ΠK is the generalized projection from E onto K and J is the duality mapping on E. The coefficient sequence {αn},{βn}⊂(0,1), {rn}⊂(0,∞) satisfying limn→∞αn=0, limsupn→∞βn<1, liminfn→∞rn>0 and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, 1/c is the 2-uniformly convexity constant of E. Then, the sequence {xn} converges strongly to ΠFx0.Proof.
As in the proof Lemma 7.1.1 of Takahashi in [44], we have VI(C,A)=CP(K,A). So, we obtain the desired result.Theorem 5.2.
LetE be a 2-uniformly convex and uniformly smooth Banach space whose duality mapping J is weakly sequentially continuous. Let T:E→E* be a maximal monotone operator and let Jr=(J+rT)-1J for r>0. Let K be a nonempty closed convex subset of E such that D(T)⊂K⊂J-1(⋂r>0R(J+rT)), let Θ be a bifunction from K×K to ℝ satisfying (A1)–(A4), let φ:K→ℝ be a proper lower semicontinuous and convex function, and let A be an α-inverse-strongly monotone operator of K into E* with F:=CP(K,A)∩T-1(0)∩MEP(Θ,φ)≠∅ and ∥Ay∥≤∥Ay-Au∥ for all y∈K and u∈F. Let {xn} be a sequence generated by x1=x∈K and
(5.4)un=Krnxn,zn=ΠKJ-1(Jun-λnAun),yn=J-1(βnJ(xn)+(1-βn)J(Jrnzn)),xn+1=ΠKJ-1(αnJ(x1)+(1-αn)J(yn)),
for n∈ℕ∪{0}, where ΠK is the generalized projection from E onto K, J is the duality mapping on E. The coefficient sequence {αn},{βn}⊂[0,1], {rn}⊂(0,∞) satisfying ∑n=0∞αn<∞, limsupn→∞βn<1liminfn→∞rn>0 and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, 1/c is the 2-uniformly convexity constant of E. Then, the sequence {xn} converges weakly to an element v of F, where v=limn→∞ΠFxn.Proof.
It follows by Lemma 7.1.1 of Takahashi in [44], we have VI(C,A)=CP(K,A). Hence, Theorem 4.2, {xn} converges weakly to an element v of F, where v=limn→∞ΠFxn.



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*Source: 174796-2011-02-17.xml*