POINTS FOR NONEXPANSIVE NONSELF-MAPPING

POINTS FOR NONEXPANSIVE NONSELF-MAPPING

RUDONG CHEN AND ZHICHUAN ZHU Received 17 May 2006; Accepted 22 June 2006

LetCbe a closed convex subset of a uniformly smooth Banach spaceE, andT:C→E a nonexpansive nonself-mapping satisfying the weakly inwardness condition such that F(T)= ∅, and f :C→Ca fixed contractive mapping. Fort∈(0, 1), the implicit itera- tive sequence{xt}is defined byxt=P(t f(xt) + (1−t)Txt), the explicit iterative sequence {xn}is given byxn+1=P(αnf(xn) + (1−αn)Txn), whereαn∈(0, 1) andPis a sunny non- expansive retraction ofEontoC. We prove that{xt}strongly converges to a fixed point ofT ast→0, and{xn}strongly converges to a fixed point ofT asαnsatisfying appro- priate conditions. The results presented extend and improve the corresponding results of Hong-Kun Xu (2004) and Yisheng Song and Rudong Chen (2006).

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

1. Introduction

LetCbe a nonempty closed convex subset of a Banach spaceE, and LetT:C→C be a nonexpansive mapping (i.e.,Tx−T y ≤ x−yfor allx,y∈C). We use Fix(T) to denote the set of fixed points ofT; that is , Fix(T)= {x∈C:x=Tx}. Recall that a self- mapping f :C→Cis a contraction onCif there exists a constantβ∈(0, 1) such that

f(x)−f(y)≤βx−y, x,y∈C. (1.1) Xu (see [6]) defined the following two viscosity iterations for nonexpansive mappings:

xt=t fxt

+ (1−t)Txt, x∈C, (1.2)

xn+1=αnfxn

+1−αnTxn, (1.3)

whereαnis a sequence in (0,1). Xu proved the strong convergence of{xt}defined by (1.2) ast→0 and{xn}defined by (1.3) in both Hilbert space and uniformly smooth Banach space.

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 16470, Pages1–12

DOI 10.1155/IJMMS/2006/16470

Recently, Song and Chen [2] proved ifCis a closed subset of a real reflexive Banach spaceEwhich admits a weakly sequentially continuous duality mapping fromEtoE^∗, and ifT:C→Eis a nonexpansive nonself-mapping satisfying the weakly inward condi- tion,F(T)=φ, f :C→Cis a fixed contractive mapping, andPis a sunny nonexpansive retraction ofEontoC, then the sequences{xt}and{xn}defined by

xt=Pt fxt

+ (1−t)Txt

, (1.4)

xn+1=Pαnfxn

+1−αn Txn

(1.5) strongly converge to a fixed point ofT.

In this paper, we establish the strong convergence of both{xt}defined by (1.4) and {xn}defined by (1.5) for a nonexpansive nonself-mappingTin a uniformly smooth Ba- nach space. Our results extend and improve the results in [2,6].

2. Preliminaries

LetEbe a real Banach space and letJdenote the normalized duality mapping fromEinto 2^E∗given by

J(x)=

f ∈E^∗: x,f = xf,x = f

∀x∈E, (2.1)

whereE^∗denotes the dual space ofEand ·,·denotes the generalized duality pairing.

In the sequence, we will denote the single-valued duality mapping by j, andxn→xwill denote strong convergence of the sequence{xn}tox. In Banach spaceE, the following result is well known [1,3] for allx,y∈E, for all j(x+y)∈J(x+y), for all j(x)∈J(x),

x²+ 2y,j(x)≤ x+y²≤ x²+ 2y,j(x+y). (2.2) Recall that the norm ofEis said to be Gˆateaux differentiable (andEis said to be smooth) if

limt→0

x+ty − x

t (2.3)

exists for eachx, yin its unit sphereU= {x∈E:x =1}. It is said to be uniformly Gˆateaux differentiable if, for eachy∈U, this limit is attained uniformly forx∈U. Fi- nally, the norm is said to be uniformly Fr´echet differentiable (andEis said to be uniformly smooth) if the limit in (2.3) is attained uniformly for (x,y)∈U×U. A Banach spaceE is said to be smooth if and only ifJis single valued. It is also well known that ifEis uni- formly smooth,Jis uniformly norm-to-norm continuous. These concepts may be found in [3].

IfCandDare nonempty subsets of a Banach spaceEsuch thatCis nonempty closed convex andD⊂C, then a mappingP:C→Dis called a retraction fromCtoDifP²=P.

It is easily known that a mappingP:C→Dis retraction, thenPx=x, for allx∈D. A mappingP:C→Dis called sunny if

PPx+t(x−Px)=Px ∀x∈C, (2.4)

wheneverPx+t(x−Px)∈Candt >0. A subsetDofCis said to be a sunny nonexpansive retract ofCif there exists a sunny nonexpansive retraction ofContoD. For more detail, see [1,3–5].

The following lemma is well known [3].

Lemma 2.1. LetCbe a nonempty convex subset of a smooth Banach spaceE,D∈C,J:E→ E^∗the (normalized) duality mapping ofE, andP:C→Da retraction. Then the following are equivalent:

(i) x−Px,j(y−Px) ≤0 for allx∈Candy∈D;

(ii)Pis both sunny and nonexpansive.

LetCbe a nonempty convex subset of a Banach spaceE, then forx∈C, we define the inward set [4,5]:

IC(x)=

y∈E:y=x+λ(z−x),z∈Candλ≥0. (2.5) A mappingT:C→Eis said to be satisfying the inward condition ifTx∈IC(x) for all x∈C.Tis also said to be satisfying the weakly inward condition if for eachx∈C,Tx∈ IC(x) (IC(x) is the closure ofIC(x)). ClearlyC⊂IC(x) and it is not hard to show thatIC(x) is a convex set asCis. Using above these results and definitions, we can easily show the following lemma.

Lemma 2.2 ([2], Lemma 1.2). LetCbe a nonempty closed subset of a smooth Banach space E, letT:C→Ebe nonexpansive nonself-mapping satisfying the weakly inward condition, and letPbe a sunny nonexpansive retraction ofEontoC. ThenF(T)=F(PT).

Lemma 2.3 ([2], Lemma 2.1). Let Ebe a Banach space and letCbe a nonempty closed convex subset ofE. Suppose thatT:C→Eis a nonexpansive mapping such that for each fixed contractive mapping f :C→C, andPis a sunny nonexpansive retraction ofEontoC.

For eacht∈(0, 1),{xt}is defined by (1.4). Supposeu∈Cis a fixed point ofT, then (i) xt−f(xt),j(xt−u) ≤0;

(ii){xt}is bounded.

Definition 2.4. μis called a Banach limit if μis a continuous linear functional on l^∞ satisfying

(i)μ(e) =1=μ(1),e=(1, 1, 1,...);

(ii)μn(an)=μn(an+1), for allan∈(a0,a1,...)∈l^∞;

(iii) lim infn→∞an≤μ(an)≤lim sup_n→∞an, for allan∈(a0,a1,...)∈l^∞. According to time and circumstances, we useμn(an) instead ofμ(a0,a1,...).

Further, we know the following result.

Lemma 2.5 ([3], Lemma 4.5.4). LetCbe a nonempty closed convex subset of a Banach space Ewith a uniformly Gˆateaux differentiable norm and let{xn}be a bounded sequence inE.

Letμbe a Banach limit andu∈C. Then μnxn−u²=min

y∈Cμnxn−y² (2.6)

if and only if

μn

x−u,Jxn−u≤0 (2.7)

for allx∈C.

3. Main results

Theorem 3.1. LetEbe a uniformly smooth Banach, suppose thatCis a nonempty closed convex subset ofEandT:C→Eis a nonexpansive nonself-mapping satisfying the weakly inward condition andF(T)= ∅. Let f :C→Cbe a fixed contractive mapping, and let{xt} be defined by (1.4), wherePis a sunny nonexpansive retraction ofEontoC. Then ast→0 {xt}converges strongly to some fixed pointqofTthatqis the unique solution inF(T) to the following variational inequality:

(I−f)q,j(q−u)≤0 ∀u∈F(T). (3.1) Proof. For allu∈F(T) byLemma 2.3(ii),{xt}is bounded, therefore the sets{Txt:t∈ (0, 1)} and{f(xt) :t∈(0, 1)} are also bounded. From xt=P(t f(xt) + (1−t)Txt), we have

xt−PTxt=Pt fxt

+ (1−t)Txt

−PTxt

≤t fxt

+ (1−t)Txt−Txt

=tTxt−fxt−→0 ast−→0.

(3.2)

This implies that

lim_t

→0

xt−PTxt=0. (3.3)

Assumetn→0, setxn:=xtn, and defineg:C→Rbyg(x)=μnxn−x²,x∈C, whereμn

is a Banach limit on^∞. Let

K= x∈C:g(x)=min

y∈Cμnxn−y². (3.4) It is easily seen that K is a nonempty closed convex bounded subset of E, since (note xn−Txn →0)

g(Tx)=μnxn−Tx²=μnTxn−Tx²≤μnxn−x²=g(x). (3.5) It follows thatT(K)⊂K, that is,K is invariant underT. Since a uniformly smooth Ba- nach space has the fixed point property for nonexpansive mappings,Thas a fixed point, sayq, inK. FromLemma 2.5we get

μn

x−q,jxn−q≤0, x∈C. (3.6)

For allq∈F(T), we havet f(xt) + (1−t)q=P[t f(xt) + (1−t)q], then xt−

t fxt

+ (1−t)q

=Pt fxt

+ (1−t)Txt

−Pt fxt

+ (1−t)q

≤(1−t)Txt−q≤(1−t)xt−q.

(3.7)

Hence from (2.2) and the above inequality we get xt−

t fxt

+ (1−t)q²

=(1−t)xt−q+txt−fxt²

≥(1−t)²xt−q²+ 2t(1−t)xt−fxt

,jxt−q.

(3.8)

Therefore

xt−fxt

,jxt−q≤0. (3.9)

Then

0≥

xt−fxt

,jxt−q

=xt−q²+q−f(q),jxt−q+f(q)−fxt

,jxt−q

≥(1−β)xt−q²+q−f(q),jxt−q.

(3.10)

We get

xt−q²≤ 1 1−β

f(q)−q,jxt−q. (3.11)

Now applying Banach limit to the above inequality, we get μnxt−q²≤μn

1 1−β

f(q)−q,jxt−q. (3.12)

Letx= f(q) in (3.6), and noting (3.12), we have

μnxt−q²≤0, (3.13)

that is,

μnxn−q²=0 (3.14)

and then exists a subsequence which is still denoted by{xn}such that

xn−→q, n−→ ∞. (3.15)

We have proved that for any sequence{xtn}in {xt:t∈(0, 1)}, there exists a subse- quence which is still denoted by{xtn}that converges to some pointqofT. To prove that

the entire net{xt}converges toq, suppose that there exists another sequence{xsk} ⊂ {xt} such thatxsk→p, assk→0, then we also havep∈F(T) (using limt→0xt−PTxt =0).

Next we show p=q andq is the unique solution inF(T) to the following variational inequality:

(I−f)q,j(q−u) ∀u∈F(T). (3.16) Since the sets{xt−u}and{xt−f(xt)}are bounded and the uniform smoothness ofE implies that the duality mapJ is norm-to-norm uniformly continuous on bounded sets ofE, for anyu∈F(T), byxsk→p(sk→0), we have

(I−f)xsk−(I−f)p−→0, sk−→0, xsk−fxsk

,jxsk−u−

(I−f)p,j(p−u)

=xsk−fxsk

−(I−f)p,jxsk−u−

(I−f)p,jxsk−u−j(p−u)

≤(I−f)xsk−(I−f)pxsk−u

+(I−f)p,jxsk−u−j(p−u)−→0 assk−→0.

(3.17) Therefore, notingLemma 2.3(i), for anyu∈F(T), we get

(I−f)p,j(p−u)=lim

sk→0

xsk−fxsk

,jxsk−u≤0. (3.18)

Similarly, we also can show

(I−f)q,j(q−u)=

xtn−fxtn

,jxtn−u≤0. (3.19) Interchangeqanduto obtain

(I−f)p,j(p−q)≤0. (3.20)

Interchangepanduto obtain

(I−f)q,j(q−p)≤0. (3.21)

This implies that

(p−q)−

f(p)−f(q),j(p−q)≤0, (3.22) that is,

p−q²≤βp−q². (3.23)

This is a contradiction, so we must haveq=p.

The proof is complete.

FromTheorem 3.1we can get the following corollary directly.

Corollary 3.2. LetEbe a uniformly smooth space, supposeCis a nonempty closed convex subset ofE,T:C→Eis a nonexpansive mapping satisfying the weakly inward condition, andF(T)= ∅. Let f :C→Cbe a fixed contractive mapping fromCtoC.{xt}is defined by

xt=t fxt

+ (1−t)PTxt, (3.24)

wherePis a sunny nonexpansive retraction ofEontoC, thenxt converges strongly to some fixed pointqofT ast→0 andqis the unique solution inF(T) to the following variational inequality:

(I−f)q,j(q−u) ∀u∈F(T). (3.25) Lemma 3.3 ([6], Lemma 2.1). Let{αn}be a sequence of nonnegative real numbers satisfying the property

αn+1≤ 1−γn

αn+δn ∀n≥0, (3.26)

where{γn} ∈(0, 1) andδnis a sequence inRsuch that:

(i) limn→∞γn=0 and^∞_n=0γn= ∞;

(ii) either^∞_n=0δn<+∞or lim sup_n→∞(δn/γn)≤0, then limn→∞αn=0.

Theorem 3.4. LetEbe a uniformly smooth Banach space, suppose thatCis a nonempty closed convex subset ofE,T:C→Eis a nonexpansive nonself-mapping satisfying the weakly inward condition, andF(T)= ∅. Let f :C→Cbe a fixed contractive mapping, and{xn} is defined by (1.5), wherePis a sunny nonexpansive retraction ofEontoC, andαn∈(0, 1) satisfies the following conditions:

(i)αn→0, asn→ ∞; (ii)^∞_n=0αn= ∞;

(iii) either^∞_n=0|αn+1−αn|<∞or limn→∞(αn+1/αn)=1.

Thenxnconverges strongly to a fixed pointqofTsuch thatqis the unique solution inF(T) to the following variational inequality:

(I−f)q,j(q−u)≤0 ∀u∈F(T). (3.27) Proof. First we show{xn}is bounded. Takeu∈F(T), it follows that

xn+1−u=P1−αn

Txn+αnfxn

−Pu

≤1−αn

Txn+αnfxn

−u

≤

1−αnTxn−u+αnfxn

−f(u)+f(u)−u

≤

1−αnxn−u+αnβxn−u+f(u)−u

=

1−(1−β)αnxn−u+αnf(u)−u

≤max xn−u, 1

1−βf(u)−u.

(3.28)

By induction,

xn−u≤max x0−u, 1

1−βf(u)−u, n≥0, (3.29) and{xn}is bounded, so are{Txn}and{f(xn)}. We claim that

xn+1−xn−→0 asn−→ ∞. (3.30)

Indeed we have (for some appropriate constantM >0) xn+1−xn=Pαnfxn

+1−αnTxn

−Pαn−1fxn−1

+1−αn−1Txn−1

≤αnfxn

+1−αn

Txn−αn−1fxn−1

− 1−αn−1

Txn−1

≤1−αn

Txn−Txn−1

+αn−αn−1

fxn−1

−Txn−1 +αnfxn

−fxn−1

≤

1−αnxn−xn−1[3pt] +Mαn−αn−1+βαnxn−xn−1

=

1−(1−β)αnxn−xn−1[3pt] +Mαn−αn−1.

(3.31) ByLemma 3.3we havexn+1−xn →0, asn→ ∞. We now show that

xn−PTxn−→0. (3.32)

In fact,

xn+1−PTxn=Pαnfxn

+1−αn Txn

−PTxn

≤αnfxn

−Txn. (3.33)

This follows from (3.30) that

xn−PTxn≤xn−xn+1+xn+1−PTxn

≤xn−xn+1+αnfxn

−Txn−→0 asn−→ ∞. (3.34) Letq=limt→0xt, where{xt}is defined inCorollary 3.2, we get thatqis the unique solu- tion inF(T) to the following variational inequality:

(I−f)q,j(q−u)≤0 ∀u∈F(T). (3.35) We next show that

lim sup

n→∞

f(q)−q,jxn−q≤0. (3.36)

FormCorollary 3.2, letxt=t f(xt) + (1−t)PTxt, indeed we can write xt−xn=tfxt

−xn

+ (1−t)PTxt−xn

. (3.37)

Noting (3.32), putting

an(t)=xn−PTxnxn−PTxn+ 2xn−xt−→0 asn−→ ∞, (3.38) and using (2.2), we obtain

xt−xn²

≤(1−t)²PTxt−xn²+ 2tfxt

−xn,jxt−xn

≤(1−t)²PTxt−PTxn+PTxn−xn²+ 2tfxt

−xt,jxt−xn + 2txt−xn²≤(1−t)²xt−xn²+ (1−t)²xn−PTxn² + 2(1−t)²PTxn−xnxt−xn+ 2tfxt

−xt,jxt−xn

+ 2txt−xn²

≤

1 +t²xt−xn²+an(t) + 2tfxt

−xt,jxt−xn .

(3.39) The last inequality implies

fxt

−xt,jxn−xt

≤t

2xt−xn²+ 1

2tan(t). (3.40) Froman(t)→0 asn→ ∞we get

lim sup

n→∞

fxt

−xt,jxn−xt

≤M·t

2, (3.41)

whereM >0 is a constant such thatM≥ xt−xn²for alln≥0 andt∈(0, 1). By letting t→0 in (3.41) we have

limt→0

lim sup

n→∞

fxt

−xt,jxn−xt

≤0. (3.42)

On the one hand, for allε >0,∃δ1such thatt∈(0,δ1), lim sup

n→∞

fxt

−xt,jxn−xt

≤ε

2. (**)

On the other hand,{xt}strongly converges toq, ast→0, the set{xt−xn}is bounded, and the duality mapJis norm-to-norm uniformly continuous on bounded sets of uniformly smooth spaceE; fromxt→q(t→0), we get

f(q)−q− fxt

−xt−→0, t−→0, f(q)−q,jxn−q−

fxt

−xt,jxn−xt

=f(q)−q,jxn−q−jxn−xt

+f(q)−q− fxt

−xt

,jxn−xt

≤f(q)−qjxn−q−jxn−xt +f(q)−q−

fxt

−xtxn−xt−→0, t−→0.

(3.43) Hence for the aboveε >0,∃δ2, such that for allt∈(0,δ2), for alln, we have

f(q)−q,jxn−q− fxt

−xt,jxn−xt≤ ε

2. (3.44)

Therefore, we have

f(q)−q,jxn−q≤ fxt

−xt,jxn−xt +ε

2. (3.45)

Noting (**) and takingδ=min{δ1,δ2}, for allt∈(0,δ), we have lim sup

n→∞

f(q)−q,jxn−q

≤lim sup

n→∞

fxt

−xt,jxn−xt + ε

2

≤ε 2+ε

2⁼ε.

(3.46)

Sinceεis arbitrary, we get

lim sup

n→∞

f(q)−q,jxn−q≤0. (3.47)

Finally we showxn→q. Indeed xn+1−

αnfxn

+1−αn q=

xn+1−q−αn fxn

−q. (3.48)

By (2.2) we have

xn+1−q²=xn+1−

αnfxn

+1−αn q+αn

fxn

−q²

≤xn+1−Pαnfxn

+1−αn

q²+ 2αn fxn

−q,jxn+1−q

≤Pαnfxn

+1−αnTxn

−Pαnfxn

+1−αnq² + 2αn

fxn

−q,jxn+1−q

≤

1−αn2Txn−q²+ 2αn fxn

−f(q),jxn+1−q + 2αn

f(q)−q,jxn+1−q

≤

1−αn2xn−q²+ 2αnf(q)−fxnxn+1−q + 2αn

f(q)−q,jxn+1−q

≤

1−αn2xn−q²+αnf(q)−fxn²+xn+1−q² + 2αn

f(q)−q,jxn+1−q.

(3.49)

Therefore, we have 1−αnxn+1−q²

≤

1−αn2xn−q²+αnβ²xn−q²+ 2αn

f(q)−q,jxn+1−q. (3.50) That is,

xn+1−q²≤

1−1−β²

1−αnαnxn−q+ α²_n 1−αn

xn−q² + 2αn

1−αn

f(q)−q,jxn+1−q

≤

1−γnxn−q²+λγnαn+ 2 1−β²γn

f(q)−q,jxn+1−q,

(3.51)

whereγn=((1−β²)/(1−αn))αnandλis a constant such thatλ >(1/(1−β²))xn−q². Hence,

xn+1−q²≤

1−γnxn−q²+γn

λαn+ 2 1−β²

f(q)−q,jxn+1−q. (3.52)

It is easily seen thatγn→0,^∞_n=1γn= ∞, and (noting (3.36)) lim sup

n→∞

λαn+ 2 1−β²

f(q)−q,jxn+1−q≤0. (3.53)

ApplyingLemma 3.3onto (3.52), we havexn→q.

The proof is complete.

Acknowledgment

This work is supported by the National Science Foundation of China, Grants 10471033 and 10271011.

References

[1] N. Shahzad, Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear Analysis 61 (2005), no. 6, 1031–1039.

[2] Y. Song and R. Chen, Viscosity approximation methods for nonexpansive nonself-mappings, Jour- nal of Mathematical Analysis and Applications 321 (2006), no. 1, 316–326.

[3] W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, 2000.

[4] W. Takahashi and G.-E. Kim, Strong convergence of approximants to fixed points of nonexpan- sive nonself-mappings in Banach spaces, Nonlinear Analysis. Theory, Methods & Applications 32 (1998), no. 3, 447–454.

[5] H.-K. Xu, Approximating curves of nonexpansive nonself-mappings in Banach spaces, Comptes Rendus de l’Acad´emie des Sciences. S´erie I. Math´ematique 325 (1997), no. 2, 151–156.

[6] , Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical Analysis and Applications 298 (2004), no. 1, 279–291.

Rudong Chen: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

E-mail address:[email protected]

Zhichuan Zhu: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

E-mail address:[email protected]