# A Hybrid Iterative Scheme for a Maximal Monotone Operator and Two Countable Families of Relatively Quasi-Nonexpansive Mappings for Generalized Mixed Equilibrium and Variational Inequality Problems

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

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## 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 for anα-inverse-strongly monotone operator, the set of solutions of the
generalized mixed equilibrium problem and zeros of a maximal monotone operator 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. The results presented in this paper improve
and extend some recent results.


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## Body

## 1. Introduction

LetE be a Banach space with norm ∥·∥, C  a nonempty closed convex subset of E, and let E* denote the dual of E. Let θ:C×C→ℝ be a bifunction, φ:C→ℝ be a real-valued function, and B:C→E* a mapping. The generalized mixed equilibrium problem, is to find x∈C such that
(1.1)θ(x,y)+〈Bx,y-x〉+φ(y)-φ(x)≥0,∀y∈C.
The set of solutions to (1.1) is denoted by GMEP(θ,B,φ), that is,
(1.2)GMEP(θ,B,φ)={x∈C:θ(x,y)+〈Bx,y-x〉+φ(y)-φ(x)≥0,∀y∈C}.
If B≡0, the problem (1.1) reduces into the mixed equilibrium problem forθ, denoted by MEP(θ,φ), which  is to find x∈C such that(1.3)θ(x,y)+φ(y)-φ(x)≥0,∀y∈C.
If θ≡0, the problem (1.1) reduces into the mixed variational inequality of Browder type, denoted by VI(C,B,φ), which  is to find x∈C such that(1.4)〈Bx,y-x〉+φ(y)-φ(x)≥0,∀y∈C.
If B≡0 and φ≡0 the problem (1.1) reduces into the equilibrium problem forθ, denoted by EP(θ), which  is to find x∈C such that(1.5)θ(x,y)≥0,∀y∈C.
If θ≡0, the problem (1.3) reduces into the minimize problem, denoted by Argmin(φ), is to find x∈C such that(1.6)φ(y)-φ(x)≥0,∀y∈C.
The above formulation (1.4) was shown in [1] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, 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, [1, 2] and references therein. Some solution methods have been proposed to solve the EP(θ); see, for example, [1, 3–11] and references therein.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 S(E)={x∈E:∥x∥=1} be the unit sphere of E. Then a Banach space E is said to be smooth if the limit (1.7)limt→0∥x+ty∥-∥x∥t
exists for each x,y∈S(E). It is also said to be uniformly smooth if the limit exists uniformly for x,y∈S(E). Let E be a Banach space. The modulus of convexity of E is the function δ:[0,2]→[0,1] defined by (1.8)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 [12, 13] for more details. Observe that every p-uniformly convex is uniformly convex. One should note that no Banach space is p-uniformly 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 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.A set valued mappingT:E⇉E* with graph G(T)={(x,x*):x*∈Tx}, domain D(T)={x∈E:Tx≠∅}, and rang R(T)=∪{Tx:x∈D(T)}. T is said to be monotone if 〈x*-y*,x-y〉≥0, whenever x*∈Tx,y*∈Ty. A monotone operator T is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone operator on the same space. We know that if T is maximal monotone, then the solution set T-10={x∈D(T):0∈Tx} is closed and convex. It is knows that T is a maximal monotone if and only if R(J+rT)=E* for all r>0 when E is a reflexive, strictly convex and smooth Banach space (see Rockafellar [14]).LetE be a smooth, strictly convex and reflexive Banach space, let C be a nonempty closed convex subset of E and let T:E⇉E* be a monotone operator satisfying D(T)⊂C⊂J-1(⋂r>0R(J+rT)). Then we define the resolventT by Jrx={z∈D(T):Jx∈Jz+rTz},  for all x∈E. In other words, Jr=(J+rT)-1J for all r>0. Jr is a single-valued mapping from E to D(T). Also, we know that T-10=F(Jr) for all r>0, where F(Jr) is the set of all fixed points of Jr. We can define, for r>0, the Yosida approximation of T by Trx=(Jx-JJrx)/r for all x∈E. We know that Trx∈T(Jrx) for all r>0 and x∈E.It is also known that ifE is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. We consider the problem of finding:(1.9)v∈Esuchthat0∈Tv,
where T is an operator from E into E*. Such v∈E is called a zero point of T. Such a problem contains numerous problems in economics, optimization and physics. When T is a maximal monotone operator, a well-know method for solving (1.9) in a Hilbert space H is the proximal point algorithm: x1=x∈H and,(1.10)xn+1=Jrnxn,n=1,2,3,…,
where {rn}⊂(0,∞) and Jrn=(I+rnT)-1, then Rockafellar [15] proved that the sequence {xn} converges weakly to an element of T-10. Let E be a real Banach space and let C be a nonempty closed convex subset of E and A:C→E* be an operator. The classical variational inequality problem for an operator A is to find x¯∈C such that(1.11)〈Ax¯,y-x¯〉≥0,∀y∈C.
The set of solution of (1.11) is denote by VI(A,C). Recall that let A:C→E* be a mapping. Then A is called (i)
monotone if
(1.12)〈Ax-Ay,x-y〉≥0,∀x,y∈C,(ii)
α-inverse-strongly monotone if there exists a constant α>0 such that
(1.13)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C.The class of inverse-strongly monotone mappings has been studied by many researchers to approximating a common fixed point; see [6, 7, 16, 17] for more details.LetC be a closed convex subset of E, a mapping S:C→C is said to be nonexpansive if ∥Sx-Sy∥≤∥x-y∥, for all x,y∈C. A point x∈C is a fixed pointof S provided Sx=x. Denote by F(S) the set of fixed points of S; that is, F(S)={x∈C:Sx=x}.Consider the functional defined by(1.14)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,forx,y∈E.
Recall that a point p in C is said to be an asymptotic fixed point of S [18] 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 [19–21] 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 [22–24]. 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 (or quasi-ϕ-nonexpansive) if F(S)≠∅ and ϕ(p,Sx)≤ϕ(p,x) for x∈C and p∈F(S). We note that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [22–26] which requires the strong restriction: F(S)=F(S)̃.LetC be a nonempty closed 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 [27] recently introduced a generalized projectionΠC from E in to C as follows:(1.15)ΠC(x)=argminy∈Cϕ(y,x),∀x∈E.
It is obvious from the definition of function ϕ that(1.16)(∥y∥-∥x∥)2≤ϕ(y,x)≤(∥y∥+∥x∥)2,∀x,y∈E.
If E is a Hilbert space, then ϕ(y,x)=∥y-x∥2 and ΠC becomes the metric projection of E onto C. Let ΠC be the generalized projection from a smooth, strictly convex and reflexive Banach space E onto a nonempty closed convex subset C of E. Then, ΠC is a closed relatively quasi-nonexpansive mapping from E onto C with F(ΠC)=C. On the author hand, the generalized projection ΠC: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
(1.17)ϕ(x¯,x)=infy∈Cϕ(y,x).
The existence and uniqueness of the operator ΠC follows from the properties of the functional ϕ(y,x) and strict monotonicity of the mapping J (see, e.g., [27–31]).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 (1.14), 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 [29, 31] for more details.In 2004, Matsushita and Takahashi [32] introduced the following iteration: a sequence {xn} defined by(1.18)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 [25] 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:(1.19)x0∈C,chosenarbitrarily,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=ΠCn∩Qnx0.They proved that{xn} converges strongly to ΠF(T)x0, where ΠF(T) is the generalized projection from C onto F(T). In 2008, Iiduka and Takahashi [33] introduced the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly monotone operator A in a 2-uniformly convex and uniformly smooth Banach space E: x1=x∈C and(1.20)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 (1.20) converges weakly to some element of VI(A,C).Recently, Takahashi and Zembayashi [34, 35], studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of Banach spaces. In 2008, Cholamjiak [36], proved the following iteration(1.21)zn=ΠCJ-1(Jxn-λnAxn),yn=J-1(αnJxn+βnJTxn+γnJSzn),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+1x0,
where J is the duality mapping on E. Assume that {αn},{βn} and {γn} are sequences in [0,1] such that αn+βn+γn=1, liminfn→∞αnβn>0 and liminfn→∞αnγn>0. Then {xn} converges strongly to q=ΠFx0, where F:=F(T)∩F(S)∩EP(θ)∩VI(A,C). In 2009, Wei et al. [37] proved the following iteration for two relatively nonexpansive mappings in a Banach space E:(1.22)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=ΠHn∩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 (1.22) converges strongly to a point Πℱ:=F(T)∩F(S)x0. Where the mapping Πℱ of E onto ℱ is the generalized projection operator. Inoue et al. [38] proved strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping by using the hybrid method. After that, Klin-eam et al. [2], extend Inoue et al. [38] to obtain the strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of two relatively nonexpansive mappings in a Banach space by using a new hybrid method.On the other hand, Nakajo et al. [39] introduced the following condition. Let C be a nonempty closed convex subset of a Hilbert space H, let {Sn} be a family of mappings of C into itself with F:=⋂n=1∞F(Sn)≠∅ and ωw(zn) denotes the set of all weak subsequential limits of a bounded sequence {zn} in C. {Sn} is said to satisfy the NST-condition if for every bounded sequence {zn} in C, (1.23)limn→∞∥zn-Snzn∥=0impliesthatωw(zn)⊂F.Recall that a mappingS:C→C is closed if for each {xn} in C, if xn→x and Sxn→y, then Sx=y. Let {Sn} be a family of mappings of C in to itself with F:=⋂n=1∞F(Sn)≠∅, {Sn} is said to satisfy the (*)-condition if for each bounded sequence {zn} in C,(1.24)limn→∞∥zn-Snzn∥=0,zn→zimplyz∈F.
It follows directly from the definitions above that if {Sn} satisfies NST-condition, then {Sn} satisfies (*)-condition. If Sn≡S and S is closed, then {Sn} satisfies (*)-condition (see [40] for more details).In this paper, we introduce a new hybrid projection method for finding a common solution of the set of common fixed points of two countable families of relatively quasi nonexpansive mappings, the set of the variational inequality for anα-inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in a real uniformly smooth and 2-uniformly convex Banach space.


## 2. Preliminaries

We also need the following lemmas for the proof of our main results.Lemma 2.1 (Xu [41]).
IfE be a 2-uniformly convex Banach space and 0<c≤1, then for all x,y∈E, one has
(2.1)∥x-y∥≤2c2∥Jx-Jy∥,
where J is the normalized duality mapping of E.The best constant1/c in lemma is called the 2-uniformly convex constant of E.Lemma 2.2 (Chidume [42, Corollary 4.17, pages 36-37]).
IfE be a p-uniformly convex Banach space and let p be a given real number with p≥2, then for all x,y∈E,  jx∈Jpx and jy∈Jpy(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 [30]).
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 [27]).
LetC be a nonempty closed 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 [27]).
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
(2.4)ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x),∀y∈C.Lemma 2.6 (Qin et al. [9]).
LetE be a real uniformly smooth and strictly convex Banach space, and C be 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.LetE be a reflexive, strictly convex, smooth Banach space and J the duality mapping from E into E*. Then J-1 is also single valued, 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 [27](2.5)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-1x*).Lemma 2.7 (Kohsaka and Takahashi [43, Lemma 3.2]).
LetE be a reflexive, strictly convex smooth Banach space and let V be as in (2.5). Then
(2.6)V(x,x*)+2〈J-1x*-x,y*〉≤V(x,x*+y*),
for all x∈E and x*,y*∈E*.Lemma 2.8 (Kohsaka and Takahashi [44]).
LetE be a smooth, strictly convex and reflexive Banach space, let C be a nonempty closed convex subset of E and let T:E⇉E* be a monotone operator satisfying D(T)⊂C⊂J-1(⋂r>0R(J+rT)). Let r>0, let Jr and Tr be the resolvent and the Yosida approximation of A, respectively. Then the following hold:  (i)
ϕ(u,Jrx)+ϕ(Jrx,x)≤ϕ(u,x), for all x∈C,u∈T-10;(ii)
(Jrx,Trx)∈T, for all x∈C;(iii)
F(Jr)=T-10.LetC be a nonempty closed convex subset of a Banach space E and let A 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,(2.7)NC(v)={x*∈E*:〈v-y,x*〉≥0,∀y∈C}.Lemma 2.9 (Rockafellar [14]).
LetC be a nonempty, closed convex subset of a Banach space E and A a monotone, hemicontinuous operator of C into E*. Let U:E⇉E* be an operator defined as follows:
(2.8)Uv={Av+NC(v),v∈C,∅,otherwise.
Then U is maximal monotone and U-10=VI(A,C).For solving the equilibrium problem for a bifunctionθ:C×C→ℝ, let us assume that θ 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,
(2.9)limt↓0θ(tz+(1-t)x,y)≤θ(x,y);(A4)
for eachx∈C, y↦θ(x,y) is convex and lower semicontinuous.For example, letB be a continuous and monotone operator of C into E* and define (2.10)θ(x,y)=〈Bx,y-x〉,∀x,y∈C.
Then, θ satisfies (A1)–(A4).The following result is in Takahashi and Zembayashi ([34, 35]).Lemma 2.10 (see [34, 35]).
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
(2.11)θ(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Motivated by Combettes and Hirstoaga [4] in a Hilbert space and Takahashi and Zembayashi [34] in a Banach space, Zhang [45] obtain the following lemma.Lemma 2.11 (Zhang [45, Lemma 1.5]).
LetC be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. Let B:C→E* be a continuous and monotone mapping, 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
(2.12)θ(u,y)+〈Bu,y-u〉+φ(y)-φ(u)+1r〈y-u,Ju-Jx〉≥0,∀y∈C.
Define a mapping Kr:C→C as follows:
(2.13)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∈C. 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)=F(Kr)̃=GMEP(θ,B,φ);(4)
GMEP(θ,B,φ) is closed and convex;(5)
ϕ(p,Krz)+ϕ(Krz,z)≤ϕ(p,z), for all p∈F(Kr) and z∈E.


## 3. Main Results

In this section, by using the(*)-condition, we prove the new convergence theorems for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of two countable families of relatively quasi-nonexpansive mappings, zeros of maximal monotone operators and the solution set of variational inequalities for an α-inverse strongly monotone mapping 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 T:E⇉E* be a maximal monotone operator satisfying D(T)⊂C and let Jr=(J+rT)-1J for all r>0, where J is the duality mapping on E. Let θ be a bifunction from C×C to ℝ satisfying (A1)–( A4), and let φ:C→ℝ be a proper lower semicontinuous and convex function. Let A be an α-inverse-strongly monotone mapping of C into E* satisfy ∥Ay∥≤∥Ay-Au∥, for all y∈C and u∈VI(A,C)≠∅ and let B:C→E* be a continuous and monotone mapping. Let Sn,Tn:C→C be two families of relatively quasi-nonexpansive mappings with satisfy the (*)-condition such that
(3.1)Θ:=(⋂n=1∞F(Sn))∩(⋂n=1∞F(Tn))∩T-10∩GMEP(θ,B,φ)∩VI(A,C)≠∅.
For an initial point x0∈E with x1=ΠC1x0 and C1=C, we define the sequence {xn} as follows:
(3.2)wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JTnJrnwn),yn=J-1(αnJxn+(1-αn)JSnzn),un∈Csuchthatθ(un,y)+〈Bun,y-un〉+φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,zn)≤ϕ(z,xn)},xn+1=ΠCn+1x0,∀n≥1,
where {αn},{βn} are sequences in [0,1] and {rn}⊂[d,∞) for some d>0 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. If liminfn→∞(1-αn)>0 and liminfn→∞(1-βn)>0, then {xn} converges strongly to p∈Θ, where p=ΠΘx0.Proof.
We split the proof into seven steps.Step 1.
We first show thatCn+1 is closed and convex for each n≥1.
By Lemma2.6, we know that (⋂n=1∞F(Sn))∩(⋂n=1∞F(Tn)) is closed and convex. We also know that if T-10 and VI(A,C) are closed and convex. From Lemma 2.11 (4), we have GMEP(θ,B,φ) is closed and convex. Hence Θ:=(⋂n=1∞F(Sn))∩(⋂n=1∞F(Tn))∩T-10∩GMEP(θ,B,φ)∩VI(A,C) is a nonempty, closed and convex subset of C. Consequently, ΠΘx0 is well defined.
Next, we prove thatCn is closed and convex for each n≥1. It is obvious that C1=C is closed and convex. Suppose that Cn is closed and convex for each n∈ℕ. Since for any z∈Cn, we know 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.Step 2.
We show thatΘ⊂Cn for all n≥1.
Next, we show by induction thatΘ⊂Cn for all n∈ℕ. Indeed, let un=Krnyn and vn=Jrnwn for all n≥1. On the other hand, from Lemma 2.11 one has Krn is relatively quasi-nonexpansive mapping and Θ⊂C1=C. Suppose that Θ⊂Cn for some n≥1. Let q∈Θ⊂Cn. Since Sn is relatively quasi-nonexpansive mapping, we have(3.4)ϕ(q,yn)=ϕ(q,J-1(αnJxn+(1-αn)JSnzn))=∥q∥2-2〈q,αnJxn+(1-αn)JSnzn〉+∥αnJxn+(1-αn)JSnzn∥2≤∥q∥2-2αn〈q,Jxn〉-2(1-αn)〈q,JSnzn〉+αn∥xn∥2+(1-αn)∥Snzn∥2=αnϕ(q,xn)+(1-αn)ϕ(q,Snzn)≤αnϕ(q,xn)+(1-αn)ϕ(q,zn),
by nonexpansiveness of Jrn (see [31, Theorem 4.6.3, page 130]) and Tn is relatively quasi-nonexpansive mappings, we also have
(3.5)ϕ(q,zn)=ϕ(q,J-1(βnJxn+(1-βn)JTnvn))=∥q∥2-2〈q,βnJxn+(1-βn)JTnvn〉+∥βnJxn+(1-βn)JTnvn∥2≤∥q∥2-2βn〈q,Jxn〉-2(1-βn)〈q,JTnvn〉+βn∥xn∥2+(1-βn)∥Tnvn∥2=βnϕ(q,xn)+(1-βn)ϕ(q,Tnvn)≤βnϕ(q,xn)+(1-βn)ϕ(q,vn)=βnϕ(q,xn)+(1-βn)ϕ(q,Jrnwn)≤βnϕ(q,xn)+(1-βn)ϕ(q,wn).
So, it follows that
(3.6)ϕ(q,un)=ϕ(q,Krnyn)≤ϕ(q,yn)≤αnϕ(q,xn)+(1-αn)ϕ(q,zn),≤αnϕ(q,xn)+(1-αn)[βnϕ(q,xn)+(1-βn)ϕ(q,wn)].
It follows from Lemmas 2.5 and 2.7, that
(3.7)ϕ(q,wn)=ϕ(q,ΠCJ-1(Jxn-λnAxn))≤ϕ(q,J-1(Jxn-λnAxn))=V(q,Jxn-λnAxn)≤V(q,(Jxn-λnAxn)+λnAxn)-2〈J-1(Jxn-λnAxn)-q,λnAxn〉=V(q,Jxn)-2λn〈J-1(Jxn-λnAxn)-q,Axn〉=ϕ(q,xn)-2λn〈xn-q,Axn〉+2〈J-1(Jxn-λnAxn)-xn,-λnAxn〉.
Thus, since q∈VI(A,C) and A is α-inverse-strongly monotone, we have
(3.8)-2λn〈xn-q,Axn〉=-2λn〈xn-q,Axn-Aq〉-2λn〈xn-q,Aq〉≤-2λn〈xn-q,Axn-Aq〉=-2αλn∥Axn-Aq∥2.
By Lemma 2.1 and the fact that ∥Ay∥≤∥Ay-Au∥ for all y∈C and u∈Θ, we obtain
(3.9)2〈J-1(Jxn-λnAxn)-xn,-λnAxn〉=2〈J-1(Jxn-λnAxn)-J-1(Jxn),-λnAxn〉≤2∥J-1(Jxn-λnAxn)-J-1(Jxn)∥∥λnAxn∥≤4c2∥JJ-1(Jxn-λnAxn)-JJ-1(Jxn)∥∥λnAxn∥=4c2∥Jxn-λnAxn-Jxn∥∥λnAxn∥=4c2∥λnAxn∥2=4c2λn2∥Axn∥2≤4c2λn2∥Axn-Aq∥2.
Substituting (3.8) and (3.9) into (3.7), we have
(3.10)ϕ(q,wn)≤ϕ(q,xn)-2αλn∥Axn-Aq∥2+4c2λn2∥Axn-Aq∥2=ϕ(q,xn)+2λn(2c2λn-α)∥Axn-Aq∥2≤ϕ(q,xn).
Substituting (3.10) into (3.6), we get
(3.11)ϕ(q,un)≤αnϕ(q,xn)+(1-αn)[βnϕ(q,xn)+(1-βn)ϕ(q,xn)]=ϕ(q,xn).
This shows thatq∈Cn+1 which implies that Θ⊂Cn+1 and hence, Θ⊂Cn for all n≥1. This implies that the sequence {xn} is well defined.
Step 3.
We prove that{xn} is bounded.
Sincexn=ΠCnx0 and xn+1=ΠCn+1x0⊂Cn+1⊂Cn, we have(3.12)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n≥1.
By Lemma 2.5, we get
(3.13)ϕ(xn,x0)=ϕ(ΠCn(x0),x0)≤ϕ(p,x0)-ϕ(p,xn)≤ϕ(p,x0),∀p∈Θ.
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 (1.16), the sequence {(∥xn∥-∥x0∥)2} is bounded. This implies {xn} is also bounded. So, we have {un}, {zn}, {yn} and {wn} are bounded.Step 4.
We show that{xn} is a Cauchy sequence in C. Since xm=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. From Lemma 2.3, we get ∥xn-xm∥→0. Hence {xn} is a Cauchy sequence and by the completeness of E and the closedness of C, we can assume that there exists p∈C such that xn→p∈C as n→∞.Step 5.
We show that∥Jun-Jxn∥→0, as n→∞. We taking m=n+1 in Step 4, we also have
(3.15)limn→∞ϕ(xn+1,xn)=0.
From Lemma 2.3, that
(3.16)limn→∞∥xn+1-xn∥=0.
Since J is uniformly norm-to-norm continuous on bounded subsets of E, we have
(3.17)limn→∞∥Jxn+1-Jxn∥=0.
Since xn+1=ΠCn+1x0∈Cn+1⊂Cn and the definition of Cn+1, we have
(3.18)ϕ(xn+1,un)≤ϕ(xn+1,xn)∀n∈ℕ.
By (3.15), we obtain
(3.19)limn→∞ϕ(xn+1,un)=0.
Again applying Lemma 2.3, we get
(3.20)limn→∞∥xn+1-un∥=0.
From
(3.21)∥un-xn∥=∥un-xn+1+xn+1-xn∥≤∥un-xn+1∥+∥xn+1-xn∥
It follows that
(3.22)limn→∞∥un-xn∥=0.
Since J is uniformly norm-to-norm continuous on bounded subsets of E, we also have
(3.23)limn→∞∥Jun-Jxn∥=0.Step 6.
We will show thatxn→p∈Θ, where
(3.24)Θ:=(⋂n=1∞F(Tn))∩(⋂n=1∞F(Sn))∩GMEP(θ,B,φ)∩VI(A,C)∩T-10(a)
We show thatxn→p∈(⋂n=1∞F(Tn))∩(⋂n=1∞F(Sn)). From definition of Cn+1, for any z∈Cn, we have(3.25)ϕ(z,zn)≤ϕ(z,xn).
Since xn+1=ΠCn+1x0∈Cn+1, we get ϕ(xn+1,zn)≤ϕ(xn+1,xn). It follows from (3.15), that
(3.26)limn→∞ϕ(xn+1,zn)=0
again from Lemma 2.3, that
(3.27)limn→∞∥xn+1-zn∥=0
it follows that since J is uniformly norm-to-norm continuous, we also have
(3.28)limn→∞∥Jxn+1-Jzn∥=0.
Since
(3.29)∥zn-xn∥≤∥zn-xn+1∥+∥xn+1-xn∥
from (3.16) and (3.27), we also have
(3.30)limn→∞∥zn-xn∥=0.
Since J is uniformly norm-to-norm continuous, we obtain
(3.31)limn→∞∥Jzn-Jxn∥=0.
From (3.4), (3.5) and (3.10), we get ϕ(p,yn)≤ϕ(p,xn). By Lemma 2.11 (5) and un=Krnyn, we observe that
(3.32)ϕ(un,yn)=ϕ(Krnyn,yn)≤ϕ(p,yn)-ϕ(p,Krnyn)≤ϕ(p,xn)-ϕ(p,Krnyn)=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∥.
Since {xn}, {yn} and {un} are bounded, it follows from (3.22), (3.23), and Lemma 2.3, we also have
(3.33)limn→∞∥un-yn∥=0.
Since J is uniformly norm-to-norm continuous, we have
(3.34)limn→∞∥Jun-Jyn∥=0.
By using the triangle inequality, we obtain
(3.35)∥xn+1-yn∥=∥xn+1-un+un-yn∥≤∥xn+1-un∥+∥un-yn∥.
By (3.20) and (3.33), we get
(3.36)limn→∞∥xn+1-yn∥=0.
Since J is uniformly norm-to-norm continuous, we obtain
(3.37)limn→∞∥Jxn+1-Jyn∥=0.
Since
(3.38)∥yn-zn∥≤∥yn-xn+1∥+∥xn+1-zn∥.
From (3.27) and (3.36), we have
(3.39)limn→∞∥yn-zn∥=0.
Since J is uniformly norm-to-norm continuous, we also have
(3.40)limn→∞∥Jyn-Jzn∥=0.
From (3.2), we get
(3.41)∥Jyn-Jzn∥=∥αn(Jxn-Jzn)+(1-αn)(JSnzn-Jzn)∥=∥(1-αn)(JSnzn-Jzn)-αn(Jzn-Jxn)∥≥(1-αn)∥JSnzn-Jzn∥-αn∥Jzn-Jxn∥,
and hence
(3.42)(1-αn)∥JSnzn-Jzn∥≤∥Jyn-Jzn∥+αn∥Jzn-Jxn∥,
it follows that
(3.43)∥JSnzn-Jzn∥≤11-αn(∥Jyn-Jzn∥+αn∥Jzn-Jxn∥).
Since liminfn→∞(1-αn)>0, (3.31) and (3.40), one has limn→∞∥JSnzn-Jzn∥=0. Since J-1 is uniformly norm-to-norm continuous, we get
(3.44)limn→∞∥Snzn-zn∥=0.
Since ∥xn-zn∥→0 and xn→p, then we get zn→p, hence it follows from (*)-condition, that p∈⋂n=1∞F(Sn).
Sincevn=Jrnwn, we compute (3.45)∥Jxn+1-Jzn∥=∥Jxn+1-(βnJxn+(1-βn)JTnvn)∥=∥βnJxn+1-βnJxn+(1-βn)Jxn+1-(1-βn)JTnvn∥=∥βn(Jxn+1-Jxn)+(1-βn)(Jxn+1-JTnvn)∥=∥(1-βn)(Jxn+1-JTnvn)-βn(Jxn-Jxn+1)∥≥(1-βn)∥Jxn+1-JTnvn∥-βn∥Jxn-Jxn+1∥
and hence
(3.46)∥Jxn+1-JTnvn∥≤11-βn(∥Jxn+1-Jzn∥+βn∥Jxn-Jxn+1∥).
From (3.17), (3.28) and liminfn→∞(1-βn)>0, we obtain that
(3.47)limn→∞∥Jxn+1-JTnvn∥=0.
Since J-1 is uniformly norm-to-norm continuous on bounded sets, we have
(3.48)limn→∞∥xn+1-Tnvn∥=0.
Using the triangle inequality, we have
(3.49)∥xn-Tnvn∥=∥xn-xn+1+xn+1-Tnvn∥≤∥xn-xn+1∥+∥xn+1-Tnvn∥.
From (3.16) and (3.48), we have
(3.50)limn→∞∥xn-Tnvn∥=0.
On the other hand, for q∈Θ, we note that
(3.51)ϕ(q,xn)-ϕ(q,un)=∥xn∥2-∥un∥2-2〈q,Jxn-Jun〉≤∥xn-un∥(∥xn∥+∥un∥)+2∥q∥∥Jxn-Jun∥.
Since {xn} and {un} are bounded, it follows from ∥xn-un∥→0 and ∥Jxn-Jun∥→0, that
(3.52)ϕ(q,xn)-ϕ(q,un)→0.
Furthermore, from (3.4), (3.5), (3.6) and (3.10), that
(3.53)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
(3.54)2a(α-2bc2)∥Axn-Aq∥2≤2λn(α-2c2λn)∥Axn-Aq∥2≤1(1-αn)(1-βn)(ϕ(q,xn)-ϕ(q,un)).
From (3.52), liminfn→∞(1-αn)>0 and liminfn→∞(1-βn)>0, obtain that
(3.55)limn→∞∥Axn-Aq∥=0.
From Lemmas 2.5, 2.7, and (3.9), we compute
(3.56)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 Lemma2.3 and (3.55) it follows that(3.57)limn→∞∥xn-wn∥=0.
Since J is uniformly norm-to-norm continuous, we also have
(3.58)limn→∞∥Jxn-Jwn∥=0.
Again by the triangle inequality, we get
(3.59)∥wn-Tnvn∥=∥wn-xn+xn-Tnvn∥≤∥wn-xn∥+∥xn-Tnvn∥.
From (3.50) and (3.57), we have
(3.60)limn→∞∥wn-Tnvn∥=0.
From (3.5), we have ϕ(q,vn)≥(1/(1-βn))(ϕ(q,zn)-βnϕ(q,xn)), it follows from Lemma 2.8 and (3.10), we note that
(3.61)ϕ(vn,wn)=ϕ(Jrnwn,wn)≤ϕ(q,wn)-ϕ(q,Jrnwn)=ϕ(q,wn)-ϕ(q,vn)≤ϕ(q,wn)-11-βn(ϕ(q,zn)-βnϕ(q,xn))≤ϕ(q,xn)-11-βn(ϕ(q,zn)-βnϕ(q,xn))=11-βn(ϕ(q,xn)-ϕ(q,zn))=11-βn(∥xn∥2-∥zn∥2-2〈q,Jxn-Jzn〉)≤11-βn(∥xn∥2-∥zn∥2+2|〈q,Jxn-Jzn〉|)≤11-βn(|∥xn∥-∥zn∥|(∥xn∥+∥zn∥)+2∥q∥∥Jxn-Jzn∥)≤11-βn(∥xn-zn∥(∥xn∥+∥zn∥)+2∥q∥∥Jxn-Jzn∥).
It follows from liminfn→∞(1-βn)>0, (3.30) and (3.31), we get
(3.62)limn→∞ϕ(vn,wn)=0.
From Lemma 2.3, it follows that
(3.63)limn→∞∥vn-wn∥=0.
By using the triangle inequality, we get
(3.64)∥vn-Tnvn∥=∥vn-wn+wn-Tnvn∥≤∥vn-wn∥+∥wn-Tnvn∥.
From (3.60) and (3.63), we have
(3.65)limn→∞∥vn-Tnvn∥=0.
Since ∥xn-vn∥≤∥xn-wn∥+∥wn-vn∥, (3.57) and (3.63), then
(3.66)limn→∞∥xn-vn∥=0.
From (3.66) and since xn→p, then vn→p. By (3.60) it follows from (*)-condition, that p∈⋂n=1∞F(Tn). Hence p∈(⋂n=1∞F(Sn))∩(⋂n=1∞F(Tn)).(b)
We show thatxn→p∈GMEP(θ,B,φ). Indeed, it follows from (A2), that(3.67)〈Bun,y-un〉+φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥-θ(un,y)≥θ(y,un),∀y∈C,
and hence
(3.68)〈Bun,y-un〉+φ(y)-φ(un)+〈y-un,(Jun-Jyn)rn〉≥θ(y,un),∀y∈C.
For t with 0<t≤1 and y∈C, let yt=ty+(1-t)p. Then, we get that yt∈C. From (3.68), it follows that
(3.69)〈Byt,yt-un〉≥〈Byt,yt-un〉-〈Bun,yt-un〉-φ(yt)+φ(un)-〈yt-un,(Jun-Jyn)rn〉+θ(yt,un)=〈Byt-Bun,yt-un〉-φ(yt)+φ(un)-〈yt-un,(Jun-Jyn)rn〉+θ(yt,un),∀yt∈C.
By the fact that yn,un→p as n→∞, and ∥Jun-Jyn∥/rn→0 as n→∞. Since B is monotone, we know that 〈Byt-Bun,yt-un〉≥0. Thus, it follows from (A4) that
(3.70)θ(yt,p)-φ(yt)+φ(p)≤liminfn→∞θ(yt,un)-φ(yt)+φ(un)≤limn→∞〈Byt,yt-un〉=〈Byt,yt-p〉.
By the conditions (A1), (A4) and convexity ofφ, we have (3.71)0=θ(yt,yt)+φ(yt)-φ(yt)≤tθ(yt,y)+(1-t)θ(yt,p)+tφ(y)+(1-t)φ(p)-φ(yt)=t[θ(yt,y)+φ(y)-φ(yt)]+(1-t)[θ(yt,p)+φ(p)-φ(yt)]≤t[θ(yt,y)+φ(y)-φ(yt)]+(1-t)[〈Byt,yt-p〉]=t[θ(yt,y)+φ(y)-φ(yt)]+(1-t)t[〈Byt,y-p〉]
and hence
(3.72)0≤θ(yt,y)+φ(y)-φ(yt)+(1-t)〈Byt,y-p〉.
From (A3) and the weakly lower semicontinuity of φ, letting t→0, we also have
(3.73)θ(p,y)+〈Bp,y-p〉+φ(y)-φ(p)≥0,∀y∈C.
This implies that p∈GMEP(θ,B,φ).(c)
We show thatxn→p∈VI(A,C). Indeed, define a set-valued U:E⇉E* by Lemma 2.9, U is maximal monotone and U-10=VI(A,C). Let (v,w)∈G(U). Since w∈Uv=Av+NC(v), we get w-Av∈NC(v).From wn∈C, we have
(3.74)〈v-wn,w-Av〉≥0.
On the other hand, since wn=ΠCJ-1(Jxn-λnAxn). Then by Lemma 2.4, we have
(3.75)〈v-wn,Jwn-(Jxn-λnAxn)〉≥0,
and thus
(3.76)〈v-wn,Jxn-Jwnλn-Axn〉≤0.
It follows from (3.74) and (3.76), that
(3.77)〈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∥. Take the limit as n→∞, (3.57) and (3.58), we obtain 〈v-p,w〉≥0. By the maximality of U, we have p∈U-10 and hence p∈VI(A,C).(d)
We show thatxn→p∈T-10. Since J is uniformly norm-to-norm continuous on bounded sets, from (3.63), we get(3.78)limn→∞∥Jwn-Jvn∥=0.
From rn≥d, we have
(3.79)limn→∞1rn∥Jwn-Jvn∥=0.
Since Jrnwn=vn, therefore,
(3.80)limn→∞∥Trnwn∥=limn→∞1rn∥Jwn-JJrnwn∥=limn→∞1rn∥Jwn-Jvn∥=0.
For (w,w*)∈G(T), from the monotonicity of T, we have 〈w-vn,w*-Trnwn〉≥0 for all n≥0. Letting n→∞, we get 〈w-p,w*〉≥0. From the maximality of T, we have p∈T-10. Hence, from (a), (b), (c) and (d), we obtain p∈Θ.Step 7.
we show thatp=ΠΘx0.
Fromxn=ΠCnx0, we have 〈Jx0-Jxn,xn-z〉≥0, for all z∈Cn. Since Θ⊂Cn, we also have (3.81)〈Jx0-Jxn,xn-y〉≥0,∀y∈Θ.
Taking limit n→∞, we obtain
(3.82)〈Jx0-Jp,p-y〉≥0,∀y∈Θ.
By Lemma 2.4, we can conclude that p=ΠΘx0 and xn→p as n→∞. This completes the proof.Theorem 3.2.
LetC be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let T:E⇉E* be a maximal monotone operator satisfying D(T)⊂C and let Jr=(J+rT)-1J for all r>0, where J is the duality mapping on E. Let θ be a bifunction from C×C to ℝ satisfying (A1)–(A4), and let φ:C→ℝ be a proper lower semicontinuous and convex function. Let A be an α-inverse-strongly monotone mapping of C into E* satisfy ∥Ay∥≤∥Ay-Au∥, for all  y∈C and u∈VI(A,C)≠∅ and let B:C→E* be a continuous and monotone mapping. Let Sn,Tn:C→C be two families of relatively quasi-nonexpansive mappings with satisfy the NST-condition such that Θ:=(⋂n=1∞F(Sn))∩(⋂n=1∞F(Tn))∩T-10∩GMEP(θ,B,φ)∩VI(A,C)≠∅. For an initial point x0∈E with x1=ΠC1x0 and C1=C, we define the sequence {xn} by (3.2) where {αn},{βn} are sequences in [0,1] and {rn}⊂[d,∞) for some d>0 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. If liminfn→∞(1-αn)>0 and liminfn→∞(1-βn)>0, then {xn} converges strongly to p∈Θ, where p=ΠΘx0.Proof.
If{Tn},{Sn} satisfy NST-condition, then {Tn},{Sn} satisfy (*)-condition.SettingSn≡S and Tn≡T in Theorem 3.1, then we obtain the following result.Corollary 3.3.
LetC be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let T̃:E⇉E* be a maximal monotone operator satisfying D(T̃)⊂C and let Jr=(J+rT̃)-1J for all r>0. Let θ be a bifunction from C×C to ℝ satisfying (A1)–(A4), and let φ:C→ℝ be a proper lower semicontinuous and convex function. Let A be an α-inverse-strongly monotone mapping of C into E* satisfying ∥Ay∥≤∥Ay-Au∥, for all  y∈C and u∈VI(A,C)≠∅ and let B:C→E* be a continuous and monotone mapping. Let T,S:C→C be two closed relatively quasi-nonexpansive mappings such that Θ:=F(S)∩F(T)∩T̃-10∩GMEP(θ,B,φ)∩VI(A,C)≠∅. For an initial point x0∈E with x1=ΠC1x0 and C1=C, we define the sequence {xn} as follows:
(3.83)wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JTJrnwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatθ(un,y)+φ(y)-φ(un)+〈Bun,y-un〉+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,zn)≤ϕ(z,xn)},xn+1=ΠCn+1x0,∀n≥1,
where J is the duality mapping on E, {αn},{βn} are sequences in [0,1] and {rn}⊂[d,∞) for some d>0 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. If liminfn→∞(1-αn)>0 and liminfn→∞(1-βn)>0, then {xn} converges strongly to p∈Θ, where p=ΠΘx0.Next, we consider the problem of finding a zero point of an inverse-strongly monotone operator ofE into E*. Assume that A satisfies the conditions:  (C1)
A is α-inverse-strongly monotone,(C2)
A-10={u∈E:Au=0}≠∅.Hence, settingT̃x≡0, for all x∈C in Corollary 3.3, then Jr=I, we also have the following result.Corollary 3.4.
LetE be a 2-uniformly convex and uniformly smooth Banach space. Let θ be a bifunction from E×E to ℝ satisfying (A1)–(A4), and let φ:E→ℝ be a proper lower semicontinuous and convex function. Let A be an operator of E into E* satisfying (C1) and (C2), and let B:E→E* be a continuous and monotone mapping. Let T,S:E→E be two closed relatively quasi-nonexpansive mappings such that
(3.84)Θ:=F(S)∩F(T)∩GMEP(θ,B,φ)∩A-10≠∅.
For an initial point x0∈E with x1=ΠE1x0 and E1=E, we define the sequence {xn} as follows: wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JTwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatθ(un,y)+φ(y)-φ(un)+〈Bun,y-un〉+1rn〈y-un,Jun-Jyn〉≥0,∀y∈E,En+1={z∈En:ϕ(z,un)≤ϕ(z,zn)≤ϕ(z,xn)},xn+1=ΠEn+1x0,∀n≥1, (3.52) where J is the duality mapping on E, {αn},{βn}  are sequences in [0,1] and {rn}⊂[d,∞) for some d>0 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. If liminfn→∞(1-αn)>0 and liminfn→∞(1-βn)>0, then {xn} converges strongly to p∈Θ, where p=ΠΘx0.Proof.
SettingT̃x≡0, for all x∈C, then, D(T̃)=E and hence C=E in Corollary 3.3, we also get ΠE=I. We also have VI(A,C)=VI(A,E)={x∈E:Ax=0}≠∅ and then the condition ∥Ay∥≤∥Ay-Au∥ holds for all y∈E and u∈A-10. So, we obtain the result.SettingA≡0 in Corollary 3.4, then we get wn=xn. Hence we obtain the following Corollary.Corollary 3.5.
LetC be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let θ be a bifunction from C×C to ℝ satisfying (A1)–(A4), and let φ:C→ℝ be a proper lower semicontinuous and convex function. Let B:C→E* be a continuous and monotone mapping. Let T,S:C→C be two closed relatively quasi-nonexpansive mappings such that
(3.86)Θ:=F(S)∩F(T)∩GMEP(θ,B,φ)≠∅.
For an initial point x0∈E with x1=ΠC1x0 and C1=C, we define the sequence {xn} as follows:
(3.87)zn=J-1(βnJxn+(1-βn)JTxn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatθ(un,y)+φ(y)-φ(un)+〈Bun,y-un〉+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,zn)≤ϕ(z,xn)},xn+1=ΠCn+1x0,∀n≥1,
where J is the duality mapping on E, {αn},{βn} are sequences in [0,1] and {rn}⊂[d,∞) for some d>0 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. If liminfn→∞(1-αn)>0 and liminfn→∞(1-βn)>0, then {xn} converges strongly to p∈Θ, where p=ΠΘx0.Remark 3.6.
Theorem3.1, Corollaries 3.4 and 3.5 improve and extend the corresponding results in Cholamjiak [36], Wei et al. [37] and Saewan et al. [26].


## 4. Application to Complementarity Problem

LetK be a nonempty, closed convex cone in E. We define the polarK* of K as follows:(4.1)K*={y*∈E*:〈x,y*〉≥0,∀x∈K}.
If A:K→E* is an operator, then an element u∈K is called a solution of the complementarity problem ([31]) if(4.2)Au∈K*,〈u,Au〉=0.
The set of solutions of the complementarity problem is denoted by C(A,K).Theorem 4.1.
LetK be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let Ã:E⇉E* be a maximal monotone operator satisfying D(Ã)⊂K and let Jr=(J+rÃ)-1J for all r>0. Let θ be a bifunction from K×K to ℝ satisfying (A1)–(A4), and let φ:K→ℝ be a proper lower semicontinuous and convex function. Let A:K→E* be an α-inverse-strongly monotone mapping of E into E* satisfying ∥Ay∥≤∥Ay-Au∥, for all y∈K and u∈C(A,K)≠∅ and let B:K→E* be a continuous and monotone mapping. Let S,T:K→K be two closed relatively quasi-nonexpansive mappings such that Θ:=F(S)∩F(T)∩Ã-10∩GMEP(θ,B,φ)∩C(A,K)≠∅. For an initial point x0∈E with x1=ΠK1x0 and K1=K, we define the sequence {xn} as follows:
(4.3)wn=ΠKJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JTJrnwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Ksuchthatθ(un,y)+φ(y)-φ(un)+〈Bun,y-un〉+1rn〈y-un,Jun-Jyn〉≥0,∀y∈K,Kn+1={z∈Kn:ϕ(z,un)≤ϕ(z,zn)≤ϕ(z,xn)},xn+1=ΠKn+1x0,∀n≥1,
where J is the duality mapping on E, {αn} and {βn} are sequences in [0,1] and {rn}⊂[d,∞) for some d>0 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. If liminfn→∞(1-αn)>0 and liminfn→∞(1-βn)>0, then {xn} converges strongly to p∈Θ, where p=ΠΘx0.Proof.
As in the proof of Takahashi in [31, Lemma 7.11], we get that VI(A,K)=C(A,K). So, we obtain the result.



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*Source: 123027-2010-11-07.xml*