# The Modified Block Iterative Algorithms for Asymptotically Relatively Nonexpansive Mappings and the System of Generalized Mixed Equilibrium Problems

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

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## Abstract

The propose of this paper is to present a modified block iterative algorithm for finding a common element between the set of solutions of the fixed points of two countable families of asymptotically relatively nonexpansive mappings and the set of solution of the system of generalized mixed equilibrium problems in a uniformly smooth and uniformly convex Banach space. Our results extend many known recent results in the literature.


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## Body

## 1. Introduction

The equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, networks, elasticity, and optimization, and it has been extended and generalized in many directions.In the theory of equilibrium problems, the development of an efficient and implementable iterative algorithm is interesting and important. This theory combines theoretical and algorithmic advances with novel domain of applications. Analysis of these problems requires a blend of techniques from convex analysis, functional analysis, and numerical analysis.LetE be a Banach space with norm ∥·∥, C be a nonempty closed convex subset of E, and let E* denote the dual of E. Let fi:C×C→ℝ be a bifunction, ψi:C→ℝ be a real-valued function, where ℝ is denoted by the set of real numbers, and Ai:C→E* be a nonlinear mapping. The goal of the system of generalized mixed equilibrium problem is to find u∈C such that
(1.1)f1(u,y)+〈A1u,y-u〉+ψ1(y)-ψ1(u)≥0,∀y∈C,f2(u,y)+〈A2u,y-u〉+ψ2(y)-ψ2(u)≥0,∀y∈C,⋮fN(u,y)+〈ANu,y-u〉+ψN(y)-ψN(u)≥0,∀y∈C.
If fi=f, Ai=A, and ψi=ψ, the problem (1.1) is reduced to the generalized mixed equilibrium problem, denoted by GEMP(f,A,ψ), to find u∈C such that
(1.2)f(u,y)+〈Au,y-u〉+ψ(y)-ψ(u)≥0,∀y∈C.
The set of solutions to (1.2) is denoted by Ω, that is,
(1.3)Ω={x∈C:f(u,y)+〈Au,y-u〉+φ(y)-φ(u)≥0,∀y∈C}.
If A=0, the problem (1.2) is reduced to the mixed equilibrium problem forf, denoted by MEP(f,ψ), to find u∈C such that
(1.4)f(u,y)+ψ(y)-ψ(u)≥0,∀y∈C.
If f≡0, the problem (1.2) is reduced to the mixed variational inequality of Browder type, denoted by VI(C,A,ψ), is to find u∈C such that
(1.5)〈Au,y-u〉+ψ(y)-ψ(u)≥0,∀y∈C.
If A=0 and ψ=0, the problem (1.2) is reduced to the equilibrium problem forf, denoted by EP(f), to find u∈C such that
(1.6)f(u,y)≥0,∀y∈C.The above formulation (1.6) was shown in [1] to cover monotone inclusion problems, saddle-point problems, variational inequality problems, minimization problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed-point problem, and optimization problem, which can also be written in the form of an EP(f). In other words, the EP(f) is a unifying model for several problems arising in physics, engineering, science, economics, and so forth. In the last two decades, many papers have appeared in the literature on the existence of solutions to EP(f); see, for example [1–4] and references therein. Some solution methods have been proposed to solve the EP(f); see, for example, [2, 4–15] and references therein. In 2005, Combettes and Hirstoaga [5] introduced an iterative scheme of finding the best approximation to the initial data when EP(f) is nonempty, and they also proved a strong convergence theorem.A Banach spaceE is said to be strictly convex if ∥(x+y)/2∥<1 for all x,y∈E with ∥x∥=∥y∥=1 and x≠y. Let U={x∈E:∥x∥=1} be the unit sphere of E. Then the Banach space E is said to be smooth, provided
(1.7)lim⁡t→0‖x+ty‖-‖x‖t
exists for each x,y∈U. It is also said to be uniformly smooth if the limit is attained uniformly for x,y∈E. The modulus of convexity of E is the function δ:[0,2]→[0,1] defined by
(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].LetE be a Banach space, C be a closed convex subset of E, a mapping T:C→C is said to be nonexpansive if
(1.9)‖Tx-Ty‖≤‖x-y‖
for all x,y∈C. We denote by F(T) the set of fixed points of T. If C is a bounded closed convex set and T is a nonexpansive mapping of C into itself, then F(T) is nonempty (see [16]). A point p in C is said to be an asymptotic fixed point of T [17] if C contains a sequence {xn} which converges weakly to p such that limn→∞∥xn-Txn∥=0. The set of asymptotic fixed points of T will be denoted by  F(T)~. A point p∈C is said to be a strong asymptotic fixed point of T, if there exists a sequence {xn}⊂C such that xn→p and ∥xn-Txn∥→0. The set of strong asymptotic fixed points of T will be denoted by F^(T). A mapping T from C into itself is said to be relatively nonexpansive [18–20] if F(T)~=F(T) and ϕ(p,Tx)≤ϕ(p,x) for all x∈C and p∈F(T). The asymptotic behavior of a relatively nonexpansive mapping was studied in [21, 22]. T is said to be ϕ-nonexpansive, if ϕ(Tx,Ty)≤ϕ(x,y) for x,y∈C. T is said to be quas-ϕ-nonexpansive if F(T)≠∅ and ϕ(p,Tx)≤ϕ(p,x) for x∈C and p∈F(T). A mapping T is said to be asymptotically relatively nonexpansive, if F(T)≠∅, and there exists a real sequence {kn}⊂[1,∞) with kn→1 such that ϕ(p,Tnx)≤knϕ(p,x), foralln≥1,x∈C, and p∈F(T). {Tn}n=0∞ is said to be a countable family of weak relatively nonexpansive mappings [23] if the following conditions are satisfied:  (i)
F({Tn}n=0∞)≠∅;(ii)
ϕ(u,Tnx)≤ϕ(u,x),forallu∈F(Tn),x∈C,n≥0;(iii)
F^({Tn}n=0∞)=∩n=0∞F(Tn). A mapping T:C→C is said to be uniformly L-Lipschitz continuous, if there exists a constant L>0 such that
(1.10)‖Tnx-Tny‖≤L‖x-y‖,∀x,y∈C,∀n≥1.
A mapping T:C→C is said to be closed if for any sequence {xn}⊂C with xn→x and Txn→y, then Tx=y. Let {Ti}i=1∞:C→C be a sequence of mappings. {Ti}i=1∞ is said to be a countable family of uniformly asymptotically relatively nonexpansive mappings, if ∩n=1∞F(Tn)≠∅, and there exists a sequence {kn}⊂[1,∞) with kn→1 such that for each i>1(1.11)ϕ(p,Tinx)≤knϕ(p,x),∀p∈⋂n=1∞F(Tn),x∈C,∀n≥1.In 2009, Petrot et al. [24], introduced a hybrid projection method for approximating a common element of the set of solutions of fixed points of hemirelatively nonexpansive (or quasi-ϕ-nonexpansive) mappings in a uniformly convex and uniformly smooth Banach space:(1.12)x0∈C,C0=C,yn=J-1(αnJxn+(1-αn)JTnzn),zn=J-1(βnJxn+(1-βn)JTnxn),Cn+1={v∈Cn:ϕ(v,yn)≤ϕ(v,xn)},xn+1=ΠCn+1(x0).
They proved that the sequence {xn} converges strongly to p∈F(T), where p∈ΠF(T)x and ΠC is the generalized projection from E onto F(T). Kumam and Wattanawitoon [25], introduced a hybrid iterative scheme for finding a common element of the set of common fixed points of two quasi-ϕ-nonexpansive mappings and the set of solutions of an equilibrium problem in Banach spaces, by the following manner:
(1.13)x0∈C,C0=Cyn=J-1(αnJxn+(1-αn)JSzn),zn=J-1(βnJxn+(1-βn)JTxn),un∈Csuch  thatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1(x0).
They proved that the sequence {xn} converges strongly to p∈F(T)∩F(S)∩EP(f), where p∈ΠF(T)∩F(S)∩EP(f)x under the assumptions (C1) limsupn→∞αn<1, (C2) limn→∞βn<1, and (C3) liminfn→∞(1-αn)βn(1-βn)>0.Recently, Chang et al. [26], introduced the modified block iterative method to propose an algorithm for solving the convex feasibility problems for an infinite family of quasi-ϕ-asymptotically nonexpansive mappings,(1.14)x0∈Cchosenarbitrary,C0=C,yn=J-1(αn,0Jxn+∑i=1∞‍αn,iJSinxn),Cn+1={v∈Cn:ϕ(v,yn)≤ϕ(v,xn)+ξn},xn+1=ΠCn+1x0,∀n≥0,where ξn=supu∈F(kn-1)ϕ(u,xn). Then, they proved that under appropriate control conditions the sequence {xn} converges strongly to Π∩n=1∞F(Si)x0.Very recently, Tan and Chang [27], introduced a new hybrid iterative scheme for finding a common element between set of solutions for a system of generalized mixed equilibrium problems, set of common fixed points of a family of quasi-ϕ-asymptotically nonexpansive mappings (which is more general than quasi-ϕ-nonexpansive mappings), and null spaces of finite family of γ-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space.In this paper, motivated and inspired by Petrot et al. [24], Kumam and Wattanawitoon [25], Chang et al. [26], and Tan and Chang [27], we introduce the new hybrid block algorithm for two countable families of closed and uniformly Lipschitz continuous and uniformly asymptotically relatively nonexpansive mappings in a Banach space. Let {xn} be a sequence defined by x0∈C, C0=C and
(1.15)yn=J-1(βn,0J(xn)+∑i=1∞‍βn,iJ(Tinxn)),zn=J-1(αn,0J(xn)+∑i=1∞‍αn,iJ(Sinyn)),un(i)=Kfi,riKfi-1,ri-1⋯Kf1,r1(zn),i=1,2,…,N,Cn+1={z∈Cn:maxi=1,2,…,Nϕ(z,un(i))≤ϕ(z,xn)+θn,ϕ(z,yn)≤ϕ(z,xn)+ξn},xn+1=ΠCn+1x0,∀n≥0.
Under appropriate conditions, we will prove that the sequence {xn} generated by algorithms (1.15) converges strongly to the point Π(∩i=1NΩi)∩(∩i=1∞F(Ti))∩(∩i=1∞F(Si))x0. Our results extend many known recent results in the literature.


## 2. Preliminaries

LetE be a real Banach space with norm ∥·∥, and let J be the normalized duality mapping from E into 2E* given by
(2.1)Jx={x*∈E*:〈x,x*〉=‖x‖‖x*‖,‖x‖=‖x*‖}
for all x∈E, where E* denotes the dual space of E and 〈·,·〉 the generalized duality pairing between E and E*. It is 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 [28, 29]):(i)
ifE is smooth, then J is single valued;(ii)
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;(iii)
ifE is reflexive, then J is surjective;(iv)
ifE is uniformly convex, then it is reflexive;(v)
ifE is a reflexive and strictly convex, then J-1 is norm-weak-continuous;(vi)
E is uniformly smooth if and only if E* is uniformly convex;(vii)
ifE* is uniformly convex, then J is uniformly norm-to-norm continuous on each bounded subset of E;(viii)
each uniformly convex Banach spaceE has the Kadec-Klee property, that is, for any sequence {xn}⊂E, if xn⇀x∈E and ∥xn∥→∥x∥, then xn→x.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
(2.2)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2,forx,y∈E.
Following Alber [30], the generalized projectionΠC: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
(2.3)ϕ(x¯,x)=inf⁡y∈Cϕ(y,x).
Existence and uniqueness of the operator ΠC follows from the properties of the functional ϕ(x,y) and strict monotonicity of the mapping J. It is obvious from the definition of function ϕ that (see [30])
(2.4)(‖y‖-‖x‖)2≤ϕ(y,x)≤(‖y‖+‖x‖)2,∀x,y∈E.
If E is a Hilbert space, then ϕ(x,y)=∥x-y∥2.IfE is a reflexive, strictly convex, and smooth Banach space, then for x,y∈E, ϕ(x,y)=0 if and only if x=y. It is sufficient to show that if ϕ(x,y)=0, then x=y. From (2.4), 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 [28, 29] for more details.We also need the following lemmas for the proof of our main results.Lemma 2.1 (see Kamimura and Takahashi [31]).
LetE be a uniformly convex and smooth real Banach space, and let {xn},{yn} be two sequences of E. If ϕ(xn,yn)→0 and either {xn} or {yn} is bounded, then ∥xn-yn∥→0.Lemma 2.2 (see Alber [30]).
LetC be a nonempty closed convex subset of a smooth Banach space E and x∈E. Then, x0=ΠCx if and only if
(2.5)〈x0-y,Jx-Jx0〉≥0,∀y∈C.Lemma 2.3 (see Alber [30]).
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
(2.6)ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x),∀y∈C.Lemma 2.4 (see Chang et al. [26]).
LetE be a uniformly convex Banach space, r>0 a positive number, and Br(0) a closed ball of E. Then, for any given sequence {xi}i=1∞⊂Br(0) and for any given sequence {λi}i=1∞ of positive number with ∑n=1∞‍λn=1, there exists a continuous, strictly increasing, and convex function g:[0,2r)→[0,∞) with g(0)=0 such that for any positive integers i,j with i<j,
(2.7)‖∑n=1∞‍λnxn‖2≤∑n=1∞‍λn‖xn‖2-λiλjg(‖xi-xj‖).Lemma 2.5 (see Chang et al. [26]).
LetE be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed convex subset of E. Let T:C→C be a closed and asymptotically relatively nonexpansive mapping with a sequence {kn}⊂[1,∞),kn→1. Then F(T) is closed and convex subset of C.For solving the generalized mixed equilibrium problem (or a system of generalized mixed equilibrium problem), let us assume that the bifunctionf:C×C→ℝ and ψ:C→ℝ is convex and lower semicontinuous satisfies the following conditions:  (A1)
f(x,x)=0 for all x∈C;(A2)
f is monotone, that is, f(x,y)+f(y,x)≤0 for all x,y∈C;(A3)
for eachx,y,z∈C,
(2.8)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 2.6 (see Chang et al. [26]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Let A:C→E* be a continuous and monotone mapping, ψ:C→ℝ is convex and lower semicontinuous and f be a bifunction from C×C to ℝ satisfying (A1)–(A4). For r>0 and x∈E, then there exists u∈C such that
(2.9)f(u,y)+〈Au,y-u〉+ψ(y)-ψ(u)+1r〈y-u,Ju-Jx〉≥0,∀y∈C.
Define a mapping Kf,r:C→C as follows:
(2.10)Kf,r(x)={u∈C:f(u,y)+〈Au,y-u〉+ψ(y)-ψ(u)+1r〈y-u,Ju-Jx〉≥0,∀y∈C}
for all x∈E. Then, the following hold:  (i)
Kf,r is singlevalued;(ii)
Kf,r is firmly nonexpansive, that is, for all x,y∈E, 〈Kf,rx-Kf,ry,JKf,rx-JKf,ry〉≤〈Kf,rx-Kf,ry,Jx-Jy〉;(iii)
F(Kf,r)=F(Kf,r)~;(iv)
u∈C is a solution of variational equation (2.9) if and only if u∈C is a fixed point of Kf,r;(v)
F(Kf,r)=Ω;(vi)
Ω is closed and convex;(vii)
ϕ(p,Kf,rz)+ϕ(Kf,rz,z)≤ϕ(p,z), forallp∈F(Kf,r), z∈E.


## 3. Main Results

Theorem 3.1.
LetE be a uniformly smooth and uniformly convex Banach space, let C be a nonempty, closed, and convex subset of E. Let Ai:C→E* be a continuous and monotone mapping, ψi:C→ℝ be a lower semi-continuous and convex function, fi be a bifunction from C×C to ℝ satisfying (A1)–(A4), Kfi,ri is the mapping defined by (2.10) where ri≥r>0, and let {Ti}i=1∞, {Si}i=1∞ be countable families of closed and uniformly Li, μi-Lipschitz continuous and asymptotically relatively nonexpansive mapping with sequence {kn},{ζn}⊂[1,∞);  kn→1,ζn→1 such that ℱ:=(∩i=1NΩi)∩(∩i=1∞F(Ti))∩(∩i=1∞F(Si))≠∅. Let {xn} be a sequence generated by x0∈C and C0=C,
(3.1)yn=J-1(βn,0J(xn)+∑i=1∞‍βn,iJ(Tinxn)),zn=J-1(αn,0J(xn)+∑i=1∞‍αn,iJ(Sinyn)),un(i)=Kfi,riKfi-1,ri-1⋯Kf1,r1(zn),i=1,2,…,N,Cn+1={z∈Cn:maxi=1,2,…,Nϕ(z,un(i))≤ϕ(z,xn)+θn,ϕ(z,yn)≤ϕ(z,xn)+ξn},xn+1=ΠCn+1x0,∀n≥0,
where ξn=supp∈ℱ(kn-1)ϕ(p,xn), θn=δn+ξnζn, and δn=supp∈ℱ(ζn-1)ϕ(p,xn). The coefficient sequences {αn,i} and {βn,i}⊂[0,1] satisfy the following:  (i)
∑i=0∞‍αn,i=1;(ii)
∑i=0∞‍βn,i=1;(iii)
liminfn→∞αn,0αn,i>0, foralli≥1;(iv)
liminfn→∞βn,0βn,i>0, foralli≥1,Ωi,i=1,2,…,N is the set of solutions to the following generalized mixed equilibrium problem:
(3.2)fi(z,y)+〈Aiz,y-z〉+ψi(y)-ψi(z)≥0,∀y∈C,i=1,2,…,N.
Then the sequence {xn} converges strongly to Πℱx0.Proof.
We first show thatCn, foralln≥0 is closed and convex. Clearly C0=C is closed and convex. Suppose that Ck is closed and convex for some k>1. For each z∈Ck, we see that ϕ(z,uk(i))≤ϕ(z,xk) is equivalent to
(3.3)2(〈z,xk〉-〈z,uk(i)〉)≤‖xk‖2-‖uk(i)‖2.
By the set of Ck+1, we have
(3.4)Cn+1={z∈Cn:max⁡i=1,2,…,Nϕ(z,un(i))≤ϕ(z,xn)+θn}=⋂i=1N{z∈C:ϕ(z,un(i))≤ϕ(z,xn)+θn}.
Hence, Cn+1 is also closed and convex.
By takingΘnj=Krj,fiKrj-1,fj-1⋯Kr1,f1 for any j∈{1,2,…,i} and Θn0=I for all n≥1. We note that un(i)=Θnizn.
Next, we show thatℱ⊂Cn,foralln≥1. For n≥1, we have ℱ⊂C=C1. For any given p∈ℱ:=(∩i=1NΩi)∩(∩i=1∞F(Ti))∩(∩i=1∞F(Si)). By (3.1) and Lemma 2.4, we have
(3.5)ϕ(p,yn)=ϕ(p,J-1(∑i=0∞‍βn,iJTinxn))=‖p‖2-∑i=0∞‍βn,i2〈p,JTinxn〉+‖∑i=0∞‍βn,iJTinxn‖2≤‖p‖2-∑i=0∞‍βn,i2〈p,JTinxn〉+∑i=0∞‍βn,i‖JTinxn‖2-βn,0βn,ig(‖JT0nxn-JTinxn‖)=‖p‖2-∑i=0∞‍βn,i2〈p,JTinxn〉+∑i=0∞‍βn,i‖Tinxn‖2-βn,0βn,ig(‖Jxn-JTinxn‖)=∑i=0∞‍βn,iϕ(p,Tinxn)-βn,0βn,ig(‖Jxn-JTinxn‖)≤knϕ(p,xn)-βn,0βn,ig(‖Jxn-JTinxn‖)≤ϕ(p,xn)+sup⁡p∈F(kn-1)ϕ(p,xn)-βn,0βn,ig(‖Jxn-JTinxn‖)≤ϕ(p,xn)+ξn-βn,0βn,ig(‖Jxn-JTinxn‖)≤ϕ(p,xn)+ξn,
where ξn=supp∈ℱ(kn-1)ϕ(p,xn).
By (3.1) and (3.5), we note that
(3.6)ϕ(p,un(i))=ϕ(p,Θnizn)≤ϕ(p,zn)≤ϕ(p,J-1(αn,0Jxn+∑i=1∞‍JSinyn))=‖p‖2-2〈p,αn,0Jxn+∑i=1∞‍JSinyn〉+‖αn,0Jxn+∑i=1∞‍JSinyn‖2≤‖p‖2-2αn,0〈p,Jxn〉-2∑i=1∞‍αn,i〈p,JSinyn〉+αn,0‖xn‖2+∑i=1∞‍‖Sinyn‖2-αn,0αn,ig‖Jxn-JSinyn‖≤αn,0ϕ(p,xn)+∑i=1∞‍αn,iϕ(p,Sinyn)-αn,0αn,ig‖Jxn-JSinyn‖≤αn,0ϕ(p,xn)+ζn∑i=1∞‍αn,iϕ(p,yn)-αn,0αn,ig‖Jxn-JSinyn‖≤αn,0ϕ(p,xn)+ζn∑i=1∞‍αn,i(ϕ(p,xn)+ξn)-αn,0αn,ig‖Jxn-JSinyn‖≤αn,0ϕ(p,xn)+ζn∑i=1∞‍αn,iϕ(p,xn)+ξnζn∑i=1∞‍αn,i-αn,0αn,ig‖Jxn-JSinyn‖≤ζnϕ(p,xn)+ξnζn∑i=1∞‍αn,i-αn,0αn,ig‖Jxn-JSinyn‖≤ϕ(p,xn)+sup⁡p∈F(ζn-1)ϕ(p,xn)+ξnζn∑i=1∞‍αn,i-αn,0αn,ig‖Jxn-JSinyn‖≤ϕ(p,xn)+δn+ξnζn-αn,0αn,ig‖Jxn-JSinyn‖≤ϕ(p,xn)+θn,
where δn=supp∈ℱ(ζn-1)ϕ(p,xn), θn=δn+ξnζn. By assumptions on {kn} and {ζn}, we have
(3.7)ξn=sup⁡p∈F(kn-1)ϕ(p,xn)≤sup⁡p∈F(kn-1)(‖p‖+M)2⟶0asn⟶∞,(3.8)δn=supp∈ℱ(ζn-1)ϕ(p,xn)≤supp∈ℱ(ζn-1)(∥p∥+M)2⟶0asn⟶∞,
where M=supn≥0∥xn∥.
So, we havep∈Cn+1. This implies that ℱ∈Cn,foralln≥0 and also {xn} is well defined.
From Lemma2.2 and xn=ΠCnx0, we have
(3.9)〈xn-z,Jx0-Jxn〉≥0,∀z∈Cn,〈xn-p,Jx0-Jxn〉≥0,∀p∈Cn.
From Lemma 2.3, one has
(3.10)ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(p,x0)-ϕ(p,xn)≤ϕ(p,x0)
for all p∈ℱ⊂Cn and n≥1. Then, the sequence {ϕ(xn,x0)} is also bounded. Thus {xn} is bounded. Since xn=ΠCnx0 and xn+1=ΠCn+1x0∈Cn+1⊂Cn, we have
(3.11)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n∈N.
Therefore, {ϕ(xn,x0)} is nondecreasing. Hence, the limit of {ϕ(xn,x0)} exists. By the construction of Cn, one has that Cm⊂Cn and xm=ΠCmx0∈Cn for any positive integer m≥n. It follows that
(3.12)ϕ(xm,xn)=ϕ(xm,ΠCnx0)≤ϕ(xm,x0)-ϕ(ΠCnx0,x0)=ϕ(xm,x0)-ϕ(xn,x0).
Letting m,n→0 in (3.12), we get ϕ(xm,xn)→0. It follows from Lemma 2.1, that ∥xm-xn∥→0 as m,n→∞. That is, {xn} is a Cauchy sequence.
Since{xn} is bounded and E is reflexive, there exists a subsequence {xni}⊂{xn} such that xni⇀u. Since Cn is closed and convex and Cn+1⊂Cn, this implies that Cn is weakly closed and u∈Cn for each n≥0. since xn=ΠCnx0, we have
(3.13)ϕ(xni,x0)≤ϕ(u,x0),∀ni≥0.
Since
(3.14)lim⁡⁡inf⁡ni→∞ϕ(xni,x0)=lim⁡⁡inf⁡ni→∞{‖xni‖2-2〈xni,Jx0〉+‖x0‖2}≤‖u‖2-2〈u,Jx0〉+‖x0‖2=ϕ(u,x0).
We have
(3.15)ϕ(u,x0)≤lim⁡⁡inf⁡ni→∞ϕ(xni,x0)≤lim⁡⁡sup⁡ni→∞ϕ(xni,x0)≤ϕ(u,x0).
This implies that limni→∞ϕ(xni,x0)=ϕ(u,x0). That is, ∥xni∥→∥u∥. Since xni⇀u, by the Kadec-klee property of E, we obtain that
(3.16)lim⁡n→∞xni=u.
If there exists some subsequence {xnj}⊂{xn} such that xnj→q, then we have
(3.17)ϕ(u,q)=lim⁡ni→∞,nj→∞ϕ(xni,xnj)≤lim⁡ni→∞,nj→∞(ϕ(xni,x0)-ϕ(ΠCnjx0,x0))=lim⁡ni→∞,nj→∞(ϕ(xni,x0)-ϕ(xnjx0,x0))=0.
Therefore, we have u=q. This implies that
(3.18)lim⁡n→∞xn=u.
Since
(3.19)ϕ(xn+1,xn)=ϕ(xn+1,ΠCnx0)≤ϕ(xn+1,x0)-ϕ(ΠCnx0,x0)=ϕ(xn+1,x0)-ϕ(xn,x0)
for all n∈ℕ, we also have
(3.20)lim⁡n→∞ϕ(xn+1,xn)=0.
Since xn+1=ΠCn+1x0∈Cn+1 and by the definition of Cn+1, for i=1,2,…,N, we have
(3.21)ϕ(xn+1,uni)≤ϕ(xn+1,xn)+θn.
Noticing that limn→∞ϕ(xn+1,xn)=0, we obtain
(3.22)lim⁡n→∞ϕ(xn+1,uni)=0,fori=1,2,…,N.
It then yields that limn→∞(∥xn+1∥-∥uni∥)=0,foralli=1,2,…,N. Since limn→∞∥xn+1∥=∥u∥, we have
(3.23)lim⁡n→∞‖uni‖=‖u‖,∀i=1,2,…,N.
Hence,
(3.24)lim⁡n→∞‖Juni‖=‖Ju‖,∀i=1,2,…,N.
From Lemma 2.1 and (3.22), we have
(3.25)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞‖xn+1-uni‖=0,∀i=1,2,…,N.
By the triangle inequality, we get
(3.26)lim⁡n→∞‖xn-uni‖=0,∀i=1,2,…,N.
Since J is uniformly norm-to-norm continuous on bounded sets, we note that
(3.27)lim⁡n→∞‖Jxn-Juni‖=lim⁡n→∞‖Jxn+1-Juni‖=0,∀i=1,2,…,N.
Now, we prove thatu∈(∩i=1∞F(Ti))∩(∩i=1∞F(Si)). From the construction of Cn, we obtain that
(3.28)ϕ(xn+1,yn)≤ϕ(xn+1,xn)+ξn.
From (3.7) and (3.20), we have
(3.29)lim⁡n→∞ϕ(xn+1,yn)=0.
By Lemma 2.1, we also have
(3.30)lim⁡n→∞‖xn+1-yn‖=0.
Since J is uniformly norm-to-norm continuous on bounded sets, we note that
(3.31)lim⁡n→∞‖Jxn+1-Jyn‖=0.
From (2.4) and (3.29), we have (∥xn+1∥-∥yn∥)2→0. Since ∥xn+1∥→∥u∥, it yields that
(3.32)‖yn‖⟶‖u‖asn⟶∞.
Since J is uniformly norm-to-norm continuous on bounded sets, it follows that
(3.33)‖Jyn‖⟶‖Ju‖asn⟶∞.
This implies that {Jyn} is bounded in E*. Since E is reflexive, there exists a subsequence {Jyni}⊂{Jyn} such that Jyni⇀r∈E*. Since E is reflexive, we see that J(E)=E*. Hence, there exists x∈E such that Jx=r. We note that
(3.34)ϕ(xni+1,yni)=‖xni+1‖2-2〈xni+1,Jyni〉+‖yni‖2=‖xni+1‖2-2〈xni+1,Jyni〉+‖Jyni‖2.
Taking the limit interior of both side and in view of weak lower semicontinuity of norm ∥·∥, we have
(3.35)0≥‖u‖2-2〈u,r〉+‖r‖2=‖u‖2-2〈u,Jx〉+‖Jx‖2=‖u‖2-2〈u,Jx〉+‖x‖2=ϕ(u,x),
that is, u=x. This implies that r=Ju and so Jyn⇀Jp. It follows from limn→∞∥Jyn∥=∥Ju∥, as n→∞ and the Kadec-Klee property of E* that Jyni→Ju as n→∞. Note that J-1:E*→E is hemicontinuous, it yields that yni⇀u. It follows from limn→∞∥un∥=∥u∥, as n→∞ and the Kadec-Klee property of E that limni→∞yni=u.
By similar, we can prove that(3.36)lim⁡n→∞yn=u.
By (3.20) and (3.30), we obtain
(3.37)lim⁡n→∞‖xn-yn‖=0.
Since J is uniformly norm-to-norm continuous on bounded sets, we note that
(3.38)lim⁡n→∞‖Jxn-Jyn‖=0.
So, from (3.27) and (3.31), by the triangle inequality, we get
(3.39)lim⁡n→∞‖Jyn-Juni‖=0,fori=1,2,…,N.
Since J-1 is uniformly norm-to-norm continuous on bounded sets, we note that
(3.40)lim⁡n→∞‖yn-uni‖=0,fori=1,2,…,N.
Since
(3.41)ϕ(p,xn)-ϕ(p,yn)=‖xn‖2-‖yn‖2-2〈p,Jxn-Jyn〉≤‖xn‖2-‖yn‖2+2‖p‖‖Jxn-Jyn‖≤‖xn-yn‖(‖xn‖+‖yn‖)+2‖p‖‖Jxn-Jyn‖.
From (3.37) and (3.38), we obtain
(3.42)ϕ(p,xn)-ϕ(p,yn)⟶0,n⟶∞.
On the other hand, we observe that, for i=1,2,…,N. 							(3.43)ϕ(p,xn)-ϕ(p,uni)=‖xn‖2-‖uni‖2-2〈p,Jxn-Juni〉≤‖xn‖2-‖uni‖2+2‖p‖‖Jxn-Juni‖≤‖xn-uni‖(‖xn‖+‖uni‖)+2‖p‖‖Jxn-Juni‖.
From (3.22) and (3.27), we have
(3.44)ϕ(p,xn)-ϕ(p,uni)⟶0,n⟶∞,∀i=1,2,…,N.
For any p∈∩i=1NΩi∩(∩i=1∞F(Ti))∩(∩i=1∞F(Si)), it follows from (3.5) that
(3.45)βn,0βn,ig(‖Jxn-JTinxn‖)≤ϕ(p,xn)+ξn-ϕ(p,yn).From condition, liminfn→∞βn,0βn,i>0, property of g, (3.7), and (3.42), we have that
(3.46)‖Jxn-JTinxn‖⟶0,n⟶∞,∀i=1,2,…,N.
Since xn→u and J is uniformly norm-to-norm continuous. It yields Jxn→Jp. Hence from (3.46), we have
(3.47)‖xn-Tinxn‖⟶0,n⟶∞,∀i=1,2,…,N.
Since xn→u, this implies that limn→∞JTinxn→Ju as n→∞. Since J-1:E*→E is hemicontinuous, it follows that
(3.48)Tinxn⇀u,foreachi≥1.
On the other hand, for each i≥1, we have
(3.49)‖Tinxn‖-‖u‖=|‖Tinxn‖-‖u‖|≤‖Tinxn-u‖⟶0,n⟶∞.
from this, together with (3.48) and the Kadec-Klee property of E, we obtain
(3.50)Tinxn⟶u,foreachi≥1.
On the other hand, by the assumption that Ti is uniformly Li-Lipschitz continuous, we have
(3.51)‖Tin+1xn-Tinxn‖≤‖Tin+1xn-Tin+1xn+1‖+‖Tin+1xn+1-xn+1‖+‖xn+1-xn‖+‖xn-Tinxn‖≤(Li+1)‖xn+1-xn‖+‖Tin+1xn+1-xn+1‖+‖xn-Tinxn‖.
By (3.18) and (3.50), we obtain
(3.52)lim⁡n→∞‖Tin+1xn-Tinxn‖=0,∀i≥1,
and limn→∞Tin+1xn=u, that is, TiTnxn→u, foralli≥1. By the closeness of Ti, we have Tiu=u, foralli≥1. This implies that u∈∩i=1∞F(Ti).
By the similar way, we can prove that for eachi≥1(3.53)‖Jxn-JSinyn‖⟶0,n⟶∞.
Since xn→u and J is uniformly norm-to-norm continuous. it yields Jxn→Jp. Hence from (3.53), we have
(3.54)‖xn-Sinyn‖⟶0,n⟶∞.
Since xn→u, this implies that limn→∞JSinyn→Ju as n→∞. Since J-1:E*→E is hemicontinuous, it follows that
(3.55)Sinyn⇀u,foreachi≥1.
On the other hand, for each i≥1, we have
(3.56)‖Sinyn‖-‖u‖=|‖Sinyn‖-‖u‖|≤‖Sinyn-u‖⟶0,n⟶∞.
From this, together with (3.54) and the Kadec-Klee property of E, we obtain
(3.57)Sinyn⟶u,foreachi≥1.
On the other hand, by the assumption that Si is uniformly μi-Lipschitz continuous, we have
(3.58)‖Sin+1yn-Sinyn‖≤‖Sin+1yn-Sin+1yn+1‖+‖Sin+1yn+1-yn+1‖+‖yn+1-yn‖+‖yn-Sinyn‖≤(μi+1)‖yn+1-yn‖+‖Sin+1yn+1-yn+1‖+‖yn-Sinyn‖.
By (3.36) and (3.57), we obtain
(3.59)lim⁡n→∞‖Sin+1yn-Sinyn‖=0
and limn→∞Sin+1yn=u, that is, SiTnyn→u. By the closeness of Si, we have Siu=u, foralli≥1. This implies that u∈∩i=1∞F(Si). Hence u∈(∩i=1∞F(Ti))∩(∩i=1∞F(Si)).
Next, we prove thatu∈∩i=1NΩi. For any p∈ℱ, for each i=1,2,…,N, we have
(3.60)ϕ(uni,zn)=ϕ(Θnizn,zn)≤ϕ(p,zn)-ϕ(p,Θnizn)=ϕ(p,zn)-ϕ(p,uni)≤ϕ(p,xn)+θn-ϕ(p,uni)→0,asn→∞.
It then yields that limn→∞(∥uni∥-∥zn∥)=0. Since limn→∞∥uni∥=∥u∥, foralli≥1, we have
(3.61)lim⁡n→∞‖zn‖=‖u‖.
Hence,
(3.62)lim⁡n→∞‖Jzn‖=‖Ju‖.
This together with limn→∞∥uni∥=∥u∥ show that for each i=1,2,…,N,
(3.63)lim⁡n→∞‖uni-uni-1‖=lim⁡n→∞‖Juni-Juni-1‖=0,
where un0=zn. On the other hand, we have
(3.64)uni=Kfi,riuni-1,foreachi=2,3,…,N,
and uni is a solution of the following variational equation
(3.65)fi(uni,y)+〈Aiuni,y-uni〉+ψi(y)-ψi(uni)+1ri〈y-uni,Juni-Juni-1〉≥0,∀y∈C.
By condition (A2), we note that
(3.66)〈Aiuni,y-uni〉+ψi(y)-ψi(uni)+1ri〈y-uni,Juni-Juni-1〉≥-fi(uni,y)≥fi(y,uni),∀y∈C.
By (A4), (3.63), and uni→u for each i=2,3,…,N, we have
(3.67)〈Aiu,y-u〉+ψi(y)-ψi(u)≥fi(y,u),∀y∈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
(3.68)〈Aiu,yt-u〉+ψi(yt)-ψi(u)≥fi(yt,u).
In view of the convexity of ϕ it yields
(3.69)t〈Aiu,y-u〉+t(ψi(y)-ψi(u))≥fi(yt,u).
It follows from (A1) and (A4) that
(3.70)0=fi(yt,yt)≤tfi(yt,y)+(1-t)fi(yt,u)≤tfi(yt,y)+(1-t)t[〈Aiu,y-u〉+(ψi(y)-ψi(u))].
Let t→0, from (A3), we obtain the following:
(3.71)fi(u,y)+〈Aiu,y-u〉+ψi(y)-ψi(u)≥0,∀y∈C,i=1,2,…,N.
This implies that u is a solution of the system of generalized mixed equilibrium problem (3.2), that is, u∈∩i=1NΩi. Hence, u∈ℱ:=(∩i=1NΩi)∩(∩i=1∞F(Ti))∩(∩i=1∞F(Si)).
Finally, we show thatxn→u=ΠFx0. Indeed from w∈F⊂Cn and xn=ΠCnx0, we have the following:
(3.72)ϕ(xn,x0)≤ϕ(w,x0),∀n≥0.
This implies that
(3.73)ϕ(u,x0)=lim⁡n→∞ϕ(xn,x0)≤ϕ(w,x0).
From the definition of ΠFx0 and (3.73), we see that u=w. This completes the proof.Since every asymptotically relatively nonexpansive mappings is quasi-ϕ-nonexpansive mappings, hence we obtain the following corollary.Corollary 3.2.
LetE be a uniformly convex and uniformly smooth Banach space, let C be a nonempty, closed, and convex subset of E. Let Ai:C→E* be a continuous and monotone mapping, ψi:C→ℝ be a lower semi-continuous and convex function, fi be a bifunction from C×C to ℝ satisfying (A1)–(A4), Kfi,ri is the mapping defined by (2.10) where ri≥r>0, and let {Ti}i=1∞, {Si}i=1∞ be countable families of closed and quasi-ϕ-nonexpansive mapping such that ℱ:=(∩i=1NΩi)∩(∩i=1∞F(Ti))∩(∩i=1∞F(Si))≠∅. Let {xn} be a sequence generated by x0∈C and C0=C, such that
(3.74)yn=J-1(βn,0J(xn)+∑i=1∞‍βn,iJ(Tixn)),zn=J-1(αn,0J(xn)+∑i=1∞‍αn,iJ(Siyn)),un(i)=Kfi,riKfi-1,ri-1⋯Kf1,r1(zn),i=1,2,…,N,Cn+1={z∈Cn:maxi=1,2,…,Nϕ(z,un(i))≤ϕ(z,xn),ϕ(z,yn)≤ϕ(z,xn)},xn+1=ΠCn+1x0,∀n≥0,
where ΠC is the generalized projection from E onto C, J is the duality mapping on E. The coefficient sequences {αn,i} and {βn,i}⊂[0,1], satisfying: (i)
∑i=0∞‍αn,i=1;(ii)
∑i=0∞‍βn,i=1;(iii)
liminfn→∞αn,0αn,i>0, foralli≥1;(iv)
liminfn→∞βn,0βn,i>0, foralli≥1.Ωi,i=1,2,…,N is the set of solutions to the following generalized mixed equilibrium problem:
(3.75)fi(z,y)+〈Aiz,y-z〉+ψi(y)-ψi(z)≥0,∀y∈C,i=1,2,…,N.
Then the sequence {xn} converges strongly to Πℱx0.IfAi=A,ψi=ψ, and fi=f for all i≥1 in Theorem 3.1, we obtain the following corollary.Corollary 3.3.
LetE be a uniformly smooth and uniformly convex Banach space, let C be a nonempty, closed, and convex subset of E. Let A:C→E* be a continuous and monotone mapping, ψ:C→ℝ be a lower semicontinuous and convex function, f be a bifunction from C×C to ℝ satisfying (A1)–(A4), Kf,r be the mapping define by (2.10) where r>0, and let {Ti}i=1∞, {Si}i=1∞ be countable families of closed and uniformly Li, μi-Lipschitz continuous, and asymptotically relatively nonexpansive mappings with sequence {kn},{ζn}⊂[1,∞);  kn→1,ζn→1 such that ℱ:=Ω∩(∩i=1∞F(Ti))∩(∩i=1∞F(Si))≠∅. Let {xn} be a sequence generated by x0∈C and C0=C, such that
(3.76)yn=J-1(βn,0J(xn)+∑i=1∞‍βn,iJ(Tinxn)),zn=J-1(αn,0J(xn)+∑i=1∞‍αn,iJ(Sinyn)),un=Kf,rzn,Cn+1={z∈Cn:maxi=1,2,…,Nϕ(z,un(i))≤ϕ(z,xn)+θn,ϕ(z,yn)≤ϕ(z,xn)+ξn},xn+1=ΠCn+1x0,∀n≥0,
where ξn=supp∈ℱ(kn-1)ϕ(p,xn), θn=δn+ξnζn, and δn=supp∈ℱ(ζn-1)ϕ(p,xn). The coefficient sequences {αn,i} and {βn,i}⊂[0,1], satisfying: (i)
∑i=0∞‍αn,i=1;(ii)
∑i=0∞‍βn,i=1;(iii)
liminfn→∞αn,0αn,i>0, foralli≥1;(iv)
liminfn→∞βn,0βn,i>0, foralli≥1. Then the sequence {xn} converges strongly to Πℱx0.Ifi=1 in Theorem 3.1, then we obtain the following corollary.Corollary 3.4.
LetE be a uniformly smooth and uniformly convex Banach space, let C be a nonempty, closed, and convex subset of E. Let A:C→E* be a continuous and monotone mapping, ψ:C→ℝ be a lower semicontinuous and convex function, f be a bifunction from C×C to ℝ satisfying (A1)–(A4), Kf,r is the mapping define by (2.10) where r>0, and let T,S are two closed and uniformly L, μ-Lipschitz continuous and asymptotically relatively nonexpansive mappings with sequence {kn},{ζn}⊂[1,∞);  kn→1,ζn→1 such that ℱ:=Ω∩F(T)∩F(S)≠∅. Let {xn} be a sequence generated by x0∈C and C0=C, we have(3.77)yn=J-1(βnJxn+(1-βn)JTnxn),zn=J-1(αnJxn+(1-αn)JSnyn),un=Kf,rzn,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+θn,ϕ(z,yn)≤ϕ(z,xn)+ξn},xn+1=ΠCn+1x0,∀n≥0,
where ξn=supp∈ℱ(kn-1)ϕ(p,xn), θn=δn+ξnζn, and δn=supp∈ℱ(ζn-1)ϕ(p,xn). The coefficient sequences {αn} and {βn}⊂[0,1], satisfying  (D1)
liminfn→∞αn(1-αn)>0;(D2)
liminfn→∞βn(1-βn)>0. Then the sequence {xn} converges strongly to Πℱx0.Remark 3.5.
Theorem3.1 and Corollary 3.3 improve and extend the corresponding results of Petrot et al. [24], Kumam and Wattanawitoon [25], and Chang et al. [26] in the following senses:  (i)
for the mappings, we extend the mappings from nonexpansive mappings, hemirelatively nonexpansive mappings to two infinite family of closed asymptotically relatively nonexpansive mappings;(ii)
from a solution of the classical equilibrium problem to a system of generalized mixed equilibrium problems and the generalized mixed equilibrium problem with an infinite family of closed relatively nonexpansive mappings.Remark 3.6.
Corollary3.4 improves and extends the corresponding results of Theorem 3.1 in Kumam and Wattanawitoon [25] and Corollary 3.3 in Saewan et al. [11] in the following senses:(i)
the mapping in [11] and [25](ii)
the conditions (D1) and (D2) of the parameters{αn} and {βn} are weaker and not complicated than the conditions (C1)–(C3) in [[25], Theorem  3.1] and [[11], Theorem  3.1] which are easy to compute.



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*Source: 395760-2012-07-11.xml*