# A New Hybrid Method for Equilibrium Problems, Variational Inequality Problems, Fixed Point Problems, and Zero of Maximal Monotone Operators

Authors: Yaqin Wang
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality, the set of solutions of the generalized mixed equilibrium problem, and zeros of maximal monotone operators in a 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. The results obtained in this paper improve and extend the result of Zeng et al. (2010) and many others.


---

## Body

## 1. Introduction

In 1994, Blum and Oettli [1] introduced equilibrium problems, which have had a great impact and influence on the development of several branches of pure and applied sciences. It has been shown that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, physics, image reconstruction, ecology, transportation, network, elasticity, and optimization.LetE be a real Banach space, E* the dual space of E, and C a nonempty closed convex subset of E. Let Θ:C×C→R be a bifunction and φ:C→R a real-valued function. The generalized mixed equilibrium problem (GMEP) of finding x∈C is such that
(1.1)Θ(x,y)+φ(y)-φ(x)+〈Ax,y-x〉≥0,∀y∈C.
Recently, Zhang [2] considered this problem. Here some special cases of problem (1.1) are stated as follows.IfA=0, then problem (1.1) reduces to the following mixed equilibrium problem of finding x∈C such that
(1.2)Θ(x,y)+φ(y)-φ(x)≥0,∀y∈C,
which was considered by Ceng and Yao [3]. The set of solutions of this problem is denoted by MEP.Ifφ=0, then problem (1.1) reduces to the following generalized equilibrium problem of finding x∈C such that
(1.3)Θ(x,y)+〈Ax,y-x〉≥0,∀y∈C,
which was studied by S. Takahashi and W. Takahashi [4].Ifφ=0 and A=0, then problem (1.1) reduces to the following equilibrium problem of finding x∈C such that
(1.4)Θ(x,y)≥0,∀y∈C.
The set of solutions of problem (1.4) is denoted by EP.IfΘ=0,φ=0, then problem (1.1) reduces to the following classical variational inequality problem of finding x∈C such that
(1.5)〈Ax,y-x〉≥0,∀y∈C.
The set of solutions of problem (1.5) is denoted by VI(C,A).The problem (1.1) is very general in the sense that it includes, as special cases, numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, for instance, [1, 3–7].The normalized duality mapping fromE to 2E* is defined by
(1.6)Jx={f∈E*:〈x,f〉=‖x‖2=‖f‖2},x∈E,
where 〈·,·〉 denotes the generalized duality pairing. It is well known that if E is smooth then J is single valued and if E is uniformly smooth then J is uniformly continuous on bounded subsets of E. Moreover, if E is a reflexive and strictly convex Banach space with a strictly convex dual, then J-1 is single valued, one to one, surjective, and it is the duality mapping from E* into E and thus JJ-1=IE* and J-1J=IE (see [8]).On the other hand, letW:E⇉E* be a set-valued mapping. The problem of finding v∈E satisfying 0∈Wv contains numerous problems in economics, optimization, and physics. Such v∈E is called a zero point of W.A set-valued mappingW:E⇉E* with graph G(W)={(x,x*):x*∈Wx}, domain D(W)={x∈E:Wx≠∅}, and range R(W)=∪{Wx:x∈D(W)} is said to be monotone if 〈x-y,x*-y*〉≥0 for all x*∈Wx,y*∈Wy. W is said to be maximal monotone if the graph G(W) of W is not properly contained in the graph of any other monotone operator. It is known that W is a maximal monotone if and only if R(J+rW)=E* for all r>0 when E is a reflexive, strictly convex, and smooth Banach space (see [9]).LetE be a smooth, strictly convex, and reflexive Banach space, let C be a nonempty closed convex subset of E, and let W:E⇉E* be a monotone operator satisfying D(W)⊂C⊂J-1(∩r>0R(J+rW)). Then the resolvent of W defined by Jr=(J+rW)-1J is a single-valued mapping from E to D(W) for all r>0. For r>0, the Yosida approximation of W is defined by Wrx=(Jx-JJrx)/r for all x∈E.A mappingA:C→E* is said to be monotone if, for each x,y∈C,
(1.7)〈x-y,Ax-Ay〉≥0.A is said to be γ-inverse strongly monotone if there exists a positive real number γ>0 such that
(1.8)〈x-y,Ax-Ay〉≥γ‖Ax-Ay‖2,∀x,y∈C.
If A is γ-inverse strongly monotone, then it is Lipschitz continuous with constant 1/γ, that is,
(1.9)‖Ax-Ay‖≤(1γ)‖x-y‖,x,y∈C.LetE be a smooth Banach space. The function ϕ:E×E→R defined by
(1.10)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2,∀x,y∈E,
is studied by Alber [10], Kamimura and Takahashi [11], and Reich [12]. It follows from the definition of the function ϕ that
(1.11)(‖x‖-‖y‖)2≤ϕ(x,y)≤(‖x‖+‖y‖)2,∀x,y∈E.
Observe that, in a Hilbert space H, ϕ(x,y)=∥x-y∥2,forallx,y∈H.Lemma 1.1 (see [10]).
LetC be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space E, and let x∈E. Then there exists a unique element x0∈C such that ϕ(x0,x)=min{ϕ(z,x):z∈C}.LetE be a reflexive, strictly convex, and smooth Banach space and C a nonempty closed and convex subset of E. The generalized projection mapping, introduced by Alber [10], is a mapping ΠC:E→C that assigns to an arbitrary point x∈E, the minimum point of the functional ϕ(x,y), that is, ΠCx=x0 due to Lemma 1.1, where x0 is the solution to the minimization problem ϕ(x0,x)=min{ϕ(z,x):z∈C}.LetT be a mapping from C into itself. F(T) denotes the set of fixed points of T. A point p in C is said to be an asymptotic fixed point of T if C contains a sequence {xn} which converges weakly to p such that limn→∞∥Txn-xn∥=0. The set of asymptotic fixed points of T will be denoted by F̂(T). A mapping T from C into itself is called nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C and relatively nonexpansive (see [13, 14]) if F̂(T)=F(T) and ϕ(p,Tx)≤ϕ(p,x) for all x∈C and p∈F(T). T is said to be ϕ-nonexpansive if ϕ(Tx,Ty)≤ϕ(x,y) for all x,y∈C. T is said to be relatively quasi-nonexpansive if F(T)≠∅ and ϕ(p,Tx)≤ϕ(p,x) for all x∈C and p∈F(T). Note that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings which requires the strong restriction: F̂(T)=F(T).WhenW is a maximal monotone operator, a well-known method for solving the equation 0∈Wv in a Hilbert space H is the proximal point algorithm (see [15]): x1=x∈H and
(1.12)xn+1=Jrnxn,n=1,2,…,
where {rn}⊂(0,∞) and Jr=(I+rW)-1 for all r>0 is the resolvent operator for W, then Rockafellar proved that the sequence {xn} converges weakly to an element of W-10.The modifications of the proximal point algorithm for different operators have been investigated by many authors. Kohsaka and Takahashi [16] considered the following Algorithm (1.13) in a smooth and uniformly convex Banach space:
(1.13)xn+1=J-1(βnJ(x1)+(1-βn)J(Jrnxn)),n=1,2,…,
and Kamimura et al. [17] considered Algorithm (1.14) in a uniformly smooth and uniformly convex Banach space:
(1.14)xn+1=J-1(βnJ(xn)+(1-βn)J(Jrnxn)),n=1,2,….
They showed that Algorithm (1.13) converges strongly and Algorithm (1.14) converges weakly provided that the sequences {βn}, {rn} of real numbers are chosen appropriately.Recently, Saewan and Kumam [18] proposed the following iterative scheme: for an initial x0∈E with x1=ΠC1x0 and C1=C(1.15)ωn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JTnJrnωn),yn=J-1(αnJxn+(1-αn)JSnzn),un=Krnyn,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,zn)≤ϕ(z,xn)},xn+1=ΠCn+1x0,n≥0,
where Kr is the same as in Lemma 2.8 and they obtained a strong convergence theorem.In 2010, Zeng et al. [19] introduced the following hybrid iterative process: let x0∈E be chosen arbitrarily,
(1.16)x̃n=J-1(αnJx0+(1-αn)(βnJxn+(1-βn)JJrnvn)),yn=J-1(α̃nJx0+(1-α̃n)(β̃nJx̃n+(1-β̃n)JJ̃rnx̃n)),un=Krnyn,Hn={z∈C:ϕ(z,un)≤(α̃n+(1-α̃n)αn)ϕ(z,x0)+(1-α̃n)(1-αn)ϕ(z,xn)},Wn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠHn∩Wnx0,n≥0.
Then they proved some strong and weak convergence theorems.Very recently, for mixed equilibrium problems, variational inequality problems, fixed point problems, and zeros of maximal monotone operators, many authors have studied them and obtained many new results, see, for instance, [20–23].On the other hand, Nakajo et al. [24] introduced the following condition. Let C be a nonempty closed convex subset of a Hilbert space H, let {Tn} be a family of mappings of C into itself with F=∩n=1∞F(Tn)≠∅, and ωw(zn) denotes the set of all weak subsequential limits of a bounded sequence {zn} in C. {Tn} is said to satisfy the NST-condition if, for every bounded sequence {zn} in C,
(1.17)lim⁡n→∞‖zn-Snzn‖=0impliesthatωw(zn)⊂F.Motivated and inspired by the above work, the purpose of this paper is to introduce a new hybrid projection iterative scheme which converges strongly to a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings and zero of maximal monotone operators in Banach spaces.


## 2. Preliminaries

LetE be a normed linear space with dim E≥2. The modulus of smoothness of E is the function ρE:[0,+∞)→[0,+∞) defined by
(2.1)ρE(τ):=sup⁡{‖x+y‖+‖x-y‖2-1:‖x‖=1,‖y‖=τ}.
The space E is said to be smooth if ρE(τ)>0,forallτ>0, and E is called uniformly smooth if and only if limt→0+(ρE(t)/t)=0.The modulus of convexity ofE is the function δE:(0,2]→[0,1] defined by
(2.2)δE(ε):=inf⁡{1-‖x+y2‖:‖x‖=‖y‖=1;ε=‖x-y‖}.E is called uniformly convex if and only if δE(ε)>0 for every ε∈(0,2]. Let p>1, then E is said to be p-uniformly convex if there exists a constant c>0 such that δE(ε)≥cεp for every ε∈(0,2]. Observe that every p-uniformly convex is uniformly convex. It is well known (see, e.g., [7]) that
(2.3)Lp(lp)orWmpis{p-uniformlyconvexifp≥2;2-uniformlyconvexif1<p≤2.In what follows, we will make use of the following lemmas.Lemma 2.1 (see [7]).
LetE be a 2-uniformly convex and smooth 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 1/c(0<c≤1) is the 2-uniformly convex constant of E.Lemma 2.2 (see [10, 11]).
LetE be a real smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset. Then the following conclusions hold: (1)
ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x),∀x∈E,y∈C;(2)
supposex∈E and z∈C, then
(2.5)z=ΠCx⟺〈z-y,Jx-Jz〉≥0,∀y∈C.Lemma 2.3 (see [11]).
LetE be a real smooth and uniformly convex Banach space, and let {xn} and {yn} be two sequences of E. If either {xn} or {yn} is bounded and ϕ(xn,yn)→0 as n→∞, then xn-yn→0 as n→∞.Lemma 2.4 (see [25]).
LetE be a real smooth Banach space, and let A:E⇉E* be a maximal monotone mapping, then A-1(0) is a closed and convex subset of E.We denote byNC(v) the normal cone for C at a point v∈C, that is, NC(v):={x*∈E*:〈v-y,x*〉≥0forally∈C}. In the following, we will use the following Lemma.Lemma 2.5 (see [15]).
LetC be a nonempty closed convex subset of a Banach space E, and let A be a monotone and hemicontinuous operator of C into E*. Let T⊂E×E* be an operator defined as follows:
(2.6)Tv={Av+NC(v),v∈C,∅,v∉C.
Then T is maximal monotone and T-10=VI(C,A).We make use of the functionV:E×E*→R defined by
(2.7)V(x,x*)=‖x‖2-2〈x,x*〉+‖x‖2,∀x∈E,x*∈E*,
studied by Alber [10]; that is, V(x,x*)=ϕ(x,J-1x*) for all x∈E and x*∈E*. We know the following lemma.Lemma 2.6 (see [10]).
LetE be a reflexive strictly convex and smooth Banach space with E* as its dual. Then
(2.8)V(x,x*)+2〈J-1x*-x,y*〉≤V(x,x*+y*),
for all x∈E and x*,y*∈E*.Lemma 2.7 (see [26]).
LetE be a uniformly convex Banach space, and let Br(0)={x∈E:∥x∥≤r} be a closed ball of E. Then there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(2.9)‖λx+μy+γz‖2≤λ‖x‖2+μ‖y‖2+γ‖z‖2-λμg(‖x-y‖),
for all x,y,z∈Br(0) and λ,μ,γ∈[0,1] with λ+μ+γ=1.For solving the equilibrium problem, let us assume thatΘ 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,
(2.10)lim⁡t→0Θ(tz+(1-t)x,y)≤Θ(x,y);(A4)
for eachx∈C,y↦Θ(x,y) is convex and lower semicontinuous.Lemma 2.8 (see [2]).
LetC be a closed subset of a smooth, strictly convex, and reflexive Banach space E. Let B:C→E* be a continuous and monotone mapping, let φ:C→R be a lower semicontinuous and convex function, and let 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
(2.11)Θ(u,y)+〈Bu,y-u〉+φ(y)-φ(u)+1r〈y-u,Ju-Jx〉≥0,∀y∈C.
Define a mapping Kr:E→C as follows:
(*)Kr(x)={u∈C:Θ(u,y)+〈Bu,y-u〉+φ(y)-φ(u)+1r〈y-u,Ju-Jx〉≥0,∀y∈C},
for all x∈E. Then, the following conclusions 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)=GMEP;(4)
GMEP is closed and convex;(5)
ϕ(p,Krz)+ϕ(Krz,z)≤ϕ(p,z),∀p∈F(Kr),z∈E.Lemma 2.9 (see [27]).
LetE be a smooth, strictly convex, and reflexive Banach space, let C be a nonempty closed convex subset of E, and let W:E⇉E* be a monotone operator satisfying D(W)⊂C⊂J-1(∩r>0R(J+rW)). Let r>0, and let Jr and Wr be the resolvent and the Yosida approximation of W, respectively. Then the following hold:(i)
ϕ(u,Jrx)+ϕ(Jrx,x)≤ϕ(u,x), for all x∈C,u∈W-10;(ii)
(Jrx,Wrx)∈G(W), for all x∈C;(iii)
F(Jr)=W-10.Lemma 2.10 (see [28]).
LetE be a real uniformly smooth and strictly convex Banach space and C a nonempty closed convex subset of E. Let S:C→C be a relatively quasi-nonexpansive mapping. Then F(S) is a closed convex subset of C.


## 3. Strong Convergence Theorems

In this section, letT,T̃:E⇉E* be two maximal monotone operators satisfying D(T),D(T̃)⊂C. We denote the resolvent operators of T and T̃ by Jr=(J+rT)-1J and J̃r=(J+rT̃)-1J for each r>0, respectively. For each r>0, the Yosida approximations of T and T̃ are defined by Ar=(J-JJr)/r and Ãr=(J-JJ̃r)/r, respectively. It is known that
(3.1)Arx∈T(Jrx),Ãrx∈T̃(J̃rx),foreachr>0,x∈E.Theorem 3.1.
LetE be a real uniformly smooth and 2-uniformly convex Banach space and C a nonempty, closed, and convex subset of E. Let A be a γ-inverse strongly monotone mapping of C into E* satisfying ∥Ax∥≤∥Ax-Ap∥ for all x∈C and p∈VI(C,A). Let B:C→E* be a monotone continuous mapping, and let {Tn},{Sn} be two countable families of relatively quasi-nonexpansive mappings from C into itself satisfying NST-conditions such that Ω:=(∩n=0∞F(Tn))∩(∩n=0∞F(Sn))∩VI(C,A)∩GMEP∩T-10∩T̃-10≠∅. Suppose that 0<a<λn<b<(c2γ)/2, where c is the constant in (2.4). Let {tn}⊂[c*,+∞) for some c*>0 and {rn}⊂(0,+∞) satisfy liminfn→∞rn>0. Let {xn} be the sequence generated by
(3.2)x0∈C,chosenarbitrarily,vn=ΠCJ-1(Jxn-λnAxn),zn=J-1(αnJx0+(1-αn)(βnJxn+(1-βn)JSnJrnvn)),yn=J-1(α̃nJx0+(1-α̃n)(β̃nJzn+(1-β̃n)JTnJ̃rnzn)),un=Ktnyn,Hn={z∈C:ϕ(z,un)≤(α̃n+(1-α̃n)αn)ϕ(z,x0)+(1-α̃n)(1-αn)ϕ(z,xn)},Wn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠHn∩Wnx0,n≥0,
where J is the normalized duality mapping, {αn},{βn},{α̃n}, and {β̃n} are four sequences in [0,1], and Kr is defined by (*). The following conditions hold:(i)
limn→∞αn=limn→∞α̃n=0;(ii)
liminfn→∞βn(1-βn)>0;(iii)
liminfn→∞β̃n(1-β̃n)>0.Then{xn} converges strongly to ΠΩx0.Proof.
We have the following steps.Step 1.
First we prove thatHn and Wn are both closed and convex and Ω⊂Hn∩Wn,foralln≥0.
In fact, it follows from Lemmas2.4, 2.5, 2.8, and 2.10 that Ω is closed and convex. It is obvious that Wn is closed and convex for each n≥0. Let γn=α̃n+(1-α̃n)αn,wn=Jrnvn,z̃n=J̃rnzn. For any z∈C,
(3.3)ϕ(z,un)≤γnϕ(z,x0)+(1-γn)ϕ(z,xn)
is equivalent to
(3.4)-2〈z,Jun〉+‖un‖2≤-2γn〈z,Jx0〉+γn‖x0‖2+(1-γn)(-2〈z,Jxn〉+‖xn‖2),
which implies that Hn is closed and convex for each n≥0. Next, we prove that Ω⊂Hn∩Wn,foralln≥0. For any given p∈Ω, by Lemma 2.9 we have
(3.5)ϕ(p,wn)=ϕ(p,Jrnvn)≤ϕ(p,vn),ϕ(p,z̃n)=ϕ(p,J̃rnzn)≤ϕ(p,zn).
Since Tn,Sn are relatively quasi-nonexpansive, from the definition of ϕ(x,y), the convexity of ∥·∥2, and (3.5), we have
(3.6)ϕ(p,yn)=ϕ(p,J-1(α̃nJx0+(1-α̃n)(β̃nJzn+(1-β̃n)JTnz̃n)))=‖p‖2-2α̃n〈p,Jx0〉-2(1-α̃n)β̃n〈p,Jzn〉-2(1-α̃n)(1-β̃n)〈p,JTnz̃n〉+‖α̃nJx0+(1-α̃n)(β̃nJzn+(1-β̃n)JTnz̃n)‖2≤‖p‖2-2α̃n〈p,Jx0〉-2(1-α̃n)β̃n〈p,Jzn〉-2(1-α̃n)(1-β̃n)〈p,JTnz̃n〉+α̃n‖x0‖2+(1-α̃n)(β̃n‖zn‖2+(1-β̃n)‖Tnz̃n‖2)=α̃nϕ(p,x0)+(1-α̃n)β̃nϕ(p,zn)+(1-α̃n)(1-β̃n)ϕ(p,Tnz̃n)≤α̃nϕ(p,x0)+(1-α̃n)β̃nϕ(p,zn)+(1-α̃n)(1-β̃n)ϕ(p,z̃n)≤α̃nϕ(p,x0)+(1-α̃n)ϕ(p,zn),(3.7)ϕ(p,zn)=ϕ(p,J-1(αnJx0+(1-αn)(βnJxn+(1-βn)JSnwn)))≤‖p‖2-2αn〈p,Jx0〉-2(1-αn)βn〈p,Jxn〉-2(1-αn)(1-βn)〈p,JSnwn〉+αn‖x0‖2+(1-αn)(βn‖xn‖2+(1-βn)‖Snwn‖2)=αnϕ(p,x0)+(1-αn)βnϕ(p,xn)+(1-αn)(1-βn)ϕ(p,Snwn)≤αnϕ(p,x0)+(1-αn)βnϕ(p,xn)+(1-αn)(1-βn)ϕ(p,wn)≤αnϕ(p,x0)+(1-αn)βnϕ(p,xn)+(1-αn)(1-βn)ϕ(p,vn).
Moreover, it follows from Lemmas 2.2 and 2.6 that
(3.8)ϕ(p,vn)≤ϕ(p,J-1(Jxn-λnAxn))≤V(p,Jxn-λnAxn)≤V(p,Jxn-λnAxn+λnAxn)-2〈J-1(Jxn-λnAxn)-p,λnAxn〉=ϕ(p,xn)-2λn〈xn-p,Axn-Ap〉-2λn〈xn-p,Ap〉-2λn〈J-1(Jxn-λnAxn)-xn,Axn〉.
Since p∈VI(C,A), A is γ-inverse strongly monotone, from the above inequality, Lemma 2.1, and the fact that ∥Ax∥≤∥Ax-Ap∥ for all x∈C and p∈VI(C,A), we obtain
(3.9)ϕ(p,vn)≤ϕ(p,xn)-2λnγ‖Axn-Ap‖2+2λn‖J-1(Jxn-λnAxn)-xn‖‖Axn‖≤ϕ(p,xn)-2λnγ‖Axn-Ap‖2+4c2λn2‖Axn-Ap‖2=ϕ(p,xn)+2λn(2c2λn-γ)‖Axn-Ap‖2≤ϕ(p,xn).
From (3.7)-(3.9), we have
(3.10)ϕ(p,zn)≤αnϕ(p,x0)+(1-αn)ϕ(p,xn).
So from (3.6) and (3.10), we have
(3.11)ϕ(p,yn)≤(α̃n+(1-α̃n)αn)ϕ(p,x0)+(1-α̃n)(1-αn)ϕ(p,xn).
By Lemma 2.8(5) and (3.11), we have
(3.12)ϕ(p,un)=ϕ(p,Ktnyn)≤ϕ(p,yn)≤(α̃n+(1-α̃n)αn)ϕ(p,x0)+(1-α̃n)(1-αn)ϕ(p,xn).
Therefore, p∈Hn; that is Ω⊂Hn for each n≥0.
Next we prove thatΩ⊂Hn∩Wn for each n≥0 by induction. For n=0,W0=C, thus Ω⊂H0∩W0. Suppose that Ω⊂Hn∩Wn for some n≥1. Since xn+1=ΠHn∩Wnx0, by Lemma 2.2, for any q∈Hn∩Wn, we have(3.13)〈xn+1-q,Jx0-Jxn+1〉≥0.
Since Ω⊂Hn∩Wn, for any p∈Ω, we have
(3.14)〈xn+1-p,Jx0-Jxn+1〉≥0,
which implies that p∈Wn+1, that is, Ω⊂Hn∩Wn for each n≥0.Step 2.
Next we prove that∥wn-Snwn∥→0,∥z̃n-Tnz̃n∥→0(n→∞).
Similar to the proof of Step  3 in [19, Theorem  3.1], we have that {ϕ(xn,x0)} is nondecreasing and bounded and ϕ(xn+1,xn)→0(n→∞). So {xn} is bounded, then, by (3.5)–(3.12), we obtain that {un},{zn},{vn},{wn}, {z̃n}, and {yn} are all bounded. Furthermore, {Snwn} and {Tnz̃n} are both bounded. By Lemma 2.3, we have(3.15)‖xn-xn+1‖⟶0(n⟶∞).
Since xn+1=ΠHn∩Wnx0∈Hn and by condition (i), we have
(3.16)ϕ(xn+1,un)≤(α̃n+(1-α̃n)αn)ϕ(xn+1,x0)+(1-α̃n)(1-αn)ϕ(xn+1,xn)⟶0(n⟶∞),
which together with Lemma 2.3 implies that ∥xn+1-un∥→0(n→∞). So we obtain
(3.17)‖xn-un‖⟶0(n⟶∞).
Let an=J-1(βnJxn+(1-βn)JSnJrnvn),bn=J-1(β̃nJzn+(1-β̃n)JTnJ̃rnzn). It follows from the boundedness of {xn} that {Snwn} and {Tnz̃n} are bounded. Let r=sup{∥xn∥,∥zn∥,∥Tnz̃n∥,∥Snwn∥}. By Lemma 2.7, (3.5), and (3.9), for any p∈Ω, we obtain
(3.18)ϕ(p,an)=ϕ(p,J-1(βnJxn+(1-βn)JSnJrnvn))=‖p‖2-2〈p,βnJxn+(1-βn)JSnJrnvn〉+‖βnJxn+(1-βn)JSnJrnvn‖2≤‖p‖2-2βn〈p,Jxn〉-2(1-βn)〈p,JSnJrnvn〉+βn‖xn‖2+(1-βn)‖SnJrnvn‖2-βn(1-βn)g(‖Jxn-JSnwn‖)≤βnϕ(p,xn)+(1-βn)ϕ(p,Snwn)-βn(1-βn)g(‖Jxn-JSnwn‖)≤βnϕ(p,xn)+(1-βn)ϕ(p,wn)-βn(1-βn)g(‖Jxn-JSnwn‖)≤ϕ(p,xn)-βn(1-βn)g(‖Jxn-JSnwn‖),(3.19)ϕ(p,bn)=ϕ(p,J-1(β̃nJzn+(1-β̃n)JTnz̃n))≤β̃nϕ(p,zn)+(1-β̃n)ϕ(p,Tnz̃n)-β̃n(1-β̃n)g(‖Jzn-JTnz̃n‖)≤β̃nϕ(p,zn)+(1-β̃n)ϕ(p,z̃n)-β̃n(1-β̃n)g(‖Jzn-JTnz̃n‖)≤ϕ(p,zn)-β̃n(1-β̃n)g(‖Jzn-JTnz̃n‖).
It follows from (3.10), (3.12), and (3.19) that
(3.20)ϕ(p,un)≤ϕ(p,yn)=ϕ(p,J-1(α̃nJx0+(1-α̃n)Jbn))≤α̃nϕ(p,x0)+(1-α̃n)ϕ(p,bn)≤α̃nϕ(p,x0)+(1-α̃n)(ϕ(p,zn)-β̃n(1-β̃n)g(‖Jzn-JTnz̃n‖))≤(α̃n+αn-α̃nαn)ϕ(p,x0)+(1-αn)(1-α̃n)ϕ(p,xn)-(1-α̃n)β̃n(1-β̃n)g(‖Jzn-JTnz̃n‖).
From (3.17) and (3.20), we have
(3.21)(1-α̃n)β̃n(1-β̃n)g(‖Jzn-JTnz̃n‖)≤(α̃n+αn-α̃nαn)ϕ(p,x0)+(αnα̃n-αn-α̃n)ϕ(p,xn)+ϕ(p,xn)-ϕ(p,un)≤(α̃n+αn-α̃nαn)(ϕ(p,x0)-ϕ(p,xn))+2p‖Jun-Jxn‖+(‖xn‖-‖un‖)(∥xn∥+∥un∥)⟶0(n⟶∞),
which together with condition (iii) implies that
(3.22)‖Jzn-JTnz̃n‖⟶0(n⟶∞).
Since J is uniformly continuous on bounded sets, then
(3.23)‖zn-Tnz̃n‖⟶0(n⟶∞).
By (3.18) and (3.19), we have
(3.24)ϕ(p,bn)≤ϕ(p,zn)=ϕ(p,J-1(αnJx0+(1-αn)Jan))≤αnϕ(p,x0)+(1-αn)ϕ(p,an)≤αnϕ(p,x0)+(1-αn)(ϕ(p,xn)-βn(1-βn)g(‖Jxn-JSnwn‖)).
It follows from (3.20) and (3.24) thatss
(3.25)ϕ(p,un)≤α̃nϕ(p,x0)+(1-α̃n)ϕ(p,bn)≤α̃nϕ(p,x0)+(1-α̃n)(αnϕ(p,x0)+(1-αn)(ϕ(p,xn)-βn(1-βn)g(‖Jxn-JSnwn‖)))=(α̃n+αn-α̃nαn)ϕ(p,x0)+(1-α̃n)(1-αn)ϕ(p,xn)-(1-α̃n)βn(1-βn)g(‖Jxn-JSnwn‖).
So by (3.17) and (3.25), we obtain
(3.26)(1-α̃n)βn(1-βn)g(‖Jxn-JSnwn‖)≤(α̃n+αn-α̃nαn)(ϕ(p,x0)-ϕ(p,xn))+(ϕ(p,xn)-ϕ(p,un))≤(α̃n+αn-α̃nαn)(ϕ(p,x0)-ϕ(p,xn))+2p‖Jun-Jxn‖+(‖xn‖-‖un‖)(‖xn‖+‖un‖)⟶0(n⟶∞),
which together with conditions (i) and (ii) implies that
(3.27)‖Jxn-JSnwn‖⟶0(n⟶∞).
Since J is uniformly continuous on bounded sets, then
(3.28)‖xn-Snwn‖⟶0(n⟶∞).
It follows from (3.2) that
(3.29)Jzn=αnJx0+(1-αn)(βnJxn+(1-βn)JSnJrnvn).
Therefore, from (3.27) and condition (i), we have
(3.30)‖Jzn-Jxn‖≤αn‖Jx0-Jxn‖+(1-αn)(1-βn)‖JSnwn-Jxn‖⟶0(n⟶∞).
Thus,
(3.31)‖zn-xn‖⟶0(n→∞).
From (3.7) and (3.9), we have
(3.32)ϕ(p,zn)≤αnϕ(p,x0)+(1-αn)βnϕ(p,xn)+(1-αn)(1-βn)(ϕ(p,xn)+2λn(2c2λn-γ)‖Axn-Ap‖2)=αnϕ(p,x0)+(1-αn)ϕ(p,xn)+2(1-αn)(1-βn)λn(2c2λn-γ)‖Axn-Ap‖2,
which together with (3.31) and condition (i) implies that
(3.33)2(1-αn)(1-βn)λn(γ-2c2λn)‖Axn-Ap‖2≤αn(ϕ(p,x0)-ϕ(p,xn))+ϕ(p,xn)-ϕ(p,zn)⟶0(n⟶∞).
Hence,
(3.34)‖Axn-Ap‖⟶0(n⟶∞).
From Lemmas 2.1, 2.2, and 2.6, (3.34), and the fact that ∥Ax∥≤∥Ax-Ap∥ for all x∈C and p∈VI(C,A), we have
(3.35)ϕ(xn,vn)=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)+2λn‖J-1(Jxn-λnAxn)-J-1Jxn‖‖Axn‖≤2λn22c2‖Axn‖2≤2λn22c2‖Axn-Ap‖2⟶0(n⟶∞).
This implies that
(3.36)‖xn-vn‖⟶0(n⟶∞).
Combining (3.28) and (3.36), we have
(3.37)‖vn-Snwn‖⟶0(n⟶∞).
It follows from (3.6), (3.7), and (3.12) that
(3.38)ϕ(p,wn)≥ϕ(p,un)(1-α̃n)(1-αn)(1-βn)-(α̃n+αn-α̃nαn)ϕ(p,x0)(1-α̃n)(1-αn)(1-βn)-(1-α̃n)(1-αn)βnϕ(p,xn)(1-α̃n)(1-αn)(1-βn).
By Lemma 2.9, (3.9), and (3.38), we have
(3.39)ϕ(wn,vn)=ϕ(Jrnvn,vn)≤ϕ(p,vn)-ϕ(p,wn)≤ϕ(p,xn)-ϕ(p,wn)≤ϕ(p,xn)-ϕ(p,un)(1-α̃n)(1-αn)(1-βn)+α̃n+αn-α̃nαn(1-α̃n)(1-αn)(1-βn)(ϕ(p,x0)-ϕ(p,xn)).
So, by (3.17) and conditions (i) and (ii), we have
(3.40)ϕ(wn,vn)⟶0(n⟶∞),
which implies that
(3.41)‖wn-vn‖⟶0(n⟶∞).
Combining (3.37) and (3.41), we obtain
(3.42)‖wn-Snwn‖⟶0(n⟶∞).
It follows from (3.6), (3.10), and (3.12) that
(3.43)ϕ(p,un)≤ϕ(p,yn)≤α̃nϕ(p,x0)+(1-α̃n)β̃nϕ(p,zn)+(1-α̃n)(1-β̃n)ϕ(p,z̃n)≤(α̃n+(1-α̃n)β̃nαn)ϕ(p,x0)+(1-α̃n)(1-αn)β̃nϕ(p,xn)+(1-α̃n)(1-β̃n)ϕ(p,z̃n),
which implies that
(3.44)ϕ(p,z̃n)≥ϕ(p,un)(1-α̃n)(1-β̃n)-(α̃n+(1-α̃n)β̃nαn)ϕ(p,x0)(1-α̃n)(1-β̃n)-(1-α̃n)(1-αn)β̃nϕ(p,xn)(1-α̃n)(1-β̃n).
Combining the above inequality, (3.10), and Lemma 2.9, we have
(3.45)ϕ(z̃n,zn)=ϕ(J̃rnzn,zn)≤ϕ(p,zn)-ϕ(p,z̃n)≤αnϕ(p,x0)+(1-αn)ϕ(p,xn)-ϕ(p,z̃n)≤ϕ(p,xn)-ϕ(p,un)(1-α̃n)(1-β̃n)+(αn+α̃n-αnα̃n)(1-α̃n)(1-β̃n)(ϕ(p,x0)-ϕ(p,xn)).
By conditions (i) and (iii), (3.17), and (3.45), we have
(3.46)ϕ(z̃n,zn)⟶0(n⟶∞),
which implies that
(3.47)‖z̃n-zn‖⟶0(n⟶∞).
It follows from (3.23) and (3.47) that
(3.48)‖z̃n-Tnz̃n‖⟶0(n⟶∞).Step 3.
Now we show thatωw({xn})⊂Ω:=(∩n=0∞F(Tn))∩(∩n=0∞F(Sn))∩VI(C,A)∩GMEP∩T-10∩T̃-10, where
(3.49)ωw({xn})={x*∈C:xnk⇀x*forsomesequence{nk}⊂{n}withnk↑∞}.
Indeed, since{xn} is bounded and X is reflexive, we know that ωw({xn})≠∅. For any arbitrary x*∈ωw({xn}), there exists a subsequence {xnk} of {xn} such that xnk⇀x*. From (3.36), we have vnk⇀x*. Since {Tn},{Sn} satisfy NST-conditions, from (3.42) and (3.48) we have x*∈(∩n=0∞F(Tn))∩(∩n=0∞F(Sn)).
LetS⊂E×E* be an operator as follows:(3.50)Sv={Av+NC(v),v∈C,∅,v∉C.
By Lemma 2.5, S is maximal monotone and S-1(0)=VI(C,A). Let (v,w)∈G(S). Since w∈Sv=Av+NC(v), we have w-Av∈NC(v). Moreover, vn∈C implies that
(3.51)〈v-vn,w-Av〉≥0.
On the other hand, it follows fromvn=ΠCJ-1(Jxn-λnAxn) and Lemma 2.2 that(3.52)〈v-vn,Jvn-(Jxn-λnAxn)〉≥0,
and hence
(3.53)〈v-vn,Jxn-Jvnλn-Axn〉≤0.
So, from (3.51), (3.53), and A being 1/γ-Lipschitz continuous, we obtain
(3.54)〈v-vn,w〉≥〈v-vn,Av〉≥〈v-vn,Av〉+〈v-vn,Jxn-Jvnλn-Axn〉=〈v-vn,Av-Axn+Jxn-Jvnλn〉≥〈v-vn,Av-Avn〉+〈v-vn,Avn-Axn〉+〈v-vn,Jxn-Jvnλn〉≥-‖v-vn‖‖Avn-Axn‖-‖v-vn‖‖Jxn-Jvna‖≥-1γ‖v-vn‖‖vn-xn‖-‖v-vn‖‖Jxn-Jvna‖.
Since J is uniformly continuous on bounded sets, by (3.36) and replacing n by nk in (3.54), as k→∞ we have 〈v-x*,w〉≥0. Thus, x*∈S-1(0), and hence x*∈VI(C,A).
Next we show thatx*∈GMEP=F(Kr). Let H(un,y)=Θ(un,y)+〈Bun,y-un〉+φ(y)-φ(un), forally∈C. It follows from (3.2) that(3.55)Jyn=α̃nJx0+(1-α̃n)(β̃nJzn+(1-β̃n)JTnJ̃rnzn),
hence, from (3.22) and condition (i), we have
(3.56)‖Jyn-Jzn‖≤α̃n‖Jx0-Jzn‖+(1-α̃n)(1-β̃n)‖JTnJ̃rnzn-Jzn‖⟶0(n⟶∞),
which implies that
(3.57)‖yn-zn‖⟶0(n⟶∞).
From (3.17), (3.31), and (3.57), we obtain
(3.58)‖yn-un‖⟶0(n⟶∞).
Thus, unk⇀x*,ynk⇀x*(k→∞). Since J is uniformly continuous on bounded sets, from (3.58) we have limk→∞∥Junk-Jynk∥=0. Therefore, it follows from tnk≥c* that ∥Junk-Jynk∥/tnk→0(n→∞). Since un=Ktnyn, we have
(3.59)H(un,y)+1tn〈y-un,Jun-Jyn〉≥0,∀y∈C.
Combining the above inequality, (A2) and (A4), we get
(3.60)‖y-un‖‖Jun-Jyn‖tn≥1tn〈y-un,Jun-Jyn〉≥-H(un,y)≥H(y,un),∀y∈C.
Replacing n by nk and taking the limit as k→∞ in the above inequality and by (A4), we have H(y,x*)≤0,forally∈C. For any t∈(0,1) and y∈C, define yt=ty+(1-t)x*∈C. So H(yt,x*)≤0. From (A1) and (A4), we have
(3.61)0=H(yt,yt)≤tH(yt,y)+(1-t)H(yt,x*)≤tH(yt,y),
that is, H(yt,y)≥0. Thus, from (A3), let t→0, we have H(x*,y)≥0,forally∈C, which implies that x*∈GMEP.
From (3.31), (3.36), (3.41), and (3.47), similar to the proof of Step  5 in Theorem  3.1 of [19], we have x*∈T-10∩T̃-10. Therefore, ωw({xn})⊂Ω.Step 4.
Finally we prove that{xn} converges strongly to x̃=ΠΩx0.
It follows from Step  6 in Theorem  3.1 of [19] that we have the conclusion. This completes the proof.IfA≡0, then we have the following result from Theorem 3.1.Corollary 3.2.
LetE be a real uniformly smooth and uniformly convex Banach space and C a nonempty, closed, and convex subset of E. Let B:C→E* be a monotone continuous mapping. Let {Tn},{Sn} be two countable families of relatively quasi-nonexpansive mappings from C into itself satisfying NST-conditions such that Ω:=(∩n=0∞F(Tn))∩(∩n=0∞F(Sn))∩GMEP∩T-10∩T̃-10≠∅. Let {tn}⊂[c*,+∞) for some c*>0 and {rn}⊂(0,+∞) satisfy liminfn→∞rn>0. Let {xn} be the sequence generated by
(3.62)x0∈C,chosenarbitrarily,zn=J-1(αnJx0+(1-αn)(βnJxn+(1-βn)JSnJrnxn)),yn=J-1(α̃nJx0+(1-α̃n)(β̃nJzn+(1-β̃n)JTnJ̃rnzn)),un=Ktnyn,Hn={z∈C:ϕ(z,un)≤(α̃n+(1-α̃n)αn)ϕ(z,x0)+(1-α̃n)(1-αn)ϕ(z,xn)},Wn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠHn∩Wnx0,n≥0,
where J is the normalized duality mapping, {αn},{βn},{α̃n}, and {β̃n} are four sequences in [0,1], and Kr is defined by (*). The following conditions hold:(i)
limn→∞αn=limn→∞α̃n=0;(ii)
liminfn→∞βn(1-βn)>0;(iii)
liminfn→∞β̃n(1-β̃n)>0.Then {xn} converges strongly to ΠΩx0.Proof.
From the proof of Theorem3.1, (2.4) is used in proving (3.9), (3.34), and (3.36). Since A≡0,vn=xn for all n≥0, so uniformly smooth and 2-uniformly convex Banach space E can be weakened to uniformly smooth and uniformly convex Banach space. Thus, from Theorem 3.1, the proof is completed.Remark 3.3.
IfSn=Tn=I,rn=tn for all n≥0, and φ=0, then Corollary 3.2 reduces to Theorem  3.1 in [19].



---

*Source: 712651-2011-12-08.xml*