Mann Type Implicit Iteration Approximation for Multivalued Mappings in Banach Spaces

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Volume 2010, Article ID 140530,8pages doi:10.1155/2010/140530

Research Article

Mann Type Implicit Iteration Approximation for

Multivalued Mappings in Banach Spaces

Huimin He,

1

_{Sanyang Liu,}

1

_{and Rudong Chen}

2

1_{Department of Mathematics, Xidian University, Xi’an 710071, China}

2_{Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China}

Correspondence should be addressed to Huimin He,[email protected]

Received 16 March 2010; Accepted 5 July 2010

Academic Editor: Mohamed Amine Khamsi

Copyrightq2010 Huimin He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

LetKbe a nonempty compact convex subset of a uniformly convex Banach spaceEand letTbe a multivalued nonexpansive mapping. For the implicit iteratesx0∈K,xn αnxn−1 1−αnyn,

yn∈Txn,n≥1. We proved that{xn}converges strongly to a fixed point ofTunder some suitable

conditions. Our results extended corresponding ones and revised a gap in the work of Panyanak 2007.

1. Introduction

Let K be a nonempty subset of a Banach space E. We will denote 2E _{by the family of all} subsets of E, CBE the family of nonempty closed and bounded subsets of E, CE the family of nonempty compact subsets of E. Let CKE symbolize the family of nonempty compact convex subsets ofE. LetH·,·be Hausdorﬀmetric onCBE; that is,

HA, B max

sup x∈A

dx, B,sup x∈B

dx, A

, ∀A, B∈CBE, 1.1

wheredx, B inf{x−y : y ∈ B}. A multivalued mappingT : K → CBEis called nonexpansiveresp., contractive, if for anyx, y∈K, there holds

HTx, Ty≤x−y,

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A pointxis called a fixed point ofT ifx∈Tx. In this paper,FTstands for the fixed point set of a mappingT.

The fixed point theory of multivalued nonexpansive mappings is much more complicated and diﬃcult than the corresponding theory of single-valued nonexpansive mappings. However, some classical fixed point theorems for single-valued nonexpansive mappings have already been extended to multivalued mappings.

In 1968, Markin 1 firstly established the nonexpansive multivalued convergence results in Hilbert space. Banach’s Contraction Principle was extended to a multivalued contraction in 1969.Below is stated in a Banach space setting.

Theorem 1.1see2 . LetKbe a nonempty closed subset of a Banach spaceEandT :K → CBK

a multivalued contraction. ThenThas a fixed point.

In 1974, one breakthrough was achieved by Lim using Edelstein’s method of asymptotic centers3 .

Theorem 1.2see Lim 3 . LetK be a nonempty closed bounded convex subset of a uniformly convex Banach spaceEandT :K → CEa multivalued nonexpansive mapping. ThenThas a fixed point.

In 1990, Kirk and Massa 4 obtained another important result for multivalued nonexpansive mappings.

Theorem 1.3see Kirk and Massa 4 . LetK be a nonempty closed bounded convex subset of a Banach space Eand T : K → CKE a multivalued nonexpansive mapping. Suppose that the asymptotic center in E of each bounded sequence of X is nonempty and compact. Then T has a fixed point.

In 1999, Sahu5 obtained the strong convergence theorems of the nonexpansive type and nonself multivalued mappings for the following1.3iteration process:

xntnu 1−tnyn, n≥0, 1.3

whereyn∈Txn, u∈K, tn∈0,1and limn→ ∞tn0. He proved that{xn}converges strongly to some fixed points of T. Xu 6 extended Theorem 1.3 to a multivalued nonexpansive nonself mapping and obtained the fixed theorem in 2001. The recent fixed point results for nonexpansive mappings can be found in7–12 and references therein.

Recently, Panyanak 13 studied the Mann iteration 1.4 and Ishikawa iterative processes1.5forx0∈Kas follows:

xn1αnxn 1−αnyn, n≥0, 1.4

whereαn∈0,1 , yn∈Txn, and fixedp∈FTare such thatyn−p ≤dp, Txn,

yn

1−βn

xnβnzn,

xn1 1−αnxnαnzn, n≥0,

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whereαn ∈ 0,1 , βn ∈ 0,1 , zn ∈ Txn, zn ∈ Tyn, and fixedp ∈ FTare such thatzn −

p ≤ dp, Txnand zn −p ≤ dp, Tynand proved the strong convergence theorems for multivalued nonexpansive mappingsTin Banach spaces.

In this paper, motivated by Panyanak13 and the previous results, we will study the following iteration process1.6. LetKbe a nonempty convex subset ofE, αn∈0,1 ,

x0∈K,

xnαnxn−1 1−αnyn, yn∈Txn, n≥1,

1.6

and we prove some strong convergence theorems of the sequence{xn}defined by1.6for nonexpansive multivalued mappings in Banach spaces. The results presented in this paper establish a new type iteration convergence theorems for multivalued nonexpansive mappings in Banach spaces and extend the corresponding results of Panyanak13 .

2. Preliminaries

LetEbe a real Banach space and letJ denote the normalized duality mapping fromEto 2E∗ defined by

Jx f∈E∗,x, f xf,xf, ∀x∈E, 2.1

whereE∗denotes the dual space ofEand·,·denotes the generalized duality pair. It is well known that ifE∗ is strictly convex, thenJ is single valued. And if Banach space Eadmits sequentially continuous duality mappingJfrom weak topology to weak star topology, then, by14, Lemma 1 , we know that the duality mappingJ is also single valued. In this case, the duality mappingJis also said to be weakly sequentially continuous; that is, if{xn}is a subject ofEwithxn x, thenJxn J∗ x. By Theorem 1 of14 , we know that ifEadmits a weakly sequentially continuous duality mapping, thenEsatisfies Opial’s condition, andE

is smooth; for the details, see14 . In the sequel, we will denote the single-valued duality mapping byj.

Throughout this paper, we writexn xresp.,xn x∗ to indicate that the sequence

xnweaklyresp., weak∗converges tox, as usualxn → xwill symbolize strong convergence. In order to show our main results, the following concepts and lemmas are needed.

A Banach spaceEis called uniformly convex if for each > 0 there is aδ > 0 such that forx, y∈Ewithx,y ≤1 andx−y ≥,xy ≤21−δholds. The modulus of convexity ofEis defined by

δE inf

1−1 2

xy:x,y≤1,x−y≥

, 2.2

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Lemma 2.1see10 . In Banach spaceE, the following result (subdiﬀerential inequality) is well known: for allx, y∈E, for alljx∈Jx, for alljxy∈Jxy,

x22y, jx ≤xy2≤ x22y, jxy . 2.3

Definition 2.2. A Banach spaceEis said to satisfyOpial’s conditionif for any sequence{xn}in

E, xn xn → ∞implies

lim sup

n→ ∞ xn−x<lim supn→ ∞

_x_n₋_y_, _∀_y_∈_E_with_{x /}_y. _2.4

We know that Hilbert spaces, lp₁ _{< p <} _∞_{, and Banach space with weakly} sequentially continuous duality mappings satisfy Opial’s condition; for the details, see

14,15 .

Definition 2.3. A multivalued mappingT:K → CBKis said to satisfyConditionIif there is a nondecreasing functionf :0,∞ → 0,∞withf0 0, fr>0 forr ∈0,∞such that

dx, Tx≥fdx, FT, ∀x∈K. 2.5

Example of mappings that satisfyConditionIcan be founded in13 .

3. Main Results

Now, we prove our results.

Theorem 3.1. LetKbe a nonempty compact convex subset of a uniformly convex Banach spaceEand letT :K → CBKbe a multivalued nonexpansive mapping, whereαn∈0,1andlimn→ ∞αn0, the sequence{xn}∞n1is generated by1.6.

Then,

iby the compactness ofK, there exists a subsequence{xni}of{xn}such thatxni → pfor

somep∈K. In addition ifyn−p ≤dp, Txn,then

iipis a fixed point ofTand the sequence{xn}converges strongly top.

Proof. Partiis trivial. And partiiremains to be proved.

Due to the compactness ofK and boundness of CBK, there exists a real number

M >0 such that

_x_n₋₁₋_y_n_≤_M. 3.1

It follows from1.6, that

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thus

_x_n₋_y_n_−→0, asn−→ ∞, 3.3

therefore

dp, Tp≤p−xndxn, Txn H

Txn, Tp

≤2p−xndxn, Txn ≤2p−xnxn−yn,

3.4

so

dp, Tp−→0, asn−→ ∞. 3.5

Hence,pis a fixed point ofT.

Next we show that limn→ ∞xn−pexists.

For alln≥1, there existjxn−p∈Jxn−p, usingLemma 2.1, we obtain _x_n₋_p2

αnxn−1 1−αnyn−p, j

xn−p

1−αn

yn−p, j

xn−p αnxn−1−p, jxn−p

≤1−αnyn−p·xn−pαnxn−1−p·xn−p ≤1−αnH

Txn, Tp

·xn−pαnxn−1−p·xn−p

≤1−αnxn−p2αnxn−1−p·xn−p,

3.6

so

_x_n₋_p2_≤

xn−1−p·xn−p. 3.7

Ifxn−p0, then limn→ ∞xn−p0 apparently holds. Letxn−p>0, from3.7, we have

_x_n₋_p_≤_x_n₋₁₋_p_. _3.8

We get that{xn−p}is a decreasing sequence, so

lim

n→ ∞xn−pexists. 3.9

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Remark 3.2. The above result modified the gap in the proof of Theorem 3.1 in13 by a new method; the gap discovered by Song and Wang16 is as follows.

Panyanak13 introduced the Ishikawa iterates1.5of a multivalued mappingT. It is obvious thatxndepends onpandT. Forp∈FT, we have

_z_n₋_p_d

p, Txn

≤HTp, Txn

≤xn−p, _z_n₋_p_d

p, Tyn

≤HTp, Tyn

≤yn−p.

3.10

Clearly, ifq∈FTand_{q /}p, then the above inequalities cannot be assured. Indeed, from the monotone decreasing sequence of{xn−p}in the proof ofTheorem 3.113 , we cannot obtain that{xn−q}is a decreasing sequence. Hence, the conclusion of Theorem 3.1 in13 cannot be achieved.

Theorem 3.3. LetEbe a Banach space satisfying Opial’s condition and letKbe a nonempty weakly compact convex subset ofE. Suppose thatT :K → CBKis a multivalued nonexpansive mapping, whereαn∈0,1andlimn→ ∞αn0, the sequence{xn}∞n1is generated by1.6.

Then,

iby the weak compactness ofK, there exists a subsequence{xni}of{xn}such thatxni p

for somep∈K. In addition if,yn−p ≤dp, Txn,then

iipis a fixed point ofTand the sequence{xn}converges weakly top.

Proof. Partiis trivial. Now we prove partii. It follows from3.3ofTheorem 3.1that

lim

n→ ∞dxn, Txn 0. 3.11

SinceKis weakly compact, from parti, there exists a subsequence{xni}of{xn}such that

xni p, for somep∈K. 3.12

Suppose thatpdoes not belong toTp. By the compactness ofTp, for any givenxni, there exist

zi∈Tpsuch thatxni−zidxni, Tpandzi → z∈Tp.

Thus_{p /}z, from

lim sup i→ ∞

xni−z ≤lim sup

i→ ∞

xni−zizi−z

lim sup i→ ∞

xni−zi

≤lim sup i→ ∞

dxni, Txni H

Txni, Tp

≤lim sup i→ ∞

_x_n

i−p<lim sup

i→ ∞ xni−z.

3.13

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Hence,pis a fixed point ofT.

It follows from3.7ofTheorem 3.1that

lim

n→ ∞xn−pexists. 3.14

Next we showxn p. Suppose not. There exists another subsequence{xnk}of{xn}

such thatxnk q /p.

Then, we also obtainq∈Tq. From Opial’s condition, we have

lim

i→ ∞xn−plim sup_i_{→ ∞} xni−p

<lim sup i→ ∞

_x_n

i−qlim sup

k→ ∞

_x_n

k−q

<lim sup k→ ∞

_x_n

k−p lim

i→ ∞xn−p.

3.15

Which is a contradiction, so the conclusion of the theorem follows. The proof is completed.

Corollary 3.4. Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mappingJfromEtoE∗, and letKbe a nonempty weakly compact convex subset ofE. Suppose thatT :K → CBKis a multivalued nonexpansive mapping, whereαn∈0,1andlimn→ ∞αn 0, the sequence{xn}∞n1is generated by1.6.

Then,

iby the weak compactness ofK, there exists a subsequence{xni}of{xn}such thatxni p

for somep∈K. In addition if,yn−p ≤dp, Txn,then

iipis a fixed point ofTand the sequence{xn}converges weakly top.

Proposition 3.5. LetKbe a nonempty compact convex subset of a uniformly convex Banach spaceE

and letT :K → CBKbe a multivalued nonexpansive mapping. ThenFTis a closed subset ofK.

Proof. Supposeqn⊂FT, n≥1, such that limn→ ∞qn q, then we have

dq, Tq≤p−qnd

qn, Tqn

HTqn, Tq

≤2q−qnd

qn, Tqn

,

3.16

so

dq, Tq−→0, asn−→ ∞. 3.17

Hence,qis a fixed point ofT.

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Theorem 3.6. LetK be a nonempty compact convex subset of a uniformly convex Banach spaceE

and letT : K → CBKbe a multivalued nonexpansive mapping satisfying Condition I, where

αn∈0,1andlimn→ ∞αn0, then the sequence{xn}∞n1generated by1.6converges strongly to a fixed point.

Proof. It follows from3.3ofTheorem 3.1that

lim

n→ ∞dxn, Txn 0. 3.18

The proof of remained part is omitted because it is similar to the proof of Theorem 3.8 in13 .

Acknowledgment

The work was supported by the Fundamental Research Funds for the Central Universities, No. JY10000970006, and National Nature Science Foundation, No. 60974082.

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