# Convergence of a Viscosity Iterative Method for Multivalued Nonself-Mappings in Banach Spaces

Authors: Jong Soo Jung
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

LetE be a reflexive Banach space having a weakly sequentially continuous duality mapping Jφ with gauge function φ, C a nonempty closed convex subset of E, and T:C→𝒦(E) a multivalued nonself-mapping such that PT is nonexpansive, where PT(x)={ux∈Tx:∥x-ux∥=d(x,Tx)}. Let f:C→C be a contraction with constant k. Suppose that, for each v∈C and t∈(0,1), the contraction defined by Stx=tPTx+(1-t)v has a fixed point xt∈C. Let {αn}, {βn}, and {γn} be three sequences in (0,1) satisfying approximate conditions. Then, for arbitrary x0∈C, the sequence {xn} generated by xn∈αnf(xn-1)+βnxn-1+γnPT(xn)
for all
n≥1 converges strongly to a fixed point of T.


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## Body

## 1. Introduction

LetE be a Banach space and C a nonempty closed subset of E. We will denote by ℱ(E) the family of nonempty closed subsets of E, by 𝒞ℬ(E) the family of nonempty closed bounded subsets of E, by 𝒦(E) the family of nonempty compact subsets of E, and by 𝒦𝒞(E) the family of nonempty compact convex subsets of E. Let H(·,·) be the Hausdorff distance on 𝒞ℬ(E); that is,
(1)H(A,B)=max{supa∈Ad(a,B),supb∈Bd(b,A)},
for all A,B∈𝒞ℬ(E), where d(a,B)=inf{∥a-b∥:b∈B} is the distance from the point a to the subset B. Recall that a mapping f:C→C is a contraction on C if there exists a constant k∈(0,1) such that ∥f(x)-f(y)∥≤k∥x-y∥,x,y∈C.A multivalued mappingT:C→ℱ(E) is said to be a contraction if there exists a constant k∈[0,1) such that
(2)H(Tx,Ty)≤k∥x-y∥∀x,y∈C.
If (2) is valid when k=1, the T is called nonexpansive. A point x is a fixed point for a multivalued mapping T if x∈Tx. Banach's contraction principle was extended to a multivalued contraction by Nadler [1] in 1969. The set of fixed points of T is denoted by F(T).Given a contractionf with constant k and t∈(0,1), we can define a contraction Gt:C→𝒦(C) by
(3)Gtx:=tTx+(1-t)f(x),x∈C.
Then Gt is a multivalued, and hence it has a (nonunique, in general) fixed point xt:=xtf∈C (see [1]); that is,(4)xt∈tTxt+(1-t)f(xt).
If T is single valued, we have
(5)xt=tTxt+(1-t)f(xt),
which was studied by Moudafi [2] (see also Xu [3]). As a special case of (5),
(6)xt=txt+(1-t)u,forgivenu∈C,
has been considered by Browder [4], Halpern [5], Jung and Kim [6, 7], Kim and Takahashi [8], Reich [9], Singh and Watson [10], Takahashi and Kim [11], Xu [12], and Xu and Yin [13] in a Hilbert space and Banach spaces.In 2007, Jung [14] established the strong convergence of {xt} defined by xt∈tTxt+(1-t)u,u∈C for the multivalued nonexpansive nonself-mapping T in a reflexive Banach space having a uniformly Gâteaux differentiable norm under the assumption Ty={y}.In order to give a partial answer to Jung's open question [14] Can the assumptionTy={y}be omitted?, in 2008, Shahzad and Zegeye [15] considered a class of multivalued mapping under some mild conditions as follows.LetC be a closed convex subset of a Banach space E. Let T:C→𝒦(E) be a multivalued nonself-mapping and
(7)PTx={ux∈Tx:∥x-ux∥=d(x,Tx)}.
Then PT:C→𝒦(E) is multivalued, and PTx is nonempty and compact for every x∈C. Instead of(8)Gtx=tTx+(1-t)u,u∈C,
we consider, for t∈(0,1), that (9)Stx=tPTx+(1-t)u,u∈C.It is clear thatStx⊆Gtx, and if PT is nonexpansive and T is weakly inward, then St is weakly inward contraction. Theorem 1 of Lim [16] guarantees that St has a fixed point, point xt; that is,
(10)xt∈tPTxt+(1-t)u⊆tTxt+(1-t)u.
If T is single valued, then (10) is reduced to (6).Shahzad and Zegeye [15] also gave the strong convergence result of {xt} defined by (10) in a reflexive Banach space having a uniformly Gâteaux differentiable norm, which unified, extended, and complemented several known results including those of Jung [14], Jung and Kim [6, 7], Kim and Jung [17], López Acedo and Xu [18], Sahu [19], and Xu and Yin [13].In 2009, motivated by the results of Rafiq [20] and Yao et al. [21], Ceng and Yao [22] considered the following iterative scheme.Theorem CY (see [22, Theorem 3.1]).
LetE be a uniformly convex Banach space E having a uniformly Gâteaux differentiable norm, C a nonempty closed convex subset of E, and T:C→𝒦(E) a multivalued nonself-mapping such that PT is nonexpansive. Suppose that C is a nonexpansive retract of E and that for each v∈C and t∈(0,1) the contraction St defined by Stx=tPTx+(1-t)v has a fixed point xt∈C. Let {αn}, {βn}, and {γn} be three sequences in (0,1) satisfying the following conditions:(i)
αn+βn+γn=1;(ii)
limn→∞αn=0 and limn→∞(βn/αn)=0. For arbitrary initial value x0∈C and a fixed element u∈C, let the sequence {xn} be generated by
(11)xn∈αnu+βnxn-1+γnPTxn∀n≥1.
Then {xn} converges strongly to a fixed point of T.Theorem CY also improves, develops, and complements the corresponding results in Jung [14], Jung and Kim [6, 7], Kim and Jung [17], López Acedo and Xu [18], Shahzad and Zegeye [15], and Xu and Yin [13] to the iterative scheme (11). For convergence of related iterative schemes for several nonlinear mappings, we can refer to [23–26] and the references therein.In this paper, inspired and motivated by the above-mentioned results, we consider a viscosity iterative method for a multivalued nonself-mapping in a reflexive Banach space having a weakly sequentially continuous duality mapping and establish the strong convergence of the sequence generated by the proposed iterative method. Our results improve and develop the corresponding results of Ceng and Yao [22], as well as some known results in the earlier and recent literature, including those of Jung [14], Jung and Kim [6, 7], Kim and Jung [17], López Acedo and Xu [18], Sahu [19], Shahzad and Zegeye [15], Xu [12], and Xu and Yin [13], to the viscosity iterative scheme in different Banach space.


## 2. Preliminaries

LetE be a real Banach space with norm ∥·∥, and let E* be its dual. The value of x*∈E* at x∈E will be denoted by 〈x,x*〉.A Banach spaceE is called uniformly convex if δ(ε)>0 for every ε>0, where the modulus δ(ε) of convexity of E is defined by
(12)δ(ε)=inf{1-∥x+y2∥:∥x∥≤1,∥y∥≤1,∥x-y∥≥ε}
for every ε with 0≤ε≤2. It is well known that if E is uniformly convex, then E is reflexive and strictly convex (cf. [27]).By a gauge function we mean a continuous strictly increasing functionφ defined on ℝ+:=[0,∞) such that φ(0)=0 and limr→∞φ(r)=∞. The mapping Jφ:E→2E* defined by
(13)Jφ(x)={f∈E*:〈x,f〉=∥x∥∥f∥,∥f∥=φ(∥x∥)}∀x∈E
is called the duality mapping with gauge function φ. In particular, the duality mapping with gauge function φ(t)=t denoted by J is referred to as the normalized duality mapping. It is known that a Banach space E is smooth if and only if the normalized duality mapping J is single valued. The following property of duality mapping is also well known:
(14)Jφ(λx)=signλ(φ(|λ|·∥x∥)∥x∥)J(x)∀x∈E∖{0},λ∈ℝ,
where ℝ is the set of all real numbers; in particular, J(-x)=-J(x) for all x∈E ([28]).We say that a Banach spaceE has a weakly sequentially continuous duality mapping if there exists a gauge function φ such that the duality mapping Jφ is single valued and continuous from the weak topology to the weak* topology; that is, for any {xn}∈E with xn⇀x,Jφ(xn)⇀*Jφ(x). For example, every lp space (1<p<∞) has a weakly continuous duality mapping with gauge function φ(t)=tp-1. It is well known that if E is a Banach space having a weakly sequentially continuous duality mapping Jφ with gauge function φ, then E has the opial condition [29]; this is, whenever a sequence {xn} in E converges weakly to x∈E, then (15)limsupn→∞∥xn-x∥<limsupn→∞∥xn-y∥∀y∈E,y≠x.
A mapping T:C→𝒞ℬ(E) is *-nonexpansive [30] if, for all x,y∈C and ux∈Tx with ∥x-ux∥=inf{∥x-z∥:z∈Tx}, there exists uy∈Ty with ∥y-uy∥=inf{∥y-w∥:w∈Ty} such that
(16)∥ux-uy∥≤∥x-y∥.
It is known that *-nonexpansiveness is different from nonexpansiveness for multivalued mappings. There are some *-nonexpansiveness multivalued mappings which are not nonexpansive and some nonexpansive multivalued mappings which are not *-nonexpansive [31].We introduce some terminology for boundary conditions for nonself-mappings. Theinward set of C at x is defined by
(17)IC(x)={z∈E:z=x+λ(y-x):y∈C,λ≥0}.
Let I-C(x)=x+TC(x) with
(18)TC(x)={y∈E:liminfλ→0+d(x+λy,C)λ=0}
for any x∈C. Note that, for a convex set C, we have I-C(x)=IC(x)¯, the closure of IC(x). A multivalued mapping T:C→ℱ(E) is said to satisfy the weak inwardness condition if Tx⊂I-C(x) for all x∈C.Finally, the following lemma was given by Xu [32] (also see Xu [33]).Lemma 1.
IfC is a closed bounded convex subset of a uniformly convex Banach space E and T:C→𝒦(E) is a nonexpansive mapping satisfying the weak inwardness condition, then T has a fixed point.


## 3. Main Results

Now, we establish strong convergence of a viscosity iterative scheme for a multivalued nonself-mapping.Theorem 2.
LetE be a reflexive Banach space having a weakly sequentially continuous duality mapping Jφ with gauge function φ. Let C be a nonempty closed convex subset of E and T:C→𝒦(E) a multivalued nonself-mapping such that F(T)≠∅ and PT is nonexpansive. Let f:C→C be a contraction with constant k. Suppose that for each v∈C and t∈(0,1), the contraction St defined by Stx=tPTx+(1-t)v has a fixed point xt∈C. Let {αn}, {βn}, and {γn} be three sequences in (0,1) satisfying the following conditions:(i)
αn+βn+γn=1;(ii)
limn→∞αn=0 and limn→∞(βn/αn)=0. For arbitrary initial value x0∈C, let the sequence {xn} be defined by
(19)xn∈αnf(xn-1)+βnxn-1+γnPTxn∀n≥1.
Then {xn} converges strongly to a fixed point of T.Proof.
First, observe that, for eachn≥1,
(20)αnf(xn-1)+βnxn-1=(1-γn)(αn1-γnf(xn-1)+βn1-γnxn-1).
From xn-1, f(xn-1)∈C, it follows that (αn/(1-γn))f(xn-1)+(βn/(1-γn))xn-1∈C. Also, notice that, for each v∈C and t∈(0,1), the contraction St defined by Stx=tPTx+(1-t)v has a fixed point xt∈C. Thus, for (αn/(1-γn))f(xn-1)+(βn/(1-γn))xn-1∈C and γn∈(0,1), there exists xn∈C such that
(21)xn∈(1-γn)(αn1-γnf(xn-1)+βn1-γnxn-1)+γnPT(xn).
This shows that the sequence {xn} can be defined well via the following:
(22)xn∈αnf(xn-1)+βnxn-1+γnPTxn∀n≥1.
Therefore, for any n≥1, we can find some zn∈PT(xn) such that
(23)xn=αnf(xn-1)+βnxn-1+γnzn.
Next, we divide the proof into several steps.
Step1. We show that {xn} is bounded. Indeed, notice that PT(y)={y} whenever y is a fixed point of T. Let p∈F(T). Then p∈PT(p) and we have
(24)∥zn-p∥=d(zn,PT(p))≤H(PT(xn),PT(p))≤∥xn-p∥.
It follows that
(25)∥xn-p∥=∥αnf(xn-1)+βnxn-1+γnzn-p∥≤αn∥f(xn-1)-f(p)∥+αn∥f(p)-p∥+βn∥xn-1-p∥+γn∥zn-p∥≤αn(k∥xn-1-p∥+∥f(p)-p∥)+βn∥xn-1-p∥+γn∥xn-p∥,
and so
(26)∥xn-p∥≤αn1-γn(k∥xn-1-p∥+∥f(p)-p∥)+βn1-γn∥xn-1-p∥=kαn+βn1-γn∥xn-1-p∥+(1-k)αn1-γn∥f(p)-p∥1-k≤max{∥xn-1-p∥,∥f(p)-p∥1-k}.
By induction, we have
(27)∥xn-p∥≤max{∥x0-p∥,11-k∥f(p)-p∥}forn≥1.
Hence {xn} is bounded and so are {zn} and {f(xn)}.
Step2. We show that limn→∞∥xn-zn∥=0. In fact, since xn=αnf(xn-1)+βnxn-1+γnzn for some zn∈PT(xn), by conditions (i) and (ii), we have
(28)∥xn-zn∥≤αn∥f(xn-1)-zn∥+βn∥xn-1-zn∥→0asn→∞.Step3. We show that there exists p∈PT(p)⊂Tp. In fact, since {xn} and {zn} are bounded and E is reflexive, there exists a subsequence {xnk} of {xn} such that xnk⇀p. For this p, by compactness of PT(p), we can find wn∈PT(p), ∀n≥1, such that
(29)∥zn-wn∥=d(zn,PT(p))≤H(PT(xn),PT(p))≤∥xn-p∥.
Now suppose that xn:=xnk and xn⇀p. The sequence {wn} has a convergent subsequence, which is denoted again by {wn} with wn→w∈PT(p). Assume that w≠p. Since a Banach space having the weakly sequentially continuous duality mapping satisfies the opial condition [29], by Step 2, we have
(30)limsupn→∞∥xn-w∥≤limsupn→∞(∥xn-zn∥+∥zn-wn∥+∥wn-w∥)≤limsupn→∞∥zn-wn∥≤limsupn→∞∥xn-p∥<limsupn→∞∥xn-w∥,
which is a contradiction. Hence we have w=p, and so p=w∈PT(p)⊂Tp.
Step4. We show that limn→∞〈zn-f(xn-1),Jφ(xn-y)〉≤0 for y∈F(T). Indeed, for y∈F(T), by (24), we have
(31)〈xn-zn,Jφ(xn-y)〉=〈xn-y+y-zn,Jφ(xn-y)〉≥∥xn-y∥φ(∥xn-y∥)-∥zn-y∥φ(∥xn-y∥)≥∥xn-y∥φ(∥xn-y∥)-∥xn-y∥φ(∥xn-y∥)=0.
So,
(32)0≤〈xn-zn,Jφ(xn-y)〉=αn〈f(xn-1)-zn,Jφ(xn-y)〉+βn〈xn-1-zn,Jφ(xn-y)〉≤αn(βnαn〈f(xn-1)-zn,Jφ(xn-y)〉+βnαn∥xn-1-zn∥φ(∥xn-y∥)).
Thus it follows that
(33)〈zn-f(xn-1),Jφ(xn-y)〉≤βnαn∥xn-1-zn∥φ(∥xn-y∥).
Hence, from condition (ii), we conclude that
(34)limn→∞〈zn-f(xn-1),Jφ(xn-y)〉≤0.Step5. We show that limn→∞∥xn-p∥=0, where p is defined as in Step 3. Indeed, for y∈F(T),
(35)∥xn-y∥φ(∥xn-y∥)=αn〈f(xn-1)-y,Jφ(xn-y)〉+βn〈xn-1-y,Jφ(xn-y)〉+γn〈zn-y,Jφ(xn-y)〉=(αn+γn)〈f(xn-1)-y,Jφ(xn-y)〉+βnαn〈xn-1-y,Jφ(xn-y)〉αn+γn〈zn-f(xn-1),Jφ(xn-y)〉.
Interchanging p and y in (35), we obtain
(36)∥xn-p∥φ(∥xn-p∥)=(αn+γn)〈f(xn-1)-f(p),Jφ(xn-p)〉+(αn+γn)〈f(p)-p,Jφ(xn-p)〉+βnαn∥xn-1-p∥φ(∥xn-p∥)+γn〈zn-f(xn-1),Jφ(xn-p)〉,
and so
(37)(∥xn-p∥-k∥xn-1-p∥)φ(∥xn-p∥)≤βnk∥xn-1-p∥φ(∥xn-p∥)+(1-βn)〈f(p)-p,Jφ(xn-p)〉+βnαn∥xn-1-p∥φ(∥xn-p∥)+γn〈zn-f(xn-1),Jφ(xn-p)〉.
Using the fact that Jφ is weakly sequentially continuous, Step 4, and condition (ii), we have
(38)(∥xn-p∥-k∥xn-1-p∥)φ(∥xn-p∥)→0asn→∞.
This implies that xn→p as n→∞. In fact, if limn→∞∥xn-p∥=η≠0, then limn→∞φ(∥xn-p∥)≠0, and by (38) limn→∞(∥xn-p∥-k∥xn-1-p∥)=0. This means that limn→∞∥xn-p∥=η=kη, which is a contradiction. Thus we proved that there exists a subsequence {xnk} of {xn} which converges strongly to a fixed point p of T.
Step6. We show that the entire sequence {xn} converges strongly to p∈F(T). Suppose that there exists another subsequence {xnj} of {xn} such that xnj→q as j→∞. Since d(xnj,PT(xnj))≤∥xnj-znj∥→0 as j→∞, it follows that d(q,PT(q))=0 and so q∈PT(q)⊂T(q); that is, q∈F(T). Notice that PT(q)={q}. Since {xn} is bounded and the duality mapping Jφ is single valued and weakly sequentially continuous from E to E*, we have
(39)|〈xnj-f(q),Jφ(xnj-p)〉-〈q-f(q),Jφ(q-p)〉|=|〈xnj-q,Jφ(xnj-p)〉+〈q-f(q),Jφ(xnj-p)-Jφ(q-p)〉|≤∥xnj-q∥φ(∥xnj-p∥)+|〈q-f(q),Jφ(xnj-p)-Jφ(q-p)〉|→0asj→∞.
Thus, from Steps 2 and 4, it follows that(40)〈q-f(q),Jφ(q-p)〉=limj→∞〈xnj-f(q),Jφ(xnj-p)〉=limj→∞〈xnj-znj,Jφ(xnj-p)〉+limj→∞〈znj-f(xnj-1),Jφ(xnj-p)〉+limj→∞〈f(xnj-1)-f(q),Jφ(xnj-p)〉≤∥xnj-znj∥φ(∥xnj-p∥)+klimj→∞∥xnj-1-q∥φ(∥xnj-p∥)+limj→∞〈znj-f(xnj-1),Jφ(xnj-p)〉≤0.
By the same argument, we also have(41)〈p-f(p),Jφ(p-q)〉≤0.
Therefore, from (40) and (41), we obtain
(42)∥p-q∥φ(∥p-q∥)=〈p-q,Jφ(p-q)〉=〈p-f(p),Jφ(p-q)〉+〈f(p)-f(q),Jφ(p-q)〉+〈q-f(q),Jφ(q-p)〉≤〈f(p)-f(q),Jφ(p-q)〉≤k∥p-q∥φ(∥p-q∥),
and so (1-k)∥p-q∥φ(∥p-q∥)≤0. Thus p=q. This completes the proof.Remark 3.
(1) In Theorem2, if f(x)=u∈C,x∈E, is a constant mapping, then the iterative scheme (19) is reduced to the iterative scheme (11) in Theorem CY of Ceng and Yao [22] in the Introduction section. Therefore Theorem 2 improves Theorem CY to the viscosity iterative scheme in different Banach space.
(2) In Theorem2, we remove the assumption that C is a nonexpansive retract of E in Theorem CY.
(3) In Theorem2, if βn=0 for n≥0, then the iterative scheme (19) becomes the following scheme:
(43)xn∈αnf(xn-1)+(1-αn)PT(xn),
which is a viscosity iterative scheme for those in Shahzad and Zegeye [15]. Therefore Theorem 2 develops Theorem 3.1 of Shahzad and Zegeye [15], as well as Theorem 1 of Jung [14], to the viscosity iterative method in different Banach space.
(4) Theorem2 also improves and complements the corresponding results of Kim and Jung [17] and Sahu [19] as well as Jung and Kim [6, 7], López Acedo and Xu [18], and Xu and Yin [13].By definition of the Hausdorff metric, we obtain that ifT is *-nonexpansive, then PT is nonexpansive. Hence, as a direct consequence of Theorem 2, we have the following result.Corollary 4.
LetE be a reflexive Banach space having a weakly sequentially continuous duality mapping Jφ with gauge function φ. Let C be a nonempty closed convex subset of E and T:C→𝒦(E) a multivalued *-nonexpansive nonself-mapping such that F(T)≠∅. Let f:C→C be a contraction with constant k. Suppose that, for each v∈C and t∈(0,1), the contraction St defined by Stx=tPTx+(1-t)v has a fixed point xt∈C. Let {αn}, {βn}, and {γn} be three sequences in (0,1) satisfying the following conditions:(i)
αn+βn+γn=1;(ii)
limn→∞αn=0 and limn→∞(βn/αn)=0. For arbitrary initial value x0∈C, let the sequence {xn} be generated by  (19). Then {xn} converges strongly to a fixed point of T.It is well known that every nonempty closed convex subsetC of a strictly convex and reflexive Banach space E is Chebyshev; that is, for any x∈E, there is a unique element u∈C such that ∥x-u∥=inf{∥x-v∥:v∈C}. Thus, we have the following corollary.Corollary 5.
LetE be a strictly convex and reflexive Banach space having a weakly sequentially continuous duality mapping Jφ with gauge function φ. Let C be a nonempty closed convex subset of E and T:C→𝒦𝒞(E) a multivalued nonself-mapping such that F(T)≠∅ and PT is nonexpansive. Let f:C→C be a contraction with constant k. Suppose that, for each v∈C and t∈(0,1), the contraction St defined by Stx=tPTx+(1-t)v has a fixed point xt∈C. Let {αn}, {βn}, and {γn} be three sequences in (0,1) satisfying the following conditions:(i)
αn+βn+γn=1;(ii)
limn→∞αn=0 and limn→∞(βn/αn)=0. For arbitrary initial value x0∈C, let the sequence {xn} be generated by  (19). Then {xn} converges strongly to a fixed point of T.Proof.
In this case,Tx is Chebyshev for each x∈C. So PT is a selector of T and PT is single valued. Thus the result follows from Theorem 2.Corollary 6.
LetE be a strictly convex and reflexive Banach space having a weakly sequentially continuous duality mapping Jφ with gauge function φ. Let C be a nonempty closed convex subset of E and T:C→𝒦𝒞(E) a multivalued *-nonexpansive nonself-mapping such that F(T)≠∅. Let f:C→C be a contraction with constant k. Suppose that, for each v∈C and t∈(0,1), the contraction St defined by Stx=tPTx+(1-t)v has a fixed point xt∈C. Let {αn}, {βn}, and {γn} be three sequences in (0,1) satisfying the following conditions:(i)
αn+βn+γn=1;(ii)
limn→∞αn=0 and limn→∞(βn/αn)=0. For arbitrary initial value x0∈C, let the sequence {xn} be generated by  (19). Then {xn} converges strongly to a fixed point of T.Corollary 7.
LetE be a uniformly convex Banach space having a weakly sequentially continuous duality mapping Jφ with gauge function φ. Let C be a nonempty closed convex subset of E and T:C→𝒦(E) a multivalued nonself-mapping satisfying the weak inwardness condition such that PT is nonexpansive. Let f:C→C be a contraction with constant k. Let {αn}, {βn}, and {γn} be three sequences in (0,1) satisfying the following conditions:(i)
αn+βn+γn=1;(ii)
limn→∞αn=0 and limn→∞(βn/αn)=0. For arbitrary initial value x0∈C, let the sequence {xn} be generated by  (19). Then {xn} converges strongly to a fixed point of T.Proof.
Define, for eachv∈C and t∈(0,1), the contraction St:C→𝒦(E) by
(44)Stx:=tPTx+(1-t)v,x∈C.
As it is easily seen that St also satisfies the weak inwardness condition: Stx⊂I-C(x) for all x∈C, it follows from Lemma 1 that St has a fixed point denoted by xt. Thus the result follows from Theorem 2.Remark 8.
(1) As in [31], Shahzad and Zegeye [15] gave the following example of a multivalued T such that PT is nonexpansive. Let C=[0,∞), and let T be defined by Tx=[x,2x] for x∈C. Then PTx={x} for x∈C. Also T is *-nonexpansive but not nonexpansive (see [31]).
(2) Corollaries4–7 develop Corollaries 3.3–3.6 of Ceng and Yao [22] to the viscosity iterative method in different Banach spaces.
(3) By replacing the iterative scheme (11) in Theorem CY with the iterative scheme (19) in Theorem 2 and using the same proof lines as Theorem CY together with our method, we can also establish the viscosity iteration version of Theorem CY.



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*Source: 369412-2013-02-19.xml*