# A New Iterative Scheme for Generalized Mixed Equilibrium, Variational Inequality Problems, and a Zero Point of Maximal Monotone Operators

Authors: Kriengsak Wattanawitoon; Poom Kumam
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The purpose of this paper is to introduce a new iterative scheme for finding a common
element of the set of solutions of generalized mixed equilibrium problems, the set of solutions of
variational inequality problems, the zero point of maximal monotone operators, and the set of two
countable families of quasi-ϕ-nonexpansive mappings in Banach spaces. Moreover, the strong convergence
theorems of this method are established under the suitable conditions of the parameter imposed
on the algorithm. Finally, we apply our results to finding a zero point of inverse-strongly monotone
operators and complementarity problems. Our results presented in this paper improve and extend
the recently results by many others.


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## Body

## 1. Introduction

Equilibrium problem theory is the most important area of mathematical sciences and widely popular among mathematicians and researchers in other fields due to its applications in a wide class of problems which arise in economics, finance, optimization, network and transportation, image reconstruction, ecology, and many others. It has been improved and extended in many directions. Furthermore, equilibrium problems are related to the problem of finding fixed point of nonexpansive mappings. In this way, they have been extensively studied by many authors; see [1–9]. They introduced new iterative schemes for finding a common element of the set of the solutions of equilibrium problems and the set of fixed points. In this paper, we are interested a new hybrid iterative method for finding a common elements of the set of solutions of generalized mixed equilibrium problems, the set of solutions of variational inequality problems, the zero point of maximal monotone operators, and the set of two countable families of quasi-ϕ-nonexpansive mappings in the framework of Banach spaces.LetE be a Banach space with norm ∥·∥ and C a nonempty closed convex subset of E and let E* denote the dual of E.A mappingS:C→C is said to be(1)
nonexpansive [1] if ∥Sx-Sy∥≤∥x-y∥ for all x,y∈C,(2)
relatively nonexpansive [10–12] if F(S)̂=F(S) and ϕ(p,Sx)≤ϕ(p,x) for all x∈C and p∈F(S), where the functional ϕ defined by (2.6). The asymptotic behavior of a relatively nonexpansive mapping was studied in [13, 14],(3)
ϕ-nonexpansive, if ϕ(Sx,Sy)≤ϕ(x,y) for x,y∈C,(4)
quasi-ϕ-nonexpansive if F(S)≠∅ and ϕ(p,Sx)≤ϕ(p,x) for x∈C and p∈F(S).In the sequel, we denoteF(T) as the set of fixed points of S. If C is a bounded closed convex set and S is a nonexpansive mapping of C into itself, then F(S) is nonempty (see [15]).A pointp in C is said to be an asymptotic fixed point of S [16] 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)̂.LetB be an operator from C into E*, and B is said to be α-inverse-strongly monotone if there exists a positive real number α such that
(1.1)〈x-y,Bx-By〉≥α‖Bx-By‖2,∀x,y∈C.If an operatorB is an α-inverse-strongly monotone, then we can said that B is Lipschitz continuous; that is, ∥Bx-By∥≤(1/α)∥x-y∥ for all x,y∈C.Letf:C×C→ℝ be a bifunction, φ:C→ℝ a real-valued function, and B:C→E* be a nonlinear mapping. The generalized mixed equilibrium problem is to find x∈C such that(1.2)f(x,y)+〈Bx,y-x〉+φ(y)-φ(x)≥0,∀y∈C.We denoteΩ as the set of solutions to (1.2) that is,(1.3)Ω={x∈C:f(x,y)+〈Bx,y-x〉+φ(y)-φ(x)≥0,∀y∈C}.IfB≡0, the problem (1.2) reduced into the mixed equilibrium problem forf, denoted by MEP(f,φ), is to find x∈C such that(1.4)f(x,y)+φ(y)-φ(x)≥0,∀y∈C.Iff≡0, the problem (1.2) reduced into the mixed variational inequality of Browder type, denoted by VI(C,B,φ), is to find x∈C such that(1.5)〈Bx,y-x〉+φ(y)-φ(x)≥0,∀y∈C.IfB≡0 and φ≡0 the problem (1.2), reduced into the equilibrium problem forF, denoted by EP(f), is to find x∈C such that(1.6)f(x,y)≥0,∀y∈C.In addition, fixed point problem, optimization problem, and many problems can be written in the form ofEP(f). There are the development of researches in this area as seen in many papers which appeared in the literature on the existence of the solutions of EP(f); see, for example [17–21] and reference therein. Furthermore, there are many solution methods proposed continuously to solve the EP(f) as shown in [2, 3, 18, 20, 22–26] and many others.Next, we letB be a monotone operator of C into E*. The so-called variational inequality problem is to find a point x∈C such that(1.7)〈Bx,y-x〉≥0∀y∈C.
The set of solutions of the variational inequality problem is denoted by VI(C,B).As we know that the classical variational inequality was first introduced and studied by Stampacchia [27] in 1964. Its solution can be computed by using iterative projection method. There are many results with corresponding to variational inequality; for example, Yao et al. [28] proposed the strong convergence theorem for a system of nonlinear variational inequalities in Banach spaces, and then, they studied the two-step projection methods, and they established the convergence theorem for a system of variational inequality problems in the framework of Banach spaces. Moreover, the important generalized variational inequalities called variational inclusion also have been extensively studied and extended in many different directions. Yao et al. [29] considered the algorithm and proved the strong convergence of common solutions for variational inclusions, mixed equilibrium problems, and fixed point problems.The one classical way to approximate a fixed point of a nonlinear self mappingT on C was firstly introduced by Halpern [30], and then, Aoyama et al. [31] extended the mapping in the Halpern-type iterative sequence to be a countable family of nonexpansive mappings. They introduced the following iterative sequence: let x1=x∈C and(1.8)xn+1=αnx+(1-αn)Tnxn,
for all n∈ℕ, where C is a nonempty closed convex subset of a Banach space, {αn} is a sequence in [0,1], and {Tn} is a sequence of nonexpansive mappings with some conditions. They proved that {xn} converges strongly to a common fixed point of {Tn}.Recently, Nakajo et al. [32] introduced the more general condition so-called the NST*-condition, and {Tn} is said to satisfy the NST*-condition if for every bounded sequence {zn} in C,(1.9)lim⁡n→∞‖zn-Tnzn‖=lim⁡n→∞‖zn-zn+1‖=0impliesωw(zn)⊂F.
They also prove strong convergence theorems by the hybrid method for families of mappings in a uniformly convex Banach space E whose norm is Gâteaux differentiable.In Hilbert spaceH, Iiduka et al. [33] introduced an iterative scheme and proved that the sequence {xn} generated by the following algorithm: x1=x∈C, and
(1.10)xn+1=PC(xn-λnBxn),
where PC is the metric projection of H onto C and {λn} is a sequence of positive real numbers, converges weakly to some element of VI(C,B).Later, Iiduka and Takahashi [34] are interested in the similar problem in the framework of Banach spaces, they introduced the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly monotone operator B:x1=x∈C, and(1.11)xn+1=ΠCJ-1(Jxn-λnBxn),
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 (1.11) converges weakly to some element of VI(C,B).In 1974, Rockafellar interested in the following problem of finding:(1.12)v∈Esuchthat0∈A(v),
where A is an operator from E into E*. Such v∈E is called a zero point of A. He introduced a well-known method, proximal point algorithm, for solving (1.12) in a Hilbert space H as shown in the following: x1=x∈H and(1.13)xn+1=Jrnxn,n=1,2,3,…,
where {rn}⊂(0,∞), A is a maximal monotone and Jrn=(I+rnA)-1. He proved that the sequence {xn} converges weakly to an element of A-1(0).In 2004, Kamimura et al. [35] considered the algorithm (1.14) in a uniformly smooth and uniformly convex Banach space E; namely,(1.14)xn+1=J-1(αnJxn+(1-αn)J(Jrnxn)),n=1,2,3,….
They proved that the algorithm {xn} generated by (1.14) converges weakly to some element of A-10.In 2008, Li and Song [36] established a strong convergence theorem in a Banach space. They introduced the following algorithm: x1=x∈E and
(1.15)yn=J-1(βnJ(xn)+(1-βn)J(Jrnxn)),xn+1=J-1(αnJx+(1-αn)J(yn)).
Under the suitable conditions of the coefficient sequences {αn}, {βn}, and {rn}, they proved that the sequence {xn} generated by the above scheme converges strongly to ΠCx, where ΠC is the generalized projection from E onto C.In 2010, Petrot et al. [37] introduced a hybrid projection iterative scheme for approximating a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of two quasi-ϕ-nonexpansive mappings in a real uniformly convex and uniformly smooth Banach space by the following manner:
(1.16)x0=x∈C,yn=J-1(δnJxn+(1-δn)Jzn),zn=J-1(αnJxn+βnJTxn+γnJSxn),un=Krnyn,Cn={z∈C:ϕ(z,un)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx-Jxn〉≥0},xn+1=ΠCn∩Qnx.
They proved that {xn} converges strongly to p∈F(T)∩F(S)∩Ω, where p∈ΠF(T)∩F(S)∩Ωx.Recently, Klin-eam et al. [38], obtained 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. Saewan and Kumam [39] introduced 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 an α-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. Wattanawitoon and Kumam [40] proved the strong convergence theorem by using modified hybrid projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of solution of variational inequality operators of an inverse strongly monotone, the zero point of a maximal monotone operator, and the set of fixed point of two relatively quasi-nonexpansive mappings in Banach space.Motivated and inspired by the ongoing research and the above-mentioned results, we are also interested in generalized mixed equilibrium problem, variational inequality problems, and the zero point of maximal monotone operators. In this paper, we extend the fixed point problems of two relatively quasi-nonexpansive mappings in [40] to the countable families of two quasi-ϕ-nonexpansive mappings and improve the iterative scheme to be more general as shown in the following: x1=x∈C,
(1.17)wn=ΠCJ-1(Jxn-λnBxn),zn=J-1(δnJ(xn)+(1-δn)J(Jrnwn)),yn=J-1(αnJxn+βnJTnxn+γnJSnzn),un∈CsuchthatF(un,y)+〈Yun,y-un〉+φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈C:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x.
By the new iterative scheme, we will prove the strong convergence theorems of the sequence {xn} which could be converged to the point Π(∩n=1∞F(Tn))∩(∩n=1∞F(Sn))∩Ω∩VI(C,B)∩A-1(0)x. Furthermore, we propose the new better appropriate conditions of the coefficient sequences {αn},{βn},{γn}, and {rn}. Finally, we will apply our result to find a zero point of inverse-strongly monotone operators and complementarity problem in the last section. The results presented in this paper extend and improve the corresponding ones announced by Kamimura et al. [35], Petrot et al. [37], Wattanawitoon and Kumam [40], and some authors in the literature.


## 2. Preliminaries

In this section, we propose the following preliminaries and lemmas which will be used in our proof.Throughout this paper, we letE be a Banach space with norm ∥·∥, and C a nonempty closed convex subset of E, and let E* denote the dual of E. We write xn⇀x to indicate that the sequence {xn} converges weakly to x and xn→x implies that the sequence {xn} converges strongly to x.LetU={x∈E:∥x∥=1} be the unit sphere of E. A Banach space E is said to be strictly convex if for any x,y∈U,(2.1)x≠yimplies‖x+y2‖<1.It is also said to beuniformly convex if for each ϵ∈(0,2], there exists δ>0 such that for any x,y∈U(2.2)‖x-y‖≥ϵimplies‖x+y2‖<1-δ.
We know that a uniformly convex Banach space is reflexive and strictly convex; see [41, 42] for more details.Themodulus of convexity of E is the function δ:[0,2]→[0,1] defined by(2.3)δ(ɛ)=inf⁡{1-‖x+y2‖:x,y∈E,‖x‖=‖y‖=1,‖x-y‖≥ɛ}.Furthermore, it is said to besmooth, provided that
(2.4)lim⁡t→0‖x+ty‖-‖x‖t
exists for all x,y∈U. It is also said to be uniformly smooth if the limit is attained uniformly for x,y∈E.Letp be a fixed real number with p≥2. Observe that every p-uniformly convex is uniformly convex. One should note that no a Banach space is p-uniformly convex for 1<p<2. It is well known that a Hilbert space is 2-uniformly convex and uniformly smooth. For each p>1, the generalized duality mappingJp:E→2E* is defined by(2.5)Jp(x)={x*∈E*:〈x,x*〉=‖x‖p,‖x*‖=‖x‖p-1},
for all x∈E.In particular,J=J2 is called the normalized duality mapping. If E is a Hilbert space, then J=I, where I is the identity mapping. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.We know the following (see [43]):(1)
ifE is smooth, then J is single-valued,(2)
ifE is strictly convex, then J is one-to-one and 〈x-y,x*-y*〉>0 holds for all (x,x*),(y,y*)∈J with x≠y,(3)
ifE is reflexive, then J is surjective,(4)
ifE is uniformly convex, then it is reflexive,(5)
ifE* is uniformly convex, then J is uniformly norm-to-norm continuous on each bounded subset of E.The dualityJ from a smooth Banach space E into E* is said to be weakly sequentially continuous [44] if xn⇀x implies Jxn⇀*Jx, where ⇀* implies the weak* convergence.LetE be a smooth, strictly convex and reflexive Banach space, and let C be a nonempty closed convex subset of E. Throughout this paper, we denote by ϕ the function defined by(2.6)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2,forx,y∈E.Remark 2.1.
We know the following: for eachx,y,z∈E,(i)
(∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2,(ii)
ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2〈x-z,Jz-Jy〉,(iii)
ϕ(x,y)=∥x-y∥2 in a real Hilbert space.Thegeneralized projection, introduced by Alber [45], ΠC:E→C is a map that assigns to an arbitrary point x∈E the minimum point of the function ϕ(x,y); that is, ΠCx=x¯, where x¯ is the solution to the minimization problem(2.7)ϕ(x¯,x)=inf⁡y∈Cϕ(y,x),
existence and uniqueness of the operator ΠC follows from the properties of the functional ϕ(x,y) and strict monotonicity of the mapping J.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 Remark 2.1 (i), 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 [43, 46] for more details.Lemma 2.2 (see [47, 48]).
IfE be a 2-uniformly convex Banach space, then for all x,y∈E, one has
(2.8)‖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 2-uniformly convex constant of E; see [41].Lemma 2.3 (see [47, 49]).
IfE is a p-uniformly convex Banach space and p a given real number with p≥2, then for all x,y∈E,Jx∈Jp(x) and Jy∈Jp(y)(2.9)〈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 2.4 (Xu [48]).
LetE be a uniformly convex Banach space, then for each r>0, there exists a strictly increasing, continuous and convex function g:[0,∞)→[0,∞) such that g(0)=0 and
(2.10)‖λx+(1-λy)‖2≤λ‖x‖2+(1-λ)‖y‖2-λ(1-λ)g(‖x-y‖),
for all x,y∈{z∈E:∥z∥≤r} and λ∈[0,1].Lemma 2.5 (Kamimura and Takahashi [50]).
LetE be a uniformly convex and smooth real Banach space and {xn},{yn} two sequences of E. If ϕ(xn,yn)→0 and either {xn} or {yn} is bounded, then ∥xn-yn∥→0.Lemma 2.6 (Alber [45]).
LetC be a nonempty closed convex subset of a smooth Banach space  E and  x∈E. Then,  x0=ΠCx if and only if
(2.11)〈x0-y,Jx-Jx0〉≥0,∀y∈C.Lemma 2.7 (Alber [45]).
LetE be a reflexive, strictly convex and smooth Banach space and  C a nonempty closed convex subset of  E and let  x∈E. Then,
(2.12)ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x),∀y∈C.LetE be a strictly convex, smooth, and reflexive Banach space and J the duality mapping from E into E*. Then, J-1 is also single-valued, one-to-one, and surjective, and it is the duality mapping from E* into E. Define a function V:E×E*→ℝ as follows (see [51]):(2.13)V(x,x*)=‖x‖2-2〈x,x*〉+‖x*‖2,
for all x∈E and x*∈E*. Then, it is obvious that V(x,x*)=ϕ(x,J-1(x*)) and V(x,J(y))=ϕ(x,y).Lemma 2.8 (Kohsaka and Takahashi [51, Lemma  3.2]).
LetE be a strictly convex,  smooth, and reflexive Banach space and  V as in (2.13). Then,
(2.14)V(x,x*)+2〈J-1(x*)-x,y*〉≤V(x,x*+y*),
for all x∈E and x*,y*∈E*.For solving the generalized mixed equilibrium problem, let us assume that the bifunctionF:C×C→ℝ and φ:C→ℝ is convex and lower semicontinuous, satisfying 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,
(2.15)lim⁡⁡sup⁡t↓0F(tz+(1-t)x,y)≤F(x,y),(A4)
for eachx∈C, y↦F(x,y) is convex and lower semicontinuous.Lemma 2.9 (Blum and Oettli [17]).
LetC be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space E and F a bifunction of C×C into ℝ satisfying (A1)–(A4).  Let r>0 and x∈E. Then, there exists z∈C such that
(2.16)F(z,y)+1r〈y-z,z-x〉≥0∀y∈C.Lemma 2.10 (Takahashi and Zembayashi [52]).
LetC be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space  E and  F a bifunction from  C×C to  ℝ satisfying (A1)–(A4). For all  r>0 and  x∈E, define a mapping  Tr:E→C as follows:
(2.17)Trx={z∈C:F(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},
for all x∈E. Then, the following hold:(1)
Tr is single-valued,(2)
Tr is a firmly nonexpansive-type mapping, that is, for all x,y∈E,
(2.18)〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉,(3)
F(Tr)=EP(F),(4)
EP(F) is closed and convex.Lemma 2.11 (Takahashi and Zembayashi [52]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, and F a bifunction from C×C to ℝ satisfying (A1)–(A4) and let r>0. Then, for x∈E and q∈F(Tr),
(2.19)ϕ(q,Trx)+ϕ(Trx,x)≤ϕ(q,x).Lemma 2.12 (Zhang [53]).
LetC be a 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 convex and lower semi-continuous, and F a bifunction from C×C to ℝ satisfying (A1)–(A4). For r>0 and x∈E, then there exists u∈C such that
(2.20)F(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:
(2.21)Kr(x)={u∈C:F(u,y)+〈Bu,y-u〉+φ(y)-φ(u)+1r〈y-u,Ju-Jx〉≥0,∀y∈C},
for all x∈E. Then, the following hold:(i)
Kr is single-valued,(ii)
Kr is firmly nonexpansive, that is, for all x,y∈E, 〈Krx-Kry,JKrx-JKry〉≤〈Krx-Kry,Jx-Jy〉,(iii)
F(Kr)=Ω,(iv)
Ω is closed and convex,(v)
ϕ(p,Krz)+ϕ(Krz,z)≤ϕ(p,z) for all p∈F(Kr), z∈E.It follows from Lemma2.10 that the mapping Kr:C→C defined by (2.21) is a relatively nonexpansive mapping. Thus, it is quasi-ϕ-nonexpansive.LetE be a reflexive, strictly convex and smooth Banach space. Let C be a closed convex subset of E. Because ϕ(x,y) is strictly convex and coercive in the first variable, we know that the minimization problem infy∈Cϕ(x,y) has a unique solution. The operator ΠCx:=argminy∈Cϕ(x,y) is said to be the generalized projection of x on C.LetA be a set-valued mapping from E to E* with graph G(A)={(x,x*):x*∈Ax}, domain D(A)={x∈E:A(x)≠∅}, and range R(A)={x*∈E*:x*∈A(x),x∈D(A)}. We denote a set-valued operator A from E to E* by A⊂E×E*. A is said to be monotone if 〈x-y,x*-y*〉≥0 for all (x,x*),(y,y*)∈A. A monotone operator A⊂E×E* is said to be maximal monotone if it graph is not properly contained in the graph of any other monotone operator. We know that if A is maximal monotone, then the solution set A-10={z∈D(A):0∈Az} is closed and convex.LetE be a reflexive, strictly convex and smooth Banach space, it is known that A is maximal monotone if and only if R(J+rA)=E* for all r>0.Define theresolvent of A by Jrx=xr. In other words, Jr=(J+rA)-1J for all r>0.  Jr is a single-valued mapping from E to D(A). Also, A-1(0)=F(Jr) for all r>0, where F(Jr) is the set of all fixed points of Jr. Define, for r>0, the Yosida approximation of A by Ar=(J-JJr)/r. We know that Arx∈A(Jrx) for all r>0 and x∈E.Lemma 2.13 (Kohsaka and Takahashi [51, Lemma  3.1]).
LetE be a smooth, strictly convex, and reflexive Banach space, let A⊂E×E* be a maximal monotone operator with A-10≠∅, r>0, and  Jr=(J+rT)-1J. Then,
(2.22)ϕ(x,Jry)+ϕ(Jry,y)≤ϕ(x,y)
for all x∈A-10 and y∈E.LetB 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)=B(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,(2.23)NC(v)={x*∈E*:〈v-y,x*〉≥0,∀y∈C}.Theorem 2.14.
(Rockafellar [54]). Let C be a nonempty, closed convex subset of a Banach space E and B a monotone, hemicontinuous operator of C into E*. Let T⊂E×E* be an operator defined as follows:
(2.24)Tv={Bv+NC(v),v∈C,∅,otherwise.
Then T, is maximal monotone and T-10=VI(C,B).Lemma 2.15 (Tan and Xu [55]).
Let{an} and {bn} be two sequences of nonnegative real numbers satisfying
(2.25)an+1≤an+bn,∀n≥0.
If ∑n=1∞bn<∞, then limn→∞an exists.


## 3. The Main Result

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions of mixed equilibrium problems, the set of solutions of the variational inequality problem, the zero point of a maximal monotone operator, and the set of two families of quasi-ϕ-nonexpansive mappings in a Banach space by using the shrinking hybrid projection method.Theorem 3.1.
LetE be a 2-uniformly convex and uniformly smooth Banach space and C a nonempty closed convex subset of E. Let F be a bifunction from C×C to ℝ satisfying (A1)–(A4),  let φ:C→ℝ be a proper lower semicontinuous and convex function, and let A:E→E* be a maximal monotone operator satisfying D(A)⊂E. Let Jr=(J+rA)-1J for r>0, let B be an α-inverse-strongly monotone operator of E into E*, and let Y:E→E* be a continuous and monotone mapping. Let {Tn} and {Sn} be two families of quasi-ϕ-nonexpansive mappings of E into itself satisfies the NST*-condition, with Θ:=(∩n=1∞F(Tn))∩(∩n=1∞F(Sn))∩Ω∩VI(C,B)∩A-1(0)≠∅ and ∥By∥≤∥By-Bu∥ for all y∈C and u∈Θ. Let {xn} be a sequence generated by x1=x∈E, and
(3.1)wn=ΠCJ-1(Jxn-λnBxn),zn=J-1(δnJ(xn)+(1-δn)J(Jrnwn)),yn=J-1(αnJxn+βnJTnxn+γnJSnzn),un∈CsuchthatF(un,y)+〈Yun,y-un〉+φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈C:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x,
for all n∈ℕ. If the coefficient sequence {αn},{βn}, {γn}, and {δn}⊂[0,1], {rn}⊂(0,∞) satisfy αn+βn+γn=1, liminfn→∞αnβn>0, liminfn→∞αnγn>0, liminfn→∞γn(1-δn)>0, liminfn→∞rn>0 and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, 1/c is the 2-uniformly convexity constant of E. Then the sequence {xn} converges strongly to ΠΘx.Proof.
We first show that{xn} is bounded. Let p∈Θ:=(∩n=1∞F(Tn))∩(∩n=1∞F(Sn))∩Ω∩VI(C,B)∩A-1(0), and let
(3.2)H(un,y)=F(un,y)+〈Yun,y-un〉+φ(y)-φ(un),y∈C,Krn={u∈C:H(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C}.
Put vn=J-1(Jxn-λnBxn) and un=Krnyn.
With its relatively nonexpansiveness ofJrn and by Lemma 2.8, the convexity of the function V in the second variable, we have
(3.3)ϕ(p,wn)=ϕ(p,ΠCvn)≤ϕ(p,vn)=ϕ(p,J-1(Jxn-λnBxn))≤V(p,Jxn-λnBxn+λnBxn)-2〈J-1(Jxn-λnBxn)-p,λnBxn〉=V(p,Jxn)-2λn〈vn-p,Bxn〉=ϕ(p,xn)-2λn〈xn-p,Bxn〉+2〈vn-xn,-λnBxn〉.
Since p∈VI(C,B) and B is α-inverse-strongly monotone, we consider
(3.4)-2λn〈xn-p,Bxn〉=-2λn〈xn-p,Bxn-Bp〉-2λn〈xn-p,Bp〉≤-2αλn‖Bxn-Bp‖2.
Therefore, by Lemma 2.2, we obtain
(3.5)2〈vn-xn,-λnBxn〉=2〈J-1(Jxn-λnBxn)-J-1(Jxn),-λnBxn〉≤2‖J-1(Jxn-λnBxn)-J-1(Jxn)‖‖λnBxn‖≤4c2‖Jxn-λnBxn-Jxn‖‖λnBxn‖=4c2λn2‖Bxn‖2≤4c2λn2‖Bxn-Bp‖2.
We can rewrite (3.3), which yield that
(3.6)ϕ(p,wn)≤ϕ(p,xn)-2αλn‖Bxn-Bp‖2+4c2λn2‖Bxn-Bp‖2≤ϕ(p,xn)+2λn(2c2λn-α)‖Bxn-Bp‖2≤ϕ(p,xn).
Apply the Lemma 2.8, Lemma 2.13 and (3.6), we consider
(3.7)ϕ(p,zn)=ϕ(p,J-1(δnJ(xn)+(1-δn)J(Jrnwn)))=V(p,δnJ(xn)+(1-δn)J(Jrnwn))≤δnV(p,J(xn))+(1-δn)V(p,J(Jrnwn))=δnϕ(p,xn)+(1-δn)ϕ(p,Jrnwn)≤δnϕ(p,xn)+(1-δn)(ϕ(p,wn)-ϕ(Jrnwn,wn))≤δnϕ(p,xn)+(1-δn)ϕ(p,wn)≤δnϕ(p,xn)+(1-δn)ϕ(p,xn)=ϕ(p,xn),
hence, we obtain
(3.8)ϕ(p,yn)=ϕ(p,J-1(αnJxn+βnJTnxn+γnJSnzn))=‖p‖2-2αn〈p,Jxn〉-2βn〈p,JTnxn〉-2γn〈p,JSnzn〉+‖αnJxn+βnJTnxn+γnJSnzn‖2≤‖p‖2-2αn〈p,Jxn〉-2βn〈p,JTnxn〉-2γn〈p,JSnzn〉+αn‖Jxn‖2+βn‖JTnxn‖2+γn‖JSnzn‖2=αnϕ(p,xn)+βnϕ(p,Tnxn)+γnϕ(p,Snzn)≤αnϕ(p,xn)+βnϕ(p,xn)+γnϕ(p,zn)≤αnϕ(p,xn)+βnϕ(p,xn)+γnϕ(p,xn)=ϕ(p,xn).
By (3.1), again,
(3.9)ϕ(p,un)=ϕ(p,Krnyn)≤ϕ(p,yn)≤ϕ(p,xn).
This shows that p∈Cn+1. Consequently, Θ⊂Cn, for all n∈ℕ.
Next, we show thatlimn→∞ϕ(xn,x0) exists. Since xn=ΠCnx, it follows from Lemma 2.7 that
(3.10)ϕ(xn,x)≤ϕ(p,x)-ϕ(p,xn)≤ϕ(p,x),
for each p∈Θ⊂Cn. Then, ϕ(xn,x) is bounded. It implies that {xn} is bounded and {yn}, {zn}, {wn}, and {Jrnwn} are also bounded.
Fromxn=ΠCnx and xn+1∈Cn+1⊂Cn, we have
(3.11)ϕ(xn,x)≤ϕ(xn+1,x),∀n∈N.
Therefore, {ϕ(xn,x)} is nondecreasing. It follows that the limit of {ϕ(xn,x)} exists, and from Lemma 2.7, we have
(3.12)ϕ(xn+1,xn)=ϕ(xn+1,ΠCnx)≤ϕ(xn+1,x)-ϕ(ΠCnx,x)=ϕ(xn+1,x)-ϕ(xn,x),
for all n∈ℕ. Thus, we have
(3.13)lim⁡n→∞ϕ(xn+1,xn)=0.
Since xn+1=ΠCn+1x∈Cn+1, it follows from the definition of Cn+1 that
(3.14)ϕ(xn+1,un)≤ϕ(xn+1,xn)⟶0.
By Lemma 2.5, (3.13), and (3.14), we note that
(3.15)lim⁡n→∞‖xn+1-un‖=lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞‖xn-un‖=0.
Since J is uniformly norm-to-norm continuous on the bounded set, we obtain
(3.16)lim⁡n→∞‖Jxn+1-Jun‖=lim⁡n→∞‖Jxn+1-Jxn‖=lim⁡n→∞‖Jxn-Jun‖=0.
Since xm=ΠCm⊂Cn for any positive integer m≥n, it follows from Lemma 2.7 that
(3.17)ϕ(xm,xn)=ϕ(xm,ΠCnxn)≤ϕ(xm,x)-ϕ(ΠCnxn,x)=ϕ(xm,x)-ϕ(xn,x).
Taking m,n→∞, we have ϕ(xm,xn)→0 as n→∞. It follows from Lemma 2.5, that ∥xm-xn∥→0 as m,n→∞. Hence, {xn} is a Cauchy sequence. Since E is a Banach space and C is closed and convex, we can assume that xn→u∈C as n→∞.
Next, we show thatu∈(∩n=1∞F(Tn))∩(∩n=1∞F(Sn)).
SinceE is a uniformly smooth Banach space, we know that E* is a uniformly convex Banach space. Let r=supn∈ℕ{∥xn∥,∥Tnxn∥,∥Snzn∥}. From Lemma 2.4, we have
(3.18)ϕ(p,un)≤ϕ(p,yn)=ϕ(p,J-1(αnJxn+βnJTnxn+γnJSnzn))=‖p‖2-2αn〈p,Jxn〉-2βn〈p,JTnxn〉-2γn〈p,JSnzn〉+‖αnJxn+βnJTnxn+γnJSnzn‖2≤‖p‖2-2αn〈p,Jxn〉-2βn〈p,JTnxn〉-2γn〈p,JSnzn〉+αn‖Jxn‖2+βn‖JTnxn‖2+γn‖JSnzn‖2-αnβng(‖JTnxn-Jxn‖)=αnϕ(p,xn)+βnϕ(p,Tnxn)+γnϕ(p,Snzn)-αnβng(‖JTnxn-Jxn‖)≤αnϕ(p,xn)+βnϕ(p,xn)+γnϕ(p,zn)-αnβng(‖JTnxn-Jxn‖)≤αnϕ(p,xn)+βnϕ(p,xn)+γnϕ(p,xn)-αnβng(‖JTnxn-Jxn‖)=ϕ(p,xn)-αnβng(‖JTnxn-Jxn‖).
This implies that
(3.19)αnβng(‖JTnxn-Jxn‖)≤ϕ(p,xn)-ϕ(p,un).
On the other hand, we have
(3.20)ϕ(p,xn)-ϕ(p,un)=‖xn‖2-‖un‖2-2〈p,Jxn-Jun〉=‖xn-un‖(‖xn‖+‖un‖)+2‖p‖‖Jxn-Jun‖.
Noticing (3.15) and (3.16), we obtain
(3.21)ϕ(p,xn)-ϕ(p,un)⟶0,asn⟶∞.
Since liminfn→∞αnβn>0 and (3.21), it follows from (3.19) that
(3.22)g(‖JTnxn-Jxn‖)⟶0,asn⟶∞.
It follows from the property of g that
(3.23)‖JTnxn-Jxn‖⟶0,asn⟶∞.
Since J-1 is uniformly norm-to-norm continuous on bounded sets, we see that
(3.24)lim⁡n→∞‖Tnxn-xn‖=0.
Similarly, using the condition limsupn→∞αnγn>0, one can obtain
(3.25)lim⁡n→∞‖Snzn-xn‖=0.
By (3.6), (3.8), and (3.18), we have
(3.26)ϕ(p,un)≤αnϕ(p,xn)+βnϕ(p,xn)+γnϕ(p,zn)-αnβng(‖JTnxn-Jxn‖)≤αnϕ(p,xn)+βnϕ(p,xn)+γn[δnϕ(p,xn)+(1-δn)ϕ(p,wn)]-αnβng(‖JTnxn-Jxn‖)≤αnϕ(p,xn)+βnϕ(p,xn)+γnδnϕ(p,xn)+γn(1-δn)ϕ(p,wn)-αnβng(‖JTnxn-Jxn‖)≤αnϕ(p,xn)+βnϕ(p,xn)+γnδnϕ(p,xn)+γn(1-δn)[ϕ(p,xn)+2λn(2c2λn-α)‖Bxn-Bp‖2]-αnβng(‖JTnxn-Jxn‖)≤αnϕ(p,xn)+βnϕ(p,xn)+γnδnϕ(p,xn)+γn(1-δn)ϕ(p,xn)+γn(1-δn)2λn(2c2λn-α)‖Bxn-Bp‖2-αnβng(‖JTnxn-Jxn‖)≤ϕ(p,xn)+γn(1-δn)2λn(2c2λn-α)‖Bxn-Bp‖2.
This implies that
(3.27)2λn(α-2c2λn)‖Bxn-Bp‖2≤1γn(1-δn)[ϕ(p,xn)-ϕ(p,un)].
By assumption, liminfn→∞γn(1-δn)>0 and (3.21), we get that
(3.28)lim⁡n→∞‖Bxn-Bp‖=0.
From Lemma 2.7, Lemma 2.8, and (3.5), we have
(3.29)ϕ(xn,wn)=ϕ(xn,ΠCvn)≤ϕ(xn,vn)=ϕ(xn,J-1(Jxn-λnBxn))=V(xn,Jxn-λnBxn)≤V(xn,(Jxn-λnBxn)+λnBxn)-2〈J-1(Jxn-λnBxn)-xn,λnBxn〉=V(xn,Jxn)-2λn〈vn-xn,Bxn〉=ϕ(xn,xn)+2〈vn-xn,λnBxn〉≤4c2λn2‖Bxn-Bp‖2.
By Lemma 2.8 and Lemma 2.13, we have
(3.30)ϕ(xn,zn)=ϕ(xn,J-1(δnJxn+(1-δn)J(Jrnwn)))=V(xn,δnJxn+(1-δn)J(Jrnwn))≤δnV(xn,Jxn)+(1-δn)V(xn,J(Jrnwn))=δnϕ(xn,xn)+(1-δn)ϕ(xn,Jrnwn)=δnϕ(xn,xn)+(1-δn)(ϕ(xn,wn)-ϕ(Jrnwn,wn))=(1-δn)ϕ(xn,wn)≤(1-δn)4c2λn2‖Bxn-Bp‖2.
From Lemma 2.5 and (3.28), we obtain
(3.31)lim⁡n→∞‖xn-wn‖=lim⁡n→∞‖xn-zn‖=0.
Since J is uniformly norm-to-norm continuous on bounded sets, we note that
(3.32)lim⁡n→∞‖Jxn-Jwn‖=lim⁡n→∞‖Jxn-Jzn‖=0.
Since xn→u as n→∞, zn→u as n→∞. Combining (3.15), (3.25), and (3.28), we also obtain
(3.33)‖Snzn-zn‖≤‖Snzn-xn‖+‖xn-zn‖⟶0asn⟶∞.
By (3.15) and (3.31), we obtain that
(3.34)‖zn+1-zn‖≤‖zn+1-xn+1‖+‖xn+1-xn‖+‖xn-zn‖⟶0asn⟶∞.
By (3.15), (3.24), (3.33), and (3.34), and {Tn}, {Sn} satisfies the NST*-condition and xn→p, then we have p∈(∩n=1∞F(Tn))∩(∩n=1∞F(Sn)).
Since{xn} is bounded, there exists a subsequence {xni} of {xn} such that xni⇀u∈C. It follows from (3.31) that we have wni⇀u as i→∞. Next, we show that u∈A-10.
By (3.6), (3.8), and (3.9), we obtain
(3.35)ϕ(p,un)≤ϕ(p,yn)≤αnϕ(p,xn)+βnϕ(p,xn)+γnϕ(p,zn)≤αnϕ(p,xn)+βnϕ(p,xn)+γn[δnϕ(p,xn)+(1-δn)(ϕ(p,wn)-ϕ(Jrnwn,wn))]≤αnϕ(p,xn)+βnϕ(p,xn)+γn[δnϕ(p,xn)+(1-δn)(ϕ(p,xn)-ϕ(Jrnwn,wn))]≤ϕ(p,xn)-γn(1-δn)ϕ(Jrnwn,wn).
This implies that
(3.36)γn(1-δn)ϕ(Jrnwn,wn)≤ϕ(p,xn)-ϕ(p,un).
By (3.21), we have
(3.37)lim⁡n→∞‖Jrnwn-wn‖=0.
Since J is uniformly norm-to-norm continuous on bounded sets, we note that
(3.38)lim⁡n→∞‖JJrnwn-Jwn‖=0.
Indeed, since liminfn→∞rn>0, it follows from (3.38) that
(3.39)lim⁡n→∞‖Arnwn‖=lim⁡n→∞1rn‖Jwn-J(Jrnwn)‖=0.
If (w,w*)∈A, then it holds from the monotonicity of A that
(3.40)〈w-wni,w*-Arniwni〉≥0,
for all i∈ℕ. Letting i→∞, we get 〈w-u,w*〉≥0. Then, the maximality of A implies u∈A-10.
Next, we show thatu∈VI(C,B). Let K⊂E×E* be an operator as follows:
(3.41)Kv={Bv+NC(v),v∈C,∅,otherwise.
By Theorem 2.14, K is maximal monotone and K-10=VI(C,B).
Let(v,w)∈G(K). Since w∈Kv=Bv+NC(v), we get w-Bv∈NC(v). From wn∈C, we have
(3.42)〈v-wn,w-Kv〉≥0.
On the other hand, since wn=ΠCJ-1(Jxn-λnBxn), then by Lemma 2.6, we have
(3.43)〈v-wn,Jwn-(Jxn-λnBxn)〉≥0,
thus
(3.44)〈v-wn,Jxn-Jwnλn-Bxn〉≤0.
It follows from (3.42) and (3.44) that
(3.45)〈v-wn,w〉≥〈v-wn,Bv〉≥〈v-wn,Bv〉+〈v-wn,Jxn-Jwnλn-Bxn〉=〈v-wn,Bv-Bxn〉+〈v-wn,Jxn-Jwnλn〉=〈v-wn,Bv-Bwn〉+〈v-wn,Bwn-Bxn〉+〈v-wn,Jxn-Jwnλn〉≥-‖v-wn‖‖wn-xn‖α-‖v-wn‖‖Jxn-Jwn‖a≥-M(‖wn-xn‖α+‖Jxn-Jwn‖a),
where M=supn≥1{∥v-wn∥}. From (3.31) and (3.32), we obtain 〈v-u,w〉≥0. By the maximality of K, we have u∈K-10 and hence u∈VI(C,B).
Next, we show thatu∈Ω. From un=Krnyn and Lemma 2.12, we obtain
(3.46)ϕ(un,yn)=ϕ(Krnyn,yn)≤ϕ(u,yn)-ϕ(u,Krnyn)≤ϕ(u,xn)-ϕ(u,Krnyn)≤ϕ(u,xn)-ϕ(u,un).
On the other hand, we have
(3.47)ϕ(u,xn)-ϕ(u,un)=‖xn‖2-‖un‖2-2〈u,Jxn-Jun〉=‖xn-un‖(‖xn‖+‖un‖)+2‖u‖‖Jxn-Jun‖.
Noticing (3.15) and (3.16), we obtain
(3.48)ϕ(u,xn)-ϕ(u,un)⟶0,asn⟶∞.
It follows that
(3.49)ϕ(un,yn)⟶0,asn⟶∞.
By Lemma 2.5, we have
(3.50)lim⁡n→∞‖un-yn‖=0.
Since J is uniformly norm-to-norm continuous on bounded sets, we get
(3.51)lim⁡n→∞‖Jun-Jyn‖=0.
From the assumption liminfn→∞rn>a, we get
(3.52)lim⁡n→∞‖Jun-Jyn‖rn=0.
Noticing that un=Krnyn, we have
(3.53)H(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C.
Hence,
(3.54)H(uni,y)+1rni〈y-uni,Juni-Jyni〉≥0,∀y∈C.
From the (A2), we note that
(3.55)‖y-uni‖‖Juni-Jyni‖rni≥1rni〈y-uni,Juni-Jyni〉≥-H(uni,y)≥H(y,uni),∀y∈C.
Taking the limit as n→∞ in the above inequality, and from (A4) and uni⇀u, we have H(y,u)≤0,forally∈C. For 0<t<1 and y∈C, define yt=ty+(1-t)u. Noticing that y,u∈C, we obtain yt∈C, which yields that H(yt,u)≤0. It follows from (A1) that
(3.56)0=H(yt,yt)≤tH(yt,y)+(1-t)H(yt,u)≤tH(yt,y).
That is, H(yt,y)≥0.
Lett↓0, from (A3), we obtain H(u,y)≥0,forally∈C. This implies that u∈Ω. Hence, u∈Θ:=(∩n=1∞F(Tn))∩(∩n=1∞F(Sn))∩Ω∩VI(C,B)∩A-1(0).
Finally, we show thatu=ΠΘx. Indeed, from xn=ΠCnx and Lemma 2.6, we have
(3.57)〈Jx-Jxn,xn-z〉≥0,∀z∈Cn.
Since Θ⊂Cn, we also have
(3.58)〈Jx-Jxn,xn-p〉≥0,∀p∈Θ.
Taking limit n→∞, we obtain
(3.59)〈Jx-Ju,u-p〉≥0,∀p∈Θ.
By again Lemma 2.6, we can conclude that u=ΠΘx0. This completes the proof.Corollary 3.2.
LetE be a 2-uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let F be a bifunction from C×C to ℝ satisfying (A1)–(A4), let φ:C→ℝ be a proper lower semicontinuous and convex function, and let A:E→E* be a maximal monotone operator satisfying D(A)⊂E. Let Jr=(J+rA)-1J for r>0, let B be an α-inverse-strongly monotone operator of E into E*, and let Y:C→E* be a continuous and monotone mapping. Let T and S be two quasi-ϕ-nonexpansive mappings of E into itself with F:=F(T)∩F(S)∩Ω∩VI(C,B)∩A-1(0)≠∅ and ∥By∥≤∥By-Bu∥ for all y∈C and u∈Θ. Let {xn} be a sequence generated by x1=x∈E, and
(3.60)wn=ΠCJ-1(Jxn-λnBxn),zn=J-1(δnJ(xn)+(1-δn)J(Jrnwn)),yn=J-1(αnJxn+βnJTxn+γnJSzn),un∈CsuchthatF(un,y)+〈Yun,y-un〉+φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈C:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x,
for all n∈ℕ. If the coefficient sequence {αn},{βn}, and {γn} and {δn}⊂[0,1], {rn}⊂(0,∞) satisfy αn+βn+γn=1, liminfn→∞αnβn>0, liminfn→∞αnγn>0, liminfn→∞γn(1-δn)>0, liminfn→∞rn>0 and {λn}⊂[a,b] for some a,b with 0<a<b<(c2α)/2, 1/c is the 2-uniformly convexity constant of E. Then, the sequence {xn} converges strongly to ΠΘx.Remark 3.3.
Theorem3.1 and Corollary 3.2 extended and improved the results of [40] by extending the mapping from two-relatively quasi-nonexpansive mappings to two countable families of quasi-ϕ-nonexpansive mappings and improving the iterative scheme to be more general, and finally, we proposed the better new conditions for the coefficient sequences which was imposed in our algorithm.


## 4. Applications

### 4.1. A Zero Point of Inverse-Strongly Monotone Operators

Next, we consider the problem of finding a zero point of an inverse-strongly monotone operator ofE into E*. Assume that B satisfies the following conditions:(C1)
B is α-inverse-strongly monotone,(C2)
B-10={u∈E:Bu=0}≠∅.Hence, we also have the following result.Corollary 4.1.
LetE be a 2-uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let F be a bifunction from C×C to ℝ satisfying (A1)–(A4), let φ:C→ℝ be a proper lower semicontinuous and convex function, and let A:E→E* be a maximal monotone operator satisfying D(A)⊂E. Let Jr=(J+rA)-1J for r>0, let B be an α-inverse-strongly monotone operator of E into E*, and let Y:C→E* be a continuous and monotone mapping. Let T and S be two quasi-ϕ-nonexpansive mappings of E into itself with
(4.1)Θ:=F(T)∩F(S)∩Ω∩B-1(0)∩A-1(0)≠∅.
Let {xn} be a sequence generated by x1=x∈E, and
(4.2)wn=ΠCJ-1(Jxn-λnBxn),zn=J-1(δnJ(xn)+(1-δn)J(Jrnwn)),yn=J-1(αnJxn+βnJTxn+γnJSzn),un∈CsuchthatF(un,y)+〈Yun,y-un〉+φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈C:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x,
for all n∈ℕ, where ΠC is the generalized projection from E onto C, J is the duality mapping on E. If the coefficient sequence {αn},{βn}, {γn}, and {δn}⊂[0,1], {rn}⊂(0,∞) satisfy αn+βn+γn=1, liminfn→∞αnβn>0, liminfn→∞αnγn>0, liminfn→∞γn(1-δn)>0, liminfn→∞rn>0, and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, 1/c is the 2-uniformly convexity constant of E, then the sequence {xn} converges strongly to ΠΘx.Proof.
SettingB̃x≡0,forallx∈C, then D(B̃)=E and hence C=E in Corollary 3.2, we also get ΠE=I. We also have VI(B,C)=VI(B,E)={x∈E:Bx=0}≠∅, and then, the condition ∥By∥≤∥By-Bu∥ holds for all y∈E and u∈B-10. So, we obtain the result.


### 4.2. Complementarity Problems

LetK be a nonempty, closed convex cone in E. We define the polarK* of K as follows:
(4.3)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 ([43]) if(4.4)Au∈K*,〈u,Au〉=0.
The set of solutions of the complementarity problem is denoted by C(A,K).Corollary 4.2.
LetE be a 2-uniformly convex and uniformly smooth Banach space, and let K be a nonempty closed convex subset of E. Let F be a bifunction from K×K to ℝ satisfying (A1)–(A4), let φ:K→ℝ be a proper lower semicontinuous and convex function, and let A:E→E* be a maximal monotone operator satisfying D(A)⊂E. Let Jr=(J+rA)-1J for r>0, let B be an α-inverse-strongly monotone operator of E into E*, and let Y:K→E* be a continuous and monotone mapping. Let T and S be two quasi-ϕ-nonexpansive mappings of E into itself with
(4.5)Θ:=F(T)∩F(S)∩Ω∩C(B,K)∩A-1(0)≠∅,
and ∥By∥≤∥By-Bu∥ for all y∈K and u∈Θ. Let {xn} be a sequence generated by x1=x∈E, and
(4.6)wn=ΠKJ-1(Jxn-λnBxn),zn=J-1(δnJ(xn)+(1-δn)J(Jrnwn)),yn=J-1(αnJxn+βnJTxn+γnJSzn),un∈KsuchthatF(un,y)+〈Yun,y-un〉+φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈K,Kn+1={z∈K:ϕ(z,un)≤ϕ(z,xn)}xn+1=ΠKn+1x,
for all n∈ℕ, where ΠK is the generalized projection from E onto K, J is the duality mapping on E. If the coefficient sequence {αn},{βn}, {γn}, and {δn}⊂[0,1], {rn}⊂(0,∞) satisfy αn+βn+γn=1, liminfn→∞αnβn>0, liminfn→∞αnγn>0, liminfn→∞γn(1-δn)>0, liminfn→∞rn>0, and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, 1/c is the 2-uniformly convexity constant of E, then the sequence {xn} converges strongly to ΠΘx.Proof.
As in the proof of Takahashi in [43, Lemma  7.11], we have VI(B,K)=C(B,K). So, we obtain the above result.


## 4.1. A Zero Point of Inverse-Strongly Monotone Operators

Next, we consider the problem of finding a zero point of an inverse-strongly monotone operator ofE into E*. Assume that B satisfies the following conditions:(C1)
B is α-inverse-strongly monotone,(C2)
B-10={u∈E:Bu=0}≠∅.Hence, we also have the following result.Corollary 4.1.
LetE be a 2-uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let F be a bifunction from C×C to ℝ satisfying (A1)–(A4), let φ:C→ℝ be a proper lower semicontinuous and convex function, and let A:E→E* be a maximal monotone operator satisfying D(A)⊂E. Let Jr=(J+rA)-1J for r>0, let B be an α-inverse-strongly monotone operator of E into E*, and let Y:C→E* be a continuous and monotone mapping. Let T and S be two quasi-ϕ-nonexpansive mappings of E into itself with
(4.1)Θ:=F(T)∩F(S)∩Ω∩B-1(0)∩A-1(0)≠∅.
Let {xn} be a sequence generated by x1=x∈E, and
(4.2)wn=ΠCJ-1(Jxn-λnBxn),zn=J-1(δnJ(xn)+(1-δn)J(Jrnwn)),yn=J-1(αnJxn+βnJTxn+γnJSzn),un∈CsuchthatF(un,y)+〈Yun,y-un〉+φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈C:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x,
for all n∈ℕ, where ΠC is the generalized projection from E onto C, J is the duality mapping on E. If the coefficient sequence {αn},{βn}, {γn}, and {δn}⊂[0,1], {rn}⊂(0,∞) satisfy αn+βn+γn=1, liminfn→∞αnβn>0, liminfn→∞αnγn>0, liminfn→∞γn(1-δn)>0, liminfn→∞rn>0, and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, 1/c is the 2-uniformly convexity constant of E, then the sequence {xn} converges strongly to ΠΘx.Proof.
SettingB̃x≡0,forallx∈C, then D(B̃)=E and hence C=E in Corollary 3.2, we also get ΠE=I. We also have VI(B,C)=VI(B,E)={x∈E:Bx=0}≠∅, and then, the condition ∥By∥≤∥By-Bu∥ holds for all y∈E and u∈B-10. So, we obtain the result.


## 4.2. Complementarity Problems

LetK be a nonempty, closed convex cone in E. We define the polarK* of K as follows:
(4.3)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 ([43]) if(4.4)Au∈K*,〈u,Au〉=0.
The set of solutions of the complementarity problem is denoted by C(A,K).Corollary 4.2.
LetE be a 2-uniformly convex and uniformly smooth Banach space, and let K be a nonempty closed convex subset of E. Let F be a bifunction from K×K to ℝ satisfying (A1)–(A4), let φ:K→ℝ be a proper lower semicontinuous and convex function, and let A:E→E* be a maximal monotone operator satisfying D(A)⊂E. Let Jr=(J+rA)-1J for r>0, let B be an α-inverse-strongly monotone operator of E into E*, and let Y:K→E* be a continuous and monotone mapping. Let T and S be two quasi-ϕ-nonexpansive mappings of E into itself with
(4.5)Θ:=F(T)∩F(S)∩Ω∩C(B,K)∩A-1(0)≠∅,
and ∥By∥≤∥By-Bu∥ for all y∈K and u∈Θ. Let {xn} be a sequence generated by x1=x∈E, and
(4.6)wn=ΠKJ-1(Jxn-λnBxn),zn=J-1(δnJ(xn)+(1-δn)J(Jrnwn)),yn=J-1(αnJxn+βnJTxn+γnJSzn),un∈KsuchthatF(un,y)+〈Yun,y-un〉+φ(y)-φ(un)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈K,Kn+1={z∈K:ϕ(z,un)≤ϕ(z,xn)}xn+1=ΠKn+1x,
for all n∈ℕ, where ΠK is the generalized projection from E onto K, J is the duality mapping on E. If the coefficient sequence {αn},{βn}, {γn}, and {δn}⊂[0,1], {rn}⊂(0,∞) satisfy αn+βn+γn=1, liminfn→∞αnβn>0, liminfn→∞αnγn>0, liminfn→∞γn(1-δn)>0, liminfn→∞rn>0, and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, 1/c is the 2-uniformly convexity constant of E, then the sequence {xn} converges strongly to ΠΘx.Proof.
As in the proof of Takahashi in [43, Lemma  7.11], we have VI(B,K)=C(B,K). So, we obtain the above result.



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*Source: 692829-2012-01-29.xml*