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Volume 2009, Article ID 520976, 16 pagesdoi:10.1155/2009/520976 Research Article Approximate Fixed Points for Nonexpansive and Quasi-Nonexpansive Mappings in Hyperspaces Zeqing Liu,1 Jeo

Volume 2009, Article ID 520976, 16 pages

doi:10.1155/2009/520976

Research Article

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

Zeqing Liu,1 Jeong Sheok Ume,2 and Shin Min Kang3

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

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

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

Correspondence should be addressed to Jeong Sheok Ume,jsume@changwon.ac.kr

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

Copyrightq 2009 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 in C Since then, many researchers have studied, under various conditions, the

convergence of the Mann and Ishikawa iteration methods dealing with nonexpansive and quasi-nonexpansive mappingssee 3 11 and the references therein Rhoades 9 pointed 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 Ishikawa7 established 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, Ishikawa 7, and Senter and Dotson10

2 Fixed Point Theory and Applications

Assume that X is a nonempty subset of a normed linear space E,  ·  and CCX denotes the family of all nonempty convex compact subsets of X, and H is the Hausdorff

metric induced by the norm ·  For x ∈ E, X ⊂ E, A, B ∈ CCX, I ⊆ CCX, T : I, H →

CCX, H, and t ∈ R  −∞, ∞, let

d x, , A  inf{x − a : a ∈ A}, D A, I  inf{HA, C : C ∈ I},

IX  {{x} : x ∈ X}, A  B  {a  b : a ∈ A, b ∈ B}, tA  {ta : a ∈ A},

coI 

n

i1

t i A i : ti ≥ 0,n

i1

t i  1, Ai ∈ I, n ≥ 1



, F T  {A ∈ I : TA  A}.

1.1

It is easy to see that tA  1 − tA  A and tA  1 − tB ∈ CCE for all t ∈ 0, 1 and

A, B ∈ CCE Hence CCE is convex Hu and Huang 12 proved that if E,  ·  is a Banach

space, thenCCX, H is a complete metric space Now we introduce the following concepts

in hyperspaces

Definition 1.1 Let I be a nonempty subset of CCE and let T : I, H → CCE, H be a

mapping Assume that{tn} n≥0,{t

n}n≥0,{sn} n≥0, and{s

n}n≥0are arbitrary real sequences in

0, 1 satisfying tn  t

n ≤ 1 and sn  s

n ≤ 1 for n ≥ 1 and {Pn} n≥0and{Qn} n≥0are any bounded

sequences of the elements in CCE.

i For A0∈ I, the sequence {An} n≥0defined by

B n1− sn − s

n



A n  sn TA n  s

n P n ,

A n11− tn − t

n



A n  tn TB n  t

is called the Ishikawa iteration sequence with errors provided that{An , B n : n ≥

0} ⊆ I

ii If s

n  t

n  0 for all n ≥ 0 in 1.2, the sequence {An} n≥0defined by

B n  1 − snAn  sn TA n , A n1  1 − tnAn  tn TB n , n ≥ 0, 1.3

is called the Ishikawa iteration sequence provided that{An , B n : n ≥ 0} ⊆ I.

iii If sn  s

n  0 for all n ≥ 0 in 1.2, the sequence {An} n≥0defined by

A n11− tn − t

n



A n  tn TA n  t

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

iv If s

n  t

n  sn  0 for all n ≥ 0 in 1.2, the sequence {An} n≥0defined by

A n1  1 − tnAn  tn TA n , n ≥ 0, 1.5

is called the Mann iteration sequence provided that{An : n ≥ 0} ⊆ I.

Definition 1.2 Let I be a nonempty subset of CCE A mapping T : I, H → CCE, H is

said to be

i nonexpansive if HTA, TB ≤ HA, B for all A, B ∈ I;

ii quasi-nonexpansive if FT / ∅ and HTA, P ≤ HA, P for all A ∈ I and P ∈ FT.

Definition 1.3 Let I be a nonempty subset of CCE A mapping T : I, H → CCE, H

withFT / ∅ is said to be satisfy the following

i Condition A if there is a continuous function f : 0, ∞ → 0, ∞ with f0  0 and

ft > 0 for t ∈ 0, ∞, such that HA, TA ≥ fDA, FT for all A ∈ I.

ii Condition B if there is a nondecreasing function f : 0, ∞ → 0, ∞ with f0  0 and ft > 0 for t ∈ 0, ∞, such that HA, TA ≥ fDA, FT for all A ∈ I.

Remark 1.4 In caseI  IX, where X is a nonempty subset of E, and T : IX → IE ⊆ CCE is

a mapping, then Definitions1.1,1.2, and1.3ii reduce to the corresponding concepts in 1

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 1.5 see 12 Let E,  ·  be a Banach space and I a compact subset of CCE, H Then

coI, H is compact, where coI stands for the closure of coI.

Lemma 1.6 see 4 Suppose that {an} n≥0 , {bn} n≥0 , and {cn} n≥0 are three sequences of nonnegative numbers such that a n1 ≤ 1  bnan  cn for all n ≥ 0 If∞

n0 b n and ∞

n0 c n converge, then

limn → ∞a n exists.

Lemma 1.7 see 14 Let X, d be a metric space Let A and B be compact subsets of X Then for

any a ∈ A, there exists b ∈ B such that da, b ≤ HA, B, where H is the Hausdorff metric induced

by d.

Lemma 1.8 Let E,  ·  be a normed linear space Then

H 1 − t − sAtBsC, 1 − t − sL  tM  sN ≤ 1 − t − sHA, L  tHB, M  sHC, N

1.6

for all A, B, C, L, M, N ∈ CCE and t, s ∈ 0, 1 with s  t ≤ 1.

Proof Set

r  1 − t − sHA, L  tHB, M  sHC, N. 1.7

For any a ∈ A, b ∈ B, c ∈ C, byLemma 1.7we infer that there exist l ∈ L, m ∈ M, n ∈ N such

that

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

4 Fixed Point Theory and Applications

which imply that

1 − t − sa  tb  sc − 1 − t − sl − tm − sn ≤ 1 − t − sa − l  tb − m  sc − n ≤ r.

1.9 That is,

sup{d1 − t − sa  tb  sc, 1 − t − sL  tM  sN : a ∈ A, b ∈ B, c ∈ C} ≤ r 1.10 Similarly we have

sup{d1 − t − sl  tm  sn, 1 − t − sA  tB  sC : l ∈ L, m ∈ M, n ∈ N} ≤ r 1.11 Thus1.6 follows from 1.10 and 1.11 This completes the proof

Lemma 1.9 Let E,  ·  be a normed linear space and I a nonempty closed subset of CCE, H.

If T : I, H → CCE, H is quasi-nonexpansive, then FT is closed.

Proof Let {Pn} n≥0 be in FT with limn → ∞ HP n , P   0 Since T is quasi-nonexpansive, it

follows that

H P, TP ≤ HPn , P   HPn , TP  ≤ 2HPn , P −→ 0 1.12

as n → ∞ Hence P ∈ FT That is, FT is closed This completes the proof.

2 Main Results

Our results are as follows

Theorem 2.1 Let E, · be a normed linear space and let I be a nonempty subset of CCE Assume

that T : I, H → CCE, H is nonexpansive and A0 ∈ I Suppose that there exists a constant t

satisfying

0 < tn  t

n ≤ t < 1, n ≥ 0, 2.1

∞



n0

t n  ∞, ∞

n0

s n < ∞,

∞



n0

s n < ∞,

∞



n0

t n < ∞,

∞



n0

t n

t n  t

n

−1

< ∞. 2.2

If the Ishikawa iteration sequence with errors {An} n≥0 is bounded, then lim n → ∞ HA n , TA n  0.

Proof Since T is nonexpansive, {A n} n≥0,{Pn} n≥0, and{Qn} n≥0are bounded, it follows that

a : sup {HA, B : A ∈ {An , B n , P n , Q n : n ≥ 0}, B ∈ {An , B n , TA n , TB n : n ≥ 0}} < ∞ 2.3

Let n and k be arbitrary nonnegative integers In view of 1.2, 2.3,Lemma 1.8, and the

nonexpansiveness of T, we conclude that

H Bn , A n ≤ sn H An , TA n  as

H TBn , A n ≤ HTBn , TA n  HTAn , A n ≤ 1  snHAn , TA n  as

n , 2.5

H An1 , A n ≤ tn H TBn , A n  at

n ≤ tn1  snHAn , TA n  at n s n  t

n



, 2.6

H An1 , TA k ≤1− tn − t

n



H An , TA k  tn H TBn , TA k  at

n

≤1− tn − t

n



H An , TA k  tn H Bn , A k  at

n ,

2.7

which yields that

H An , TA k ≥1− tn − t

n

−1

H An1 , TA k − tn H Bn , A k − at

n



. 2.8 Using1.2, 2.3–2.6,Lemma 1.8, and the nonexpansiveness of T, we have

H Bn , A nk1 ≤ HBn , A n1 k

i1

H Ani , A ni1

≤1− sn − s

n



H An , A n1  sn H TAn , A n1  as

n

k

i1



1  snitni H Ani , TA ni  at ni s ni  t

ni



≤1− sn − s

n



t n1  snHTAn , A n  at n s n  t

n



 sn1− tn − t

n



H TAn , A n  tn H TBn , TA n  at

n

 as

n

k

i1

1  snitni H Ani , TA ni  ak

i1



t ni s ni  t

ni



≤ t n  sn − sn t n − s2

n t n − s

n t n − sn t n − s

n t n s n

H An , TA n

 a1− sn − s

n



t n s n  t

n



 sn t n



s n H An , TA n  as

n



 asn t n  as

nk

i1

1  snitni H Ani , TA ni  ak

i1



t ni s ni  t

ni



≤k

i0

tni  sniHAni , TA ni  a

s nk

i0



t ni s ni  t

ni



,

2.9

6 Fixed Point Theory and Applications

H TAn1 , A n1 ≤1− tn − t

n



H An , TA n1  tn H TBn , TA n1  t

n H Qn , TA n1

≤1− tn − t

n



HAn1 , TA n1  HAn , A n1  tn H Bn , A n1  at

n

≤1− tn − t

n



H An1 , TA n1

1− tn − t

n



1  sntn H An , TA n  at n s n  t

n



 tntn  snHAn , TA n  as n  tn s n  t

n



 at

n

≤1− tn − t

n



H An1 , TA n1  tn1  2snHAn , TA n  2at n s n  t

n



2.10 which implies that

H An1 , TA n1 ≤t n  t

n

−1

t n1  2snHAn , TA n  2at n s n  t

n



≤ 1  2snHAn , TA n  2a s n  tnt n  t

n

−1

.

2.11

lim

n → ∞ H An , TA n  r, 2.12

which implies that for any ε > 0 there exists a positive integer N such that

r − ε ≤ H An , TA n ≤ r  ε for n ≥ N. 2.13

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

H

A p , TA pn



≥ r  ε



1n−1

i0

t pi



− 2εn−1 i0

1− tpi − t

pi

−1

− r  ε n−1

i0

⎡

⎣tpi

⎛

⎝n−1

ji

s pj

⎞

⎠i

k0

1− tpk − t

pk

−1⎤

⎦

− an−1 i0

⎧

⎨

⎩

⎡

⎣tpi

⎛

⎝s

pin−1

ji

t pj s pj  t

pj

⎞

⎠  t

pi

⎤

⎦

×i

k0

1− tpk − t

pk

−1

, p ≥ N.

2.14

According to1.2, 2.8, 2.9, and 2.13, we derive that

H

A p , TA p1



≥ 1− tp − t

p

−1

H

A p1 , TA p1



− tp H

B p , A p1



− at

p

≥ 1− tp − t

p

−1

r − ε − r  εtpt p  sp− atp s p  tp s p  t

p

− at

p

 1−tp−t

p

−1

r − ε− r  ε 1− 21− tp1− tp2 tp s p

− a t p s p  tp s p  t

p

 t

p

≥ r  ε1 tp− 2ε 1− tp − t

p

−1

− r  εtp s p 1− tp − t

p

−1

− at p s p  tp s p  t

p

 t

p 1− tp − t

p

−1

, p ≥ N.

2.15 Hence2.14 holds for n  1 Suppose that 2.14 holds for n  m ≥ 1 That is,

H

A p , TA pm



≥ r  ε



1m−1

i0

t pi



− 2εm−1 i0

1− tpi − t

pi

−1

− r  ε m−1

i0

⎡

⎣tpi

⎛

⎝m−1

ji

s pj

⎞

⎠i

k0

1− tpk − t

pk

−1⎤

⎦

− a m−1 i0

⎧

⎨

⎩

⎡

⎣tpi

⎛

⎝s

pim−1

ji

t pj s pj  t

pj

⎞

⎠  t

pi

⎤

⎦

×i

k0

1− tpk − t

pk

−1

, p ≥ N.

2.16

In view of1.2, 2.8, 2.9, and 2.16, we infer that

H

A p , TA pm1



≥ 1− tp − t

p

−1

H

A p1 , TA pm1



− tp H

B p , A pm1



− at

n



≥ 1− tp − t

p

−1

r  ε



1m−1

i0

t p1i



− 2εm−1 i0

1− tp1i − t

p1i

−1

− r  ε m−1

i0

⎡

⎣tp1i

⎛

⎝m−1

ji

s p1j

⎞

⎠i

k0

1− tp1k − t

p1k

−1⎤

⎦

− a m−1 i0

⎡

⎣

⎛

⎝tp1i

⎛

⎝s

p1im−1

ji

t p1j s p1j  t

p1j

⎞

⎠  t

p1i

⎞

⎠

×i

k0

1− tp1k − t

p1k

−1

− tp m−1 i0



t pi  spir  ε

−atp



s pm

i0

t pi s pi  t

pi



− at

p



8 Fixed Point Theory and Applications

 −2εm

i0

1− tpi − t

pi

−1

 1− tp − t

p

−1

r  ε

×

1m

i0

t p1i− 1  21− tp−1− tp2− tpm

i1

t pi − tpm

i0

s pi

− r  ε m−1

i0

⎡

⎣tp1i

⎛

⎝m−1

ji

s p1j

⎞

⎠i1

k0

1− tpk − t

pk

−1⎤

⎦

− a



t p



s pm

i0

t pi s pi  t

pi



 t

p 1− tp − t

p

−1

m−1

i0

⎡

⎣

⎛

⎝tp1i

⎛

⎝s

p1im−1

ji

t p1j s p1j  t

p1j

⎞

⎠  t

p1i

⎞

⎠

×i1

k0

1− tpk − t

pk

−1⎤

⎦

⎫

⎬

⎭

 −2εm

i0

1− tpi − t

pi

−1

 r  ε1− tp 1− tp − t

p

−1

1m

i0

t pi



− r  ε

⎧

⎨

⎩t p 1− tp − t

p

−1 m

i0

s pim−1

i0

⎡

⎣tp1i

⎛

⎝m−1

ji

s p1j

⎞

⎠i1

k0

1− tpk − t

pk

−1⎤

⎦

⎫

⎬

⎭

− am

i0

⎧

⎨

⎩

⎡

⎣tpi

⎛

⎝s

pim

ji

t pj s pj  t

pj

⎞

⎠  t

pi

⎤

⎦i

k0

1− tpk − t

pk

−1⎫⎬

⎭

≥ r  ε



1m

i0

t pi



− 2εm i0

1− tpi − t

pi

−1

 r  εm

i0

⎡

⎣tpi

⎛

⎝m

ji

s pj

⎞

⎠i

k0

1− tpk − t

pk

−1⎤

⎦

− am

i0

⎧

⎨

⎩

⎡

⎣tpi

⎛

⎝s

pim

ji

t pj s pj  t

pj

⎞

⎠  t

pi

⎤

⎦i

k0

1− tpk − t

pk

−1⎫⎬

⎭, p ≥ N.

2.17

That is,2.14 holds for n  m  1 Hence 2.14 holds for all n ≥ 1.

We now assert that r  0 If not, then r > 0 Let m be an arbitrary positive integer and

ε  min

!

r, 2−1rt 2r  a−11 − t m , r 1 − t m 2 at−1−1"

. 2.18

According to2.1, 2.2, and 2.12, we know that there exists a positive integer N  Nε

satisfying2.13 and

max

np



kn

s k , s inp

kn



t k s k  t

k



< ε for n, i ≥ N, p ≥ 1. 2.19

It follows from2.1, 2.2, 2.13, 2.14, and 2.19 that

H AN , TA Nm ≥ r  ε



1m−1

i0

t Ni



− 2εm−1 i0



1− tNi − t

Ni

−1

− r  ε m−1

i0

⎡

⎣tNi

⎛

⎝m−1

ji

s Nj

⎞

⎠i

k0



1− tNk − t

Nk

−1⎤⎦

− a m−1 i0

⎧

⎨

⎩

⎡

⎣tNi

⎛

⎝s

Nim−1

ji

t Nj s Nj  t

Nj

⎞

⎠  t

Ni

⎤

⎦

×i

k0



1− tNk − t

Nk

−1

≥ r  ε



1m−1

i0

t Ni



− 2ε1 − t −m

− r  εε m−1

i0

t Ni1 − t −i−1 − a m−1

i0



t Ni ε  t Ni

1 − t −i−1

≥ r  ε



1m−1

i0

t Ni



− 2ε1 − t −m − r  εε m−1

i0

⎡

⎣tNii 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

⎤

⎦

≥ r  ε



1m−1

i0

t Ni



− 2ε1 − t −m − r  εεt−11 − t −m m−1

i0

t Ni

− aεt−11 − t −m m−1

i0

t Ni − aε m−1

i0

1 − t −i−1

≥ r  ε − r  ε  aεt−11 − t −mm−1

i0

t Ni

 r  ε − 2ε 1 − t −m − aεt−11 − t −m

≥ r − 2r  aεt−11 − t −mm−1

i0

t Ni r − 2 at−1

ε 1 − t −m

≥ 2−1r

m−1

i0

t Ni−→ ∞

2.20

10 Fixed Point Theory and Applications

as m → ∞ Thus 2.3 and 2.20 yield that a  ∞, which is absurd Hence r  0 This

completes the proof

Theorem 2.2 Let E, · be a Banach space and I a nonempty closed subset of CCE Assume that

T : I, H → CCE, H is nonexpansive and there exists a compact subset Ω of CCE such that TI ∪ {P n , Q n : n ≥ 0} ⊆ Ω If 2.1 and 2.2 hold, then T has a fixed point in I Moreover, given

A0∈ I, the Ishikawa iteration sequence with errors {An} n≥0 converges to a fixed point of T.

Proof SettingI0  co{A0} ∪ Ω, byLemma 1.5and the compactnessΩ we conclude that I0

is compact It is evident that{An} n≥0 ⊆ I0, which yields that{An} n≥0is bounded SinceI is closed and{An} n≥0 ⊆ I, we conclude that there exist a subsequence {An i}i≥0of{An} n≥0and

A ∈ I such that

lim

i → ∞ H An i , A   0. 2.21

It follows from2.21,Theorem 2.1, and the nonexpansiveness of T that

H A, TA ≤ HA, An i   HAn i , TA n i   HTAn i , TA

≤ 2HA, An i   HAn i , TA n i −→ 0 2.22

as i → ∞ That is, A  TA Put

b  sup {HPn , A , HQn , A  : n ≥ 0}. 2.23

In view of1.2,Lemma 1.8and the nonexpansiveness of T, we derive that

H An1 , A ≤1− tn − t

n



H An , A   tn H TBn , A   bt

n

≤1− tn − t

n



H An , A

 tn1− sn − s

n



H An , A   sn H TAn , A   bs

n



 bt

n

2.24

for n ≥ 0 It follows fromLemma 1.6,2.2, 2.23, and 2.24 that limi → ∞HA n , A exists.

Using2.21 we get that limi → ∞HA n , A  0 This completes the proof.

Theorem 2.3 Let E,  ·  be a Banach space and I a nonempty closed subset of CCE Suppose

that T : I, H → CCE, H is a qusi-nonexpansive mapping and satisfies Condition A Assume that2.1 and 2.2 hold and A0is in I If FT is bounded, then the Ishikawa iteration sequence with

errors {An} n≥0 converges to a fixed point of T in I.

Proof Let b  sup{HP n , A, HQ n , A : n ≥ 0 and A ∈ FT} Then b < ∞ As in the

proof ofTheorem 2.2, we get that2.24 holds and limi → ∞HA n , A exists, where A ∈ FT.

Consequently,{An} n≥0is bounded and

D An1 , F T ≤ DAn , F T  bs n  t

n



∀n ≥ 0. 2.25

According to2.1, 2.2, and 2.12, we know that there exists a positive integer N  Nε

satisfying2.13 and

max

np... is,2.14 holds for n  m  Hence 2.14 holds for all n ≥ 1.

We now assert that r  If not, then r > Let m be an arbitrary positive integer and< /i>

ε  min

!... class="text_page_counter">Trang 10

10 Fixed Point Theory and Applications

as m → ∞ Thus 2.3 and 2.20 yield that a  ∞, which is absurd