# A New Hybrid Algorithm for a System of Generalized Mixed Equilibrium Problems and Fixed Point of Quasi-ϕ-Asymptotically Nonexpansive Mappings

Authors: Jinhua Zhu; Shih-Sen Chang
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

The purpose of this paper is to use a new hybrid algorithm for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for an infinite family of quasi-ϕ-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.


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## Body

## 1. Introduction

Throughout this paper, we denote byℕ and ℝ the sets of positive integers and real numbers, respectively. We also assume that E is a real Banach space, E* is the dual space of E, C is a nonempty closed convex subset of E, and 〈·,·〉 is the pairing between E and E*. In the sequel, we denote the strong convergence and weak convergence of a sequence {xn} by xn→x and xn⇀x, respectively.Letψ:C→ℝ be a real-valued function, A:C→E* a nonlinear mapping, and {Θi:C×C→ℝ,i=1,2,…,N} a finite family of equilibrium bifunctions, that is, Θi(u,u)=0 for each u∈C. The “so-called” system of generalized mixed equilibrium problems (SGMEP) for functions (Θ1,Θ2,…,ΘN,A,ψ) is to find a common element x*∈C such that
(1.1)Θ1(x*,y)+〈Ax*,y-x*〉+ψ(y)-ψ(x*)≥0,∀y∈C,Θ2(x*,y)+〈Ax*,y-x*〉+ψ(y)-ψ(x*)≥0,∀y∈C,⋮ΘN(x*,y)+〈Ax*,y-x*〉+ψ(y)-ψ(x*)≥0,∀y∈C.We denote the set of solutions of (1.1) by Ω:=⋂i=1NΩ(Θi,ψ), where Ω(Θi,ψ) is the set of solutions to the generalized mixed equilibrium problem:
(1.2)Θi(x*,y)+〈Ax*,y-x*〉+ψ(y)-ψ(x*)≥0,∀y∈C.Special examples are as follows.(i)
IfA=0 and N=1, the problem (1.1) is equivalent to finding x*∈C such that
(1.3)Θ(x*,y)+ψ(y)-ψ(x*)≥0,∀y∈C,which is called the mixed equilibrium problem (MEP) [1]. The set of solutions to (1.3) is denoted by MEP.(ii)
IfΘ=0 and N=1, the problem (1.1) is equivalent to finding x*∈C such that
(1.4)〈Ax*,y-x*〉+ψ(y)-ψ(x*)≥0,∀y∈C,which is called the mixed variational inequality of Browder type (VI) [2]. The set of solutions to (1.4) is denoted by VI(C,A,ψ).Recently, many authors studied the problems of finding a common element of the set of fixed point for a nonexpansive mapping and the set of solutions for an equilibrium problem in the setting of Hilbert space and uniformly smooth and uniformly convex Banach space, respectively (see, e.g., [3–5] and the references therein).Motivated and inspired by the researches going on in this direction, the purpose of this paper is using a hybrid algorithm for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for an infinite family of quasi-ϕ-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results improve and extend the corresponding results in [6–12].


## 2. Preliminaries

First, we recall some definitions and conclusions.The mappingJ:E→2E* defined by
(2.1)J(x)={x*∈E*:〈x,x*〉=‖x‖=‖x*‖},x∈E,
is called the normalized duality mapping. By the Hahn-Banach theorem, J(x)≠∅ for each x∈E.A Banach spaceE is said to be strictly convex if ∥x+y∥/2<1 for all x,y∈U={z∈E:∥z∥=1} with x≠y. E is said to be uniformly convex if, for each ϵ∈(0,2], there exists δ>0 such that ∥x+y∥/2<1-δ for all x,y∈U with ∥x-y∥≥ϵ. E is said to be smooth if the limit
(2.2)lim⁡t→0‖x+ty‖-‖x‖t
exists for all x,y∈U. E is said to be uniformly smooth if the above limit exists uniformly in x,y∈U.Remark 2.1.
The following basic properties can be found in Cioranescu [13]. (i)
IfE is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E.(ii)
IfE is a reflexive and strictly convex Banach space, then J-1 is norm-weak continuous.(iii)
IfE is a smooth, strictly convex, and reflexive Banach space, then J is single-valued, one-to-one, and onto.(iv)
A Banach spaceE is uniformly smooth if and only if E* is uniformly convex.(v)
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.
Next we assume thatE is a smooth, strictly convex, and reflexive Banach space and C is a nonempty closed convex subset of E. In the sequel, we always use ϕ:E×E→ℝ+ to denote the Lyapunov functional defined by
(2.3)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2,∀x,y∈E.
It is obvious from the definition ofϕ that
(2.4)(‖x‖-‖y‖)2≤ϕ(x,y)≤(‖x‖+‖y‖)2,∀x,y∈E.
Following Alber [14], the generalized projection ΠC:E→C is defined by
(2.5)ΠC(x)=arginf⁡y∈Cϕ(y,x),∀x∈E.Lemma 2.2 (see [14, 15]).
LetE be a smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset of E. Then the following conclusions hold:(a)
ϕ(x,ΠCy)+ϕ(ΠCy,y)≤ϕ(x,y), for allx∈Candy∈E;(b)
ifx∈E and z∈C, then
(2.6)z=ΠCx⟺〈z-y,Jx-Jz〉≥0,∀y∈C;(c)
forx,y∈E, ϕ(x,y)=0 if and only x=y.Remark 2.3.
IfE is a real Hilbert space H, then ϕ(x,y)=∥x-y∥2 and ΠC is the metric projection PC of H onto C.
LetE be a smooth, strictly convex, and reflexive Banach space, C a nonempty closed convex subset of E,T:C→C a mapping, and F(T) the set of fixed points of T. A point p∈C is said to be an asymptotic fixed pointof T if there exists a sequence {xn}⊂C such that xn⇀p and ∥xn-Txn∥→0. We denoted the set of all asymptotic fixed points of T by F̃(T).Definition 2.4 (see [16]).
(1) A mappingT:C→C is said to be relatively nonexpansive if F(T)≠∅,F(T)=F̃(T), and
(2.7)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).
(2) A mappingT:C→Cis said to beclosed if, for any sequence {xn}⊂Cwithxn→x andTxn→y, thenTx=y.Definition 2.5 (see [9]).
(1) A mappingT:C→C is said to be quasi-ϕ-nonexpansive if F(T)≠∅ and
(2.8)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).
(2) A mapping T:C→C is said to be quasi-ϕ-asymptotically nonexpansive if F(T)≠∅ and there exists a real sequence {kn}⊂[1,∞) with kn→1 such that
(2.9)ϕ(p,Tnx)≤knϕ(p,x),∀n≥1,x∈C,p∈F(T).
(3) A mapping T:C→C is said to be uniformly L-Lipschitz continuous if there exists a constant L>0 such that
(2.10)‖Tnx-Tny‖≤L‖x-y‖,∀x,y∈C,∀n≥1.Remark 2.6.
(1) From the definition, it is easy to know that each relatively nonexpansive mapping is closed.
(2) The class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-nonexpansive mappings as a subclass, and the class of quasi-ϕ-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.Lemma 2.7 (see [6]).
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 numbers with ∑i=1∞λi=1, then 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,
(2.11)‖∑n=1∞λnxn‖2≤∑n=1∞λn‖xn‖2-λiλjg(‖xi-xj‖).Lemma 2.8 (see [6]).
LetE be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and C a nonempty closed convex subset of E. Let T:C→C be a closed and quasi-ϕ-asymptotically nonexpansive mapping with a sequence {kn}⊂[1,∞),kn→1. Then F(T) is a closed convex subset of C.For solving the system of generalized mixed equilibrium problems (1.1), let us assume that the function ψ:C→ℝ is convex and lower semicontinuous, the nonlinear mapping A:C→E* is continuous and monotone, and the bifunction Θi:C×C→ℝ,i=1,2,…,N satisfies the following conditions:(A1)
Θi(x,x)=0,forallx∈C;(A2)
Θi is monotone, that is, Θi(x,y)+Θi(y,x)≤0,forallx,y∈C;(A3)
limsupt↓0Θi(x+t(z-x),y)≤Θi(x,y)forallx,z,y∈C;(A4)
the functiony↦Θi(x,y) is convex and lower semicontinuous.Lemma 2.9.
LetE be a smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset of E. Let Θ:C×C→ℝ be a bifunction satisfying conditions (A1)–(A4). Let r>0 and x∈E. Then, the following hold.(i)
(Blum and Oettli [17]) There exists z∈C such that
(2.12)Θ(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.(ii)
(Takahashi and Zembayashi [3]) Define a mapping Tr:E→C by
(2.13)Tr(x)={z∈C:Θ(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},x∈E.Then, the following conclusions hold:(a)
Tr is single valued;(b)
Tr is firmly nonexpansive-type mapping, that is, for all z,y∈E,
(2.14)〈Trz-Try,JTrz-JTry〉≤〈Trz-Try,Jz-Jy〉;(c)
F(Tr)=EP(Θ)=F̃(Tr);(d)
EP(Θ) is closed and convex;(e)
ϕ(q,Trx)+ϕ(Trx,x)≤ϕ(q,x),forallq∈F(Tr).Lemma 2.10 (see [7]).
LetE be a smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset of E. Let A:C→E* be a continuous and monotone mapping, ψ:C→ℝ a lower semicontinuous and convex function, and Θ:C×C→ℝ a bifunction satisfying the conditions (A1)–(A4). Let r>0 be any given number and x∈E be any given point. Then, the following hold.(i)
There existsu∈C such that
(2.15)Θ(u,y)+〈Au,y-u〉+ψ(y)-ψ(u)+1r〈y-u,Ju-Jx〉≥0,∀y∈C.(ii)
If we define a mappingKrΘ:C→C by
(2.16)KrΘ(x)={1ru∈C:Θ(u,y)+〈Au,y-u〉+ψ(y)-ψ(u)+1r〈y-u,Ju-Jx〉≥0,∀y∈C},∀x∈C.then, the mapping KrΘ has the following properties:(a)
KrΘ is single valued;(b)
KrΘ is a firmly nonexpansive-type mapping, that is,
(2.17)〈KrΘz-KrΘy,JKrΘz-JKrΘy〉≤〈KrΘz-KrΘy,Jz-Jy〉,∀z,y∈E;(c)
F(KrΘ)=Ω(Θ,ψ)=F̃(KrΘ);(d)
Ω(Θ,ψ) is closed and convex;(e)
(2.18)ϕ(q,KrΘx)+ϕ(KrΘx,x)≤ϕ(q,x),∀q∈F(KrΘ).Remark 2.11.
It follows from Lemma2.9 that the mapping KrΘ is a relatively nonexpansive mapping. Thus, it is quasi-ϕ-asymptotically nonexpansive.


## 3. Main Results

In this section, we will use the hybrid method to prove some strong convergence theorems for finding a common element of the set of solutions for a system of the generalized mixed equilibrium problems (1.1) and the set of common fixed points for an infinite family of quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces.Theorem 3.1.
LetE be a uniformly smooth and strictly convex Banach space with Kleac-Klee property and C a nonempty closed convex subset of E. Let A:C→E* be a continuous and monotone mapping, ψ:C→ℝ a lower semicontinuous and convex function, and {Θi:C×C→ℝ,i=1,2,…,N} be a finite family of bifunction satisfying conditions (A1)–(A4). Let {Si}i=1∞:C→C an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞) and kn→1. Suppose that for each i≥1,Si is uniformly Li-Lipschitz continuous and that G:=⋂i=1∞F(Si)⋂Ω is a nonempty and bounded subset in C, where Ω=⋂i=1NΩ(Θi,ψ). Let {xn},{yn},{zn}, and {un(k)},k=1,2,…,N, be the sequences generated by
(3.1)x0∈C,C0=C,zn=J-1(αn,0Jxn+∑i=1∞αn,iJSinxn),yn=J-1(αnJzn+(1-αn)Jxn),un(N)∈Csuchthat,∀y∈C,ΘN(un(N),y)+〈Aun(N),y-un(N)〉+ψ(y)-ψ(un(N))+1rn〈y-un(N),Jun(N)-Jun(N-1)〉≥0,ΘN-1(un(N-1),y)+〈Aun(N-1),y-un(N-1)〉+ψ(y)-ψ(un(N-1))+1rn〈y-un(N-1),Jun(N-1)-Jun(N-2)〉≥0,⋮Θ1(un(1),y)+〈Aun(1),y-un(1)〉+ψ(y)-ψ(un(1))+1rn〈y-un(1),Jun(1)-Jyn〉≥0,Cn+1={v∈Cn:ϕ(v,un(N))≤ϕ(v,xn)+ξn},∀n≥0,xn+1=ΠCn+1x0,
where
(3.2)un(1)=KrnΘ1yn,un(i)=KrnΘiun(i-1)=KrnΘiKrnΘi-1un(i-2)=KrnΘi⋯KrnΘ2un(1)=KrnΘi⋯KrnΘ2KrnΘ1yn,i=1,2,…,N,
and KrnΘi:C→C, i=1,2,…,N is the mapping defined by (2.16),rn∈[d,∞) for some d>0, ξn=supu∈G(kn-1)ϕ(u,xn), ΠCn+1 is the generalized projection of E onto the set Cn+1, and for each n≥0, {αn,i}, {αn} are sequences in [0,1] satisfying the following conditions:(a)
∑i=0∞αn,i=1 for all n≥0;(b)
liminfn→∞αn,0·αn,i>0 for all i≥1;(c)
0<α≤αn<1 for some α∈(0,1).
Then{xn} converges strongly to ΠGx0, where ΠG is the generalized projection from E onto G.Proof.
We divide the proof of Theorem3.1 into five steps.(I)
We first prove thatG and Cn both are closed and convex subset of C for all n≥0.
In fact, it follows from Lemmas2.8 and 2.10 that F(Si), i≥1, and Ω both are closed and convex. Therefore G is a closed and convex subset in C. Furthermore, it is obvious that C0=C is closed and convex. Suppose that Cn is closed and convex for some n≥1. Since the inequality ϕ(v,un(N))≤ϕ(v,xn)+ξn is equivalent to
(3.3)2〈v,Jxn-Jun(N)〉≤‖xn‖2-‖un(N)‖2+ξn,
therefore, we have(3.4)Cn+1={v∈Cn:2〈v,Jxn-Jun(N)〉≤‖xn‖2-‖un(N)‖2+ξn}.
This implies thatCn+1 is closed and convex. The desired conclusions are proved. These in turn show that ΠGx0 and ΠCnx0 are well defined.
(II)
We prove that{xn} is a bounded sequence in C.By the definition of Cn, we have xn=ΠCnx0 for all n≥0. It follows from Lemma 2.2(a) that
(3.5)ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(u,x0)-ϕ(u,ΠCnx0)≤ϕ(u,x0),∀n≥0,u∈G.
This implies that{ϕ(xn,x0)} is bounded. By virtue of (2.4), {xn} is bounded. Denote by
(3.6)M=sup⁡n≥0⁡{‖xn‖}<∞.
In view of the structure of{Cn}, we have Cn+1⊂Cn, xn=ΠCnx0, and xn+1=ΠCn+1x0. This implies that xn+1∈Cn and
(3.7)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n≥1.
Therefore, {ϕ(xn,x0)} is convergent. Without loss of generality, we can assume that
(3.8)lim⁡n→∞ϕ(xn,x0)=r≥0.
(III)
Next, we prove thatG:=⋂i=1∞F(Si)⋂(⋂i=1NΩ(Θi,ψ))⊂Cn for all n≥0.
Indeed, it is obvious thatG⊂C0=C. Suppose that G⊂Cn for some n≥0. Since un(N)=KrnΘNKrnΘN-1⋯KrnΘ2KrnΘ1yn, by Lemma 2.10 and Remark 2.6, KrnΘi is quasi-ϕ-asymptotically nonexpansive. Since E is uniformly smooth, E* is uniformly convex. For any given u∈G⊂Cn and for any positive integer j>0, from Lemma 2.7 we have
(3.9)ϕ(u,un(N))=ϕ(u,KrnΘNKrnΘN-1⋯KrnΘ2KrnΘ1yn)≤ϕ(u,yn)=ϕ(u,J-1(αnJzn+(1-αn)Jxn))≤‖u‖2-2〈u,αnJzn+(1-αn)Jxn〉+‖αnJzn+(1-αn)Jxn‖2≤‖u‖2-2〈u,αnJzn+(1-αn)Jxn〉+αn‖Jzn‖2+(1-αn)‖Jxn‖2=‖u‖2-2〈u,αnJzn+(1-αn)Jxn〉+αn‖zn‖2+(1-αn)‖xn‖2=αnϕ(u,zn)+(1-αn)ϕ(u,xn)=αnϕ(u,J-1(αn,0Jxn+∑i=1∞αn,iJSinxn))+(1-αn)ϕ(u,xn)=αn(‖u‖2-2αn,0〈u,Jxn〉-2∑i=1∞αn,i〈u,JSinxn〉+‖αn,0Jxn+∑i=1∞αn,iJSinxn‖2)+(1-αn)ϕ(u,xn)≤αn(‖u‖2-2αn,0〈u,Jxn〉-2∑i=1∞αn,i〈u,JSinxn〉+αn,0‖Jxn‖2+∑i=1∞αn,i‖JSinxn‖2-αn,0αn,jg(‖Jxn-JSjnxn‖)∑i=1∞αn,0〈u,JSinxn〉)+(1-αn)ϕ(u,xn)≤αn(‖u‖2-2αn,0〈u,Jxn〉-2∑i=1∞αn,i〈u,JSinxn〉+αn,0‖xn‖2+∑i=1∞αn,i∥Sinxn∥2-αn,0αn,jg(‖Jxn-JSjnxn‖)∑i=1∞αn,0〈u,JSinxn〉)+(1-αn)ϕ(u,xn)=αn(αn,0ϕ(u,xn)+∑i=1∞αn,iϕ(u,Sinxn)-αn,0αn,jg(‖Jxn-JSjnxn‖))+(1-αn)ϕ(u,xn)≤αn(αn,0ϕ(u,xn)+∑i=1∞αn,iknϕ(u,xn)-αn,0αn,jg(‖Jxn-JSjnxn‖))+(1-αn)ϕ(u,xn)≤αn(knϕ(u,xn)-αn,0αn,jg(‖Jxn-JSjnxn‖))+(1-αn)ϕ(u,xn)≤αn(ϕ(u,xn)+sup⁡u∈G⁡(kn-1)ϕ(u,xn)-αn,0αn,jg(‖Jxn-JSjnxn‖))+(1-αn)ϕ(u,xn)=ϕ(u,xn)+αnξn-αnαn,0αn,jg(‖Jxn-JSinxn‖)≤ϕ(u,xn)+ξn.
Hence  u∈Cn+1, and so G⊂Cn for all n≥0. By the way, from the definition of {ξn}, (2.4), and (3.6), it is easy to see that
(3.10)ξn=sup⁡u∈G(kn-1)ϕ(u,xn)≤sup⁡u∈G(kn-1)(‖u‖+M)2⟶0(asn⟶∞).
(IV)
Now, we prove that{xn} converges strongly to some point p∈G:=⋂i=1∞F(Si)⋂Ω.
First, we prove that{xn} converges strongly to some point p∈⋂i=1∞F(Si).
In fact, since{xn} is bounded in C and E is reflexive, there exists a subsequence {xni}⊂{xn} such that xni⇀p. Again since Cn is closed and convex for each n≥1, it is weakly closed, and so p∈Cn for each n≥0. Since xn=ΠCnx0, from the definition of ΠCn, we have
(3.11)ϕ(xni,x0)≤ϕ(p,x0),n≥0.
Since
(3.12)liminf⁡ni→∞ϕ(xni,x0)=liminf⁡ni→∞{‖xni‖2-2〈xni,Jx0〉+‖x0‖2}≥‖p‖2-2〈p,Jx0〉+‖x0‖2=ϕ(p,x0),
we have
(3.13)ϕ(p,x0)≤liminf⁡ni→∞ϕ(xni,x0)≤limsup⁡ni→∞ϕ(xni,x0)≤ϕ(p,x0).
This implies that limni→∞ϕ(xni,x0)=ϕ(p,x0), that is, ∥xni∥→∥p∥. In view of the Kadec-Klee property of E, we obtain that limn→∞xni=p.
Now we prove thatxn→p(n→∞). In fact, if there exists a subsequence {xnj}⊂{xn} such that xnj→q, then we have
(3.14)ϕ(p,q)=lim⁡ni→∞,nj→∞ϕ(xni,xnj)≤lim⁡ni→∞,nj→∞ϕ(xnix0)-ϕ(ΠCnjx0,x0)=lim⁡ni→∞,nj→∞ϕ(xni,x0)-ϕ(xnj,x0)=0(by(3.8)).
Therefore we have p=q. This  implies that
(3.15)lim⁡n→∞xn=p.
Now we prove thatp∈⋂i=1∞F(Si). In fact, by the construction of Cn, we have that Cn+1⊂Cn and xn+1=ΠCn+1x0. Therefore, by Lemma 2.2(a) we have
(3.16)ϕ(xn+1,xn)=ϕ(xn+1,ΠCnx0)≤ϕ(xn+1,x0)-ϕ(ΠCnx0,x0)=ϕ(xn+1,x0)-ϕ(xn,x0)⟶0(asn⟶∞).
In view of xn+1∈Cn and noting the construction of Cn+1 we obtain that
(3.17)ϕ(xn+1,un(N))≤ϕ(xn+1,xn)+ξn⟶0(asn⟶∞).
From (2.4) it yields (∥xn+1∥-∥un(N)∥)2→0. Since ∥xn+1∥→∥p∥, we have
(3.18)‖un(N)‖⟶‖p‖(asn⟶∞).
Hence we have(3.19)‖Jun(N)‖⟶‖Jp‖(asn⟶∞).
This implies that{Jun(N)} is bounded in E*. Since E is reflexive, and so E* is reflexive, there exists a subsequence {Juni(N)}⊂{Jun(N)} such that Juni(N)⇀p0∈E*. In view of the reflexive of E, we see that J(E)=E*. Hence there exists x∈E such that Jx=p0. Since
(3.20)ϕ(xni,uni(N))=‖xni‖2-2〈xni,Juni(N)〉+‖uni(N)‖2=‖xni‖2-2〈xni,Juni(N)〉+‖Juni(N)‖2,
taking liminfn→∞ on both sides of above equality and in view of the weak lower semicontinuity of norm ∥·∥, it yields that
(3.21)0≥‖p‖2-2〈p,p0〉+‖p0‖2=‖p‖2-2〈p,Jx〉+‖Jx‖2=‖p‖2-2〈p,Jx〉+‖x‖2=ϕ(p,x).
That is, p=x. This implies that p0=Jp, and so Jun(N)⇀Jp. It follows from (3.19) and the Kadec-Klee property of E* that Juni(N)→Jp (as n→∞). Noting that J-1:E*→E is norm-weak continuous, it yields that uni(N)⇀p. It follows from (3.18) and the Kadec-Klee property of E that limni→∞uni(N)=p.
By the similar way as given in the proof of (3.15), we can also prove that
(3.22)lim⁡n→∞un(N)=p.
From (3.15) and (3.22) we have that
(3.23)‖xn-un(N)‖⟶0(asn⟶∞).
SinceJ is uniformly continuous on any bounded subset of E, we have
(3.24)‖Jxn-Jun(N)‖⟶0(asn⟶∞).
For anyj≥1 and any u∈G, it follows from (3.9), (3.15), and (3.22) that
(3.25)αnαn,0αn,jg(‖Jxn-JSinxn‖)≤ϕ(u,xn)-ϕ(u,un(N))+αnξn.
Since(3.26)ϕ(u,xn)-ϕ(u,un(N))=‖xn‖2-‖un(N)‖2-2〈u,Jxn-Jun(N)〉≤|‖xn‖2-‖un(N)‖2|+2‖u‖⋅‖Jxn-Jun(N)‖≤‖xn-un(N)‖(‖xn‖+‖un(N)‖)+2‖u‖⋅‖Jxn-Jun(N)‖,
From (3.23) and (3.24), it follows that
(3.27)ϕ(u,xn)-ϕ(u,un(N))⟶0(asn⟶∞).
In view of conditions (b)liminfn→∞αn,0αn,j>0 and (c) αn≥α>0, we see that
(3.28)g(‖Jxn-JSinxn‖)⟶0(asn⟶∞).
It follows from the property ofg that
(3.29)‖Jxn-JSinxn‖⟶0(asn⟶∞).
Sincexn→p and J is uniformly continuous, it yields Jxn→Jp. Hence from (3.29) we have
(3.30)JSjnxn⟶Jp(asn⟶∞).
Since J-1:E*→E is norm-weak continuous, it follows that
(3.31)Sjnxn⇀p(foreachj≥1).
On the other hand, for each j≥1 we have
(3.32)|‖Sjnxn‖-‖p‖|=|‖J(Sjnxn)‖-‖Jp‖|≤‖JSjnxn-Jp‖⟶0(asn⟶∞).
This together with (3.31) shows that
(3.33)Sjnxn⟶p(foreachj≥1).
Furthermore, by the assumption that for eachj≥1, Sj is uniformly Li-Lipschitz continuous, hence we have
(3.34)‖Sjn+1xn-Sjnxn‖≤‖Sjn+1xn-Sjn+1xn+1‖+‖Sjn+1xn+1-xn+1‖+‖xn+1-xn‖+‖xn-Sjnxn‖≤(Lj+1)‖xn+1-xn‖+‖Sjn+1xn+1-xn+1‖+‖xn-Sjnxn‖.
This together with (3.15) and (3.33) yields ∥Sjn+1xn-Sjnxn∥→0 (as n→∞). Hence from (3.33) we have Sjn+1xn→p, that is, SjSjnxn→p. In view of (3.33) and the closeness of Sj, it yields that Sjp=p,for allj≥1. This implies that p∈⋂i=1∞F(Si).
Next, we prove thatp∈⋂i=1NΩ(Θi,ψ). Since un(N)=KrnΘNun(N-1), from (2.18), (3.9), and (3.27), we have
(3.35)ϕ(un(N),un(N-1))=ϕ(KrnΘNun(N-1),un(N-1))≤ϕ(u,un(N-1))-ϕ(u,KrnΘNun(N-1))≤ϕ(u,xn)+ξn-ϕ(u,KrnΘNun(N-1))=ϕ(u,xn)+ξn-ϕ(u,un(N))⟶0(asn⟶∞).
From (2.4) it yields (∥un(N)∥-∥un(N-1)∥)2→0. Since ∥un(N)∥→∥p∥, we have
(3.36)‖un(N-1)‖⟶‖p‖(asn⟶∞).
Hence we have(3.37)‖Jun(N-1)‖⟶‖Jp‖(asn⟶∞).
This implies that{Jun(N-1)} is bounded in E*. Since E is reflexive, and so E* is reflexive, there exists a subsequence {Juni(N-1)}⊂{Jun(N-1)} such that Juni(N-1)⇀p0∈E*. In view of the reflexive of E, we see that J(E)=E*. Hence there exists x∈E such that Jx=p0. Since
(3.38)ϕ(uni(N),uni(N-1))=‖uni(N)‖2-2〈uni(N),Juni(N-1)〉+‖uni(N-1)‖2=‖uni(N)‖2-2〈uni(N),Juni(N-1)〉+‖Juni(N-1)‖2,
taking liminfn→∞ on both sides of equality above and in view of the weak lower semicontinuity of norm ∥·∥, it yields that
(3.39)0≥‖p‖2-2〈p,p0〉+‖p0‖2=‖p‖2-2〈p,Jx〉+‖Jx‖2=‖p‖2-2〈p,Jx〉+‖x‖2=ϕ(p,x).
That is, p=x. This implies that p0=Jp, and so Juni(N-1)⇀Jp. It follows from (3.37) and the Kadec-Klee property of E* that Jun(N-1)→Jp (as n→∞). Note that J-1:E*→E is hemi-continuous, it yields that uni(N-1)⇀p. It follows from (3.36) and the Kadec-Klee property of E that limni→∞uni(N-1)=p.
By the similar way as given in the  proof of (3.15), we can also prove that
(3.40)lim⁡n→∞un(N-1)=p.
From (3.22) and (3.40) we have that
(3.41)∥un(N)-un(N-1)∥⟶0(asn⟶∞).
Since J is uniformly continuous on any bounded subset of E, we have
(3.42)∥Jun(N)-Jun(N-1)∥⟶0(asn⟶∞).
Since(3.43)un(i)=KrnΘiun(i-1),i=2,3,…,N,un(1)=KrnΘ1yn,
by the similar way as above, we can also prove that(3.44)∥un(i)-un(i-1)∥⟶0,∥Jun(i)-Jun(i-1)∥⟶0,∥yn-un(1)∥⟶0,∥Jyn-Jun(1)∥⟶0(asn⟶∞),i=2,3,…,N.
From (3.44) and the assumption  that rn≥d,foralln≥0, we have
(3.45)lim⁡n→∞‖Jun(i)-Jun(i-1)‖rn=0,lim⁡n→∞‖Jyn-Jun(1)‖rn=0,i=2,3,…,N.
Now, we define functions Hi:C×C→ℝ,i=1,2,…,N by
(3.46)Hi(x,y)=Θi(x,y)+〈Ax,y-x〉+ψ(y)-ψ(x),∀x,y∈C.
By Lemma 2.9, we know that the function Hi satisfies conditions (A1)–(A4). Since
(3.47)Hi(un(i),y)+1rn〈y-un(i),Jun(i)-Jun(i-1)〉≥0,H1(un(1),y)+1rn〈y-un(1),Jun(1)-Jyn〉≥0,∀y∈C,
by condition (A2), we have
(3.48)1rn〈y-un(i),Jun(i)-Jun(i-1)〉≥-Hi(un(i),y)≥Hi(y,un(i)),1rn〈y-un(1),Jun(1)-Jyn〉≥-H1(un(1),y)≥H1(y,un(1)),∀y∈C.
By the assumption that y↦Hi(x,y) is convex and lower semicontinuous, letting n→∞ in (3.48), from (3.45) and (3.48), for each i, we have Hi(y,p)≤0, for all y∈C.
Fort∈(0,1] and y∈C, letting yt=ty+(1-t)p, there are yt∈C and Hi(yt,p)≤0. By conditions (A1) and (A4), we have
(3.49)0=Hi(yt,yt)≤tHi(yt,y)+(1-t)Hi(yt,p)≤tHi(yt,y).
Dividing both sides of the above equation by t, we have Hi(yt,y)≥0,for all y∈C. Letting t↓0, from condition (A3), we have Hi(p,y)≥0, for all y∈C, that is, p∈⋂i=1NΩ(Θi,ψ), that is, p∈Ω, and p∈⋂i=1∞F(Si)⋂Ω.
(V)
Now, we provexn→Π⋂i=1∞F(Si)⋂Ωx0.
Letw=Π⋂i=1∞F(Si)⋂Ωx0. From w∈⋂i=1∞F(Si)⋂Ω⊂Cn+1 and xn+1=ΠCn+1x0, we have ϕ(xn+1,x0)≤ϕ(w,x0), for all n≥0. This implies that
(3.50)ϕ(p,x0)=lim⁡n→∞ϕ(xn,x0)≤ϕ(w,x0).
By the definition ofΠ⋂i=1∞F(Si)⋂Ωx0 and (3.50), we have p=w. Therefore, xn→Π⋂i=1∞F(Si)⋂Ωx0. This completes the proof of Theorem 3.1.Theorem 3.2.
LetE be a uniformly smooth and strictly convex Banach space with Kleac-Klee property and C a nonempty closed convex subset of E. Let A:C→E* be a continuous and monotone mapping, ψ:C→ℝ a lower semicontinuous and convex function, and {Θi:C×C→ℝ,i=1,2,…,N} a finite family of bifunction satisfying conditions (A1)–(A4). Let {Si}i=1∞:C→C be an infinite family of closed and quasi-ϕ-nonexpansive mappings. Suppose that G:=⋂i=1∞F(Si)⋂Ω≠∅, where Ω=⋂i=1NΩ(Θi,ψ). Let {xn}, {yn}, {zn}, and {un(k)}, k=1,2,…,N, be the sequences generated by
(3.51)x0∈C,C0=C,zn=J-1(αn,0Jxn+∑i=1∞αn,iJSinxn),yn=J-1(αn+(1-αn)Jzn),un(N)∈Csuchthat,∀y∈C,ΘN(un(N),y)+〈Aun(N),y-un(N)〉+ψ(y)-ψ(un(N))+1rn〈y-un(N),Jun(N)-Jun(N-1)〉≥0,ΘN-1(un(N-1),y)+〈Aun(N-1),y-un(N-1)〉+ψ(y)-ψ(un(N-1))+1rn〈y-un(N-1),Jun(N-1)-Jun(N-2)〉≥0,⋮Θ1(un(1),y)+〈Aun(1),y-un(1)〉+ψ(y)-ψ(un(1))+1rn〈y-un(1),Jun(1)-Jyn〉≥0,Cn+1={v∈Cn:ϕ(v,un(N))≤ϕ(v,xn)},∀n≥0,xn+1=ΠCn+1x0,
where rn∈[d,∞) for some d>0 and for i≥0,{αn,i},{αn} are sequences in [0,1] satisfying the following conditions:(a)
∑i=0∞αn,i=1, for all n≥0;(b)
liminfn→∞αn,0·αn,i>0 for all i≥1;(c)
0<α≤αn<1 for some α∈(0,1).Then {xn} converges strongly to ΠGx0.Proof.
Since{Si}i=1∞:C→C is an infinite family of closed quasi-ϕ-nonexpansive mappings, it is an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with sequence {kn=1}. Hence ξn=supu∈G(kn-1)ϕ(u,xn)=0. Therefore, the conditions appearing in Theorem 3.1—“G is bounded subset in C” and for each i≥1,Si is uniformly Li-Lipschitz continuous— are of no use here. In fact, by the same methods as given in the proofs of (3.15), (3.22), (3.31), (3.37), and (3.42), we can prove that xn→p, un(i)→p, yn→p, and Sjxn→p (as n→∞), for each j≥1, that is, p∈⋂i=1∞F(Si)⋂Ω. Therefore, all conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.2 is obtained from Theorem 3.1 immediately.Remark 3.3.
Theorems3.1 and 3.2 improve and extend the corresponding results in [10–12, 18].(a)
For the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property (note that each uniformly convex Banach space must have Kadec-Klee property).(b)
For the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, or quasi-ϕ-nonexpansive mapping to an infinite family of quasi-ϕ-asymptotically nonexpansive mappings.(c)
We extend a single generalized mixed equilibrium problem to a system of generalized mixed equilibrium problems.



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*Source: 595263-2011-06-22.xml*