Approximate Fixed Points for Nonexpansive and Quasi-Nonexpansive Mappings in Hyperspaces

Volume 2009, Article ID 520976,16pages doi:10.1155/2009/520976

Research Article

Approximate Fixed Points for Nonexpansive and Quasi-Nonexpansive Mappings in Hyperspaces

Zeqing Liu,

Jeong Sheok Ume,

and Shin Min Kang

1Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China

2Department of Applied Mathematics, Changwon National University, Changwon 641-773, South Korea

3Department of Mathematics, Research Institute of Natural Science, Gyeongsang National University, Chinju 660-701, South Korea

Correspondence should be addressed to Jeong Sheok Ume,[email protected] Received 9 May 2009; Accepted 14 December 2009

Recommended by W. A. Kirk

This paper provides a few convergence results of the Ishikawa iteration sequence with errors for nonexpansive and quasi-nonexpansive mappings in hyperspaces. The results presented in this paper improve and generalize some results in the literature.

Copyrightq2009 Zeqing Liu 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.

1. Introduction and Preliminaries

Browder 1 and Kirk 2 established that a nonexpansive mapping T which maps a closed bounded convex subset C of a uniformly convex Banach space into itself has a fixed point inC. Since then, many researchers have studied, under various conditions, the convergence of the Mann and Ishikawa iteration methods dealing with nonexpansive and quasi-nonexpansive mappingssee3–11and the references therein. Rhoades9pointed out that the Picard iteration schemes for nonexpansive mappings need not converge. Senter and Dotson 10 obtained conditions under which the Mann iteration schemes generated by nonexpansive and quasi-nonexpansiv mappings in uniformly convex Banach spaces, converge to fixed points of these mappings, respectively. Ishikawa7established that the Mann iteration methods can be used to approximate fixed points of nonexpansive mappings in Banach spaces. Deng 3 obtained similar results for Ishikawa iteration processes in normed linear spaces and Banach spaces.

Our aim is to prove several convergence theorems of the Ishikawa iteration sequence with errors for nonexpansive and quasi-nonexpansive mappings in hyperspaces. Our results presented in this paper extend substantially the results due to Deng3, Ishikawa7, and Senter and Dotson10.

Assume thatX is a nonempty subset of a normed linear spaceE, · and CCX denotes the family of all nonempty convex compact subsets ofX, and H is the Hausdorff metric induced by the norm · . Forx∈E,X ⊂E,A, B∈CCX,I⊆CCX,T :I, H → CCX, H, andt∈R −∞,∞, let

dx, , A inf{x−a:a∈A}, DA,I inf{HA, C:C∈I}, IX {{x}:x∈X}, AB{ab:a∈A, b∈B}, tA{ta:a∈A}, coI

_n

i1

tiAi:ti≥0, n

i1

ti1, Ai∈I, n≥1

, FT {A∈I:TAA}.

1.1

It is easy to see that tA 1−tA Aand tA 1−tB ∈ CCE for allt ∈ 0,1and A, B∈CCE. HenceCCEis convex. Hu and Huang12proved that ifE, · is a Banach space, thenCCX, His a complete metric space. Now we introduce the following concepts in hyperspaces.

Definition 1.1. LetIbe a nonempty subset ofCCEand letT : I, H → CCE, Hbe a mapping. Assume that{tn}_n≥0,{t_n}_n≥0,{sn}_n≥0, and{s_n}_n≥0 are arbitrary real sequences in 0,1satisfyingtnt_n ≤1 andsns_n≤1 forn≥1 and{Pn}_n≥0and{Qn}_n≥0are any bounded sequences of the elements inCCE.

iForA0∈I, the sequence{An}_n≥0defined by Bn

1−sn−s_n

AnsnTAns_nPn, An1

1−tn−t_n

AntnTBnt_nQn, n≥0 1.2 is called the Ishikawa iteration sequence with errors provided that{An, Bn : n ≥ 0} ⊆I.

iiIfs_nt_n0 for alln≥0 in1.2, the sequence{An}_n≥0defined by

Bn 1−snAnsnTAn, An1 1−tnAntnTBn, n≥0, 1.3 is called the Ishikawa iteration sequence provided that{An, Bn:n≥0} ⊆I.

iiiIfsns_n0 for alln≥0 in1.2, the sequence{An}_n≥0defined by An1

1−tn−t_n

AntnTAnt_nQn, n≥0, 1.4 is called the Mann iteration sequence with errors provided that{An:n≥0} ⊆I.

ivIfs_nt_nsn0 for alln≥0 in1.2, the sequence{An}_n≥0defined by

An1 1−tnAntnTAn, n≥0, 1.5 is called the Mann iteration sequence provided that{An:n≥0} ⊆I.

Definition 1.2. LetIbe a nonempty subset ofCCE. A mappingT :I, H → CCE, His said to be

inonexpansive ifHTA, TB≤HA, Bfor allA, B∈I;

iiquasi-nonexpansive ifFT/∅andHTA, P≤HA, Pfor allA∈IandP∈FT. Definition 1.3. LetIbe a nonempty subset ofCCE. A mappingT :I, H → CCE, H withFT/∅is said to be satisfy the following.

iCondition A if there is a continuous functionf:0,∞ → 0,∞withf0 0 and ft>0 fort∈0,∞, such thatHA, TA≥fDA, FTfor allA∈I.

iiCondition B if there is a nondecreasing functionf :0,∞ → 0,∞withf0 0 andft>0 fort∈0,∞, such thatHA, TA≥fDA, FTfor allA∈I.

Remark 1.4. In caseIIX, whereXis a nonempty subset ofE, andT :IX → IE ⊆CCEis a mapping, then Definitions1.1,1.2, and1.3iireduce to the corresponding concepts in1–

11,13. It is well known that every nonexpansive mapping with nonempty fixed point set is quasi-nonexpansive, but the converse is not true; see8. Examples3.1and3.4in this paper reveal that the class of nonexpansive mappings with nonempty fixed point set is a proper subclass of quasi-nonexpansive mappings with both Condition A and Condition B.

The following lemmas play important roles in this paper.

Lemma 1.5see12. LetE, · be a Banach space andIa compact subset ofCCE, H. Then coI, His compact, wherecoIstands for the closure ofcoI.

Lemma 1.6see4. Suppose that{an}_n≥0,{bn}_n≥0, and{cn}_n≥0are three sequences of nonnegative numbers such that an1 ≤ 1 bnan cn for alln ≥ 0. If_∞

n0bn and _∞

n0cn converge, then limn→ ∞anexists.

Lemma 1.7see14. LetX, dbe a metric space. LetAandBbe compact subsets ofX. Then for anya∈A, there existsb∈Bsuch thatda, b≤HA, B, whereHis the Hausdorffmetric induced byd.

Lemma 1.8. LetE, · be a normed linear space. Then

H1−t−sAtBsC,1−t−sLtMsN≤1−t−sHA, L tHB, M sHC, N 1.6 for allA, B, C, L, M, N∈CCEandt, s∈0,1withst≤1.

Proof. Set

r 1−t−sHA, L tHB, M sHC, N. 1.7

For anya∈A,b∈B,c∈C, byLemma 1.7we infer that there existl∈L,m∈M,n∈Nsuch that

a−l ≤HA, L, b−m ≤HB, M, c−n ≤HC, N, 1.8

which imply that

1−t−satbsc−1−t−sl−tm−sn ≤1−t−sa−ltb−msc−n ≤r.

1.9

That is,

sup{d1−t−satbsc,1−t−sLtMsN:a∈A, b∈B, c∈C} ≤r. 1.10

Similarly we have

sup{d1−t−sltmsn,1−t−sAtBsC:l∈L, m∈M, n∈N} ≤r. 1.11

Thus1.6follows from1.10and1.11. This completes the proof.

Lemma 1.9. LetE, · be a normed linear space andIa nonempty closed subset ofCCE, H.

IfT :I, H → CCE, His quasi-nonexpansive, thenFTis closed.

Proof. Let {Pn}_n≥0 be inFTwith limn→ ∞HPn, P 0. Since T is quasi-nonexpansive, it follows that

HP, TP≤HPn, P HPn, TP≤2HPn, P−→0 1.12

asn → ∞. HenceP ∈FT. That is,FTis closed. This completes the proof.

2. Main Results

Our results are as follows.

Theorem 2.1. LetE,·be a normed linear space and letIbe a nonempty subset ofCCE. Assume thatT :I, H → CCE, His nonexpansive andA0 ∈I. Suppose that there exists a constantt satisfying

0< tnt_n ≤t <1, n≥0, 2.1 ∞

n0

tn∞, ^∞

n0

sn<∞, ^∞

n0

s_n<∞, ^∞

n0

t_n<∞, ^∞

n0

t_n

tnt_n−1

<∞. 2.2

If the Ishikawa iteration sequence with errors{An}_n≥0is bounded, then lim_{n→ ∞}HAn, TAn 0.

Proof. SinceT is nonexpansive,{An}_n≥0,{Pn}_n≥0, and{Qn}_n≥0are bounded, it follows that a:sup{HA, B:A∈ {An, Bn, Pn, Qn:n≥0}, B∈ {An, Bn, TAn, TBn:n≥0}}<∞. 2.3

Let nand k be arbitrary nonnegative integers. In view of 1.2,2.3,Lemma 1.8, and the nonexpansiveness ofT, we conclude that

HBn, An≤snHAn, TAn as_n, 2.4

HTBn, An≤HTBn, TAn HTAn, An≤1snHAn, TAn as_n, 2.5 HA_n1, An≤tnHTBn, An at_n≤tn1snHAn, TAn a

tns_nt_n

, 2.6

HAn1, TAk≤

1−tn−t_n

HAn, TAk tnHTBn, TAk at_n

≤

1−tn−t_n

HAn, TAk tnHBn, Ak at_n,

2.7

which yields that

HAn, TAk≥

1−tn−t_n−1

HA_n1, TAk−tnHBn, Ak−at_n

. 2.8

Using1.2,2.3–2.6,Lemma 1.8, and the nonexpansiveness ofT, we have

HBn, Ank1≤HBn, An1

k i1

HAni, Ani1

≤

1−sn−s_n

HAn, A_n1 snHTAn, A_n1 as_n

^k

i1

1snitniHAni, TAni a

tnis_nit_ni

≤

1−sn−s_n

tn1snHTAn, An a

tns_nt_n sn

1−tn−t_n

HTAn, An tnHTBn, TAn at_n as_n

^k

i1

1snitniHAni, TAni a k

i1

tnis_nit_ni

≤

tnsn−sntn−s²_ntn−s_ntn−snt_n−s_ntnsn

HAn, TAn

a

1−sn−s_n

tns_nt_n sntn

snHAn, TAn as_n

asnt_nas_n^k

i1

1s_nit_niHA_ni, TA_ni a k

i1

t_nis_nit_ni

≤^k

i0

tnisniHAni, TAni a

s_n^k

i0

tnis_nit_ni ,

2.9

HTAn1, An1≤

1−tn−t_n

HAn, TAn1 tnHTBn, TAn1 t_nHQn, TAn1

≤

1−tn−t_n

HAn1, TAn1 HAn, An1 tnHBn, An1 at_n

≤

1−tn−t_n

HA_n1, TA_n1

1−tn−t_n

1sntnHAn, TAn a

tns_nt_n tn

tnsnHAn, TAn a

s_ntns_nt_n at_n

≤

1−tn−t_n

HA_n1, TAn1 tn12snHAn, TAn 2a

tns_nt_n 2.10

which implies that

HAn1, TAn1≤

tnt_n−1

tn12snHAn, TAn 2a

tns_nt_n

≤12snHAn, TAn 2a s_ntn

t_nt_n−1 .

2.11

Lemma 1.6,2.2, and2.11yield that there exists a nonnegative constantrsatisfying

nlim→ ∞HAn, TAn r, 2.12

which implies that for anyε >0 there exists a positive integerNsuch that

r−ε≤HAn, TAn≤rε forn≥N. 2.13

Now we prove by induction that the following inequality holds for alln≥1:

H

Ap, TA_pn

≥rε

1ⁿ⁻¹

i0

t_pi

−2ε n−1

i0

1−t_pi−t_pi−1

−rεⁿ⁻¹

i0

⎡

⎣tpi

⎛

⎝ⁿ⁻¹

ji

spj

⎞

⎠ⁱ

k0

1−tpk−t_pk−1⎤

⎦

−a n−1

i0

⎧⎨

⎩

⎡

⎣tpi

⎛

⎝s_piⁿ⁻¹

ji

tpjs_pjt_pj⎞

⎠t_pi

⎤

⎦

×ⁱ

k0

1−t_pk−t_pk−1

, p≥N.

2.14

According to1.2,2.8,2.9, and2.13, we derive that H

Ap, TAp1

≥

1−tp−t_p−1 H

Ap1, TAp1

−tpH

Bp, Ap1

−at_p

≥

1−tp−t_p−1

r−ε−rεtp

tpsp

−atp

s_ptps_pt_p

−at_p

1−tp−t_p−1

r−ε−rε 1−2

1−tp

1−tp

₂ tpsp

−a tp

s_ptps_pt_p t_p

≥rε 1tp

−2ε

1−tp−t_p−1

−rεtpsp

1−tp−t_p−1

−a tp

s_ptps_pt_p t_p

1−tp−t_p−1

, p≥N.

2.15 Hence2.14holds forn1. Suppose that2.14holds fornm≥1. That is,

H

Ap, TApm

≥rε

1^m−1

i0

tpi

−2ε m−1

i0

1−tpi−t_pi−1

−rε^m−1

i0

⎡

⎣tpi

⎛

⎝^m−1

ji

spj

⎞

⎠ⁱ

k0

1−tpk−t_pk−1⎤

⎦

−a

m−1

i0

⎧⎨

⎩

⎡

⎣tpi

⎛

⎝s_pi^m−1

ji

tpjs_pjt_pj⎞

⎠t_pi

⎤

⎦

×ⁱ

k0

1−tpk−t_pk−1

, p≥N.

2.16

In view of1.2,2.8,2.9, and2.16, we infer that H

Ap, TApm1

≥

1−tp−t_p−1 H

Ap1, TApm1

−tpH

Bp, Apm1

−at_n

≥

1−tp−t_p−1 rε

1^m−1

i0

tp1i

−2ε m−1

i0

1−tp1i−t_p1i−1

−rε^m−1

i0

⎡

⎣t_p1i

⎛

⎝^m−1

ji

s_p1j

⎞

⎠ⁱ

k0

1−t_p1k−t_p1k−1⎤

⎦

−a

m−1

i0

⎡

⎣

⎛

⎝t_p1i

⎛

⎝s_p1i^m−1

ji

t_p1js_p1jt_p1j⎞

⎠ t_p1i

⎞

⎠

×ⁱ

k0

1−t_p1k−t_p1k−1

−tp m−1

i0

t_pis_pi rε

−atp

s_p^m

i0

tpis_pit_pi

−at_p

−2ε^m

i0

1−tpi−t_pi−1

1−tp−t_p−1 rε

×

1^m

i0

tp1i−12 1−tp

− 1−tp

2−tp

m i1

tpi−tp

m i0

spi

−rε^m−1

i0

⎡

⎣tp1i

⎛

⎝^m−1

ji

sp1j

⎞

⎠ⁱ¹

k0

1−tpk−t_pk−1⎤

⎦

−a

tp

s_p^m

i0

t_pis_pit_pi t_p

1−tp−t_p−1

^m−1

i0

⎡

⎣

⎛

⎝tp1i

⎛

⎝s_p1i^m−1

ji

tp1js_p1jt_p1j⎞

⎠t_p1i

⎞

⎠

×ⁱ¹

k0

1−tpk−t_pk−1⎤

⎦

⎫⎬

⎭ −2ε^m

i0

1−t_pi−t_pi−1

rε 1−tp

1−tp−t_p−1 1^m

i0

t_pi

−rε

⎧⎨

⎩tp

1−tp−t_p−1^m

i0

spi^m−1

i0

⎡

⎣tp1i

⎛

⎝^m−1

ji

sp1j

⎞

⎠ⁱ¹

k0

1−tpk−t_pk−1⎤

⎦

⎫⎬

⎭

−a m

i0

⎧⎨

⎩

⎡

⎣tpi

⎛

⎝s_pi^m

ji

tpjs_pjt_pj⎞

⎠t_pi

⎤

⎦ⁱ

k0

1−tpk−t_pk−1⎫

⎬

⎭

≥rε

1^m

i0

tpi

−2ε m

i0

1−tpi−t_pi−1

rε^m

i0

⎡

⎣t_pi

⎛

⎝^m

ji

s_pj

⎞

⎠ⁱ

k0

1−t_pk−t_pk−1⎤

⎦

−a m

i0

⎧⎨

⎩

⎡

⎣t_pi

⎛

⎝s_pi^m

ji

t_pjs_pjt_pj⎞

⎠ t_pi

⎤

⎦ⁱ

k0

1−t_pk−t_pk−1⎫

⎬

⎭, p≥N.

2.17

That is,2.14holds fornm1. Hence2.14holds for alln≥1.

We now assert thatr0. If not, thenr >0. Letmbe an arbitrary positive integer and

εmin

!

r,2⁻¹rt2ra⁻¹1−t^m, r1−t^m

2at⁻¹₋₁"

. 2.18

According to2.1,2.2, and2.12, we know that there exists a positive integerN Nε satisfying2.13and

max _np

kn

sk, s_i^np

kn

tks_kt_k

< ε forn, i≥N, p≥1. 2.19

It follows from2.1,2.2,2.13,2.14, and2.19that

HAN, TA_Nm≥rε

1^m−1

i0

t_Ni

−2ε m−1

i0

1−t_Ni−t_Ni−1

−rε^m−1

i0

⎡

⎣t_Ni

⎛

⎝^m−1

ji

s_Nj

⎞

⎠ⁱ

k0

1−t_Nk−t_Nk−1⎤

⎦

−a

m−1

i0

⎧⎨

⎩

⎡

⎣tNi

⎛

⎝s_Ni^m−1

ji

tNjs_Njt_Nj⎞

⎠t_Ni

⎤

⎦

×ⁱ

k0

1−t_Nk−t_Nk−1

≥rε

1^m−1

i0

tNi

−2ε1−t^−m

−rεε^m−1

i0

tNi1−t⁻ⁱ⁻¹−a

m−1

i0

tNiεt_Ni

1−t⁻ⁱ⁻¹

≥rε

1^m−1

i0

tNi

−2ε1−t^−m−rεε^m−1

i0

⎡

⎣tNi

i j0

1−t^−j−1

⎤

⎦

−aε

m−1

i0

t_Ni i j0

1−t^−j−1−a

m−1

i0

⎡

⎣1−t⁻ⁱ⁻¹ⁱ

j0

t_Nj

⎤

⎦

≥rε

1^m−1

i0

tNi

−2ε1−t^−m−rεεt⁻¹1−t^−mm−1

i0

tNi

−aεt⁻¹1−t^−mm−1

i0

tNi−aε

m−1

i0

1−t⁻ⁱ⁻¹

≥

rε−rεaεt⁻¹1−t^−mm−1

i0

tNi

rε−2ε1−t^−m−aεt⁻¹1−t^−m

≥

r−2raεt⁻¹1−t^−mm−1

i0

tNi r−

2at⁻¹

ε1−t^−m

≥2⁻¹r

m−1

i0

tNi−→ ∞

2.20

asm → ∞. Thus2.3and 2.20 yield thata ∞, which is absurd. Hence r 0. This completes the proof.

Theorem 2.2. LetE,·be a Banach space andIa nonempty closed subset ofCCE. Assume that T :I, H → CCE, His nonexpansive and there exists a compact subsetΩofCCEsuch that TI∪ {Pn, Qn :n≥0} ⊆Ω.If 2.1and2.2hold, thenThas a fixed point inI. Moreover, given A0∈I, the Ishikawa iteration sequence with errors{An}_n≥0converges to a fixed point ofT.

Proof. SettingI0 co{A0} ∪Ω, byLemma 1.5and the compactnessΩwe conclude thatI0

is compact. It is evident that{An}_n≥0 ⊆ I0, which yields that{An}_n≥0is bounded. SinceIis closed and{An}_n≥0 ⊆I, we conclude that there exist a subsequence{Ani}_i≥0of{An}_n≥0and A∈Isuch that

ilim→ ∞HAni, A 0. 2.21

It follows from2.21,Theorem 2.1, and the nonexpansiveness ofTthat HA, TA≤HA, Ani HAni, TAni HTAni, TA

≤2HA, Ani HAni, TAni−→0 2.22

asi → ∞. That is,ATA. Put

bsup{HPn, A, HQn, A:n≥0}. 2.23

In view of1.2,Lemma 1.8and the nonexpansiveness ofT, we derive that HAn1, A≤

1−tn−t_n

HAn, A tnHTBn, A bt_n

≤

1−tn−t_n

HAn, A tn

1−sn−s_n

HAn, A snHTAn, A bs_n bt_n

2.24

forn ≥ 0. It follows fromLemma 1.6,2.2,2.23, and2.24that limi→ ∞HAn, Aexists.

Using2.21we get that lim_{i→ ∞}HAn, A 0. This completes the proof.

Theorem 2.3. LetE, · be a Banach space andIa nonempty closed subset of CCE. Suppose thatT :I, H → CCE, His a qusi-nonexpansive mapping and satisfies Condition A. Assume that2.1and2.2hold andA0is inI. IfFTis bounded, then the Ishikawa iteration sequence with errors{An}_n≥0converges to a fixed point ofT inI.

Proof. Let b sup{HPn, A, HQn, A : n ≥ 0 and A ∈ FT}. Thenb < ∞. As in the proof ofTheorem 2.2, we get that2.24holds and lim_{i→ ∞}HAn, Aexists, whereA∈FT. Consequently,{An}_n≥0is bounded and

DAn1, FT≤DAn, FT b s_nt_n

∀n≥0. 2.25

It follows fromLemma 1.6,2.2, and2.25that limn→ ∞DAn, FT s ≥ 0. In view of Theorem 2.1and Condition A, we have

n→ ∞limHAn, TAn 0, fDAn, FT≤HAn, TAn ∀n≥0. 2.26

Using the continuity off, we know thatfs 0. That is,s0 and

nlim→ ∞DAn, FT 0. 2.27

Clearly2.27ensures that for anyi≥0 there existNi≥1 andPi∈FTsuch thatHANi, Pi<

2⁻ⁱ,which implies from2.24that

HAn, Pi<2⁻ⁱb n−1 kNi

s_kt_k

forn≥Ni. 2.28

We requireNi1> Nifor alli≥0. Notice that for anyj > i≥0

H Pi, Pj

≤^j−1

ki

HPk, ANk1 HANk1, Pk1

≤^j−1

ki

2^−kb

Nk1−1 mNk

s_mt_m 2^−k−1

3

2⁻ⁱ−2^−j b

Nj−1 lNi

s_lt_l .

2.29

Thus 2.2 and 2.29 yield that {Pi}_i≥0 is a Cauchy sequence in FT. It follows from Lemma 1.9that there existsP ∈ FTsatisfying lim_{i→ ∞}Pi P. For anyε > 0 there exists i0>0 such that

max

⎧⎨

⎩2⁻ⁱ⁰, HPi0, P, bⁿ⁻¹

kNi0

s_kt_k⎫

⎬

⎭<3⁻¹ε forn > Ni0. 2.30

Using2.28and2.30we have

HAn, P≤HAn, Pi0 HPi0, P

≤2⁻ⁱ⁰b n−1 kNi0

s_kt_k

HPi0, P

< ε

2.31

forn > Ni0. That is,{An}_n≥0converges toP ∈FT. This completes the proof.

A proof similar to that ofTheorem 2.3gives the following result and is thus omitted.

Theorem 2.4. Let E, · be a Banach space and let I be a nonempty closed subset ofCCE.

Suppose thatT :I, H → CCE, His a qusi-nonexpansive mapping and satisfies Condition B.

Assume thatA0is inIand there exists a constanttsatisfying

0< tn≤t <1, n≥0, ∞ n0

tn∞, ^∞

n0

sn<∞. 2.32

Then the Ishikawa iteration sequence{An}_n≥0converges to a fixed point ofT inI.

LetX be a nonempty subset ofE, · . It is easy to see thatIX, His isometric to X, · . Thus Theorems2.1–2.4yield the following results.

Corollary 2.5. LetX be a nonempty subset of a normed linear space E, · . Assume that T : X, · → E, · is nonexpansive andA0∈X. Suppose that2.1and2.2hold. If the Ishikawa iteration sequence with errors{An}_n≥0is bounded, then lim_{n→ ∞}An−TAn0.

Remark 2.6. Corollary 2.5 extends Theorem 1 in3and Lemma 2 in 7from the Ishikawa iteration scheme and Mann iteration scheme into the Ishikawa iteration scheme with errors, respectively.

Corollary 2.7. LetX be a nonempty closed subset of a Banach space E, · . Assume that T : X, · → E, · is nonexpansive and there exists a compact subsetY ofEwithTX∪ {Pn, Qn: n ≥ 0} ⊆ Y. Suppose that2.1and 2.2 hold. ThenT has a fixed point inX. Moreover for any A0∈X, the Ishikawa iteration sequence with errors{An}_n≥0converges to a fixed point ofT.

Remark 2.8. Theorem 3 in3and Theorem 1 in7and8are special cases ofCorollary 2.7.

Corollary 2.9. LetXbe a nonempty closed subset of a Banach spaceE, · and letT :X, · → E,·be quasi-nonexpansive. Assume that2.1and2.2hold andTsatisfies Condition A. IfFT is bounded, then for anyA0 ∈X, the Ishikawa iteration sequence with errors{An}_n≥0converges to a fixed point ofT inX.

Corollary 2.10. LetXbe a nonempty closed subset of a Banach spaceE,·and letT :X,· → E, · be quasi-nonexpansive. Assume that2.32holds andA0is inX. IfT satisfies Condition B, then the Ishikawa iteration sequence{An}_n≥0converges to a fixed point ofTinX.

Remark 2.11. Corollary 2.10extends, improves, and unifies Theorem 4 in3, Theorem 2 in7 and8in the following ways:

ithe Mann iteration method in 7, 8, and Ishikawa iteration method in 3 are replaced by the more general Ishikawa iteration method with errors;

iithe nonexpansive mappings in 3, 7, 8 are replaced by the more general quasi- nonexpansive mappings.

3. Examples and Problems

Now we construct a few nontrivial examples to illustrate the results in Section 2. The following example reveals thatCorollary 2.10extends properly Theorem 4 in3, Theorem 2 in7and8.

Example 3.1. LetE Rwith the usual norm| · | and letX 0,1. DefineT : X → Eand f:0,∞ → 0,∞by

Tx

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 3

4x, forx∈

$ 0,1

2

% , 1

2x, forx∈

&

1 2,1

% ,

3.1

andft 1/4tfort≥0. Settn 2√

n⁻¹andsn 1n²⁻¹forn≥0 andA0∈X. Then FT {0}and

|Tx−0| ≤ 3

4|x−0|, |x−Tx| ≥1

4|x|fdx, FT for x∈X. 3.2

Thus the assumptions ofCorollary 2.10are satisfied. However, Theorem 4 in3, Theorem 2 in7and8are not applicable since

''''T1 2 −T17

32 '''' 7

64 > 1 32 ''

''1 2− 17

32

'''', 3.3

that is,Tis not nonexpansive.

The examples below show that Theorems2.1–2.4extend substantially Corollaries2.5–

2.10, respectively.

Example 3.2. Let E R² with the usual norm | · | and let X 0,1². For any a, b ∈ X, Δ0,0a,00, b stands for the triangle with vertices 0,0,a,0, and 0, b. Let I {Δ0,0a,00, b : a, b ∈ X}and {Pn}_n≥0 and{Qn}_n≥0 be inI. DefineT : I → CCE by

TΔ0,0a,00, b Δ0,0

2⁻¹ab,0

0,4⁻¹'''b²−a²'''

fora, b∈X. 3.4

Puttn 2√

n⁻¹,t_n 2n^7/4−1,sn s_n 3n^3/2−1forn≥0 andA0 ∈X. It follows thatIis a compact subset ofCCE,FT {Δ0,00,00,0}and

HTΔ0,0a,00, b, TΔ0,0c,00, d H

Δ0,0

2⁻¹ab,0

0,4⁻¹'''b²−a²'''

,Δ0,0

2⁻¹cd,0

0,4⁻¹'''d²−c²''' max(

2⁻¹|ab−c−d|,4⁻¹)))'''b²−a²'''−'''d²−c²''')))*

≤max{|a−c|,|b−d|}

HΔ0,0a,00, b,Δ0,0c,00, d

3.5

fora, b,c, d ∈X. That is, the conditions of Theorems2.1and2.2are fulfilled. Hence we can invoke our Theorems2.1and2.2show that the Ishikawa iteration sequence with errors {An}_n≥0converges toΔ0,00,00,0and lim_{n→ ∞}HAn, TAn 0.

Example 3.3. LetE, X,I,{Pn}_n≥0,{Qn}_n≥0,{sn}_n≥0,{s_n}_n≥0,{tn}_n≥0,{t_n}_n≥0, andA0 be as in Example 3.2. DefineT :I → CCEandf :0,∞ → 0,∞by

TΔ0,0a,00, b Δ0,0

&

a³

1a²₋₁ ,0

+&

0, b³

1b²₋₁+

fora, b∈X, ft t

1t²₋₁

fort∈0,∞.

3.6

Obviously,FT {Δ0,00,00,0},

HTΔ0,0a,00, b,Δ0,00,00,0 H

&

Δ0,0

&

a³

1a²₋₁ ,0

+&

0, b³

1b²₋₁+

,Δ0,00,00,0 +

max

! a³

1a²₋₁ , b³

1b²₋₁"

≤max{a, b}

HΔ0,0a,00, b,Δ0,00,00,0, HΔ0,0a,00, b, TΔ0,0a,0b,0

H

&

Δ0,0a,0b,0,Δ0,0

&

a³

1a²₋₁ ,0

+&

0, b³

1b²₋₁++

max,

fa, fb-

≥fmax{a, b}

fDΔ0,0a,00, b, FT

3.7

fora, b∈X. Therefore the conditions ofTheorem 2.3are fulfilled.

Example 3.4. LetE, X,I, andA0 be as inExample 3.2. DefineT :I → CCE,f : 0,∞ → 0,∞andh:0,1 → 1,2by

TΔ0,0a,00, b Δ0,0

2⁻¹aha,0

0,2⁻¹b²

fora, b∈X, ft 8⁻¹t fort≥0,

hx

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 7

4, forx∈

$ 0,1

2

% ,

1, forx∈

&

1 2,1

% .

3.8

It follows thatFT {Δ0,00,00,0},

HTΔ0,0a,00, b,Δ0,00,00,0 max(

2⁻¹aha,2⁻¹b²*

≤max{a, b}

HΔ0,0a,00, b,Δ0,00,00,0, HΔ0,0a,00, b, TΔ0,0a,00, b

H

Δ0,0a,0b,0,Δ0,0

2⁻¹aha,0

0,2⁻¹b² max(

a

1−2⁻¹ha , b

1−2⁻¹b²*

≥8⁻¹max{a, b}

fDΔ0,0a,00, b, FT

3.9

fora, b∈X. Obviously, the assumptions ofTheorem 2.4are fulfilled. On the other hand,T is not nonexpansive since

H

&

TΔ0,0

&

1 2,0

+&

0,1 2

+

, TΔ0,0

&

9 16,0

+&

0,1 2

++

1 2

''''1 2h

&

1 2

+

− 9 16h

&

9 16

+'''' 5

32 > 1 16 H

&

Δ0,0

&

1 2,0

+&

0,1 2

+

,Δ0,0

&

9 16,0

+&

0,1 2

++

.

3.10

We conclude with the following problems.

Problem 3.5. Can Condition A inTheorem 2.3be replaced by Condition B?

Problem 3.6. Can the boundedness ofFTinTheorem 2.3be removed?

Problem 3.7. CanTheorem 2.4be extended to the Ishikawa iteration method with errors?

Acknowledgment

This work was supported by the Korea Research Foundation Grant funded by the Korean GovernmentKRF-2008-313-C00042.

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