# A New Hybrid Algorithm for a Pair of Quasi-ϕ-Asymptotically Nonexpansive Mappings and Generalized Mixed Equilibrium Problems in Banach Spaces

Authors: Jinhua Zhu; Shih-Sen Chang
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

The purpose of this paper is, by using a new hybrid method, to prove a strong
convergence theorem for finding a common element of the set of solutions for a generalized
mixed equilibrium problem, the set of solutions for a variational inequality problem, and
the set of common fixed points for a pair 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*.Letψ:C→ℝ be a real-valued function, Θ:C×C→ℝ a bifunction, and A:C→E* a nonlinear mapping. The “so-called” generalized mixed equilibrium problem is to find u∈C such that (1.1)Θ(u,y)+〈Au,y-u〉+ψ(y)-ψ(u)≥0,∀y∈C.
The set of solutions for (1.1) is denoted by Ω, that is, (1.2)Ω={u∈C:Θ(u,y)+〈Au,y-u〉+ψ(y)-ψ(u)≥0,∀y∈C}.Special examples are as follows.(I)
Ifψ=0, the problem (1.1) is equivalent to finding u∈C such that
(1.3)Θ(u,y)+〈Au,y-u〉≥0,∀y∈C,
which is called the generalized equilibrium problem. The set of solutions for (1.3) is denoted by GEP.(II)
IfA=0, the problem (1.1) is equivalent to finding u∈C such that
(1.4)Θ(u,y)+ψ(y)-ψ(u)≥0,∀y∈C,
which is called the mixed equilibrium problem (MEP) [1]. The set of solutions for (1.4) is denoted by MEP.(III)
IfΘ=0, the problem (1.1) is equivalent to finding u∈C such that
(1.5)〈Au,y-u〉+ψ(y)-ψ(u)≥0,∀y∈C,
which is called the mixed variational inequality of Browder type (VI) [2]. The set of solutions for (1.5) is denoted by VI(C,A,ψ).(IV)
Ifψ=0 and A=0, the problem (1.1) is equivalent to finding u∈C such that
(1.6)Θ(u,y)≥0,∀y∈C,
which is called the equilibrium problem. The set of solutions for (1.6) is denoted by EP(Θ).(V)
Ifψ=0 and Θ=0, the problem (1.1) is equivalent to finding u∈C such that
(1.7)〈Au,y-u〉≥0,∀y∈C,
which is called the variational inequality of Browder type. The set of solutions for (1.7) is denoted by VI(C,A).The problem (1.1) is very general in the sense that numerous problems in physics, optimiztion and economics reduce to finding a solution for (1.1). Some methods have been proposed for solving the generalized equilibrium problem and the equilibrium problem in Hilbert space (see, e.g., [3–6]).A mappingS:C→E is called nonexpansive if (1.8)‖Sx-Sy‖≤‖x-y‖,∀x,y∈C.
We denote the fixed point set of S by F(S).In 2008, S. Takahashi and W. Takahashi [6] proved some strong convergence theorems for finding an element or a common element of EP, EP(f)∩F(S) or VI(C,A)∩F(S), respectively, in a Hilbert space.Recently, Takahashi and Zembayashi [7, 8] proved some weak and strong convergence theorems for finding a common element of the set of solutions for equilibrium (1.6) and the set of fixed points of a relatively nonexpansive mapping in a Banach space.In 2010, Chang et al. [9] proved a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem (1.3) and the set of common fixed points of a pair of relatively nonexpansive mappings in a Banach space.Motivated and inspired by [4–9], we intend in this paper, by using a new hybrid method, to prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem (1.1) and the set of common fixed points of a pair of quasi-ϕ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.


## 2. Preliminaries

For the sake of convenience, we first recall some definitions and conclusions which will be needed in proving our main results.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.In the sequel, we denote the strong convergence and weak convergence of a sequence{xn} by xn→x and xn⇀x, respectively.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 [10].(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 hemicontinuous.(iii)
IfE is a smooth, strictly convex, and reflexive Banach space, then J is singlevalued, 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 [11], the generalized projection ΠC:E→C is defined by (2.5)ΠC(x)=arginf⁡y∈Cϕ(y,x),∀x∈E.Lemma 2.2 (see [11, 12]).
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 all x∈C and y∈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 point of 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 [13]).
(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→C is said to be closed if, for any sequence {xn}⊂C with xn→x and Txn→y, Tx=y.Definition 2.5 (see [14]).
(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 mappingT: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 pair of mappingsT1,T2:C→C is said to be uniformly quasi-ϕ-asymptotically nonexpansive if F(T1)⋂F(T2)≠∅ and there exists a real sequence {kn}⊂[1,∞) with kn→1 such that for i=1,2(2.10)ϕ(p,Tinx)≤knϕ(p,x),∀n≥1,x∈C,p∈F(T1)∩F(T2).
(4) A mappingT:C→C is said to be uniformly L-Lipschitz continuous if there exists a constant L>0 such that
(2.11)‖Tnx-Tny‖≤L‖x-y‖,∀x,y∈C.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 [15]).
LetE be a uniformly convex Banach space, r>0 a positive number, and Br(0) a closed ball of E. Then, for any given subset {x1,x2,…,xN}⊂Br(0) and for any positive numbers {λ1,λ2,…,λN} with ∑i=1Nλi=1, there exists a continuous, strictly increasing, and convex function g:[0,2r)→[0,∞) with g(0)=0 such that, for any i,j∈{1,2,…,N} with i<j,
(2.12)‖∑n=1Nλnxn‖2≤∑n=1Nλn‖xn‖2-λiλjg(‖xi-xj‖).Lemma 2.8 (see [15]).
LetE be a real uniformly smooth and strictly convex Banach space with the 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 generalized mixed equilibrium problem (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 Θ:C×C→ℝ satisfies the following conditions:(A1)
Θ(x,x)=0, for all x∈C,(A2)
Θ is monotone, that is, Θ(x,y)+Θ(y,x)≤0, ∀x,y∈C,(A3)
limsupt↓0Θ(x+t(z-x),y)≤Θ(x,y)∀x,z,y∈C,(A4)
the functiony↦Θ(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 a bifunction satisfying the conditions (A1)–(A4). Let r>0 and x∈E. Then, the followings hold.(i)
(Blum and Oettli [3]) there exists z∈C such that
(2.13)Θ(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.(ii)
(Takahashi and Zembayashi [8]) Define a mapping Tr:E→C by
(2.14)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 a firmly nonexpansive-type mapping, that is, ∀z,y∈E,
(2.15)〈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),∀q∈F(Tr).Lemma 2.10 (see [16]).
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 conditions (A1)–(A4). Let r>0 be any given number and x∈E any given point. Then, the following hold.(i)
There existsu∈C such that
(2.16)Θ(u,y)+〈Au,y-u〉+ψ(y)-ψ(u)+1r〈y-u,Ju-Jx〉≥0,∀y∈C.(ii)
If we define a mappingKr:C→C by(2.17)Kr(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 Kr has the following properties:(a)
Kr is single valued,(b)
Kr is a firmly nonexpansive-type mapping, that is,
(2.18)〈Krz-Kry,JKrz-JKry〉≤〈Krz-Kry,Jz-Jy〉,∀z,y∈E,(c)
F(Kr)=Ω=F̃(Kr),(d)
Ω is closed and convex,(e)
(2.19)ϕ(q,Krz)+ϕ(Krz,z)≤ϕ(q,z),∀q∈F(Kr),z∈E.Remark 2.11.
It follows from Lemma2.9 that the mapping Kr is a relatively nonexpansive mapping. Thus, it is quasi-ϕ-nonexpansive.


## 3. Main Results

In this section, we will prove a strong convergence theorem for finding a common element of the set of solutions for the generalized mixed equilibrium problem (1.1) and the set of common fixed points for a pair of quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces.Theorem 3.1.
LetE be a uniformly smooth and strictly convex Banach space with the Kadec-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 Θ:C×C→ℝ a bifunction satisfying conditions (A1)–(A4). Let S,T:C→C be two closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞) and kn→1. Suppose that S and T are uniformly L-Lipschitz continuous and that G=F(T)⋂F(S)⋂Ω is a nonempty and bounded subset in C. Let {xn} be the sequence generated by
(3.1)x0∈C,C0=C,Q0=C,zn=J-1(αnJxn+(1-αn)JTnxn),yn=J-1(βnJxn+(1-βn)JSnzn),un∈Csuch  that,∀y∈C,Θ(un,y)+〈Aun,y-un〉+ψ(y)-ψ(un)+1rn〈y-un,Jun-Jyn〉≥0,Cn={v∈Cn-1:ϕ(v,zn)≤ϕ(v,xn)+ξn,ϕ(v,un)≤ϕ(v,xn)+(1+kn)(1-βn)ξn},Qn={z∈Qn-1:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0,∀n≥0,
where J:E→E* is the normalized duality mapping, {αn} and {βn} are sequences in [0,1] and {rn}⊂[a,∞) for some a>0, ξn=supu∈G(kn-1)ϕ(u,xn). Suppose that the following conditions are satisfied:(i)
liminfn→∞αn(1-αn)>0,(ii)
liminfn→∞βn(1-βn)>0.Then {xn} converges strongly to ΠF(S)∩F(T)∩Ωx0, where ΠF(S)∩F(T)∩Ω is the generalized projection of E onto F(S)∩F(T)∩Ω.Proof.
Firstly, we define two functionsH:C×C→ℝ and Kr:C→C by
(3.2)H(x,y)=Θ(x,y)+〈Ax,y-x〉+ψ(y)-ψ(x),∀x,y∈C,Kr(x)={u∈C:H(u,y)+1r〈y-u,Ju-Jx〉≥0,∀y∈C},x∈C.
By Lemma 2.10, we know that the function H satisfies conditions (A1)–(A4) and Kr has properties (a)–(e). Therefore, (3.1) is equivalent to
(3.3)x0∈C,C0=C,Q0=C,zn=J-1(αnJxn+(1-αn)JTnxn),yn=J-1(βnJxn+(1-βn)JSnzn),un∈Csuch that,∀y∈C,H(un,y)+1rn〈y-un,Jun-Jyn〉≥0,Cn={v∈Cn-1:ϕ(v,zn)≤ϕ(v,xn)+ξn,ϕ(v,un)≤ϕ(v,xn)+(1+kn)(1-βn)ξn},Qn={z∈Qn-1:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0,∀n≥0.
We divide the proof of Theorem 3.1 into five steps.
(I) First we prove thatCn and Qn are both closed and convex subsets of C for all n≥0.
In fact, it is obvious thatQn is closed and convex for all n≥0. Again we have that
(3.4)ϕ(v,zn)≤ϕ(v,xn)+ξn⟺2〈v,Jxn-Jzn〉≤‖xn‖2-‖zn‖2+ξn,ϕ(v,un)≤ϕ(v,xn)+(1+kn)(1-βn)ξn⟺2〈v,Jxn-Jun〉≤‖xn‖2-‖un‖2+(1+kn)(1-βn)ξn.
Hence Cn, ∀n≥0, is closed and convex, and so Cn∩Qn is closed and convex for all n≥0.
(II) Next we prove thatF(T)∩F(S)∩Ω⊂Cn∩Qn, ∀n≥0.
Puttingun=Krnyn, ∀n≥0, by Lemma 2.10 and Remark 2.11, Krn is relatively nonexpansive. Again since S and T are quasi-ϕ-asymptotically nonexpansive, for any given u∈F(S)∩F(T)∩Ω, we have that
(3.5)ϕ(u,zn)=ϕ(u,J-1(αnJxn+(1-αn)JTnxn))=‖u‖2-2〈u,αnJxn+(1-αn)JTnxn〉+‖αnJxn+(1-αn)JTnxn‖2≤‖u‖2-2αn〈u,Jxn〉-2(1-αn)〈u,JTnxn〉+αn‖xn‖2+(1-αn)‖Tnxn‖2-αn(1-αn)g(‖Jxn-JTnxn‖)=αnϕ(u,xn)+(1-αn)ϕ(u,Tnxn)-αn(1-αn)g(‖Jxn-JTnxn‖)≤αnϕ(u,xn)+(1-αn)knϕ(u,xn)-αn(1-αn)g(‖Jxn-JTnxn‖)≤knϕ(u,xn)-αn(1-αn)g(‖Jxn-JTnxn‖)≤ϕ(u,xn)+sup⁡p∈G⁡(kn-1)ϕ(p,xn)-αn(1-αn)g(‖Jxn-JTnxn‖)=ϕ(u,xn)+ξn-αn(1-αn)g(‖Jxn-JTnxn‖)≤ϕ(u,xn)+ξn.
From (3.5) we have that
(3.6)ϕ(u,un)=ϕ(u,Krnyn)≤ϕ(u,yn)≤ϕ(u,J-1(βnJxn+(1-βn)JSnzn))=‖u‖2-2〈u,βnJxn+(1-βn)JSnzn〉+‖βnJxn+(1-βn)JSnzn‖2≤‖u‖2-2βn〈u,Jxn〉-2(1-βn)〈u,JSnzn〉+βn‖xn‖2+(1-β)‖Snzn‖2-βn(1-βn)g(‖Jxn-JSnzn‖)=βnϕ(u,xn)+(1-βn)ϕ(u,Snzn)-βn(1-βn)g(‖Jxn-JSnzn‖)≤βnϕ(u,xn)+(1-βn)knϕ(u,zn)-βn(1-βn)g(‖Jxn-JSnzn‖)≤βnϕ(u,xn)+(1-βn)kn(ϕ(u,xn)+ξn)-βn(1-βn)g(‖Jxn-JSnzn‖)≤βnϕ(u,xn)+(1-βn)(ϕ(u,xn)+ξn)+(1-βn)knξn-βn(1-βn)g(‖Jxn-JSnzn‖)≤ϕ(u,xn)+(1-βn)ξn+(1-βn)knξn-βn(1-βn)g(‖Jxn-JSnzn‖)≤ϕ(u,xn)+(1+kn)(1-βn)ξn-βn(1-βn)g(‖Jxn-JSnzn‖)≤ϕ(u,xn)+(1+kn)(1-βn)ξn∀n≥0.
This implies that u∈Cn, ∀n≥0, and so F(T)∩F(S)∩Ω⊂Cn, ∀n≥0.
Now we prove thatF(T)∩F(S)∩Ω⊂Cn∩Qn, ∀n≥0.
In fact, fromQ0=C, we have that F(T)∩F(S)∩Ω⊂C0∩Q0. Suppose that F(T)∩F(S)∩Ω⊂Ck∩Qk, for some k≥0. Now we prove that F(T)∩F(S)∩Ω⊂Ck+1∩Qk+1. In fact, since xk+1=ΠCk∩Qkx0, we have that
(3.7)〈xk+1-z,Jx0-Jxk+1〉≥0,∀z∈Ck∩Qk.
Since F(T)∩F(S)∩Ω⊂Ck∩Qk, for any z∈F(T)∩F(S)∩Ω, we have that
(3.8)〈xk+1-z,Jx0-Jxk+1〉≥0.
This shows that z∈Qk+1, and so F(T)∩F(S)∩Ω⊂Qk+1. The conclusion is proved.
(III) Now we prove that{xn} is bounded.
From the definition ofQn, we have that xn=ΠQnx0, ∀n≥0. Hence, from Lemma 2.2(1),
(3.9)ϕ(xn,x0)=ϕ(ΠQnx0,x0)≤ϕ(u,x0)-ϕ(u,ΠQnx0)≤ϕ(u,x0),∀u∈F(T)∩F(S)∩Ω⊂Qn,∀n≥0.
This implies that {ϕ(xn,x0)} is bounded. By virtue of (2.4), {xn} is bounded. Denote
(3.10)M=sup⁡n≥0{‖xn‖}<∞.
Since xn+1=ΠCn∩Qnx0∈Cn∩Qn⊂Qn and xn=ΠQnx0, from the definition of ΠQn, we have that
(3.11)ϕ(xn,x0)≤ϕ(xn+1,x0)≤(M+‖x0‖)2,∀n≥0.
This implies that {ϕ(xn,x0)} is nondecreasing, and so the limit limn→∞ϕ(xn,x0) exists. Without loss of generality, we can assume that
(3.12)lim⁡n→∞ϕ(xn,x0)=r≥0.
By the way, from the definition of {ξn}, (2.4), and (3.10), it is easy to see that
(3.13)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=F(T)∩F(S)∩Ω.
In fact, since{xn} is bounded in C and E is reflexive, there exists a subsequence {xni}⊂{xn} such that xni⇀p. Again since Qn is closed and convex for each n≥0, it is weakly closed, and so p∈Qn for each n≥0. Since xn=ΠQnx0, from the defintion of ΠQn, we have that
(3.14)ϕ(xni,x0)≤ϕ(p,x0),n≥0.
Since
(3.15)lim⁡inf⁡ni→∞ϕ(xni,x0)=lim⁡inf⁡ni→∞{‖xni‖2-2〈xni,Jx0〉+‖x0‖2}≥‖p‖2-2〈p,Jx0〉+‖x0‖2=ϕ(p,x0),
we have that
(3.16)ϕ(p,x0)≤lim⁡inf⁡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 first prove thatxn→p(n→∞). In fact, if there exists a subsequence {xnj}⊂{xn} such that xnj→q, then we have that
(3.17)ϕ(p,q)=lim⁡ni→∞,nj→∞ϕ(xni,xnj)≤lim⁡ni→∞,nj→∞ϕ(xni,x0)-ϕ(ΠQnjx0,x0)=lim⁡ni→∞,nj→∞ϕ(xni,x0)-ϕ(xnj,x0)=0(by(3.12)).
Therefore we have that p=q. This implies that
(3.18)lim⁡n→∞xn=p.
Now we first prove thatp∈F(T)∩F(S). In fact, by the construction of Qn, we have that xn=ΠQnx0. Therefore, by Lemma 2.2(a) we have that
(3.19)ϕ(xn+1,xn)=ϕ(xn+1,ΠQnx0)≤ϕ(xn+1,x0)-ϕ(ΠQnx0,x0)=ϕ(xn+1,x0)-ϕ(xn,x0)⟶0(asn⟶∞).
In view of xn+1∈Cn∩Qn⊂Cn and noting the construction of Cn we obtain
(3.20)ϕ(xn+1,zn)≤ϕ(xn+1,xn)+ξn,ϕ(xn+1,un)≤ϕ(xn+1,xn)+(1+kn)(1-βn)ξn.
From (3.13) and (3.19), we have that
(3.21)lim⁡n→∞ϕ(xn+1,un)=0,lim⁡n→∞ϕ(xn+1,zn)=0.
From (2.4) it yields that (||xn+1||-||un||)2→0 and (||xn+1||-||zn||)2→0. Since ||xn+1||→||p||, we have that
(3.22)‖un‖⟶‖p‖,‖zn‖⟶‖p‖(asn⟶∞).
Hence, we have that
(3.23)‖Jun‖⟶‖Jp‖,‖Jzn‖⟶‖Jp‖(asn⟶∞).
This implies that{Jzn} is bounded in E*. Since E is reflexive, and so E* is reflexive, there exists a subsequence {Jzni}⊂{Jzn} such that Jzni⇀p0∈E*. In view of the reflexiveness of E, we see that J(E)=E*. Hence, there exists x∈E such that Jx=p0. Since
(3.24)ϕ(xni+1,zni)=‖xni+1‖2-2〈xni+1,Jzni〉+‖zni‖2=‖xni+1‖2-2〈xni+1,Jzni〉+‖Jzni‖2,
taking liminfn→∞ on both sides of the equality above and in view of the weak lower semicontinuity of norm ||·||, it yields that
(3.25)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 Jzn⇀Jp. It follows from (3.23) and the Kadec-Klee property of E* that Jzni→Jp (as n→∞). Noting that J-1:E*→E is hemicontinuous, it yields that zni⇀p. It follows from (3.22) and the Kadec-Klee property of E that limni→∞zni=p.
By the same way as given in the proof of (3.18), we can also prove that
(3.26)lim⁡n→∞zn=p.
From (3.18) and (3.26), we have that
(3.27)‖xn-zn‖⟶0(asn⟶∞).
Since J is uniformly continuous on any bounded subset of E, we have that
(3.28)‖Jxn-Jzn‖⟶0(asn⟶∞).
For any u∈F(T)⋂F(S)⋂Ω, it follows from (3.5) that
(3.29)αn(1-αn)g(‖Jxn-Tnxn‖)≤ϕ(u,xn)-ϕ(u,zn)+ξn.
Since
(3.30)ϕ(u,xn)-ϕ(u,zn)=‖xn‖2-‖zn‖2-2〈u,Jxn-Jzn〉≤|‖xn‖2-‖zn‖2|+2‖u‖⋅‖Jxn-Jzn‖≤‖xn-zn‖(‖xn‖+‖zn‖)+2‖u‖⋅‖Jxn-Jzn‖,
From (3.27) and (3.28), it follows that
(3.31)ϕ(u,xn)-ϕ(u,zn)⟶0(asn⟶∞).
In view of condition (i) andliminfn→∞αn(1-αn)>0, we see that
(3.32)g(‖Jxn-JTnxn‖)⟶0(asn⟶∞).
It follows from the property of g that
(3.33)‖Jxn-JTnxn‖⟶0(asn⟶∞).
Since xn→p and J is uniformly continuous, it yields that Jxn→Jp. Hence from (3.33) we have that
(3.34)JTnxn⟶Jp(asn⟶∞).
Since J-1:E*→E is hemicontinuous, it follows that
(3.35)Tnxn⇀p.
On the other hand, we have that
(3.36)|‖Tnxn‖-‖p‖|=|‖J(Tnxn)‖-‖Jp‖|≤‖JTnxn-Jp‖⟶0(asn⟶∞).
This together with (3.35) shows that
(3.37)Tnxn⟶p.
Furthermore, by the assumption thatT is uniformly L-Lipschitz continuous, we have that
(3.38)‖Tn+1xn-Tnxn‖≤‖Tn+1xn-Tn+1xn+1‖+‖Tn+1xn+1-xn+1‖+‖xn+1-xn‖+‖xn-Tnxn‖≤(L+1)‖xn+1-xn‖+‖Tn+1xn+1-xn+1‖+‖xn-Tnxn‖.
This together with (3.18) and (3.37), yields ||Tn+1xn-Tnxn||→0 (as n→∞). Hence from (3.37) we have that Tn+1xn→p, that is, TTnxn→p. In view of (3.37) and the closeness of T, it yields that Tp=p. This implies that p∈F(T).
By the same way as given in the proof of (3.23) to (3.31), we can also prove that
(3.39)lim⁡n→∞un=p,ϕ(u,xn)-ϕ(u,un)⟶0(asn⟶∞).
Sinceun=Krnyn, from (2.19), (3.6), (3.13), and (3.39), we have that
(3.40)ϕ(un,yn)=ϕ(Krnyn,yn)≤ϕ(u,yn)-ϕ(u,un)≤ϕ(u,xn)-ϕ(u,un)+(1+kn)(1-βn)ξn⟶0(asn⟶∞).
From (2.4) it yields that (||un||-||yn||)2→0. Since ||un||→||p||, we have that
(3.41)‖yn‖⟶‖p‖(asn⟶∞).
Hence we have that(3.42)‖Jyn‖⟶‖Jp‖(asn⟶∞).
By the same way as given in the proof of (3.26), we can also prove that
(3.43)lim⁡n→∞yn=p.
From (3.39) and (3.43) we have that
(3.44)‖un-yn‖⟶0(asn⟶∞).
Since J is uniformly continuous on any bounded subset of E, we have that
(3.45)‖Jun-Jyn‖⟶0(asn⟶∞).
For any u∈F(T)⋂F(S)⋂Ω, it follows from (3.6), (3.13), and (3.39) that
(3.46)βn(1-βn)g(‖Jxn-Snzn‖)≤ϕ(u,xn)-ϕ(u,un)+(1+kn)(1-βn)ξn⟶0.
In view of condition (ii) and liminfn→∞βn(1-βn)>0, we see that
(3.47)g(‖Jxn-JSnzn‖)⟶0(asn⟶∞).
It follows from the property of g that
(3.48)‖Jxn-JSnzn‖⟶0(asn⟶∞).
Since xn→p and J is uniformly continuous, it yields, Jxn→Jp. Hence from (3.48) we have that
(3.49)JSnzn⟶Jp(asn⟶∞).
Since J-1:E*→E is hemicontinuous, it follows that
(3.50)Snzn⇀p.
On the other hand, we have that
(3.51)|‖Snzn‖-‖p‖|=|‖J(Snzn)‖-‖Jp‖|≤‖JSnzn-Jp‖⟶0(asn⟶∞).
This together with (3.50) shows that
(3.52)Snzn⟶p.
Furthermore, by the assumption thatS is uniformly L-Lipschitz continuous, we have that
(3.53)‖Sn+1zn-Snzn‖≤‖Sn+1zn-Sn+1zn+1‖+‖Sn+1zn+1-zn+1‖+‖zn+1-zn‖+‖zn-Snzn‖≤(L+1)‖zn+1-zn‖+‖Sn+1zn+1-zn+1‖+‖zn-Snzn‖.
This together with (3.26) and (3.52), yields that ||Sn+1zn-Snzn||→0 (as n→∞). Hence from (3.52) we have that Sn+1zn→p, that is, SSnzn→p. In view of (3.52) and the closeness of T, it yields that Sp=p. This implies that p∈F(S).
Next we prove thatp∈Ω. From (3.45) and the assumption that rn≥a, ∀n≥0, we have that
(3.54)lim⁡n→∞‖Jun-Jyn‖rn=0.
Since un=Krnyn, we have that
(3.55)H(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C.
Replacing n by nk in (3.55), from condition (A2), we have that
(3.56)1rnk〈y-unk,Junk-Jynk〉≥-H(unk,y)≥H(y,unk),∀y∈C.
By the assumption that y↦H(x,y) is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting nk→∞ in (3.55), from (3.54) and condition (A4), we have that H(y,p)≤0, ∀y∈C.
Fort∈(0,1] and y∈C, letting yt=ty+(1-t)p, there are yt∈C and H(yt,p)≤0. By conditions (A1) and (A4), we have that
(3.57)0=H(yt,yt)≤tH(yt,y)+(1-t)H(yt,p)≤tH(yt,y).
Dividing both sides of the above equation by t, we have that H(yt,y)≥0, ∀y∈C. Letting t↓0, from condition (A3), we have that H(p,y)≥0, ∀y∈C, that is, Θ(p,y)+〈Ap,y-p〉+ψ(y)-ψ(p)≥0, ∀y∈C. Therefore p∈Ω, and so p∈F(T)⋂F(S)⋂Ω.
(V) Finally, we prove thatxn→ΠF(T)⋂F(S)⋂Ωx0.
Letw=ΠF(T)⋂F(S)⋂Ωx0. From w∈F(T)⋂F(S)⋂Ω⊂Cn∩Qn, and xn+1=ΠCn∩Qnx0, we have that
(3.58)ϕ(xn+1,x0)≤ϕ(w,x0),∀n≥0.
Since the norm is weakly lower semicontinuous, this implies that
(3.59)ϕ(p,x0)=‖p‖2-2〈p,Jx0〉+‖x0‖2≤lim⁡nk→∞inf⁡{‖xnk‖2-2〈xnk,Jx0〉+‖x0‖2}≤lim⁡nk→∞inf⁡ϕ(xnk,x0)≤lim⁡nk→∞sup⁡ϕ(xnk,x0)≤ϕ(w,x0).
It follows from the definition of ΠF(T)⋂F(S)⋂Ωx0 and (3.59) that we have p=w. Therefore, xn→ΠF(T)⋂F(S)⋂Ωx0. This completes the proof of Theorem 3.1.Remark 3.2.
Theorem3.1 improves and extends the corresponding results in [7–9].(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 the Kadec-Klee property).(b)
For the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, or weak relatively nonexpansive mappings to a pair of quasi-ϕ-asymptotically nonexpansive mappings.(c)
For the equilibrium problem, we extend the generalized equilibrium problem to the generalized mixed equilibrium problem.The following theorems can be obtained from Theorem3.1 immediately.Theorem 3.3.
LetE be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let A:C→E* be a continuous and monotone mapping and Θ:C×C→ℝ a bifunction satisfying conditions (A1)–(A4). Let S,T:C→C be two closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞) and kn→1. Suppose that S and T are uniformly L-Lipschitz continuous and that G=F(T)⋂F(S)⋂GEP is a nonempty and bounded subset in C. Let {xn} be the sequence generated by
(3.60)x0∈C,C0=C,Q0=C,zn=J-1(αnJxn+(1-αn)JTnxn),yn=J-1(βnJxn+(1-βn)JSnzn),un∈Csuch that,∀y∈C,Θ(un,y)+〈Aun,y-un〉+1rn〈y-un,Jun-Jyn〉≥0,Cn={v∈Cn-1:ϕ(v,zn)≤ϕ(v,xn)+ξn,ϕ(v,un)≤ϕ(v,xn)+(1+kn)(1-βn)ξn},Qn={z∈Qn-1:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0,∀n≥0,
where J:E→E* is the normalized duality mapping, {αn} and {βn} are sequences in [0,1], and {rn}⊂[a,∞) for some a>0, ξn=supu∈G(kn-1)ϕ(u,xn). If {αn} and {βn} satisfy conditions (i)-(ii) in Theorem 3.1, then {xn} converges strongly to ΠF(S)∩F(T)∩GEPx0, where GEP is the set for the solutions of generalized equilibrium problem (1.3).Proof.
Puttingψ=0 in Theorem 3.1, the conclusion of Theorem 3.3 can be obtained from Theorem 3.1.Theorem 3.4.
LetE be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let ψ:C→ℝ be a lower semicontinuous and convex function and Θ:C×C→ℝ a bifunction satisfying conditions (A1)–(A4). Let S,T:C→C be two closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞) and kn→1. Suppose that S and T are uniformly L-Lipschitz continuous and that G=F(T)⋂F(S)⋂MEP is a nonempty and bounded subset in C. Let {xn} be the sequence generated by
(3.61)x0∈C,C0=C,Q0=C,zn=J-1(αnJxn+(1-αn)JTnxn),yn=J-1(βnJxn+(1-βn)JSnzn),un∈Csuchthat,∀y∈C,Θ(un,y)+ψ(y)-ψ(un)+1rn〈y-un,Jun-Jyn〉≥0,Cn={v∈Cn-1:ϕ(v,zn)≤ϕ(v,xn)+ξn,ϕ(v,un)≤ϕ(v,xn)+(1+kn)(1-βn)ξn},Qn={z∈Qn-1:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0,∀n≥0,
where J:E→E* is the normalized duality mapping, {αn} and {βn} are sequences in [0,1], and {rn}⊂[a,∞) for some a>0,ξn=supu∈G(kn-1)ϕ(u,xn). If {αn} and {βn} satisfy conditions (i)-(ii) in Theorem 3.1, then {xn} converges strongly to ΠF(S)∩F(T)∩MEPx0, where MEP is the set of solutions for mixed equilibrium problem (1.4).Proof.
PuttingA=0 in Theorem 3.1, the conclusion of Theorem 3.4 can be obtained from Theorem 3.1.Theorem 3.5.
LetE be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let A:C→E* be a continuous and monotone mapping and ψ:C→ℝ a lower semicontinuous and convex function. Let S,T:C→C be two closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞) and kn→1. Suppose that S and T are uniformly L-Lipschitz continuous and that G=F(T)⋂F(S)⋂VI(C,A,ψ) is a nonempty and bounded subset in C. Let {xn} be the sequence generated by
(3.62)x0∈C,C0=C,Q0=C,zn=J-1(αnJxn+(1-αn)JTnxn),yn=J-1(βnJxn+(1-βn)JSnzn),un∈Csuchthat,∀y∈C,〈Aun,y-un〉+ψ(y)-ψ(un)+1rn〈y-un,Jun-Jyn〉≥0,Cn={v∈Cn-1:ϕ(v,zn)≤ϕ(v,xn)+ξn,ϕ(v,un)≤ϕ(v,xn)+(1+kn)(1-βn)ξn},Qn={z∈Qn-1:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0,∀n≥0,
where J:E→E* is the normalized duality mapping, {αn} and {βn} are sequences in [0,1], and {rn}⊂[a,∞) for some a>0,ξn=supu∈G(kn-1)ϕ(u,xn). If {αn} and {βn} satisfy conditions (i)-(ii) in Theorem 3.1, then {xn} converges strongly to ΠF(S)∩F(T)∩VI(C,A,ψ)x0, where VI(C,A,ψ) is the set of solutions for the mixed variational inequality (1.5).Proof.
PuttingΘ=0 in Theorem 3.1, the conclusion of Theorem 3.5 can be obtained from Theorem 3.1.Theorem 3.6.
LetE be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let Θ:C×C→ℝ be a bifunction satisfying conditions (A1)–(A4). Let S,T:C→C be two closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞) and kn→1. Suppose that S and T are uniformly L -Lipschitz continuous and that G=F(T)⋂F(S)⋂EP(Θ) is a nonempty and bounded subset in C. Let {xn} be the sequence generated by
(3.63)x0∈C,C0=C,Q0=C,zn=J-1(αnJxn+(1-αn)JTnxn),yn=J-1(βnJxn+(1-βn)JSnzn),un∈Csuchthat,∀y∈C,Θ(un,y)+1rn〈y-un,Jun-Jyn〉≥0,Cn={v∈Cn-1:ϕ(v,zn)≤ϕ(v,xn)+ξn,ϕ(v,un)≤ϕ(v,xn)+(1+kn)(1-βn)ξn},Qn={z∈Qn-1:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0,∀n≥0,
where J:E→E* is the normalized duality mapping, {αn} and {βn} are sequences in [0,1], and {rn}⊂[a,∞) for some a>0, ξn=supu∈G(kn-1)ϕ(u,xn). If {αn} and {βn} satisfy conditions (i)-(ii) in Theorem 3.1, then {xn} converges strongly to ΠF(S)∩F(T)∩EP(Θ)x0, where EP(Θ) is the set of solutions for the equilibrium problem (1.6).Proof.
Puttingψ=0 and A=0 in Theorem 3.1, the conclusion of Theorem 3.6 can be obtained from Theorem 3.1.Theorem 3.7.
LetE be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let A:C→E* be a continuous and monotone mapping and S,T:C→C two closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞) and kn→1. Suppose that S and T are uniformly L-Lipschitz continuous and that G=F(T)⋂F(S)⋂VI(C,A) is a nonempty and bounded subset in C. Let {xn} be the sequence generated by
(3.64)x0∈C,C0=C,Q0=C,zn=J-1(αnJxn+(1-αn)JTnxn),yn=J-1(βnJxn+(1-βn)JSnzn),un∈Csuchthat,∀y∈C,〈Aun,y-un〉+1rn〈y-un,Jun-Jyn〉≥0,Cn={v∈Cn-1:ϕ(v,zn)≤ϕ(v,xn)+ξn,ϕ(v,un)≤ϕ(v,xn)+(1+kn)(1-βn)ξn},Qn={z∈Qn-1:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0,∀n≥0,
where J:E→E* is the normalized duality mapping, {αn} and {βn} are sequences in [0,1], and {rn}⊂[a,∞) for some a>0, ξn=supu∈G(kn-1)ϕ(u,xn). If {αn} and {βn} satisfy conditions (i)-(ii) in Theorem 3.1, then {xn} converges strongly to ΠF(S)∩F(T)∩VI(C,A)x0, where VI(C,A) is the set of solutions for the variational inequality (1.7)Proof.
Puttingψ=0 and Θ=0 in Theorem 3.1, the conclusion of Theorem 3.7 can be obtained from Theorem 3.1.Theorem 3.8.
LetE be a uniformly smooth and strictly convex Banach space with the Kadec-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 Θ:C×C→ℝ a bifunction satisfying conditions (A1)–(A4). Let S:C→C be a closed and quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞) and kn→1. Suppose that S is uniformly L-Lipschitz continuous and that F(S)⋂Ω is a nonempty and bounded subset in C. Let {xn} be the sequence generated by
(3.65)x0∈C,C0=C,Q0=C,yn=J-1(βnJxn+(1-βn)JSnxn),un∈Csuchthat,∀y∈C,Θ(un,y)+〈Aun,y-un〉+ψ(y)-ψ(un)+1rn〈y-un,Jun-Jyn〉≥0,Cn={v∈Cn-1:ϕ(v,un)≤ϕ(v,xn)+ξn},Qn={z∈Qn-1:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0,∀n≥0,
where J:E→E* is the normalized duality mapping, {αn} and {βn} are sequences in [0,1], and {rn}⊂[a,∞) for some a>0,ξn=supu∈F(S)∩Ω(kn-1)ϕ(u,xn). If {βn} satisfy condition (ii) in Theorem 3.1, then {xn} converges strongly to ΠF(S)∩Ωx0.Proof.
TakingT=I in Theorem 3.1, we have that zn=xn, ∀n≥0. Hence, the conclusion of Theorem 3.8 is obtained.Theorem 3.9.
LetE be a uniformly smooth and strictly convex Banach space with the Kadec-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 Θ:C×C→ℝ a bifunction satisfying conditions (A1)–(A4). Suppose that Ω is a nonempty subset in C. Let {xn} be the sequence generated by
(3.66)x0∈C,C0=C,Q0=C,un∈Csuchthat,∀y∈C,Θ(un,y)+〈Aun,y-un〉+ψ(y)-ψ(un)+1rn〈y-un,Jun-Jxn〉≥0,Cn={v∈Cn-1:ϕ(v,un)≤ϕ(v,xn)},Qn={z∈Qn-1:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0,∀n≥0,
where {rn}⊂[a,∞) for some a>0. Then {xn} converges strongly to ΠΩx0.Proof.
TakingT=S=I in Theorem 3.1, the conclusion is obtained.Theorem 3.10.
LetE be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let S,T:C→C be two closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞) and kn→1. Suppose that S and T are uniformly L-Lipschitz continuous and that F(T)⋂F(S) is a nonempty and bounded subset in C. Let {xn} be the sequence generated by
(3.67)x0∈C,C0=C,Q0=C,zn=J-1(αnJxn+(1-αn)JTnxn),yn=J-1(βnJxn+(1-βn)JSnzn),un=ΠCyn,Cn={v∈Cn-1:ϕ(v,zn)≤ϕ(v,xn)+ξn,ϕ(v,un)≤ϕ(v,xn)+(1+kn)(1-βn)ξn},Qn={z∈Qn-1:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0,∀n≥0,
where J:E→E* is the normalized duality mapping, {αn} and {βn} are sequences in [0,1], and ξn=supu∈F(S)∩F(T)(kn-1)ϕ(u,xn). If {αn} and {βn} satisfy conditions (i)-(ii) in Theorem 3.1, then {xn} converges strongly to ΠF(S)∩F(T)x0.Proof.
TakingA=Θ=0 and rn=1, ∀n≥0 in Theorem 3.1, the conclusion of Theorem 3.10 is obtained.



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*Source: 956852-2011-07-13.xml*