# Strong Convergence Theorems for Modifying Halpern Iterations for Quasi-ϕ-Asymptotically Nonexpansive Multivalued Mapping in Banach Spaces with Applications

Authors: Li Yi
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

An iterative sequence for quasi-ϕ-asymptotically nonexpansive multivalued mapping for modifying Halpern's iterations is introduced. Under suitable conditions, some strong convergence theorems are proved. The results presented in the paper improve and extend the corresponding results in the work by Chang et al. 2011.


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## Body

## 1. Introduction

Throughout this paper, we denote byN and R the sets of positive integers and real numbers, respectively. Let D be a nonempty closed subset of a real Banach space X. A mapping T:D→D is said to be nonexpansive, if ∥Tx-Ty∥≤∥x-y∥, for all x,y∈D. Let N(D) and CB(D) denote the family of nonempty subsets and nonempty closed bounded subsets of D, respectively. The Hausdorff metric on CB(D) is defined by
(1.1)H(A1,A2)=max{supx∈A1d(x,A2),supy∈A2d(y,A1)}
for A1, A2∈CB(D), where d(x,A1)=inf{∥x-y∥,y∈A1}. The multivalued mapping T:D→CB(D) is called nonexpansive, if H(Tx,Ty)≤∥x-y∥, for all x,y∈D. An element p∈D is called a fixed point of T:D→N(D), if p∈T(p). The set of fixed points of T is represented by F(T).LetX be a real Banach space with dual X*. We denote by J the normalized duality mapping from X to 2X* which is defined by
(1.2)J(x)={x*∈X*:〈x,x*〉=∥x∥2=∥x*∥2},x∈X,
where 〈·,·〉 denotes the generalized duality pairing.A Banach spaceX is said to be strictly convex, if ∥(x+y)/2∥≤1 for all x,y∈X with ∥x∥=∥y∥=1 and x≠y. A Banach space is said to be uniformly convex, if limn→∞∥xn-yn∥=0 for any two sequences {xn},{yn}⊂X with ∥xn∥=∥yn∥=1 and limn→∞∥(xn+yn)/2∥=0.The norm of Banach spaceX is said to be Gâteaux differentiable, if for each x,y∈S(x), the limit
(1.3)limt→0∥x+ty∥-∥x∥t
exists, where S(x)={x:∥x∥=1,x∈X}. In this case, X is said to be smooth. The norm of Banach space X is said to be Fréchet differentiable, if for each x∈S(x), the limit (1.3) is attained uniformly, for y∈S(x), and the norm is uniformly Fréchet differentiable if the limit (1.3) is attained uniformly for x,y∈S(x). In this case, X is said to be uniformly smooth.Remark 1.1.
The following basic properties for Banach space X and for the normalized duality mappingJ can be found in Cioranescu [1].  (1)
X (X*, resp.) is uniformly convex if and only if X* (X, resp.) is uniformly smooth.(2)
IfX is smooth, then J is single-valued and norm-to-weak* continuous.(3)
IfX is reflexive, then J is onto.(4)
IfX is strictly convex, then Jx⋂‍Jy≠∅, for all x,y∈X.(5)
IfX has a Fréchet differentiable norm, then J is norm-to-norm continuous.(6)
IfX is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of X.(7)
Each uniformly convex Banach spaceX has the Kadec-Klee property, that is, for any sequence {xn}⊂X, if xn⇀x∈X and ∥xn∥→∥x∥, then xn→x∈X.(8)
IfX is a reflexive and strictly convex Banach space with a strictly convex dual X* and J*:X*→X is the normalized duality mapping in X*, then J-1=J*, JJ*=IX* and J*J=IX.Next, we assume thatX is a smooth, strictly convex, and reflexive Banach space and D is a nonempty, closed and convex subset of X. In the sequel, we always use ϕ:X×X→R+ to denote the Lyapunov functional defined by
(1.4)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,x,y∈X.
It is obvious from the definition of the function ϕ that
(1.5)(∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2,ϕ(y,x)=ϕ(y,z)+ϕ(z,x)+2〈z-y,Jx-Jz〉,x,y,z∈X,(1.6)ϕ(x,J-1(λJy+(1-λ)Jz))≤λϕ(x,y)+(1-λ)ϕ(x,z),
for all λ∈[0,1] and x,y,z∈X.Following Alber [2], the generalized projection ΠD:X→D is defined by
(1.7)ΠD(x)=arginfy∈Dϕ(y,x),∀x∈X.
Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping.Example 1.2 (see [3]).
LetΠD be the generalized projection from a smooth, reflexive and strictly convex Banach space X onto a nonempty closed convex subset D of X, then ΠD is a closed and quasi-ϕ-nonexpansive from X onto D.In 1953, Mann [4] introduced the following iterative sequence {xn}:
(1.8)xn+1=αnxn+(1-αn)Txn,
where the initial guess x1∈D is arbitrary, and {αn} is a real sequence in [0,1]. It is known that under appropriate settings the sequence {xn} converges weakly to a fixed point of T. However, even in a Hilbert space, Mann iteration may fail to converge strongly [5]. Some attempts to construct iteration method guaranteeing the strong convergence have been made. For example, Halpern [6] proposed the following so-called Halpern iteration:
(1.9)xn+1=αnu+(1-αn)Txn,
where u, x1∈D are arbitrary given and {αn} is a real sequence in [0,1]. Another approach was proposed by Nakajo and Takahashi [7]. They generated a sequence as follows:
(1.10)x1∈Xisarbitrary,yn=αnu+(1-αn)Txn,Cn={z∈D:∥yn-z∥≤∥xn-z∥},Qn={z∈D:〈xn-z,x1-xn〉≥0},xn+1=PCn⋂‍Qnx1(n=1,2,…),
where {αn} is a real sequence in [0,1] and PK denotes the metric projection from a Hilbert space H onto a closed and convex subset K of H. It should be noted here that the iteration previous works only in Hilbert space setting. To extend this iteration to a Banach space, the concept of relatively nonexpansive mappings are introduced (see [8–12]).Inspired by Matsushita and Takahashi, in this paper, we introduce modifying Halpern-Mann iterations sequence for finding a fixed point of multivalued mappingT:D→CB(D).


## 2. Preliminaries

In the sequel, we denote the strong convergence and weak convergence of the sequence{xn} by xn→x and xn⇀x, respectively.Lemma 2.1 (see [2]).
LetX be a smooth, strictly convex, and reflexive Banach space, and let D be a nonempty closed and convex subset of X. Then the following conclusions hold (a)
ϕ(x,y)=0 if and only if x=y, for all x,y∈X;(b)
ϕ(x,ΠDy)+ϕ(ΠDy,y)≤ϕ(x,y), for all x∈D, for all y∈X;(c)
ifx∈X and z∈D, then z=ΠDx⇔〈z-y,Jx-Jz〉≥0, for all y∈D.Remark 2.2.
IfH is a real Hilbert space, then ϕ(x,y)=∥x-y∥2 and ΠD is the metric projection PD of H onto D.Definition 2.3.
A pointp∈D is said to be an asymptotic fixed point of T:D→CB(D), if there exists a sequence {xn}⊂D such that xn⇀x∈X and d(xn,T(xn))→0. Denote the set of all asymptotic fixed points of T by F^(T).Definition 2.4.
(1) A multivalued mappingT:D→CB(D) is said to be relatively nonexpansive, if F(T)≠∅, F^(T)=F(T), and ϕ(p,z)≤ϕ(p,x), for all x∈D, p∈F(T), z∈T(x).
(2) A multivalued mappingT:D→CB(D) is said to be closed, if for any sequence {xn}⊂D with xn→x∈D and d(y,T(xn))→0, then d(y,T(x))=0.Next, we present an example of relatively nonexpansive multivalued mapping.Example 2.5 (see [13]).
LetX be a smooth, strictly convex, and reflexive Banach space, let D be a nonempty closed and convex subset of X, and let f:D×D→R be a bifunction satisfying the conditions: (A1) f(x,x)=0, for all x∈D; (A2) f(x,y)+f(y,x)≤0, for all x,y∈D; (A3) limt→0f(tz+(1-t)x,y)≤f(x,y), for each x,y,z∈D; (A4) the function y↦f(x,y) is convex and lower semicontinuous, for each x∈D. The “so-called” equilibrium problem for f is to find a x*∈D such that f(x*,y)≥0, for all y∈D. The set of its solutions is denoted by EP(f).
Letr>0, x∈X and define mapping Tr:X→D as follows:
(2.1)Tr(x)={x∈D,f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈D},∀x∈X,
then (1) Tr is single-valued, and so {z}=Tr(x); (2) Tr is a relatively nonexpansive mapping, therefore it is a closed and quasi-ϕ-nonexpansive mapping; (3) F(Tr)=EP(f).Definition 2.6.
(1) A multivalued mappingT:D→CB(D) is said to be quasi-ϕ-nonexpansive, if F(T)≠∅, and ϕ(p,z)≤ϕ(p,x), for all x∈D, p∈F(T), z∈T(x).
(2) A multivalued mappingT:D→CB(D) is said to be quasi-ϕ-asymptotically nonexpansive, if F(T)≠∅, and there exists a real sequence kn⊂[1,+∞), kn→1 such that
(2.2)ϕ(p,zn)≤knϕ(p,x),∀x∈D,p∈F(T),zn∈Tn(x).
(3) A multivalued mappingT:D→CB(D) is said to be totally quasi-ϕ-asymptotically nonexpansive, if F(T)≠∅, and there exist nonnegative real sequences {vn}, {μn} with vn, μn→0 (as n→∞) and a strictly increasing continuous function ζ:R+→R+ with ζ(0)=0 such that
(2.3)ϕ(p,zn)≤ϕ(p,x)+vnζ[ϕ(p,x)]+μn,∀x∈D,∀n≥1,p∈F(T),zn∈Tn(x).Remark 2.7.
From the definitions, it is obvious that a relatively nonexpansive multivalued mapping is a quasi-ϕ-nonexpansive multivalued mapping, and a quasi-ϕ-nonexpansive multivalued mapping is a quasi-ϕ-asymptotically nonexpansive multivalued mapping, but the converse is not true.Lemma 2.8.
LetX be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and let D be a nonempty closed and convex subset of X. Let {xn} and {yn} be two sequences in D such that xn→p and ϕ(xn,yn)→0, where ϕ is the function defined by (1.4), then yn→p.Proof.
Forϕ(xn,yn)→0, we have (∥xn∥-∥yn∥)2→0. This implies that ∥yn∥→∥p∥ and so ∥Jyn∥→∥Jp∥. Since D is uniformly smooth, X* is reflexive and JX=X*, therefore, there exist a subsequence {Jyni}⊂{Jyn} and a point x∈X such that Jyni⇀Jx. Because the norm ∥·∥ is weakly lower semi continuous, we have
(2.4)0=limni→∞ϕ(xni,yni)=limni→∞{∥xni∥2-2〈xni,Jyni〉+∥Jyni∥2}≥∥p∥2-2〈p,Jx〉+∥Jx∥2=ϕ(p,x).
By Lemma 2.1(a), we have p=x. Hence we have Jyni⇀Jp. Since ∥Jyni∥→∥Jp∥ and X* has the Kadec-Klee property, we have Jyni→Jp. By Remark 1.1, it follows that yni⇀p. Since ∥Jyni∥→∥Jp∥, by using the Kadec-Klee property of X, we get yni→p. If there exists another subsequence {Jynj}⊂{Jyn} such that ynj→q, then we have
(2.5)0=limnj→∞ϕ(xnj,ynj)=limnj→∞{∥xnj∥2-2〈xnj,Jynj〉+∥Jynj∥2}=∥p∥2-2〈p,Jq〉+∥q∥2=ϕ(p,q).
This implies that p=q. So yn→p. The conclusion of Lemma 2.8 is proved.Lemma 2.9.
LetX and D be as in Lemma 2.8. Let T:D→CB(D) be a closed and quasi-ϕ-asymptotically nonexpansive multivalued mapping with nonnegative real sequences {kn}⊂[1,+∞), if kn→1, then the fixed point set F(T) of T is a closed and convex subset of D.Proof.
Let{xn} be a sequence in F(T), such that xn→x*. Since T is quasi-ϕ-asymptotically nonexpansive multivalued mapping, we have
(2.6)ϕ(xn,z)≤k1ϕ(xn,x*)
for all z∈Tx* and for all n∈N. Therefore,
(2.7)ϕ(x*,z)=limn→∞ϕ(xn,z)≤limn→∞k1ϕ(xn,x*)=k1ϕ(x*,x*)=0.
By Lemma 2.1, we obtain z=x*, Hence, Tx*={x*}. So, we have x*∈F(T). This implies that F(T) is closed.
Letp, q∈F(T) and t∈(0,1), and put w=tp+(1-t)q. we prove that w∈F(T). Indeed, in view of the definition of ϕ, let zn∈Tnw, we have
(2.8)ϕ(w,zn)=∥w∥2-2〈w,Jzn〉+∥zn∥2=∥w∥2-2〈tp+(1-t)q,Jzn〉+∥zn∥2=∥w∥2+tϕ(p,zn)+(1-t)ϕ(q,zn)-t∥p∥2-(1-t)∥q∥2.
Since
(2.9)tϕ(p,zn)+(1-t)ϕ(q,zn)≤tknϕ(p,w)+(1-t)knϕ(q,w)=t{∥p∥2-2〈p,Jw〉+∥w∥2+(kn-1)ϕ(p,w)}+(1-t){∥q∥2-2〈q,Jw〉+∥w∥2+(kn-1)ϕ(q,w)}=t∥p∥2+(1-t)∥q∥2-∥w∥2+t(kn-1)ϕ(p,w)+(1-t)(kn-1)ϕ(q,w).
Substituting (2.8) into (2.9) and simplifying it, we have
(2.10)ϕ(w,zn)≤t(kn-1)ϕ(p,w)+(1-t)(kn-1)ϕ(q,w)→0,(asn→∞).
Hence, we have zn→w. This implies that zn+1(∈TTnw)→w. Since T is closed, we have Tw={w}, that is, w∈F(T). This completes the proof of Lemma 2.9.Definition 2.10.
A mappingT:D→CB(D) is said to be uniformly L-Lipschitz continuous, if there exists a constant L>0 such that ∥xn-yn∥≤L∥x-y∥, where x,y∈D, xn∈Tnx, yn∈Tny.


## 3. Main Results

Theorem 3.1.
LetX be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, let D be a nonempty, closed and convex subset of X, and let T:D→CB(D) be a closed and uniformly L-Lipschitz continuous quasi-ϕ-asymptotically nonexpansive multivalued mapping with nonnegative real sequences {kn}⊂[1,+∞) and kn→1 satisfying condition (2.2). Let {αn} be a sequence in (0,1). If {xn} is the sequence generated by
(3.1)x1∈Xisarbitrary;D1=D,yn=J-1[αnJx1+(1-αn)Jzn],zn∈Tnxn,Dn+1={z∈Dn:ϕ(z,yn)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)+ξn},xn+1=ΠDn+1x1(n=1,2,…),
where ξn=(kn-1)supp∈F(T)ϕ(p,xn), F(T) is the fixed point set of T, and ΠDn+1 is the generalized projection of X onto Dn+1. If F(T) is nonempty, then {xn} converges strongly to ΠF(T)x1.Proof.
(I)First, we prove thatDnare closed and convex subsets inD. By the assumption that D1=D is closed and convex. Suppose that Dn is closed and convex for some n≥1. In view of the definition of ϕ, we have
(3.2)Dn+1={z∈Dn:ϕ(z,yn)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)+ξn}={z∈D:ϕ(z,yn)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)+ξn}∩Dn={∥xn∥2z∈D:2αn〈z,Jx1〉+2(1-αn)〈z,Jxn〉-2〈z,Jzn〉≤αn∥x1∥2+(1-αn)∥xn∥2-∥zn∥2}∩Dn.
This shows that Dn+1 is closed and convex. The conclusions are proved.
(II)Next, we prove thatF(T)⊂Dn, for alln≥1. In fact, it is obvious that F(T)⊂D1. Suppose F(T)⊂Dn, for some n≥1. Hence, for any u∈F(T)⊂Dn, by (1.6), we have
(3.3)ϕ(u,yn)=ϕ(u,J-1(αnJx1+(1-αn)Jzn))≤αnϕ(u,x1)+(1-αn)ϕ(u,zn)≤αnϕ(u,x1)+(1-αn)knϕ(u,xn)=αnϕ(u,x1)+(1-αn){ϕ(u,xn)+(kn-1)ϕ(u,xn)}≤αnϕ(u,x1)+(1-αn){ϕ(u,xn)+(kn-1)supu∈F(T)ϕ(u,xn)}=αnϕ(u,x1)+(1-αn)ϕ(u,xn)+ξn.
This shows that u∈F(T)⊂Dn+1 and so F(T)⊂Dn.
(III)Now, we prove that{xn}converges strongly to some pointp*. In fact, since xn=ΠDnx1, from Lemma 2.1(c), we have
(3.4)〈xn-y,Jx1-Jxn〉≥0,∀y∈Dn.
Again since F(T)⊂Dn, we have
(3.5)〈xn-u,Jx1-Jxn〉≥0,∀u∈F(T).
It follows from Lemma 2.1(b) that for each u∈F(T) and for each n≥1,
(3.6)ϕ(xn,x1)=ϕ(ΠDnx1,x1)≤ϕ(u,x1)-ϕ(u,xn)≤ϕ(u,x1).
Therefore, {ϕ(xn,x1)} is bounded, and so is {xn}. Since xn=ΠDnx1 and xn+1=ΠDn+1x1∈Dn+1⊂Dn, we have ϕ(xn,x1)≤ϕ(xn+1,x1). This implies that {ϕ(xn,x1)} is nondecreasing. Hence limn→∞ϕ(xn,x1) exists. Since X is reflexive, there exists a subsequence {xni}⊂{xn} such that xni⇀p* (some point in D=D1). Since Dn is closed and convex and Dn+1⊂Dn. This implies that Dn is weakly closed and p*∈Dn for each n≥1. In view of xni=ΠDnix1, we have
(3.7)ϕ(xni,x1)≤ϕ(p*,x1),∀ni≥1.
Since the norm ∥·∥ is weakly lower semicontinuous, we have
(3.8)limni→∞infϕ(xn,x1)=limni→∞inf(∥xni∥2-2〈xni,Jx1〉+∥x1∥2)≥∥p*∥2-2〈p*,Jx1〉+∥x1∥2=ϕ(p*,x1),
and so
(3.9)ϕ(p*,x1)≤limni→∞infϕ(xn,x1)≤limni→∞supϕ(xn,x1)=ϕ(p*,x1).
This shows that limni→∞ϕ(xni,x1)=ϕ(p*,x1), and we have ∥xni∥→∥p*∥. Since xni⇀p*, by the virtue of Kadec-Klee property of X, we obtain that xni→p*. Since {ϕ(xn,x1)} is convergent, this together with limni→∞ϕ(xni,x1)=ϕ(p*,x1) shows that limni→∞ϕ(xn,x1)=ϕ(p*,x1). If there exists some subsequence {xnj}⊂{xn} such that xn→q, then from Lemma 2.1, we have
(3.10)ϕ(p*,q)=limni,nj→∞ϕ(xni,xnj)=limni,nj→∞ϕ(xni,ΠDnjx1)≤limni,nj→∞[ϕ(xni,x1)-ϕ(ΠDnjx1,x1)]=limni,nj→∞[ϕ(xni,x1)-ϕ(xnj,x1)]=ϕ(p*,x1)-ϕ(p*,x1)=0,
that is, p*=q and hence
(3.11)xn→p*.
By the way, from (3.11), it is easy to see that
(3.12)ξn=(kn-1)supp∈F(T)ϕ(p,xn)→0.
(IV)Now, we prove thatp*∈F(T). In fact, since xn+1∈Dn+1, from (3.1), (3.11), and (3.12), we have
(3.13)ϕ(xn+1,yn)≤αnϕ(xn+1,x1)+(1-αn)ϕ(xn+1,xn)+ξn→0.
Since xn→p*, it follows from (3.13) and Lemma 2.8 that
(3.14)yn→p*.
Since {xn} is bounded and T is quasi-ϕ-asymptotically nonexpansive multivalued mapping, Tnxn is bounded. In view of αn→0. Hence from (3.1), we have that
(3.15)limn→∞∥Jyn-Jzn∥=limn→∞∥Jx1-Jzn∥=0.
Since Jyn→Jp*, this implies Jzn→Jp*. From Remark 1.1, it yields that
(3.16)zn⇀p*.
Again since
(3.17)∥zn∥-∥p*∥=∥Jzn∥-∥Jp*∥≤∥Jzn-Jp*∥→0,
this together with (3.16) and the Kadec-Klee-property of X shows that
(3.18)zn→p*.
On the other hand, by the assumptions that T is L-Lipschitz continuous, thus we have
(3.19)d(Tzn,zn)≤d(Tzn,zn+1)+∥zn+1-xn+1∥+∥xn+1-xn∥+∥xn-zn∥≤(L+1)∥xn+1-xn∥+∥zn+1-xn+1∥+∥xn-zn∥.
From (3.18) and xn→p*, we have that d(Tzn,zn)→0. In view of the closeness of T, it yields that T(p*)={p*}, this implies that p*∈F(T).
(V)Finally, we prove thatp*=ΠF(T)x1  and soxn→ΠF(T)x1. Let w=ΠF(T)x1. Since w∈F(T)⊂Dn, we have ϕ(p*,x1)≤ϕ(w,x1). This implies that
(3.20)ϕ(p*,x1)=limn→∞ϕ(xn,x1)≤ϕ(w,x1),
which yields that p*=w=ΠF(T)x1. Therefore, xn→ΠF(T)x1. This completes the proof of Theorem 3.1.By Remark2.7, the following corollaries are obtained.Corollary 3.2.
LetX and D be as in Theorem 3.1, and let T:D→CB(D) be a closed and uniformly L-Lipschitz continuous a relatively nonexpansive multivalued mapping. Let {αn} in (0,1) with limn→∞αn=0. Let {xn} be the sequence generated by
(3.21)x1∈Xisarbitrary;D1=D,yn=J-1[αnJx1+(1-αn)Jzn],zn∈Txn,Dn+1={z∈Dn:ϕ(z,yn)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)},xn+1=ΠDn+1x1(n=1,2,…),
where F(T) is the set of fixed points of T, and ΠDn+1 is the generalized projection of X onto Dn+1, then {xn} converges strongly to ΠF(T)x1.Corollary 3.3.
LetX and D be as in Theorem 3.1, and let T:D→CB(D) be a closed and uniformly L-Lipschitz continuous quasi-ϕ-nonexpansive multivalued mapping. Let {αn} be a sequence of real numbers such that αn∈(0,1) for all n∈N, and satisfying: limn→∞αn=0. Let {xn} be the sequence generated by (3.21). Then, {xn} converges strongly to ΠF(T)x1.


## 4. Application

We utilize Corollary3.3 to study a modified Halpern iterative algorithm for a system of equilibrium problems.Theorem 4.1.
LetD, X, and {αn} be the same as in Theorem 3.1. Let f:D×D→R be a bifunction satisfying conditions (A1)–(A4) as given in Example 2.5. Let Tr:X→D be a mapping defined by (2.1), that is,
(4.1)Tr(x)={x∈D,f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈D},∀x∈X.
Let {xn} be the sequence generated by
(4.2)x1∈Xisarbitrary;D1=D,f(un,y)+1r〈y-un,Jun-Jxn〉≥0,∀y∈D,r>0,un∈Trxn,yn=J-1[αnJx1+(1-αn)Jun],Dn+1={z∈Dn:ϕ(z,yn)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)},xn+1=ΠDn+1x1(n=1,2,…).
If F(Tr)≠∅, then {xn} converges strongly to ΠF(T)x1 which is a common solution of the system of equilibrium problems for f.Proof.
In Example2.5, we have pointed out that un=Tr(xn), F(Tr)=EP(f), and Tr is a closed and quasi-ϕ-nonexpansive mapping. Hence (4.2) can be rewritten as follows:
(4.3)x1∈Xisarbitrary;D1=D,yn=J-1[αnJx1+(1-αn)Jun],un∈Trxn,Dn+1={z∈Dn:ϕ(z,yn)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)},xn+1=ΠDn+1x1(n=1,2,…).
Therefore the conclusion of Theorem 4.1 can be obtained from Corollary 3.3.



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