# Generalized Mixed Equilibrium Problems and Fixed Point Problem for a Countable Family of Total Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

Authors: Jinhua Zhu; Shih-Sen Chang; Min Liu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The purpose of this paper is first to introduce the concept oftotal quasi-ϕ-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases and then to use a hybrid algorithm to introduce a new iterative scheme 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 a countable family of total quasi-ϕ-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in an 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 byℝ and ℝ+ the set of all real numbers and all nonnegative 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, and J:E→2E* is the normalized duality mapping defined by (1.1)J(x)={x*∈E*:〈x,x*〉=‖x‖=‖x*‖},x∈E.
Let ψ:C→ℝ be a proper real-valued function, A:C→E* a nonlinear mapping, and F:C×C→ℝ a bifunction. The “so called” generalized mixed equilibrium problem for F,A,ψ is to find x*∈C such that (1.2)F(x*,y)+〈Ax*,y-x*〉+ψ(y)-ψ(x*)≥0,∀y∈C.
We denote the set of solutions of (1.2) by GMEP(F,A,ψ), that is, (1.3)GMEP⁡(F,A,ψ)={x*∈C:F(x*,y)+〈Ax*,y-x*〉+ψ(y)-ψ(x*)≥0,∀y∈C}.Special Examples
(i)
IfA=0, then the problem (1.2) is reduced to the mixed equilibrium problem (MEP), and the set of its solutions is denoted by
(1.4)MEP⁡(F,ψ)={x*∈C:Θ(x*,y)+ψ(y)-ψ(x*)≥0,∀y∈C}.(ii)
Ifψ≡0, then the problem (1.2) is reduced to the generalized equilibrium problem (GEP), and the set of its solutions is denoted by
(1.5)GEP⁡(F,A)={x*∈C:F(x*,y)+〈Ax*,y-x*〉≥0,∀y∈C}.(iii)
IfA=0,ψ=0, then the problem (1.2) is reduced to the equilibrium problem (EP), and the set of its solutions is denoted by
(1.6)EP⁡(F)={x*∈C:F(x*,y)≥0,∀y∈C}.(iv)
IfF=0, then the problem (1.2) is reduced to the mixed variational inequality of Browder type (VI), and the set of its solutions is denoted by
(1.7)VI⁡(C,A,ψ)={x*∈C:〈Ax*,y-x*〉+ψ(y)-ψ(x*)≥0,∀y∈C}.These show that the problem (1.2) is very general in the sense that numerous problems in physics, optimization, and economics reduce to finding a solution of (1.2). Recently, some methods have been proposed for the generalized mixed equilibrium problem in Banach space (see, e.g., [1–5]).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 (1.8)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 1.1.
The following basic properties for Banach spaceE and for the normalized duality mapping J can be found in Cioranescu [6].(i)
IfE is an arbitrary Banach space, then J is monotone and bounded;(ii)
IfE is a strictly convex Banach space, then J is strictly monotone;(iii)
IfE is a a smooth Banach space, then J is single-valued, and hemicontinuous; that is, J is continuous from the strong topology of E to the weak star topology of E*;(iv)
IfE is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E;(v)
IfE is a reflexive and strictly convex Banach space with a strictly convex dual E* and J*:E*→E is the normalized duality mapping in E*, then J-1=J*,JJ*=IE* and J*J=IE;(vi)
IfE is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping J is single valued, one to one and onto;(vii)
A Banach spaceE is uniformly smooth if and only if E* is uniformly convex. If E is uniformly smooth, then it is smooth and reflexive.Recall that a Banach spaceE has the Kadec-Klee property, if for any sequence {xn}⊂E and x∈E with xn⇀x∈E and ∥xn∥→∥x∥, then xn→x (as n→∞). It is well known that if E is a uniformly convex Banach space, then E has the Kadec-Klee property.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 (1.9)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2,∀x,y∈E.It is obvious from the definition ofϕ that (1.10)(‖x‖-‖y‖)2≤ϕ(x,y)≤(‖x‖+‖y‖)2,∀x,y∈E.Following Alber [7], the generalized projection ΠC:E→C is defined by (1.11)ΠC(x)=arginf⁡y∈Cϕ(y,x),∀x∈E.LetT:C→C be a mapping and F(T) be the set of fixed points of T.Recall that a pointp∈C is said to be an asymptotic  fixed  point  ofT 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). A point p∈C is said to be a strong asymptotic fixed point ofT, if there exists a sequence {xn}⊂C such that xn→p and ∥xn-Txn∥→0. We denoted the set of all strong asymptotic fixed points of T by F̂(T).Definition 1.2.
(1) A mappingT:C→C is said to be nonexpansive if
(1.12)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C.
(2) A mappingT:C→C is said to be relatively nonexpansive [8] if F(T)≠∅,F(T)=F̃(T) and
(1.13)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).
(3) A mappingT:C→C is said to be weak relatively nonexpansive [9] if F(T)≠∅,F(T)=F̂(T) and
(1.14)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).
(4) A mappingT:C→C is said to be closed, if for any sequence {xn}⊂C with xn→x and Txn→y, then Tx=y.Definition 1.3.
(1) A mappingT:C→C is said to be quasi-ϕ-nonexpansive [10] if F(T)≠∅ and
(1.15)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).
(2) A mappingT:C→C is said to be quasi-ϕ-asymptotically   nonexpansive[11], if F(T)≠∅ and there exists a real sequence {kn}⊂[1,∞) with kn→1 such that
(1.16)ϕ(p,Tnx)≤knϕ(p,x),∀n≥1,x∈C,p∈F(T).
(3) A mappingT:C→C is said to be uniformlyL-Lipschitz continuous, if there exists a constant L>0 such that
(1.17)‖Tnx-Tny‖≤L‖x-y‖,∀x,y∈C,∀n≥1.Definition 1.4.
(1) A mappingT:C→C is said to be total quasi-ϕ-asymptotically  nonexpansive if F(T)≠∅ and there exist nonnegative real sequences {νn},{μn} with νn→0,μn→0 (as n→∞) and a strictly increasing continuous function ζ:ℝ+→ℝ+ with ζ(0)=0 such that for all x∈C,P∈F(T)(1.18)ϕ(p,Tnx)≤ϕ(p,x)+νnζ(ϕ(p,x))+μn,∀n≥1.
(2) A countable family of mappings{Tn}:C→C is said to be uniformly total quasi-ϕ-asymptotically nonexpansive, if ⋂i=1∞F(Ti)≠∅ and there exist nonnegative real sequences {νn},{μn} with νn→0,μn→0 (as n→∞) and a strictly increasing continuous function ζ:ℝ+→ℝ+ with ζ(0)=0 such that for all x∈C,p∈⋂i=1∞F(Ti)(1.19)ϕ(p,Tinx)≤ϕ(p,x)+νnζ(ϕ(p,x))+μn,∀n≥1.Remark 1.5.
From the definition, it is easy to know that(1)
each relatively nonexpansive mapping is closed;(2)
takingζ(t)=t,t≥0,νn=(kn-1) and μn=0, then (1.16) can be rewritten as
(1.20)ϕ(p,Tinx)≤ϕ(p,x)+νnζ(ϕ(p,x))+μn,∀n≥1,x∈C,p∈F(T).
This implies that each quasi-ϕ-asymptotically nonexpansive mapping must be a total quasi-ϕ-asymptotically nonexpansive mapping, but the converse is not true;(3)
the class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-nonexpansive mappings as a subclass, but the converse is not true;(4)
the class of quasi-ϕ-nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, but the converse is not true;(5)
the class of weak relatively nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse is not true.A mappingA:C→E* is said to be α-inverse  strongly  monotone, if there exists α>0 such that (1.21)〈x-y,Ax-Ay〉≥α‖Ax-Ay‖2.Remark 1.6.
IfA is an α-inverse strongly monotone mapping, then it is 1/α-Lipschitz continuous.
Iterative approximation of fixed points for relatively nonexpansive mappings in the setting of Banach spaces has been studied extensively by many authors. In 2005, Matsushita and Takahashi [12] obtained some weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Recently, Ofoedu and Malonza [4], Zhang [5], Su et al. [13], Zegeye and Shahzad [14], Wattanawitoon and Kumam [15], Qin et al. [16], Takahashi and Zembayashi [17], Chang et al. [18, 19], Yao et al. [20, 21], Qin et al. [22], and Cho et al. [23, 24] extend the notions from relatively nonexpansive mappings, weakly relatively nonexpansive mappings or quasi-ϕ-nonexpansive mappings to quasi-ϕ-asymptotically nonexpansive mappings and also prove some strongence theorems to approximate a common fixed point of quasi-ϕ-nonexpansive mappings or quasi-ϕ-asymptotically nonexpansive mappings.The purpose of this paper is first to introduce the concept of total quasi-ϕ-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases, and then by using a hybrid algorithm to introduce a new iterative scheme 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 a countable family of total quasi-ϕ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results improve and extend the corresponding results in [8, 11–25].


## 2. Preliminaries

First, we recall some definitions and conclusions.Lemma 2.1 (see [7, 26]).
LetE be a smooth, strictly convex and reflexive Banach space and C be 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
(2.1)z=ΠCx⟺〈z-y,Jx-Jz〉≥0,∀y∈C;(c)
forx,y∈E,ϕ(x,y)=0 if and only if x=y.Remark 2.2.
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.Lemma 2.3 (see [18]).
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,
(2.2)‖∑n=1∞λnxn‖2≤∑n=1∞λn‖xn‖2-λiλjg(‖xi-xj‖).Lemma 2.4.
LetE be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and let C be a nonempty closed convex subset of E. Let T:C→C be a closed and total quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences {νn},{μn} and a strictly increasing continuous functions ζ:ℝ+→ℝ+ such that μ1=0,νn→0,μn→0 (as n→∞) and ζ(0)=0. Then F(T) is a closed convex subset of C.Proof.
Letting{pn} be a sequence in F(T) with pn→p (as n→∞), we prove that p∈F(T). In fact, from the definition of T, we have
(2.3)ϕ(pn,Tp)≤ϕ(pn,p)+ν1ζ(ϕ(pn,p))+μ1⟶0(asn⟶∞).
Therefore we have
(2.4)lim⁡n→∞ϕ(pn,Tp)=lim⁡n→∞(‖pn‖2-2〈pn,JTp〉)+‖Tp‖2=‖p‖2-2〈p,JTp〉+‖Tp‖2=ϕ(p,Tp)=0,
that is, p∈F(T).
Next we prove thatF(T) is convex. For any p,q∈F(T),t∈(0,1), putting w=tp+(1-t)q, we prove that w∈F(T). Indeed, in view of the definition of ϕ(x,y), we have
(2.5)ϕ(w,Tnw)=‖w‖2-2〈w,JTnw〉+‖Tnw‖2=‖w‖2-2t〈p,JTnw〉-2(1-t)〈q,JTnw〉+‖Tnw‖2=‖w‖2+tϕ(p,Tnw)+(1-t)ϕ(q,Tnw)-t‖p‖2-(1-t)‖q‖2≤‖w‖2+t(ϕ(p,w)+νnζ(ϕ(p,w))+μn)+(1-t)(ϕ(q,w)+νnζ(ϕ(q,w))+μn)-t‖p‖2-(1-t)‖q‖2=‖w‖2+t(‖p‖2-2〈p,Jw〉+‖w‖2)+t(νnζ(ϕ(p,w))+μn)+(1-t)×(‖q‖2-2〈q,Jw〉+‖w‖2)+(1-t)(νnζ(ϕ(q,w))+μn)-t‖p‖2-(1-t)‖q‖2=‖w‖2-2〈w,Jw〉+‖w‖2+tνnζ(ϕ(p,w))+(1-t)(νnζ(ϕ(q,w))+μn=tνnζ(ϕ(p,w))+(1-t)(νnζ(ϕ(q,w))+μn.
Since μn→0 and νn→0, we have ϕ(w,Tnw)→0 (as n→∞). From (1.10) we have ∥Tnw∥→∥w∥. Consequently ∥JTnw∥→∥Jw∥. This implies that {JTnw} is a bounded sequence. Since E is reflexive, E* is also reflexive. So we can assume that
(2.6)JTnw⇀f0∈E*.
Again since E is reflexive, we have J(E)=E*. Therefore there exists x∈E such that Jx=f0. By virtue of the weakly lower semicontinuity of norm ∥·∥, we have
(2.7)0=liminf⁡n→∞ϕ(w,Tnw)=liminf⁡n→∞(‖w‖2-2〈w,J(Tnw)〉+‖Tnw‖2)=liminf⁡n→∞(‖w‖2-2〈w,J(Tnw)〉+‖J(Tnw)‖2)≥‖w‖2-2〈w,f0〉+‖f0‖2=‖w‖2-2〈w,Jx〉+‖Jx‖2=‖w‖2-2〈w,Jx〉+‖x‖2=ϕ(w,x),
that is, w=x which implies that f0=Jw. Hence from (2.6) we have JTnw⇀Jw∈E*. Since ∥JTnw∥→∥w∥ and E* has the Kadec-Klee property, we have JTnw→Jw. Since E is uniformly smooth, E* is uniformly convex, which in turn implies that E* is smooth. From Remark 1.1(iii) it yields that J-1:E*→E is hemi-continuous. Therefore we have Tnw⇀w. Again since ∥Tnw∥→∥w∥, by using the Kadec-Klee property of E, we have Tnw→w. This implies that TTnw=Tn+1w→w. Since T is closed, we have w=Tw.
This completes the proof of Lemma2.4.Lemma 2.5.
LetE be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E. Let f:C×C→ℝ be a bifunction satisfying the following conditions:(A1)
f(x,x)=0,forallx∈C,(A2)
f is monotone, that is, f(x,y)+f(y,x)≤0,forallx,y∈C,(A3)
limsupt↓0f(x+t(z-x),y)≤f(x,y)forallx,z,y∈C,(A4)
The functiony↦f(x,y) is convex and lower semi-continuous.
Then the following conclusions hold:(1)
(Blum and Oettli [27]) for any given r>0 and x∈E, there exists a unique z∈C such that
(2.8)f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C;(2)
(Takahashi and Zembayashi [28]) for any given r>0 and x∈E, define a mapping Krf:E→C by(2.9)Krf(x)={z∈C:f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},x∈E.
Then, the following conclusions hold:(a)
Krf is single-valued;(b)
Krf is firmly nonexpansive-type mapping, that is, for allz,y∈E,
(2.10)〈Krfz-Krfy,JKrfz-JKrfy〉≤〈Krfz-Krfy,Jz-Jy〉;(c)
F(Krf)=EP(f) and Krf is quasi-ϕ-nonexpansive;(d)
EP(f) is closed and convex;(e)
ϕ(q,Krfx)+ϕ(Krfx,x)≤ϕ(q,x),forallq∈F(Krf).
For solving the generalized mixed equilibrium problem (1.2), let us assume that the following conditions are satisfied:(1)
E is a smooth, strictly convex, and reflexive Banach space and C is a nonempty closed convex subset of E;(2)
A:C→E* is β-inverse strongly monotone mapping;(3)
F:C×C→ℝ is bifunction satisfying the conditions (A1), (A3), (A4) in Lemma 2.5 and the following condition (A2)′:(A2)′
for someγ≥0 with γ≤β(2.11)F(x,y)+F(y,x)≤γ‖Ax-Ay‖2,∀x,y∈C;(4)
ψ:C→ℝ is a lower semicontinuous and convex function.
Under the assumptions as above, we have the following results.Lemma 2.6.
LetE,C,A,F,ψ satisfy the above conditions (1)–(4). Denote by
(2.12)Γ(x,y)=F(x,y)+ψ(y)-ψ(x)+〈Ax,y-x〉,∀x,y∈C.
For any given r>0 and x∈E, define a mapping KrΓ:E→C by
(2.13)KrΓ(x)={z∈C:Γ(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C}.
Then, the following hold:(a)
KrΓ is single-valued;(b)
KrΓ is a firmly nonexpansive-type mapping, that is, for all z,y∈E,
(2.14)〈KrΓ(z)-KrΓ(y),JKrΓ(z)-JKrΓ(y)〉≤〈KrΓ(z)-KrΓ(y),Jz-Jy〉;(c)
F(KrΓ)=EP(Γ)=GMEP(F,A,ψ);(d)
GMEP(F,A,ψ) is closed and convex;(e)
(2.15)ϕ(q,KrΓx)+ϕ(KrΓx,x)≤ϕ(q,x),∀q∈F(KrΓ).Proof.
It follows from Lemma2.5 that in order to prove the conclusions of Lemma 2.6 it is sufficient to prove that the function Γ:C×C→ℝ satisfies the conditions (A1)–(A4) in Lemma 2.5.
In fact, by the similar method as given in the proof of Lemma 2.4 in [1], we can prove that the function Γ satisfies the conditions (A1), (A3), and (A4). Now we prove that Γ also satisfies the conditions (A2).
Indeed, for anyx,y∈C, by condition (A2)′, we have
(2.16)Γ(x,y)+Γ(y,x)=F(x,y)+ψ(y)-ψ(x)+〈Ax,y-x〉+F(y,x)+ψ(x)-ψ(y)+〈Ay,x-y〉=F(x,y)+F(y,x)-〈Ax-Ay,x-y〉≤(γ-β)‖Ax-Ay‖2≤0.
This implies that the function Γ satisfies the conditions (A2). Therefore the conclusions of Lemma 2.6 can be obtained from Lemma 2.3 immediately.Remark 2.7.
It follows from Lemma2.5 that the mapping KrΓ is a relatively nonexpansive mapping. Thus, it is quasi-ϕ-nonexpansive.


## 3. Main Results

In this section, we shall 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 (1.2) and the set of common fixed points of a countable family of total quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces.In the sequel, we assume thatE,C,{Si}i=1∞,{Ai}i=1M,{Fi}i=1M,{ψi}i=1M satisfy the following conditions.(1)
LetE be a uniformly smooth and strictly convex Banach space with Kleac-Klee property and C a nonempty closed convex subset of E.(2)
LetSi:C→C be a countable family of closed and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences {νn},{μn} and a strictly increasing continuous functions ζ:ℝ+→ℝ+ such that νn→0,μn→0 (as n→∞) and μ1=0,ζ(0)=0. Suppose further that for each i≥1,Si is a uniformly Li-Lipschitz mapping, that is, there exists a constant Li>0 such that
(3.1)‖Sinx-Siny‖≤Li‖x-y‖,∀x,y∈C,∀n≥1.(3)
LetAi:C→E*(i=1,2,…,M) be a finite family of βi-inverse strongly monotone mappings.(4)
LetFi:C→ℝ(i=1,2,…,M) be a finite family of bifunction satisfying the conditions (A1), (A3), (A4), and the following condition (A2)′:(A2)′
For eachi=1,2,…,M there exists γi≥0 with γi≤βi such that
(3.2)Fi(x,y)+Fi(y,x)≤γi‖Aix-Aiy‖2,∀x,y∈C;(5)
Letψi:C→ℝ(i=1,2,…,M) be a finite family of lower semicontinuous and convex functions.Theorem 3.1.
LetE,C,{Si}i=1∞,{Ai}i=1M,{Fi}i=1M,{ψi}i=1M be the same as above. Suppose that
(3.3)F:=⋂i=1∞F(Ti)⋂⋂j=1MGMEP⁡(Fj,Aj,ψj)
is a nonempty and bounded subset of C. For any given x0∈C, let {xn} be the sequence generated by
(3.4)x0∈C0=C,zn=J-1(αn,0Jxn+∑i=1∞αn,iJSinxn),yn=J-1(αnJzn+(1-αn)Jxn),un=KrM,nΓMKrM-1,nΓM-1⋯Kr2,nΓ2Kr1,nΓ1yn,Cn+1={v∈Cn:ϕ(v,un)≤ϕ(v,xn)+ηn},xn+1=ΠCn+1x0,∀n≥0,
where
(3.5)ηn=νnsup⁡u∈Fζ(ϕ(u,xn))+μn,∀n≥1,Kri,nΓi:E→C,i=1,2,…,M is the mapping defined by (2.13) with Γ=Γi,r=ri,n, and
(3.6)Γi(x,y)=Fi(x,y)+〈Aix,y-x〉+ψi(y)-ψi(x),∀x,y∈C.rk,n∈[d,∞),k=1,2,…,M,n≥1 for some d>0,ΠCn+1 is the generalized projection of E onto the set Cn+1,and {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 Πℱx0, where Πℱ is the generalized projection from E onto ℱ.Proof.
We divide the proof of Theorem3.1 into five steps.
(i) We first prove thatℱ and Cn both are closed and convex subset of C for all n≥0.
In fact, it follows from Lemmas2.4 and 2.6 that F(Si),i≥1 and GMEP(Fj,Aj,ψj)(j=1,2,…,M) both are closed and convex. Therefore ℱ 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)≤ϕ(v,xn)+ηn is equivalent to
(3.7)2〈v,Jxn-Jun〉≤‖xn‖2-‖un‖2+ηn,
therefore, we have(3.8)Cn+1={v∈Cn:2〈v,Jxn-Jun〉≤‖xn‖2-‖un‖2+ηn}.
This implies thatCn+1 is closed and convex. The desired conclusions are proved. These in turn show that Πℱx0 and ΠCnx0 are well defined.
(ii) We prove that{xn} and {Sinxn}n=0∞ for all i≥1 are both bounded sequences in C.
By the definition ofCn, we have xn=ΠCnx0 for all n≥0. It follows from Lemma 2.1 (a) that
(3.9)ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(u,x0)-ϕ(u,ΠCnx0)≤ϕ(u,x0),∀n≥0,u∈F.
This implies that {ϕ(xn,x0)} is bounded. By virtue of (1.10), {xn} is bounded. Since ϕ(u,Sinxn)≤ϕ(u,xn)+νnζ(ϕ(u,xn))+μn for all u∈ℱ and i≥1,{Sinxn} is bounded for all i≥1, and so {zn}is bounded in E. Denote M by
(3.10)M=sup⁡n≥0,i≥1{‖xn‖,‖Sinxn‖,‖zn‖}<∞.
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
(3.11)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n≥1.
Therefore {ϕ(xn,x0)} is convergent. Without loss of generality, we can assume that
(3.12)lim⁡n→∞ϕ(xn,x0)=r≥0.(iii) Next, we prove that ℱ∶=⋂i=1∞F(Si)⋂⋂i=1MGMEP(Fi,Ai,ψi)⊂Cn for all n≥0.
Indeed, it is obvious thatℱ⊂C0=C. Suppose that ℱ⊂Cn for some n≥0. Since un=KrM,nΓMKrM-1,nΓM-1⋯Kr2,nΓ2Kr1,nΓ1yn, by Lemma 2.6 and Remark 2.7, Kri,nΓi is quasi-ϕ-nonexpansive. Again since E is uniformly smooth, E* is uniformly convex. Hence, For any given u∈ℱ⊂Cn and for any positive integer j>0, from Lemma 2.3 we have
(3.13)ϕ(u,un)=ϕ(u,KrM,nΓMKrM-1,nΓM-1⋯Kr2,nΓ2Kr1,nΓ1yn)≤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,i〈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,i〈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,i{ϕ(u,xn)+νnζ(ϕ(u,xn))+μn}-αn,0αn,jg(‖Jxn-JSjnxn‖)∑i=1∞αn,i〈u,JSinxn〉)+(1-αn)ϕ(u,xn)≤αn(ϕ(u,xn)+∑i=1∞αn,i(νnζ(ϕ(u,xn))+μn)-αn,0αn,jg(‖Jxn-JSjnxn‖))+(1-αn)ϕ(u,xn)≤ϕ(u,xn)+αn(νnsup⁡u∈Fζ(ϕ(u,xn))+μn)-αnαn,0αn,jg(‖Jxn-JSjnxn‖))=ϕ(u,xn)+αnηn-αnαn,0αn,jg(‖Jxn-JSjnxn‖)≤ϕ(u,xn)+ηn.
Hence u∈Cn+1 and so ℱ⊂Cn for all n≥0. By the way, from the definition of {ηn} and ζ and (3.10), it is easy to see that
(3.14)ηn=νnsup⁡u∈Fζ(ϕ(u,xn))+μn≤νnsup⁡u∈Fζ((‖u‖+M)2)+μn⟶0(asn⟶∞).(IV) Now, we prove that {xn} converges strongly to some point
(3.15)p∈F∶=⋂i=1∞F(Si)⋂⋂j=1∞GMEP⁡(Fj,Aj,ψj).
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
(3.16)ϕ(xni,x0)≤ϕ(p,x0),n≥0.
Since
(3.17)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
(3.18)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
(3.19)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(3.12)).
Therefore we have p=q. This implies that
(3.20)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 2.1(a) we have
(3.21)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
(3.22)ϕ(xn+1,un)≤ϕ(xn+1,xn)+ηn⟶0(asn⟶∞).
From (1.10) it yields (∥xn+1∥-∥un∥)2→0. Since∥xn+1∥→∥p∥, we have
(3.23)‖un‖⟶‖p‖(asn⟶∞).
Hence we have
(3.24)‖Jun‖⟶‖Jp‖(asn⟶∞).
This implies that{Jun} is bounded in E*. Since E is reflexive, and so E* is reflexive, there exists a subsequence {Juni}⊂{Jun} such that Juni⇀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
(3.25)ϕ(xni+1,uni)=‖xni+1‖2-2〈xni+1,Juni〉+‖uni‖2=‖xni+1‖2-2〈xni+1,Juni〉+‖Juni‖2
taking liminfn→∞ on the both sides of above equality and in view of the weak lower semicontinuity of norm ∥·∥, then it yields that
(3.26)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⇀Jp. It follows from (3.24) and the Kadec-Klee property of E* that Juni→Jp (as n→∞). Note that J-1:E*→E is hemi-continuous, it yields that uni⇀p. It follows from (3.23) and the Kadec-Klee property of E that limni→∞uni=p.
By the similar way as given in the proof of (3.20), we can also prove that
(3.27)lim⁡n→∞un=p.
From (3.20) and (3.27) we have that
(3.28)‖xn-un‖⟶0(asn⟶∞).
Since J is uniformly continuous on any bounded subset of E, we have
(3.29)‖Jxn-Jun‖⟶0(asn⟶∞).
For any j≥1 and any u∈ℱ, it follows from (3.13), (3.20), and (3.27) that
(3.30)αnαn,0αn,jg(‖Jxn-JSjnxn‖)≤ϕ(u,xn)-ϕ(u,un)+αnηn.
Since
(3.31)ϕ(u,xn)-ϕ(u,un)=‖xn‖2-‖un‖2-2〈u,Jxn-Jun〉≤|‖xn‖2-‖un‖2|+2‖u‖⋅‖Jxn-Jun‖≤‖xn-un‖(‖xn‖+‖un‖)+2‖u‖⋅‖Jxn-Jun‖,
from (3.28) and (3.29), it follows that
(3.32)ϕ(u,xn)-ϕ(u,un)⟶0(n⟶∞).
In view of condition (b) and condition (c), we have that
(3.33)g(‖Jxn-JSjnxn‖)⟶0(asn⟶∞).
It follows from the property of g that
(3.34)‖Jxn-JSjnxn‖⟶0,(asn⟶∞).
Since xn→p and J is uniformly continuous, it yields Jxn→Jp. Hence from (3.34) we have
(3.35)JSjnxn⟶Jp(asn⟶∞).
Since J-1:E*→E is hemicontinuous, it follows that
(3.36)Sjnxn⇀p(∀j≥1).
On the other hand, for each j≥1 we have
(3.37)|‖Sjnxn‖-‖p‖|=|‖J(Sjnxn)‖-‖Jp‖|≤‖JSjnxn-Jp‖⟶0(asn⟶∞).
This together with (3.36) shows that
(3.38)Sjnxn⟶p(foreachj≥1).
Furthermore, by the assumption that for eachj≥1,Sj is uniformly Li-Lipschitz continuous, hence we have
(3.39)‖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 (3.20) and (3.38), yields ∥Sjn+1xn-Sjnxn∥→0 (as n→∞). Hence from (3.36) we have Sjn+1xn→p, that is, SjSjnxn→p. In view of (3.38) and the closeness of Sj, it yields that Sjp=p,forallj≥1. This implies that p∈⋂i=1∞F(Si).
Next, we prove thatp∈⋂i=1MGMEP(Fi,Ai,ψi). Denote that
(3.40)un(m)=Krm,nΓmKrm-1,nΓm-1⋯Kr2,nΓ2Kr1,nΓ1yn,m=1,2,…,M-1,un(M)=un.
By the similar method as in the proof of (3.13), we can prove that
(3.41)ϕ(u,un(m))≤ϕ(u,xn)+ηn,m=1,2,…,M,u∈F,∀n≥1.
It follows from Lemma 2.6, (2.15), (3.32) that for any u∈ℱ,
(3.42)ϕ(un(M),un(M-1))=ϕ(KrnΓMun(M-1),un(M-1))≤ϕ(u,un(M-1))-ϕ(u,KrnΓMun(M-1))≤ϕ(u,xn)+ηn-ϕ(u,ΓrnMun(M-1))=ϕ(u,xn)+ηn-ϕ(u,un(M))=ϕ(u,xn)+ηn-ϕ(u,un)⟶0(asn⟶∞).
From (1.10) it yields (∥un(M)∥-∥un(M-1)∥)2→0. Since ∥un(M)∥=∥un∥→∥p∥, we have
(3.43)‖un(M-1)‖⟶‖p‖(asn⟶∞).
Hence we have(3.44)‖Jun(M-1)‖→‖Jp‖(asn⟶∞).
This implies that{Jun(M-1)} is bounded in E*. Since E is reflexive, and so E* is reflexive, there exists a subsequence {Juni(M-1)}⊂{Jun(M-1)} such that Juni(M-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
(3.45)ϕ(uni(M),uni(N-1))=‖uni(M)‖2-2〈uni(M),Juni(M-1)〉+‖uni(M-1)‖2=‖uni(M)‖2-2〈uni(M),Juni(M-1)〉+‖Juni(M-1)‖2
taking liminfni→∞ on the both sides of above equality and in view of the weak lower semicontinuity of norm ∥·∥, it yields that
(3.46)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).
This is, p=x. This implies that p0=Jp, and so Juni(M-1)⇀Jp. It follows from (3.44) and the Kadec-Klee property of E* that Juni(M-1)→Jp (as ni→∞). Note that J-1:E*→E is hemicontinuous it yields that uni(M-1)⇀p. It follows from (3.43) and the Kadec-Klee property of E that limni→∞uni(M-1)=p.
By the similar way as given in the proof of (3.20), we can also prove that
(3.47)lim⁡n→∞un(M-1)=p.
From (3.27) and (3.47) we have that
(3.48)‖un(M)-un(M-1)‖⟶0(asn⟶∞).
Since J is uniformly continuous on any bounded subset of E, we have
(3.49)‖Jun(M)-Jun(M-1)‖⟶0(asn⟶∞).
Since(3.50)un(i)=KrnΓiun(i-1),i=2,3,…,M,un(0)=yn,un(M)=un.
By the similar way as above, we can also prove that(3.51)un(i)→p,‖un(i)-un(i-1)‖⟶0,‖Jun(i)-Jun(i-1)‖⟶0,i=2,3,…,M‖yn-un(1)‖⟶0,‖Jyn-Jun(1)‖⟶0(asn⟶∞).
From (3.51) and the assumption that rn≥d,∀n≥0, we have
(3.52)lim⁡n→∞‖Jun(i)-Jun(i-1)‖ri,n=0,i=2,3,…,M;lim⁡n→∞‖Jyn-Jun(1)‖r1,n=0.
In the proof of Lemma 2.6 we have proved that the function Γi,i=1,2,…,M defined by (3.6) satisfies the condition (A1)–(A4) and
(3.53)Γi(un(i),y)+1ri,n〈y-un(i),Jun(i)-Jun(i-1)〉≥0,∀y∈C.
Therefore for any y∈C we have
(3.54)1ri,n〈y-un(i),Jun(i)-Jun(i-1)〉≥-Γi(un(i),y)≥Γi(y,un(i)).
This implies that
(3.55)Γi(y,un(i))≤1ri,n〈y-un(i),Jun(i)-Jun(i-1)〉≤(M1+‖y‖)‖Juni-Jun(i-1)‖ri,n
for some constant M1>0. Since the function y↦Γi(x,y) is convex and lower semi-continuous, letting n→∞ in (3.55), from (3.52) and (3.55), for each i, we have Γi(y,p)≤0,forally∈C.
Fort∈(0,1] and y∈C, letting yt=ty+(1-t)p, there are yt∈C and Γi(yt,p)≤0. By condition (A1) and (A4), we have
(3.56)0=Γi(yt,yt)≤tΓi(yt,y)+(1-t)Γi(yt,p)≤tΓi(yt,y).
Dividing both sides of the above equation by t, we have Γi(yt,y)≥0,forally∈C. Letting t↓0, from condition (A3), we have Γi(p,y)≥0,forally∈C,foralli=1,2,…,M, that is, for each i=1,2,…,M, we have
(3.57)Fi(p,y)+〈Aip,y-p〉+ψi(y)-ψi(p)≥0,∀y∈C.
This implies that p∈⋂j=1MGMEP(Fj,Aj,ψj). Therefore, we have that
(3.58)p∈F.
(V) Now, we provexn→Πℱx0.
Letw=Πℱx0. From w∈ℱ⊂Cn+1 and xn+1=ΠCn+1x0, we have ϕ(xn+1,x0)≤ϕ(w,x0),foralln≥0. This implies that
(3.59)ϕ(p,x0)=lim⁡n→∞ϕ(xn,x0)≤ϕ(w,x0).
By the definition ofΠℱx0 and (3.59), we have p=w. Therefore, xn→Πℱx0. This completes the proof of Theorem 3.1.Theorem 3.2.
LetE,C,{Kri,nΓi}i=1M,{Ai}i=1M,{Fi}i=1M,{ψi}i=1M,{GMEP(Fj,Aj,ψj)}j=1M be the same as above. Let {Si}i=1∞:C→C be 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
(3.60)G∶=⋂i=1∞F(Si)⋂⋂j=1MGMEP⁡(Fj,Aj,ψj)
is a nonempty and bounded subset of C. For any given x0∈C, let {xn} be the sequence generated by
(3.61)x0∈C0=C,zn=J-1(αn,0Jxn+∑i=1∞αn,iJSinxn),yn=J-1(αnJzn+(1-αn)Jxn),un=KrM,nΓMKrM-1,nΓM-1⋯Kr2,nΓ2Kr1,nΓ1yn,Cn+1={v∈Cn:ϕ(v,un)≤ϕ(v,xn)+ξn},xn+1=ΠCn+1x0,∀n≥0,
where ξn=supu∈𝒢(kn-1)ϕ(u,xn),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-asymptotically nonexpansive mappings, it is an infinite family of closed and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with sequence ζ(t)=t,t≥0,νn=kn-1,μn=0. Hence ζn=νnsupu∈𝒢ζ(ϕ(u,xn))+μn=supu∈𝒢(kn-1)ϕ(u,xn)→0. 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 [8, 11, 15, 16, 18–24, 28] and others in the following aspects.(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, quasi-ϕ-nonexpansive mapping or quasi-ϕ-asymptotically nonexpansive mappings to a countable family of total 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: 961560-2011-12-13.xml*