Single-valued and Multivalued Nonexpansive Mappings in Banach Spaces

Volume 2010, Article ID 618767,9pages doi:10.1155/2010/618767

Research Article

Ishikawa Iterative Process for a Pair of

Single-valued and Multivalued Nonexpansive Mappings in Banach Spaces

K. Sokhuma

and A. Kaewkhao

1Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand

2Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Correspondence should be addressed to A. Kaewkhao,[email protected] Received 8 August 2010; Accepted 24 September 2010

Academic Editor: T. D. Benavides

Copyrightq2010 K. Sokhuma and A. Kaewkhao. 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.

Let Ebe a nonempty compact convex subset of a uniformly convex Banach spaceX, and let t : E → EandT : E → KCEbe a single-valued nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume in addition that Fixt∩FixT/∅andTw {w}

for allw∈Fixt∩FixT. We prove that the sequence of the modified Ishikawa iteration method generated from an arbitraryx0 ∈Ebyyn 1−βnxnβnzn,xn1 1−αnxnαntyn, where zn ∈ Txnand{αn},{βn}are sequences of positive numbers satisfying 0 < a ≤αn,βn ≤ b < 1, converges strongly to a common fixed point of tand T; that is, there exists x ∈ Esuch that xtx∈Tx.

1. Introduction

Let X be a Banach space, and letEbe a nonempty subset of X. We will denote by FBE the family of nonempty bounded closed subsets ofEand byKCEthe family of nonempty compact convex subsets ofE. LetH·,·be the Hausdorffdistance onFBX, that is,

HA, B max

sup

a∈Adista, B, sup

b∈B distb, A

, A, B∈FBX, 1.1

where dista, B inf{a−b:b∈B}is the distance from the pointato the subsetB.

A mappingt:E → Eis said to be nonexpansive if

tx−ty≤x−y, ∀x, y∈E. 1.2 A pointxis called a fixed point oftiftxx.

A multivalued mappingT :E → FBXis said to be nonexpansive if H

Tx, Ty

≤x−y, ∀x, y∈E. 1.3

A pointxis called a fixed point for a multivalued mappingTifx∈Tx.

We use the notation FixTstanding for the set of fixed points of a mappingT and Fixt∩FixTstanding for the set of common fixed points oftandT. Precisely, a pointxis called a common fixed point oftandTifxtx∈Tx.

In 2006, S. Dhompongsa et al. 1 proved a common fixed point theorem for two nonexpansive commuting mappings.

Theorem 1.1 see 1, Theorem 4.2. Let E be a nonempty bounded closed convex subset of a uniformly Banach spaceX, and lett:E → E, andT:E → KCEbe a nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume thattandTare commuting; that is, if for everyx, y ∈Esuch thatx∈ Tyandty ∈ E, there holdstx∈ Tty. Then,tandT have a common fixed point.

In this paper, we introduce an iterative process in a new sense, called the modified Ishikawa iteration method with respect to a pair of single-valued and multivalued nonexpansive mappings. We also establish the strong convergence theorem of a sequence from such process in a nonempty compact convex subset of a uniformly convex Banach space.

2. Preliminaries

The important property of the uniformly convex Banach space we use is the following lemma proved by Schu2in 1991.

Lemma 2.1 see 2. LetX be a uniformly convex Banach space, let{un} be a sequence of real numbers such that 0< b≤un ≤c <1 for alln≥ 1, and let{xn}and{yn}be sequences ofXsuch that lim sup_{n→ ∞}xn ≤ a, lim sup_{n→ ∞}yn ≤ a, and limn→ ∞unxn 1−unyn afor some a≥0. Then, lim_{n→ ∞}xn−yn0.

The following observation will be used in proving our results, and the proof is straightforward.

Lemma 2.2. LetXbe a Banach space, and letEbe a nonempty closed convex subset ofX. Then,

dist y, Ty

≤y−xdistx, Tx H Tx, Ty

, 2.1

wherex, y∈EandT is a multivalued nonexpansive mapping fromEintoFBE.

A fundamental principle which plays a key role in ergodic theory is the demiclosed- ness principle. A mappingtdefined on a subsetEof a Banach spaceXis said to be demiclosed if any sequence {xn} in Ethe following implication holds:xn x and txn → y implies txy.

Theorem 2.3see3. Let Ebe a nonempty closed convex subset of a uniformly convex Banach spaceX, and lett:E → Ebe a nonexpansive mapping. If a sequence{xn}inEconverges weakly to pand{xn−txn}converges to 0 asn → ∞, thenp∈Fixt.

In 1974, Ishikawa introduced the following well-known iteration.

Definition 2.4see4. LetXbe a Banach space, letEbe a closed convex subset ofX, and let tbe a selfmap onE. Forx0∈E, the sequence{xn}of Ishikawa iterates oftis defined by

yn 1−βn

xnβntxn,

x_n1 1−α_nx_nα_nty_n, n≥0, 2.2

where{α_n}and{β_n}are real sequences.

A nonempty subsetKofEis said to be proximinal if, for anyx ∈ E, there exists an elementy∈Ksuch thatx−ydistx, K. We will denotePKby the family of nonempty proximinal bounded subsets ofK.

In 2005, Sastry and Babu5defined the Ishikawa iterative scheme for multivalued mappings as follows.

LetEbe a compact convex subset of a Hilbert spaceX, and let T : E → PEbe a multivalued mapping, and fixp∈FixT.

x0∈E, y_n

1−β_n

x_nβ_nz_n, x_n1 1−α_nx_nα_nz_n, ∀n≥0,

2.3

where{αn},{βn}are sequences in0,1withzn ∈Txnsuch thatzn−p distp, Txnand z_n−pdistp, Tyn.

They also proved the strong convergence of the above Ishikawa iterative scheme for a multivalued nonexpansive mappingTwith a fixed pointpunder some certain conditions in a Hilbert space.

Recently, Panyanak 6 extended the results of Sastry and Babu 5 to a uniformly convex Banach space and also modified the above Ishikawa iterative scheme as follows.

LetEbe a nonempty convex subset of a uniformly convex Banach spaceX, and let T :E → PEbe a multivalued mapping

x0∈E, yn

1−βn

xnβnzn, xn1 1−αnxnαnz_n, ∀n≥0,

2.4

where {αn},{βn} are sequences in0,1with zn ∈ Txn and un ∈ FixT such that zn − u_n distu_n, Tx_nandx_n−u_n distx_n,FixT, respectively. Moreover,z_n ∈ Tx_n and vn ∈FixTsuch thatz_n−vndistvn, Txnandyn−vndistyn,FixT, respectively.

Very recently, Song and Wang7,8 improved the results of5,6by means of the following Ishikawa iterative scheme.

LetT : E → FBEbe a multivalued mapping, whereαn, βn ∈ 0,1. The Ishikawa iterative scheme{xn}is defined by

x0∈E, yn

1−βn

xnβnzn, xn1 1−αnxnαnz_n, ∀n≥0,

2.5

where z_n ∈ Tx_n and z_n ∈ Ty_n such thatz_n −z_n ≤ HTx_n, Ty_n γ_n andz_n1 −z_n ≤ HTxn1, Tyn γn, respectively. Moreover,γn∈0,∞such that lim_{n→ ∞}γn0.

At the same period, Shahzad and Zegeye9modified the Ishikawa iterative scheme {x_n}and extended the result of7, Theorem 2to a multivalued quasinonexpansive mapping as follows.

LetK be a nonempty convex subset of a Banach spaceX, and letT :E → FBEbe a multivalued mapping, whereα_n, β_n∈0,1. The Ishikawa iterative scheme{x_n}is defined by

x0∈E, y_n

1−β_n

x_nβ_nz_n, xn1 1−αnxnαnz_n, ∀n≥0,

2.6

wherez_n∈Tx_nandz_n∈Ty_n.

In this paper, we introduce a new iteration method modifying the above ones and call it the modified Ishikawa iteration method.

Definition 2.5. LetEbe a nonempty closed bounded convex subset of a Banach spaceX, lett: E → Ebe a single-valued nonexpansive mapping, and letT :E → FBEbe a multivalued nonexpansive mapping. The sequence{xn}of the modified Ishikawa iteration is defined by

y_n 1−β_n

x_nβ_nz_n,

xn1 1−αnxnαntyn, 2.7

wherex0∈E,z_n ∈Tx_n, and 0< a≤α_n,β_n≤b <1.

3. Main Results

We first prove the following lemmas, which play very important roles in this section.

Lemma 3.1. LetEbe a nonempty compact convex subset of a uniformly convex Banach spaceX, and lett :E → EandT :E → FBEbe a single-valued and a multivalued nonexpansive mapping,

respectively, and Fixt∩FixT/∅satisfyingTw {w}for allw ∈Fixt∩FixT. Let{xn}be the sequence of the modified Ishikawa iteration defined by2.7. Then, lim_{n→ ∞}x_n−wexists for all w∈Fixt∩FixT.

Proof. Lettingx0∈Eandw∈Fixt∩FixT, we have

xn1−w1−αnxnαnt 1−βn

xnβnzn

−w 1−αnxnαnt

1−βn

xnβnzn

−1−αnw−αnw

≤1−α_nx_n−wα_nt 1−β_n

x_nβ_nz_n

−w

≤1−αnxn−wαn1−βn

xnβnzn−w 1−αnxn−wαn1−βn

xnβnzn− 1−βn

w−βnw

≤1−α_nx_n−wα_n 1−β_n

x_n−wα_nβ_nz_n−w 1−αnxn−wαn

1−βn

xn−wαnβndistzn, Tw

≤1−αnxn−wαn 1−βn

xn−wαnβnHTxn, Tw

≤1−α_nx_n−wα_n 1−β_n

x_n−wα_nβ_nx_n−w xn−w.

3.1

Since{xn−w}is a decreasing and bounded sequence, we can conclude that the limit of {xn−w}exists.

We can see howLemma 2.1is useful via the following lemma.

Lemma 3.2. LetEbe a nonempty compact convex subset of a uniformly convex Banach spaceX, and lett :E → EandT : E → FBEbe a single-valued and a multivalued nonexpansive mapping, respectively, and Fixt∩FixT/∅satisfyingTw{w}for allw∈Fixt∩FixT. Let{x_n}be the sequence of the modified Ishikawa iteration defined by2.7. If 0< a≤α_n≤b <1 for somea, b∈^Ê, then, lim_{n→ ∞}tyn−xn0.

Proof. Letw∈Fixt∩FixT. ByLemma 3.1, we put lim_{n→ ∞}xn−wcand consider tyn−w≤yn−w

1−β_n

x_nβ_nz_n−w

≤ 1−β_n

x_n−wβ_nz_n−w

1−βn

xn−wβndistzn, Tw

≤ 1−β_n

x_n−wβ_nHTx_n, Tw

≤ 1−βn

xn−wβnxn−w xn−w.

3.2

Then, we have

lim sup

n→ ∞

tyn−w≤lim sup

n→ ∞

yn−w≤lim sup

n→ ∞ xn−wc. 3.3

Further, we have

c lim

n→ ∞x_n1−w lim

n→ ∞1−α_nx_nα_nty_n−w lim

n→ ∞αntyn−αnwxn−αnxnαnw−w lim

n→ ∞αn

tyn−w

1−αnxn−w.

3.4

ByLemma 2.1, we can conclude that lim_{n→ ∞}ty_n−w−x_n−wlim_{n→ ∞}ty_n−x_n0.

Lemma 3.3. LetEbe a nonempty compact convex subset of a uniformly convex Banach spaceX, and lett :E → EandT :E → FBEbe a single-valued and a multivalued nonexpansive mapping, respectively, and Fixt∩FixT/∅satisfyingTw {w}for allw ∈Fixt∩FixT. Let{x_n}be the sequence of the modified Ishikawa iteration defined by2.7. If 0 < a≤αn,βn ≤b < 1 for some a, b∈^Ê, then lim_{n→ ∞}xn−zn0.

Proof. Letw∈Fixt∩FixT. We put, as inLemma 3.2, lim_{n→ ∞}x_n−wc. Forn≥0, we have

xn1−w1−αnxnαntyn−w

1−α_nx_nα_nty_n−1−α_nw−α_nw

≤1−αnxn−wαntyn−w

≤1−αnxn−wαnyn−w,

3.5

and hence

x_n1−w − x_n−w ≤ −α_nx_n−wα_ny_n−w, x_n1−w − x_n−w ≤α_ny_n−w− xn−w

, xn1−w − xn−w

α_n ≤yn−w− xn−w.

3.6

Therefore, since 0< a≤α_n ≤b <1, xn1−w − xn−w

α_n

xn−w ≤yn−w. 3.7

Thus,

lim inf

n→ ∞

x_n1−w − x_n−w αn

xn−w

≤lim inf

n→ ∞ yn−w. 3.8

It follows that

c≤lim inf

n→ ∞ yn−w. 3.9

Since, from3.3, lim sup_{n→ ∞}y_n−w ≤c, we have

c lim

n→ ∞yn−w lim

n→ ∞1−β_n

x_nβ_nz_n−w lim

n→ ∞1−βn

xn−w βnzn−w.

3.10

Recall that

z_n−wdistz_n, Tw

≤HTxn, Tw

≤ xn−w.

3.11

Hence, we have

lim sup

n→ ∞ z_n−w ≤lim sup

n→ ∞ x_n−wc. 3.12

Using the fact that 0< a≤β_n≤b <1 and by3.10, we can conclude that lim_{n→ ∞}x_n−z_n 0.

The following lemma allows us to go on.

Lemma 3.4. LetEbe a nonempty compact convex subset of a uniformly convex Banach spaceX, and lett :E → EandT : E → FBEbe a single-valued and a multivalued nonexpansive mapping, respectively, and Fixt∩FixT/∅satisfyingTw {w}for allw ∈Fixt∩FixT. Let{xn}be the sequence of the modified Ishikawa iteration defined by2.7. If 0 < a ≤ αn,βn ≤ b < 1, then lim_{n→ ∞}tx_n−x_n0.

Proof. Consider

txn−xntxn−tyntyn−xn

≤tx_n−ty_nty_n−x_n

≤x_n−y_nty_n−x_n xn−

1−βn

xn−βnzntyn−xn x_n−x_nβ_nx_n−β_nz_nty_n−x_n βnxn−zntyn−xn.

3.13

Then, we have

n→ ∞limtx_n−x_n ≤ lim

n→ ∞β_nx_n−z_n lim

n→ ∞ty_n−x_n. 3.14

Hence, by Lemmas3.2and3.3, lim_{n→ ∞}tx_n−x_n0.

We give the sufficient conditions which imply the existence of common fixed points for single-valued mappings and multivalued nonexpansive mappings, respectively, as follows Theorem 3.5. LetEbe a nonempty compact convex subset of a uniformly convex Banach spaceX, and lett:E → EandT:E → FBEbe a single-valued and a multivalued nonexpansive mapping, respectively, and Fixt∩FixT/∅satisfyingTw{w}for allw∈Fixt∩FixT. Let{xn}be the sequence of the modified Ishikawa iteration defined by2.7. If 0< a≤αn,βn≤b <1, thenxni → y for some subsequence{x_ni}of{x_n}impliesy∈Fixt∩FixT.

Proof. Assume that lim_{n→ ∞}xni−y0. FromLemma 3.4, we have 0 lim

n→ ∞txni−xni lim

n→ ∞I−txni. 3.15

SinceI−tis demiclosed at 0, we haveI−ty 0, and henceyty, that is,y∈Fixt. By Lemma 2.2and byLemma 3.4, we have

dist y, Ty

≤y−x_nidistx_ni, Tx_ni H

Tx_ni, Ty

≤y−xnixni −znixni−y−→0, asi→ ∞. 3.16

It follows thaty∈FixT. Thereforey∈Fixt∩FixTas desired.

Hereafter, we arrive at the convergence theorem of the sequence of the modified Ishikawa iteration. We conclude this paper with the following theorem.

Theorem 3.6. LetEbe a nonempty compact convex subset of a uniformly convex Banach spaceX, and lett:E → EandT:E → FBEbe a single-valued and a multivalued nonexpansive mapping, respectively, and Fixt∩FixT/∅satisfyingTw {w}for allw ∈Fixt∩FixT. Let{x_n}be

the sequence of the modified Ishikawa iteration defined by2.7with 0 < a ≤αn,βn ≤b < 1. Then {x_n}converges strongly to a common fixed point oftandT.

Proof. Since{xn}is contained inEwhich is compact, there exists a subsequence{xni}of{xn} such that {xni}converges strongly to some point y ∈ E, that is, limi→ ∞xni −y 0. By Theorem 3.5, we havey ∈Fixt∩FixT, and byLemma 3.1, we have that lim_{n→ ∞}x_n−y exists. It must be the case in which lim_{n→ ∞}xn−y lim_{i→ ∞}xni −y 0. Therefore,{xn} converges strongly to a common fixed pointyoftandT.

Acknowledgments

The first author would like to thank the Office of the Higher Education Commission, Thailand for supporting by grant fund under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree for this research. The authors would like to express their deep gratitude to Prof. Dr. Sompong Dhompongsa whose guidance and support were valuable for the completion of the paper. This work was completed with the support of the Commission on Higher Education and The Thailand Research Fund under Grant no. MRG5180213.

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