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Volume 2007, Article ID 50596, 9 pagesdoi:10.1155/2007/50596 Research Article Approximating Fixed Points of Nonexpansive Mappings in Hyperspaces Zeqing Liu, Chi Feng, Shin Min Kang, and

Volume 2007, Article ID 50596, 9 pages

doi:10.1155/2007/50596

Research Article

Approximating Fixed Points of Nonexpansive

Mappings in Hyperspaces

Zeqing Liu, Chi Feng, Shin Min Kang, and Jeong Sheok Ume

Received 29 March 2007; Revised 27 August 2007; Accepted 13 September 2007

Recommended by Wataru Takahashi

Two convergence theorems for the Ishikawa and Mann iteration sequences involving nonexpansive mappings in hyperspaces are established

Copyright © 2007 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

LetX be a nonempty compact subset of a Banach space (E, ·), and letC(X) and CC(X)

denote the families of all nonempty compact and all nonempty compact convex subsets

ofX, respectively It is well known that (C(X),H) is compact, where H is the Hausdorff

metric induced by· For A,B ∈ CC(X) and t ∈ R =(−∞, +∞), letA + B = { a + b :

a ∈ A, b ∈ B }, and lettA = { ta : a ∈ A } In the sequel, we assume thatX is a nonempty

compact convex subset ofE Hu and Huang [1] proved that (CC(X),H) is a compact

subset of (C(X),H) It is clear that tA + (1 − t)B ∈ CC(X) for all A,B ∈ CC(X) and t ∈

[0, 1] That is, CC(X) has convexity structure LetIbe a nonempty subset of CC(X).

A mappingT : (I,H) →(I,H) is said to be nonexpansive if H(TA,TB) ≤ H(A,B) for all A,B ∈I

Within the past 20 years or so, a few researchers have applied the Mann iteration method and the Ishikawa iteration method to approximate fixed points of nonexpansive mappings in several classes of subsets of Banach spaces For details we refer to [2–11] Recently, Hu and Huang [1] established the following result

Theorem 1.1 Let X be a nonempty compact convex subset of a Banach space (E, · ), and

letI be a nonempty compact convex subset of CC(X) Suppose that T : (I,H) →(I,H) is nonexpansive Then for any A0∈I, the sequence defined by

A n =2−1 

A n −1+TA n −1 

converges to a fixed point of T.

Inspired and motivated by the results in [1–11], in this paper we introduce the cepts of the Mann and Ishikawa iteration sequences in hyperspaces, and establish the con-vergence theorems for the Mann and Ishikawa iteration sequences dealing with nonex-pansive mappings in hyperspaces The results in this paper extend substantially Theorem 1.1

In order to prove our results, we need the following concepts and results

Definition 1.2 LetIbe a nonempty compact convex subset ofCC(X), and let T : (I,H) →

(I,H) be a mapping.

(1) For anyA0∈I, the sequence{ A n } n ≥0⊆Idefined by

B n =1− s n

A n+s n TA n, n ≥0,

A n+1 =1− t n

A n+t n TB n, n ≥0, (1.2)

is called the Ishikawa iteration sequence, where { t n } n ≥0and{ s n } n ≥0are real sequences in [0, 1] satisfying appropriate conditions

(2) Ifs n =0 for alln ≥0 in (1.2), the sequence{ A n } n ≥0⊆Idefined by

A n+1 =1− t n

A n+t n TA n, n ≥0, (1.3)

is called the Mann iteration sequence.

(3) Ifs n =0 andt n =1 for alln ≥0 in (1.2), the sequence{ A n } n ≥0⊆Idefined by

is called the Picard iteration sequence.

Lemma 1.3 Let A, B, U, and V be in CC(X), and let t be in [0,1] Then

HtA + (1 − t)B,tU + (1 − t)V≤ tH(A,U) + (1 − t)H(B,V). (1.5)

Proof Put r = tH(A,U) + (1 − t)H(B,V) For any a ∈ A and b ∈ B, by Nadler’s result we

know that there existu ∈ U, v ∈ V such that  a − u  ≤ H(A,U) and  b − v  ≤ H(B,V)

which yield that

ta + (1 − t)b − tu −(1− t)v  ≤ t  a − u + (1− t)  b − v  ≤ r. (1.6)

It follows that

sup

a ∈ A,b ∈ B

 inf

u ∈ U,v ∈ Vta + (1 − t)b − tu −(1− t)v≤ r. (1.7) Similarly, we have

sup

u ∈ U,v ∈ V

 inf

a ∈ A,b ∈ Bta + (1 − t)b − tu −(1− t)v≤ r. (1.8) Consequently, we infer that

HtA + (1 − t)B,tU + (1 − t)V

=max

 sup

a ∈ A,b ∈ B inf

u ∈ U,v ∈ Vta + (1 − t)b − tu −(1− t)v, sup

u ∈ U,v ∈ V inf

a ∈ A,b ∈ Bta + (1 − t)b − tu −(1− t)v≤ r.

(1.9)

Lemma 1.4 [9] Suppose that{ a n } n ≥0and { b n } n ≥0are two sequences of nonnegative num-bers such that a n+1 ≤ a n+b n for all n ≥ 0 If∞

n =0b n converges, then lim n →∞ a n exists.

2 Main results

Now we prove the following results

Theorem 2.1 Let X be a nonempty compact convex subset of a Banach space (E, · ), and

letI be a nonempty compact convex subset of CC(X) Suppose that T : (I,H) →(I,H) is nonexpansive and there exist constants a and b satisfying that

0< a ≤ t n ≤ b < 1, 0 ≤ s n ≤1, n ≥0, (2.1)

∞



n =0

Then for any A0∈I, the Ishikawa iteration sequence { A n } n ≥0converges to a fixed point of T Proof Let n and k be arbitrary nonnegative integers Note that tA + (1 − t)A = A for any

A ∈ CC(X) and t ∈[0, 1] Using (1.2), Lemma 1.3and the nonexpansiveness ofT, we

infer that

HTB n,A n

≤ HTB n,TA n

+HTA n,A n

≤ HB n,A n

+HTA n,A n

≤1 +s n

HA n,TA n

and that

HA n+1,A n

≤ t n HTB n,A n

≤ t n

1 +s n

HA n,TA n

By virtue of (1.2), (2.3), (2.4),Lemma 1.3, and the nonexpansiveness ofT, we get that

HB n,A n+k+1

≤ HB n,A n+1

+

k



i =1

HA n+i,A n+i+1

≤1− s n

HA n,A n+1

+s n HTA n,A n+1

+

k



i =1

t n+i

1 +s n+i

HA n+i,TA n+i

≤1− s2

n

t n HA n,TA n

+s n 

1− t n

HA n,TA n

+t n HTB n,TA n

+

k



i =1



t n+i+s n+i

HA n+i,TA n+i

≤t n+s n

1− t n

HA n,TA n

+

k



i =1



t n+i+s n+i

HA n+i,TA n+i

≤

k



i =0



t n+i+s n+i

HA n+i,TA n+i

,

(2.5)

and that

HTA n+1,A n+1

≤1− t n

HA n,TA n+1

+t n HTB n,TA n+1

≤1− t n

HA n+1,TA n+1

+HA n+1,A n

+t n HB n,A n+1

≤1− t n

HA n+1,TA n+1

+

1− t n

t n

1 +s n

HA n,TA n +t n

1− t n

HA n,B n

+t n HTB n,B n

,

(2.6)

which together with (2.1) implies that

HA n+1,TA n+1

≤1− t n

1 +s n

HA n,TA n +

1− t n

HA n,B n

+t n HTB n,B n

≤1− t n

1 + 2s n

HA n,TA n +t n

1− s n

HA n,TB n

+s n HTA n,TB n

≤1 + 2s n

1− t n

HA n,TA n

≤1 + 2

1− as n

HA n,TA n

.

(2.7)

Notice that the compactness ofIimplies that{ H(A n,TA k) :n ≥0,k ≥0}is bounded It follows fromLemma 1.4, (2.2), and (2.7) that

lim

n →∞ HA n,TA n

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

r − ε ≤ HA n,TA n

It follows that

HA n+1,TC≤1− t n

HA n,TC+t n HTB n,TC

≤1− t n

HA n,TC+t n HB n,C, C ∈I,n ≥0, (2.10) which yields that

HA n,TC≥1− t n−1 

HA n+1,TC− t n HB n,C, C ∈I,n ≥0. (2.11) Now we prove by induction that the following inequality holds for alln ≥1:

HA p,TA p+n

≥(r + ε)

1 +

n−1

i =0

t p+i

−2ε n −

1

i =0



1− t p+i−1

−(r + ε) n −

1



i =0



t p+i

n−1

j = i s p+j

 i

k =0



1− t p+k−1

 , p ≥ N.

(2.12)

Using (2.5), (2.9), and (2.11), we obtain that

HA p,TA p+1

≥1− t p−1 

HA p+1,TA p+1

− t p HB p,A p+1

≥1− t p−1 

r − ε −(r + ε)t p

t p+s p

=1− t p−1

r − ε −(r + ε)1−2

1− t p +

1− t p 2 +t p s p

=(r + ε)1 +t p

−2ε1− t p−1

−(r + ε)t p s p

1− t p−1

, p ≥ N.

(2.13) Hence (2.12) holds forn =1 Suppose that (2.12) holds forn = m ≥1 That is,

HA p,TA p+m

≥(r + ε)

1 +

m−1

i =0

t p+i

−2ε m −1

i =0



1− t p+i−1

−(r + ε) m−1

i =0



t p+i

m−1

j = i s p+j

 i

k =0



1− t p+k−1

 , p ≥ N.

(2.14)

According to (2.5), (2.9), (2.11), and (2.14), we infer that

HA p,TA p+m+1

≥1− t p−1 

HA p+1,TA p+m+1

− t p HB p,A p+m+1

≥1− t p−1

 (r + ε)

1 +

m−1

i =0

t p+1+i

−2ε m −

1

i =0



1− t p+1+i−1

−(r + ε)

m−1

i =0

t p+1+i

m−1

j = i s p+1+j

 i

k =0



1− t p+1+k−1



−(r + ε)t p

m



i =0



t p+i+s p+i

=(r + ε)1− t p−1



1 +

m−1

i =0

t p+1+i −

t2

p+t p

m



i =1

t p+i+t p

m



i =0

s p+i



−2ε m

i =0



1− t p+i−1

−(r + ε)1− t p−1m−1

i =0



t p+1+i

m−1

j = i s p+1+j

 i

k =0



1− t p+1+k−1



=(r + ε)

1 +

m



i =0

t p+i

−(r + ε)1− t p−1

t p

m



i =0

s p+i

−2ε m

i =0



1− t p+i−1

−(r + ε)m

i =1



t p+i

m

j = i s p+j

 i

k =0



1− t p+k−1



=(r + ε)

1 +

m



i =0

t p+i

−2ε m

i =0



1− t p+i−1

−(r + ε)m

i =0



t p+i

m

j = i s p+j

 i

k =0



1− t p+k−1

 , p ≥ N.

(2.15)

That is, (2.12) holds forn = m + 1 Hence (2.12) holds for anyn ≥1

We next assert thatr =0 Otherwiser > 0 Let m be an arbitrary positive integer, and

letε =2−1(1− b) mmin{ r,1 } It follows from (2.2) and (2.8) that there exists a positive integerN = N(ε) satisfying (2.9) and that







q



i =0

s n+i





According to (2.1), (2.2), (2.9), (2.12), and (2.16), we easily conclude that

HA N,TA N+m

≥(r + ε)1 +

m−1

i =0

t N+i



−2ε m −

1

i =0



1− t N+i−1

−(r + ε) m−1

i =0



t N+i

m−1

j = i s N+j

 i

k =0



1− t N+k−1



≥(r + ε)1 +

m−1

i =0

t N+i



−2ε(1 − b) − m −(r + ε)ε m −

1



i =0

t N+i(1− b) − i −1

≥ r + ε −2ε(1 − b) − m+ (r + ε)1− ε(1 − b) − mm−1

i =0

t N+i

≥ r + ε −2·2−1r(1 − b) m(1− b) − m

+ (r + ε)1−2−1(1− b) m(1− b) − mm−1

i =0

t N+i

≥2−1r m−1

i =0

t N+i ≥2−1rma −→ +∞ asm −→ ∞

(2.17)

That is,{ H(A n,TA k) :n ≥0,k ≥0}is unbounded, which is a contradiction Hencer =0 The compactness ofIyields that there exists a subsequence{ A n k } k ≥0of{ A n } n ≥0satisfying that

lim

k →∞ HA n k,A=0 for someA ∈I. (2.18)

In view of (2.8), (2.18) and the nonexpansiveness ofT, we have

H(A,TA) ≤ HA,A n k

 +HA n k,TA n k

 +HTA n k,TA

≤2HA,A n k

 +HA n k,TA n k



−→0 ask −→ ∞ (2.19)

That is,A = TA From (1.2) andLemma 1.3, we know that

HA n+1,A≤1− t n

HA n,A+t n HTB n,A

≤1− t n

HA n,A+t n HB n,A

≤1− t n

HA n,A+t n

1− s n

HA n,A+s n HTA n,A

≤ HA n,A, n ≥0.

(2.20)

It follows from (2.18) and (2.20) that limn →∞ H(A n,A) =0 This completes the proof 

FromTheorem 2.1we have the following

Theorem 2.2 Let X be a nonempty compact convex subset of a Banach space (E, · ), and

letI be a nonempty compact convex subset of CC(X) Suppose that T : (I,H) →(I,H) is

nonexpansive and there exist constants a and b satisfying that

Then for any A0∈I, the Mann iteration sequence { A n } n ≥0converges to a fixed point of T Remark 2.3 In case t n =1/2 for all n ≥0,Theorem 2.2reduces to [1, Theorem 3.2] by Hu and Huang The following example reveals thatTheorem 2.2extends properly the result

of Hu and Huang

Example 2.4 Let E = Rwith the usual norm|·|,X =[0, 1], and letI= {[0,x] : x ∈ X } DefineT : (I,H) →(I,H) by

ThenIis a nonempty compact convex subset ofCC(X) and

HT[0,x],T[0, y]= | x − y | = H[0,x],[0, y], x, y ∈ X. (2.23) That is,T is nonexpansive Set t n =(n + 1)/(10n + 3) for all n ≥0 anda =1/10, b =1/3.

Thus all conditions ofTheorem 2.2are fulfilled Therefore, we may invoke our Theorem 2.2to show thatT has a fixed point inI; but we cannot invoke [1, Theorem 3.2] by Hu and Huang to show thatT has fixed points inIsincet n =1/2 for all n ≥0

Remark 2.5 The example below shows that the Picard iteration sequences of

nonexpan-sive mappings in hyperspaces need not converge and the condition “t n ≤ b < 1, n ≥0” in Theorem 2.2is necessary

Example 2.6 Let E, X,I, andT be as inExample 2.4 Taket n =1 for alln ≥0 For any

A0=[0,x] with x ∈ X \ {1/2 }, the Picard iteration sequence{ A n } n ≥0⊂Idoes not con-verge sinceA2n =[0,x] for all n ≥0 andA2n −1=[0, 1− x] for all n ≥1

Acknowledgments

The authors thank the referees sincerely for their valuable and useful comments and sug-gestions This work was supported by the Science Research Foundation of Educational Department of Liaoning Province (20060467) and the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-312-C00026)

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Zeqing Liu: Department of Mathematics, Liaoning Normal University, P.O Box 200, Dalian, Liaoning 116029, China

Email address:zeqingliu@sina.com.cn

Chi Feng: Department of Science, Dalian Fisheries College, Dalian, Liaoning 116023, China

Email address:chifeng@x.cn

Shin Min Kang: Department of Mathematics and the Research Institute of Natural Science,

Gyeongsang National University, Jinju 660-701, South Korea

Email address:smkang@nongae.gsnu.ac.kr

Jeong Sheok Ume: Department of Applied Mathematics, Changwon National University,

Changwon 641-733, South Korea

Email address:jsume@sarim.changwon.ac.kr

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Proceedings of the American Mathematical Society, vol 44, no 2,... 375–380, 1974.

[8] W Takahashi and G.-E Kim, ? ?Approximating fixed points of nonexpansive mappings in Banach

spaces,” Mathematica Japonica, vol