# Convergence Theorem Based on a New Hybrid Projection Method for Finding a Common Solution of Generalized Equilibrium and Variational Inequality Problems in Banach Spaces

Authors: Siwaporn Saewan; Poom Kumam; Kriengsak Wattanawitoon
Journal: Abstract and Applied Analysis
 (2010)

---

## Abstract

The purpose of this paper is to introduce a new hybrid projection method for finding a common
element of the set of common fixed points of two relatively quasi-nonexpansive mappings, the set of
the variational inequality for anα-inverse-strongly monotone, and the set of solutions of the generalized
equilibrium problem in the framework of a real Banach space. We obtain a strong convergence theorem
for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space.
Base on this result, we also get some new and interesting results. The results in this paper generalize,
extend, and unify some well-known strong convergence results in the literature.


---

## Body

## 1. Introduction

LetE be a real Banach space, E* the dual space of E. A Banach space E 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 a Banach space E is said to be smooth if the limit(1.1)limt→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∈U. Let E be a Banach space. The modulus of convexity of E is the function δ:[0,2]→[0,1] defined by(1.2)δ(ε)=inf{1-∥x+y2∥:x,y∈E,∥x∥=∥y∥=1,∥x-y∥≥ε}.A Banach spaceE is uniformly convex if and only if δ(ε)>0 for all ε∈(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(ε)≥cεp for all ε∈[0,2]; see [1, 2] 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, uniformly smooth. For each p>1, the generalized duality mappingJp:E→2E* is defined by(1.3)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.LetE be a real Banach space with norm ∥·∥ and E* denotes the dual space of E. Consider the functional defined by(1.4)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2∀x,y∈E.Observe that, in a Hilbert spaceH, (1.4) reduces to ϕ(x,y)=∥x-y∥2, x,y∈H. 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=x̅, where x̅ is the solution to the minimization problem
(1.5)ϕ(x̅,x)=infy∈Cϕ(y,x);
existence and uniqueness of the mapping ΠC follow from the properties of the functional ϕ(x,y) and strict monotonicity of the mapping J (see, e.g., [3–7]). In Hilbert spaces, ΠC=PC. It is obvious from the definition of function ϕ that(1.6)(∥y∥-∥x∥)2≤ϕ(y,x)≤(∥y∥+∥x∥)2,∀x,y∈E.Remark 1.1.
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 (2.13), 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 [5, 7] for more details.Next, we give some examples which are closed relatively quasi-nonexpansive (see [8]).Example 1.2.
LetΠC be the generalized projection from a smooth, strictly convex and reflexive Banach space E onto a nonempty closed and convex subset C of E. Then, ΠC is a closed relatively quasi-nonexpansive mapping from E onto C with F(ΠC)=C.LetE be a real Banach space and let C be a nonempty closed and convex subset of E, and A:C→E* be a mapping. The classical variational inequality problem for a mapping A is to find x*∈C such that
(1.7)〈Ax*,y-x*〉≥0,∀y∈C.
The set of solutions of (1.4) is denoted by VI(A,C). Recall that A is called(i)
monotone if
(1.8)〈Ax-Ay,x-y〉≥0,∀x,y∈C,(ii)
anα-inverse-strongly monotone if there exists a constant α>0 such that
(1.9)〈Ax-Ay,x-y〉≥α∥x-y∥2,∀x,y∈C.Such a problem is connected with the convex minimization problem, the complementary problem, and the problem of finding a pointx*∈E satisfying Ax*=0.Letf be a bifunction from C×C to ℝ, where ℝ denotes the set of real numbers. The equilibrium problem (for short, EP) is to find x*∈C such that(1.10)f(x*,y)≥0,∀y∈C.The set of solutions of (1.10) is denoted by EP(f). Given a mapping T:C→E*, let f(x,y)=〈Tx,y-x〉 for all x,y∈C. Then x*∈EP(f) if and only if 〈Tx*,y-x*〉≥0 for all y∈C; that is, x* is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.10). Some methods have been proposed to solve the equilibrium problem; see, for instance, [9–11].LetC be a closed convex subset of E; a mapping T:C→C is said to be nonexpansive if(1.11)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.A pointx∈C is a fixed point  of T provided that Tx=x. Denote by F(T) the set of fixed points of T; that is, F(T)={x∈C:Tx=x}. Recall that a point p in C is said to be an asymptotic fixed point of T [12] if C contains a sequence {xn} which converges weakly to p such that limn→∞∥xn-Txn∥=0. The set of asymptotic fixed points of T will be denoted by F(T)̂. A mapping T from C into itself is said to be relatively nonexpansive [13–15] if F(T)̂=F(T) and ϕ(p,Tx)≤ϕ(p,x) for all x∈C and p∈F(T). The asymptotic behavior of a relatively nonexpansive mapping was studied in [16–18]. T is said to be ϕ-nonexpansive, if ϕ(Tx,Ty)≤ϕ(x,y) for x,y∈C. T is said to be relatively quasi-nonexpansive if F(T)≠∅ and ϕ(p,Tx)≤ϕ(p,x) for x∈C and p∈F(T). A mapping T in a Banach space E is closed if xn→x and Txn→y, then Tx=y.Remark 1.3.
The class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [16–19] which requires the strong restriction F(T)=F(T)̂.In Hilbert spacesH, Iiduka et al. [20] proved that the sequence {xn} defined by: x1=x∈C and(1.12)xn+1=PC(xn-λnAxn),
where PC is the metric projection of H onto C, and {λn} is a sequence of positive real numbers, and converges weakly to some element of VI(A,C).It is well know that ifC is a nonempty closed and convex subset of a Hilbert space H and PC:H→C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [4] recently introduced a generalized projection mapping ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces.In 2008, Iiduka and Takahashi [21] introduced the following iterative scheme for finding a solution of the variational inequality problem for inverse-strongly monotone A in a 2-uniformly convex and uniformly smooth Banach space E: x1=x∈C and(1.13)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 (1.13) converges weakly to some element of VI(A,C).Matsushita and Takahashi [22] introduced the following iteration: a sequence {xn} defined by(1.14)xn+1=ΠCJ-1(αnJxn+(1-αn)JTxn),
where the initial guess element x0∈C is arbitrary, {αn} is a real sequence in [0,1], T is a relatively nonexpansive mapping, and ΠC denotes the generalized projection from E onto a closed convex subset C of E. They proved that the sequence {xn} converges weakly to a fixed point of T.In 2005, Matsushita and Takahashi [19] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping T in a Banach space E:(1.15)x0∈Cchosenarbitrarily,yn=J-1(αnJxn+(1-αn)JTxn),Cn={z∈C:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=∏Cn∩Qnx0.They proved that{xn} converges strongly to ΠF(T)x0, where ΠF(T) is the generalized projection from C onto F(T).Recently, Takahashi and Zembayashi [23] proposed the following modification of iteration (1.15) for a relatively nonexpansive mapping:(1.16)x0=x∈C,yn=J-1(αnJxn+(1-αn)JSxn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Hn={z∈C:ϕ(z,un)≤ϕ(z,xn)},Wn={z∈C:〈xn-z,Jx-Jxn〉≥0},xn+1=∏Hn∩Wnx,
where J is the duality mapping on E. Then, {xn} converges strongly to ΠF(S)∩EP(f)x, where ΠF(S)∩EP(f) is the generalized projection of E onto F(S)∩EP(f). Also, Takahashi and Zembayashi [24] proved the following iteration for a relatively nonexpansive mapping:(1.17)yn=J-1(αnJxn+(1-αn)JSxn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)},xn+1=∏Cn+1x,
where J is the duality mapping on E. Then, {xn} converges strongly to ΠF(S)∩EP(f)x, where ΠF(S)∩EP(f) is the generalized projection of E onto F(S)∩EP(f). Qin and Su [25] proved the following iteration for relatively nonexpansive mappings T in a Banach space E:(1.18)x0∈C,chosenarbitrarily,yn=J-1(αnJxn+(1-αn)JTzn),zn=J-1(βnJxn+(1-βn)JTxn),Cn={v∈C:ϕ(v,yn)≤αnϕ(v,xn)+(1-αn)ϕ(v,zn)},Qn={v∈C:〈Jx0-Jxn,xn-v〉≥0},xn+1=∏Cn∩Qnx0,
the sequence {xn} generated by (1.18) converges strongly to ΠF(T)x0.In 2009, Wei et al. [26] proved the following iteration for two relatively nonexpansive mappings in a Banach space E:(1.19)x0∈C,Jzn=αnJxn+(1-αn)JTxn,Jun=(βnJxn+(1-βn)JSzn),Hn={v∈C:ϕ(v,un)≤βnϕ(v,xn)+(1-βn)ϕ(v,zn)≤ϕ(v,xn)},Wn={z∈C:〈z-xn,Jx0-Jxn〉≤0},xn+1=QHn∩Wnx0;
if {αn} and {βn} are  sequences in [0,1) such that αn≤1-δ1 and βn≤1-δ2 for some δ1,δ2∈(0,1), then {xn} generated by (1.19) converges strongly to a point QF(T)∩F(S)x0, where the mapping QC of E onto C is the generalized projection. Very recently, Cholamjiak [27] proved the following iteration:(1.20)zn=ΠCJ-1(Jxn-λnAxn),yn=J-1(αnJxn+βnJTxn+γnJSzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)},xn+1=∏Cn+1x0,
where J is the duality mapping on E. Assume that αn, βn, and γn are sequences in [0,1]. Then {xn} converges strongly to q=ΠFx0, where F:=F(T)∩F(S)∩EP(f)∩VI(A,C).Motivated and inspired by Iiduka and Takahashi [21], Takahashi and Zembayashi [23, 24], Wei et al. [26], Cholamjiak [27], and Kumam and Wattanawitoon [28], we introduce a new hybrid projection iterative scheme which is difference from the algorithm (1.20) of Cholamjiak in [27, Theorem  3.1] for two relatively quasi-nonexpansive mappings in a Banach space. For an initial point x0∈E with x1=ΠC1x0 and C1=C, define a sequence {xn} as follows:(1.21)wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JTwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E. Then, we prove that under certain appropriate conditions on the parameters, the sequences {xn} and {un} generated by (1.21) converge strongly to ΠF(S)∩F(T)∩EP(f)∩VI(A,C).The results presented in this paper improve and extend the corresponding results announced by Iiduka and Takahashi [21], Wei et al. [26], Kumam and Wattanawitoon [28], and many other authors in the literature.


## 2. Preliminaries

We also need the following lemmas for the proof of our main results.Lemma 2.1 (Beauzamy [29] and Xu [30]).
IfE is a 2-uniformly convex Banach space, then, for all x,y∈E we have
(2.1)∥x-y∥≤2c2∥Jx-Jy∥,
where J is the normalized duality mapping of E and 0<c≤1.The best constant1/c in the Lemma is called the p-uniformly convex constant of E.Lemma 2.2 (Beauzamy [29] and Zǎlinescu [31]).
IfE is a p-uniformly convex Banach space and p is a given real number with p≥2, then for all x,y∈E,Jx∈Jp(x), and Jy∈Jp(y),(2.2)〈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 (Kamimura and Takahashi [6]).
LetE be a uniformly convex and smooth Banach space and let {xn} and {yn} be two sequences of E. If ϕ(xn,yn)→0 and either {xn} or {yn} is bounded, then ∥xn-yn∥→0.Lemma 2.4 (Alber [4]).
LetC be a nonempty closed and convex subset of a smooth Banach space E and x∈E. Then, x0=ΠCx if and only if
(2.3)〈x0-y,Jx-Jx0〉≥0,∀y∈C.Lemma 2.5 (Alber [4]).
LetE be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty closed and convex subset of E, and let x∈E. Then
(2.4)ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x),∀y∈C.Lemma 2.6 (Qin et al. [8]).
LetE be a uniformly convex and smooth Banach space, let C be a closed and convex subset of E, and let T be a closed relatively quasi-nonexpansive mapping from C into itself. Then F(T) is a closed and convex subset of C.For solving the equilibrium problem for a bifunctionf:C×C→ℝ, let us assume that f 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,
(2.5)limt↓0f(tz+(1-t)x,y)≤f(x,y);(A4)
for eachx∈C, y↦f(x,y) is convex and lower semi-continuous.Lemma 2.7 (Blum and Oettli [9]).
LetC be a closed and convex subset of a smooth, strictly convex and reflexive Banach space E, let f be a bifunction from C×C to ℝ satisfying (A1)–(A4), and let r>0 and x∈E. Then, there exists z∈C such that
(2.6)f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Lemma 2.8 (Combettes and Hirstoaga [10]).
LetC be a closed and convex subset of a uniformly smooth, strictly convex and reflexive Banach space E and let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). For r>0 and x∈E, define a mapping Tr:E→C as follows:
(2.7)Trx={z∈C:f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},
for all x∈C. Then the following holds:  (1)
Tr is single-valued;(2)
Tr is a firmly nonexpansive-type mapping, for all x,y∈E,
(2.8)〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉;(3)
F(Tr)=EP(f);(4)
EP(f) is closed and convex.Lemma 2.9 (Takahashi and Zembayashi [24]).
LetC be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C×C to ℝ satisfying (A1)–(A4), and let r>0. Then, for x∈E and q∈F(Tr),(2.9)ϕ(q,Trx)+ϕ(Trx,x)≤ϕ(q,x).LetE be a reflexive, strictly convex, and smooth Banach space and J the duality mapping from E into E*. Then J-1 is also single value, one-to-one, surjective, and it is the duality mapping from E* into E. We make use of the following mapping V studied in Alber [4]:(2.10)V(x,x*)=∥x∥2-2〈x,x*〉+∥x*∥2
for all x∈E and x*∈E*, that is, V(x,x*)=ϕ(x,J-1(x*)).Lemma 2.10 (Alber [4]).
LetE be a reflexive, strictly convex, and smooth Banach space and let V be as in (2.10). Then
(2.11)V(x,x*)+2〈J-1(x*)-x,y*〉≤V(x,x*+y*)
for all x∈E and x*,y*∈E*.LetA be an inverse-strongly monotone 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,(2.12)NC(v)={x*∈E*:〈v-y,x*〉≥0,∀y∈C}.Theorem 2.11 (Rockafellar [32]).
LetC be a nonempty, closed and convex subset of a Banach space E, and A a monotone, hemicontinuous mapping of C into E*. Let T⊂E×E* be a mapping defined as follows:
(2.13)Tv={Av+NC(v),v∈C;∅,otherwise.
Then T is maximal monotone and T-10=VI(A,C).


## 3. Main Results

In this section, we establish a new hybrid iterative scheme for finding a common element of the set of solutions of an equilibrium problems, the common fixed point set of two relatively quasi-nonexpansive mappings, and the solution set of variational inequalities forα-inverse strongly monotone in a 2-uniformly convex and uniformly smooth Banach space.Theorem 3.1.
LetC be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4) and let A be an α-inverse-strongly monotone mapping of C into E* satisfying ∥Ay∥≤∥Ay-Au∥,forally∈C and u∈VI(A,C)≠∅. Let T,S:C→C be closed relatively quasi-nonexpansive mappings such that Ω:=F(T)∩F(S)∩EP(f)∩VI(A,C)≠∅. For an initial point x0∈E with x1=ΠC1x0 and C1=C, we define the sequence {xn} as follows:
(3.1)wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JTwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E, {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1), {rn}⊆(0,2α) and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2-uniformly convexity constant of E. Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.Proof.
We have several steps to prove this theorem as follows:Step 1.
We show thatCn+1 is closed and convex.
ClearlyC1=C is closed and convex. Suppose that Cn is closed and convex for each n∈ℕ. Since for any z∈Cn, we know that
(3.2)ϕ(z,un)≤ϕ(z,xn)
is equivalent to
(3.3)2〈z,Jxn-Jun〉≤∥xn∥2-∥un∥2.
So, Cn+1 is closed and convex. Then, by induction, Cn is closed and convex for all n≥1.Step 2.
We show that{xn} is well defined.
Putun=Trnyn for all n≥0. On the other hand, from Lemma 2.8 one has Trn is relatively quasi-nonexpansive mappings and Ω⊂C1=C. Supposing Ω⊂Ck for k∈ℕ, by the convexity of ∥·∥2, for each q∈Ω⊂Ck, we have
(3.4)ϕ(q,uk)=ϕ(q,Trkyk)≤ϕ(q,yk)=ϕ(q,J-1(αkJxk+(1-αk)JSzk))=∥q∥2-2〈q,αkJxk+(1-αk)JSzk〉+∥αkJxk+(1-αk)JSzk∥2≤∥q∥2-2αk〈q,Jxk〉-2(1-αk)〈q,JSzk〉+αk∥xk∥2+(1-αk)∥Szk∥2=αkϕ(q,xk)+(1-αk)ϕ(q,Szk)≤αkϕ(q,xk)+(1-αk)ϕ(q,zk),
and  so
(3.5)ϕ(q,zk)=ϕ(q,J-1(βkJxk+(1-βk)JTwk))=∥q∥2-2〈q,βkJxk+(1-βk)JTwk〉+∥βkJxk+(1-βk)JTwk∥2≤∥q∥2-2βk〈q,Jxk〉-2(1-βk)〈q,JTwk〉+βk∥Jxk∥2+(1-βk)∥JTwk∥2=βkϕ(q,xk)+(1-βk)ϕ(q,Twk)≤βkϕ(q,xk)+(1-βk)ϕ(q,wk).
For all q∈Ω, we know from Lemma 2.10, that
(3.6)ϕ(q,wk)=ϕ(q,ΠCJ-1(Jxk-λkAxk))≤ϕ(q,J-1(Jxk-λkAxk))=V(q,Jxk-λkAxk)≤V(q,(Jxk-λkAxk)+λkAxk)-2〈J-1(Jxk-λkAxk)-q,λkAxk〉=V(q,Jxk)-2λk〈J-1(Jxk-λkAxk)-q,Axk〉=ϕ(q,xk)-2λk〈xk-q,Axk〉+2〈J-1(Jxk-λkAxk)-xk,-λkAxk〉.
Since q∈VI(A,C) and from A being an α-inverse-strongly monotone, we get
(3.7)-2λk〈xk-q,Axk〉=-2λk〈xk-q,Axk-Aq〉-2λk〈xk-q,Aq〉≤-2λk〈xk-q,Axk-Aq〉=-2αλk∥Axk-Aq∥2.
From Lemma 2.1 and A being an α-inverse-strongly monotone, we obtain
(3.8)2〈J-1(Jxk-λkAxk)-xk,-λkAxk〉=2〈J-1(Jxk-λkAxk)-J-1(Jxk),-λkAxk〉≤2∥J-1(Jxk-λkAxk)-J-1(Jxk)∥∥λkAxk∥≤4c2∥JJ-1(Jxk-λkAxk)-JJ-1(Jxk)∥∥λkAxk∥=4c2∥Jxk-λkAxk-Jxk∥∥λkAxk∥=4c2∥λkAxk∥2=4c2λk2∥Axk∥2≤4c2λk2∥Axk-Aq∥2.
Substituting (3.7) and (3.8) into (3.6), we have
(3.9)ϕ(q,wk)≤ϕ(q,xk)-2αλk∥Axk-Aq∥2+4c2λk2∥Axk-Aq∥2=ϕ(q,xk)+2λk(2c2λk-α)∥Axk-Aq∥2≤ϕ(q,xk).
Replacing (3.9) into (3.5), we get
(3.10)ϕ(q,zk)≤ϕ(q,xk).
Substituting (3.10) into (3.4), we also have
(3.11)ϕ(q,uk)≤αkϕ(q,xk)+(1-αk)ϕ(q,xk),=ϕ(q,xk).
This shows thatq∈Ck+1 and hence, Ω⊂Ck+1. Hence, Ω⊂Cn for all n≥1. This implies that the sequence {xn} is well defined.Step 3.
We show thatlimn→∞ϕ(xn,x0) exists and {xn} is bounded.
Fromxn=ΠCnx0 and xn+1=ΠCn+1x0, we have
(3.12)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n≥1,
and from Lemma 2.5, we have
(3.13)ϕ(xn,x0)=ϕ(ΠCn(x0),x0)≤ϕ(p,x0)-ϕ(p,xn)≤ϕ(p,x0),∀p∈Ω.
From (3.12) and (3.13), then {ϕ(xn,x0)} are nondecreasing and bounded. So, we obtain that limn→∞ϕ(xn,x0) exists. In particular, by (1.6), the sequence {(∥xn∥-∥x0∥)2} is bounded. This implies that {xn} is also bounded.Step 4.
We show that{xn} is a Cauchy sequence in C.
Sincexm=ΠCmx0∈Cm⊂Cn, for m>n, by Lemma 2.5, we have
(3.14)ϕ(xm,xn)=ϕ(xm,ΠCnx0)≤ϕ(xm,x0)-ϕ(ΠCnx0,x0)=ϕ(xm,x0)-ϕ(xn,x0).
Takingm,n→∞, we have ϕ(xm,xn)→0. We have limn→∞ϕ(xn+1,x0)=0. From Lemma 2.3, we get limn→∞∥xn+1-x0∥=0. Thus {xn} is a Cauchy sequence.Step 5.
We cliam that∥Jun-Jxn∥→0, as n→∞.
By the completeness ofE, the closedness of C and {xn} is a Cauchy sequence (from Step 4); we can assume that there exists p∈C such that {xn}→p as n→∞.
By definition ofΠCnx0, we have
(3.15)ϕ(xn+1,xn)=ϕ(xn+1,ΠCnx0)≤ϕ(xn+1,x0)-ϕ(ΠCnx0,x0)=ϕ(xn+1,x0)-ϕ(xn,x0).
Since limn→∞ϕ(xn,x0) exists, we get
(3.16)limn→∞ϕ(xn+1,xn)=0.
It follow form Lemma 2.3, that
(3.17)limn→∞∥xn+1-xn∥=0.
Since xn+1=ΠCn+1x0∈Cn+1⊂Cn and from the definition of Cn+1, we have
(3.18)ϕ(xn+1,un)≤ϕ(xn+1,xn),∀n≥1
and so
(3.19)limn→∞ϕ(xn+1,un)=0.
Hence
(3.20)limn→∞∥xn+1-un∥=0.
By using the triangle inequality, we obtain
(3.21)∥un-xn∥=∥un-xn+1+xn+1-xn∥≤∥un-xn+1∥+∥xn+1-xn∥.
By (3.17), (3.20), we get
(3.22)limn→∞∥un-xn∥=0.
Since J is uniformly norm-to-norm continuous on bounded subsets of E, we have
(3.23)limn→∞∥Jun-Jxn∥=0.Step 6.
Show thatxn→p∈EP(f).
Applying (3.4) and (3.11), we get ϕ(p,yn)≤ϕ(p,xn). From Lemma 2.9 and un=Trnyn, we observe that
(3.24)ϕ(un,yn)=ϕ(Trnyn,yn)≤ϕ(p,yn)-ϕ(p,Trnyn)≤ϕ(p,xn)-ϕ(p,Trnyn)=ϕ(p,xn)-ϕ(p,un)=∥p∥2-2〈p,Jxn〉+∥xn∥2-(∥p∥2-2〈p,Jun〉+∥un∥2)=∥xn∥2-∥un∥2-2〈p,Jxn-Jun〉≤∥xn-un∥(∥xn+un∥)+2∥p∥∥Jxn-Jun∥.
From (3.22), (3.23) and Lemma 2.3, we get
(3.25)limn→∞∥un-yn∥=0.
Since J is uniformly norm-to-norm continuous, we obtain
(3.26)limn→∞∥Jun-Jyn∥=0.
From rn>0, we have ∥Jun-Jyn∥/rn→0 as n→∞ and
(3.27)f(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C.
By (A2), that
(3.28)∥y-un∥∥Jun-Jyn∥rn≥1rn〈y-un,Jun-Jyn〉≥-f(un,y)≥f(y,un),∀y∈C
and un→p, we get f(y,p)≤0 for all y∈C. For 0<t<1, define yt=ty+(1-t)p. Then yt∈C which implies that f(yt,p)≤0. From (A1), we obtain that
(3.29)0=f(yt,yt)≤tf(yt,y)+(1-t)f(yt,p)≤tf(yt,y).
Thus f(yt,y)≥0. From (A3), we have f(p,y)≥0 for all y∈C. Hence p∈EP(f).Step 7.
We show thatxn→p∈F(T)∩F(S).
From definition ofCn, we have
(3.30)αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)⇔ϕ(z,zn)≤ϕ(z,xn).
Since xn+1=ΠCn+1x0∈Cn+1, we have
(3.31)ϕ(xn+1,zn)≤ϕ(xn+1,xn).
It follows from (3.16) that
(3.32)limn→∞ϕ(xn+1,zn)=0,
again from Lemma 2.3, we get
(3.33)limn→∞∥xn+1-zn∥=0.
By using the triangle inequality, we get
(3.34)∥zn-xn∥≤∥zn-xn+1∥+∥xn+1-xn∥.
Again by (3.17) and (3.33), we also have
(3.35)limn→∞∥zn-xn∥=0.
Since J is uniformly norm-to-norm continuous, we obtain
(3.36)limn→∞∥Jzn-Jxn∥=0.
Since
(3.37)∥yn-zn∥≤∥yn-un∥+∥un-xn∥+∥xn-zn∥,
from (3.22), (3.25), and (3.35), we have
(3.38)limn→∞∥yn-zn∥=0.
Since J is uniformly norm-to-norm continuous, we also have
(3.39)limn→∞∥Jyn-Jzn∥=0.
From (3.1), we get
(3.40)∥Jyn-Jzn∥=∥αn(Jxn-Jzn)+(1-αn)(JSzn-Jzn)∥=∥(1-αn)(JSzn-Jzn)-αn(Jzn-Jxn)∥≥(1-αn)∥JSzn-Jzn∥-αn∥Jzn-Jxn∥;
it follows  that
(3.41)(1-αn)∥JSzn-Jzn∥≤∥Jyn-Jzn∥+αn∥Jzn-Jxn∥,
and hence
(3.42)∥JSzn-Jzn∥≤11-αn(∥Jyn-Jzn∥+αn∥Jzn-Jxn∥).
Since αn≤1-δ1 for some δ1∈(0,1), (3.36), and (3.39), one has limn→∞∥JSzn-Jzn∥=0. Since J-1 is uniformly norm-to-norm continuous, we get
(3.43)limn→∞∥Szn-zn∥=0.
Since
(3.44)∥Sxn-xn∥≤∥Sxn-Szn∥+∥Szn-zn∥+∥zn-xn∥≤∥xn-zn∥+∥Szn-zn∥+∥zn-xn∥,
from (3.35) and (3.43), we obtain
(3.45)limn→∞∥Sxn-xn∥=0.
Since S is closed and xn→p, we have p∈F(S).
On the other hand, we note that
(3.46)ϕ(q,xn)-ϕ(q,un)=∥xn∥2-∥un∥2-2〈q,Jxn-Jun〉≤∥xn-un∥(∥xn+un∥)+2∥q∥∥Jxn-Jun∥.
It follows from ∥xn-un∥→0 and ∥Jxn-Jun∥→0, that
(3.47)ϕ(q,xn)-ϕ(q,un)→0.
Furthermore, from (3.4) and (3.5),
(3.48)ϕ(q,un)≤ϕ(q,yn)≤αnϕ(q,xn)+(1-αn)ϕ(q,zn)≤αnϕ(q,xn)+(1-αn)[βnϕ(q,xn)+(1-βn)ϕ(q,wn)]=αnϕ(q,xn)+(1-αn)βnϕ(q,xn)+(1-αn)(1-βn)ϕ(q,wn)≤αnϕ(q,xn)+(1-αn)βnϕ(q,xn)+(1-αn)(1-βn)×[ϕ(q,xn)-2λn(α-2c2λn)∥Axn-Aq∥2]=αnϕ(q,xn)+(1-αn)βnϕ(q,xn)+(1-αn)(1-βn)ϕ(q,xn)-(1-αn)(1-βn)2λn(α-2c2λn)∥Axn-Aq∥2=ϕ(q,xn)-(1-αn)(1-βn)2λn(α-2c2λn)∥Axn-Aq∥2,
and hence
(3.49)δ1δ22a(α-2ac2)∥Axn-Aq∥2≤(1-αn)(1-βn)2λn(α-2c2λn)∥Axn-Aq∥2≤ϕ(q,xn)-ϕ(q,un).
From (3.47) and (3.49), we have
(3.50)∥Axn-Aq∥→0.
From Lemma 2.5, Lemma 2.10, and (3.8), we compute
(3.51)ϕ(xn,wn)=ϕ(xn,ΠCJ-1(Jxn-λnAxn))≤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〈J-1(Jxn-λnAxn)-xn,-λnAxn〉=2〈J-1(Jxn-λnAxn)-xn,-λnAxn〉≤4λn2c2∥Axn-Aq∥2≤4b2c2∥Axn-Aq∥2.
Applying Lemmas 2.3 and (3.50), we obtain that
(3.52)∥xn-wn∥→0.
Again sinceJ is uniformly norm-to-norm continuous on bounded set, we have
(3.53)∥Jxn-Jwn∥→0.
Since
(3.54)∥zn-wn∥≤∥zn-xn∥+∥xn-wn∥,
by (3.35) and (3.52), we have
(3.55)limn→∞∥zn-wn∥=0,
and hence
(3.56)limn→∞∥Jzn-Jwn∥=0.From (3.1) we obtain that(3.57)∥Jzn-Jwn∥=∥βnJxn+(1-βn)JTwn-Jwn∥≥(1-βn)∥JTwn-Jwn∥-βn∥Jwn-Jxn∥,
and hence
(3.58)(1-βn)∥JTwn-Jwn∥≤∥Jzn-Jwn∥+βn∥Jwn-Jxn∥,
so
(3.59)∥JTwn-Jwn∥≤11-βn∥Jzn-Jwn∥+βn∥Jwn-Jxn∥.By (3.53), (3.56) andcondition βn≤1-δ2 for some δ2∈(0,1), we obtain
(3.60)∥JTwn-Jwn∥→0.
Since J-1 is uniformly norm-to-norm continuous on bounded set, we obtain
(3.61)∥Twn-wn∥→0.
Since xn→wn, then ∥Txn-xn∥→0. Thus by the closedness of T and xn→p, we get p∈F(T). Hence p∈F(T)∩F(S).Step 8.
We show thatxn→p∈VI(A,C).
DefineT⊂E×E* by Theorem 2.11; T is maximal monotone and T-10=VI(A,C). Let (v,w)∈G(T). Since w∈Tv=Av+NC(v), we get w-Av∈NC(v).
Fromwn∈C, we have
(3.62)〈v-wn,w-Av〉≥0.
On the other hand, since wn=ΠCJ-1(Jxn-λnAxn), then by Lemma 2.4, we have
(3.63)〈v-wn,Jwn-(Jxn-λnAxn)〉≥0,
and hence
(3.64)〈v-wn,Jxn-Jwnλn-Axn〉≤0.
It follows from (3.62) and (3.64), that
(3.65)〈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∥. Taking the limit as n→∞ and (3.53), we obtain 〈v-p,w〉≥0. By the maximality of T, we have p∈T-10; that is, p∈VI(A,C).Step 9.
We show thatp=ΠΩx0.
Fromxn=ΠCnx0, we have 〈Jx0-Jxn,xn-z〉≥0, ∀z∈Cn. Since Ω⊂Cn, we also have
(3.66)〈Jx0-Jxn,xn-y〉≥0,∀y∈Ω.
By taking limit n→∞, we obtain that
(3.67)〈Jx0-Jp,p-y〉≥0,∀y∈Ω.
By Lemma 2.4, we can conclude that p=ΠΩx0 and xn→p as n→∞. This completes the proof.SettingS≡T in Theorem 3.1., so, we obtain the following corollary.Corollary 3.2.
LetC be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4) and let A be an α-inverse-strongly monotone mapping of C into E* satisfying ∥Ay∥≤∥Ay-Au∥,for ally∈C and u∈VI(A,C)≠∅. Let T:C→C be closed relatively quasi-nonexpansive mappings such that Ω:=F(T)∩EP(f)∩VI(A,C)≠∅. For an initial point x0∈E with x1=ΠC1x0 and C1=C, define a sequence {xn} as follows:
(3.68)wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JTwn),yn=J-1(αnJxn+(1-αn)JTzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E. Assume that {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1), {rn}⊆(0,2α), and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2-uniformly convexity constant of E. Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.IfA≡0 in Theorem 3.1, then we obtain the following corollary.Corollary 3.3.
LetC be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). Let T,S:C→C is closed relatively quasi-nonexpansive mappings such that Ω:=F(T)∩F(S)∩EP(f)≠∅. For an initial point x0∈E with x1=ΠC1x0 and C1=C, define a sequence {xn} as follows:
(3.69)zn=J-1(βnJxn+(1-βn)JTwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E. Assume that {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1) and {rn}⊆(0,2α). Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.


## 4. Application

### 4.1. Complementarity Problem

LetK be a nonempty, closed and convex cone E, A a mapping of K into E*. We define its polar in E* to be the set(4.1)K*={y*∈E*:〈x,y*〉≥0,∀x∈K}.
Then the element u∈K is called a solution of the complementarity problem if(4.2)Au∈K*,〈u,Au〉=0.
The set of solutions of the complementarity problem is denoted by C(K,A).Theorem 4.1.
LetK be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from K×K to ℝ satisfying (A1)–(A4) and let A be an α-inverse-strongly monotone of E into E* satisfying ∥Ay∥≤∥Ay-Au∥,forally∈K and u∈C(K,A)≠∅. Let T,S:K→K be closed relatively quasi-nonexpansive mappings and Ω:=F(T)∩F(S)∩EP(f)∩C(K,A)≠∅. For an initial point x0∈E with x1=ΠK1 and K1=K, we define the sequence {xn} as follows:
(4.3)wn=ΠKJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JTwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈K,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E, {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1), {rn}⊆(0,2α), and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2-uniformly convexity constant of E. Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.Proof.
As in the proof of Takahashi in [7, Lemma 7.11], we get that VI(K,A)=C(K,A). So, we obtain the result.


### 4.2. Approximation of a Zero of a Maximal Monotone Operator

LetB be a multivalued mapping from E to E* with domain D(B)={z∈E:Az≠∅} and range R(B)=∪{Bz:z∈D(B)}. A mapping B is said to be a monotone operator if 〈x1-x2,y1-y2〉≥0 for each xi∈D(B) and yi∈Axi,i=1,2. A monotone operator B is said to be maximal if its graph G(B)={(x,y):y∈Ax} is not property contained in the graph of any other monotone operator. We know that if B is a maximal monotone operator, then B-1(0) is closed and convex. Let E be a reflexive, strictly convex, and smooth Banach space, and let B be a monotone operator from E to E*, we know that B is maximal if and only if R(J+rB)=E* for all r>0. Let Jr:E→D(B)be  defined by Jr=(J+rB)-1Jand such a Jr is called the resolvent of B. We know that Jr is a relatively nonexpansive (closed relatively quasi-nonexpansive for example; see [8]), and B-1(0)=F(Jr) for all r>0 (see [7, 33–35] for more details).Theorem 4.2.
LetC be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4) and let A be α-inverse-strongly monotone of E into E* satisfying ∥Ay∥≤∥Ay-Au∥,for ally∈C and u∈VI(A,C)≠∅. Let B be a maximal monotone operator of E into E* and let Jr be a resolvent of B and a closed mapping such that Ω:=B-1(0)∩F(S)∩EP(f)∩VI(A,C)≠∅. For an initial point x0∈E with x1=ΠC1 and C1=C, we define the sequence {xn} as follows:
(4.4)wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JJrwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E, {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1), {rn}⊆(0,2α) and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2-uniformly convexity constant of E. Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.Proof.
SinceJr is a closed relatively nonexpansive mapping and B-10=F(Jr). So, we obtain the result.Corollary 4.3.
LetC be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4) and let A be α-inverse-strongly monotone of E into E* satisfying ∥Ay∥≤∥Ay-Au∥,for ally∈C and u∈VI(A,C)≠∅. Let B be a maximal monotone operator of E into E* and let Jr be a resolvent of B and closed such that Ω:=B-1(0)∩EP(f)∩VI(A,C)≠∅. For an initial point x0∈E with x1=ΠC1 and C1=C, we define the sequence {xn} as follows:
(4.5)wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JJrwn),yn=J-1(αnJxn+(1-αn)JJrzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E, {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1), {rn}⊆(0,2α) and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2-uniformly convexity constant of E. Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.


## 4.1. Complementarity Problem

LetK be a nonempty, closed and convex cone E, A a mapping of K into E*. We define its polar in E* to be the set(4.1)K*={y*∈E*:〈x,y*〉≥0,∀x∈K}.
Then the element u∈K is called a solution of the complementarity problem if(4.2)Au∈K*,〈u,Au〉=0.
The set of solutions of the complementarity problem is denoted by C(K,A).Theorem 4.1.
LetK be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from K×K to ℝ satisfying (A1)–(A4) and let A be an α-inverse-strongly monotone of E into E* satisfying ∥Ay∥≤∥Ay-Au∥,forally∈K and u∈C(K,A)≠∅. Let T,S:K→K be closed relatively quasi-nonexpansive mappings and Ω:=F(T)∩F(S)∩EP(f)∩C(K,A)≠∅. For an initial point x0∈E with x1=ΠK1 and K1=K, we define the sequence {xn} as follows:
(4.3)wn=ΠKJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JTwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈K,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E, {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1), {rn}⊆(0,2α), and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2-uniformly convexity constant of E. Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.Proof.
As in the proof of Takahashi in [7, Lemma 7.11], we get that VI(K,A)=C(K,A). So, we obtain the result.


## 4.2. Approximation of a Zero of a Maximal Monotone Operator

LetB be a multivalued mapping from E to E* with domain D(B)={z∈E:Az≠∅} and range R(B)=∪{Bz:z∈D(B)}. A mapping B is said to be a monotone operator if 〈x1-x2,y1-y2〉≥0 for each xi∈D(B) and yi∈Axi,i=1,2. A monotone operator B is said to be maximal if its graph G(B)={(x,y):y∈Ax} is not property contained in the graph of any other monotone operator. We know that if B is a maximal monotone operator, then B-1(0) is closed and convex. Let E be a reflexive, strictly convex, and smooth Banach space, and let B be a monotone operator from E to E*, we know that B is maximal if and only if R(J+rB)=E* for all r>0. Let Jr:E→D(B)be  defined by Jr=(J+rB)-1Jand such a Jr is called the resolvent of B. We know that Jr is a relatively nonexpansive (closed relatively quasi-nonexpansive for example; see [8]), and B-1(0)=F(Jr) for all r>0 (see [7, 33–35] for more details).Theorem 4.2.
LetC be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4) and let A be α-inverse-strongly monotone of E into E* satisfying ∥Ay∥≤∥Ay-Au∥,for ally∈C and u∈VI(A,C)≠∅. Let B be a maximal monotone operator of E into E* and let Jr be a resolvent of B and a closed mapping such that Ω:=B-1(0)∩F(S)∩EP(f)∩VI(A,C)≠∅. For an initial point x0∈E with x1=ΠC1 and C1=C, we define the sequence {xn} as follows:
(4.4)wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JJrwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E, {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1), {rn}⊆(0,2α) and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2-uniformly convexity constant of E. Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.Proof.
SinceJr is a closed relatively nonexpansive mapping and B-10=F(Jr). So, we obtain the result.Corollary 4.3.
LetC be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4) and let A be α-inverse-strongly monotone of E into E* satisfying ∥Ay∥≤∥Ay-Au∥,for ally∈C and u∈VI(A,C)≠∅. Let B be a maximal monotone operator of E into E* and let Jr be a resolvent of B and closed such that Ω:=B-1(0)∩EP(f)∩VI(A,C)≠∅. For an initial point x0∈E with x1=ΠC1 and C1=C, we define the sequence {xn} as follows:
(4.5)wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JJrwn),yn=J-1(αnJxn+(1-αn)JJrzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E, {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1), {rn}⊆(0,2α) and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2-uniformly convexity constant of E. Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.



---

*Source: 734126-2010-01-28.xml*