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INFINITE FAMILIES OF NONEXPANSIVEMAPPINGS IN GENERAL BANACH SPACES TOMONARI SUZUKI Received 2 June 2004 In 1979, Ishikawa proved a strong convergence theorem for finite families of sive

INFINITE FAMILIES OF NONEXPANSIVE

MAPPINGS IN GENERAL BANACH SPACES

TOMONARI SUZUKI

Received 2 June 2004

In 1979, Ishikawa proved a strong convergence theorem for finite families of sive mappings in general Banach spaces Motivated by Ishikawa’s result, we prove strongconvergence theorems for infinite families of nonexpansive mappings

in [2,4,5], and other references

Many convergence theorems for nonexpansive mappings and families of nonexpansivemappings have been studied; see [1,3,6,7,10,11,12,14,15,17,18,19,20,21] and others.For example, in 1979, Ishikawa proved the following theorem

Theorem 1.1 [12] LetC be a compact convex subset of a Banach space E Let { T1,T2, ,T k }

be a finite family of commuting nonexpansive mappings on C Let { β i } k

for n ∈ N Then { x n } converges strongly to a common fixed point of { T1,T2, ,T k }

Copyright©2005 Hindawi Publishing Corporation

Fixed Point Theory and Applications 2005:1 (2005) 103–123

DOI: 10.1155/FPTA.2005.103

The author thinks this theorem is one of the most interesting convergence theoremsfor families of nonexpansive mappings In the case ofk =4, this iteration scheme is asfollows:

We remark thatS i S j = S j S idoes not hold in general

Very recently, in 2002, the following theorem was proved in [19]

Theorem 1.2 [19] LetC be a compact convex subset of a Banach space E and let S and T

be nonexpansive mappings on C with ST = TS Let x1∈ C and define a sequence { x n } in C by

theo-Theorem 1.3 [15] LetC be a compact convex subset of a strictly convex Banach space E Let { T n:n ∈ N} be an infinite family of commuting nonexpansive mappings on C Let { β n }

be a sequence in (0, 1) Put S i x = β i T i x + (1 − β i)x for i ∈ N and x ∈ C Let f be a mapping

onNsatisfying
(f −1(i)) = ∞ for all i ∈ N Define a sequence { x n } in C by x1∈ C and

x n+1 = S f (n) ◦ S f (n −1)◦ ··· ◦ S f (1) x1 (1.5)

for n ∈ N Then { x n } converges strongly to a common fixed point of { T n:n ∈ N}

The following mapping f onNsatisfies the assumption inTheorem 1.3: ifn ∈ Nisfies

2 Lemmas

In this section, we prove some lemmas The following lemma is connected with nosel’ski˘ı and Mann’s type sequences [14,16] This is a generalization of [19, Lemma 1].See also [8,20]

Kras-Lemma 2.1 Let { z n } and { w n } be sequences in a Banach space E and let { α n } be a sequence

in [0, 1] with lim sup n α n < 1 Put

for j ∈ N Put a =(1−lim supn α n)/2 We note that 0 < a < 1 Fix k, ∈ N andε > 0.

Then there existsm  ≥  such that a ≤1− α n,
w n+1 − w n
−
z n+1 − z n
≤ ε, and
w n+ j −

z n+ j
−
w n − z n
≤ ε/2 for all n ≥ m andj =1, 2, ,k In the case of d =lim supn
w n −

forj =1, 2, ,k In both cases, such m satisfies that m ≥ , a ≤1− α n ≤1,
w n+1 − w n
−

z n+1 − z n
≤ ε for all n ≥ m, and

By usingLemma 2.1, we obtain the following useful lemma, which is a generalization

of [19, Lemma 2] and [20, Lemma 6]

Lemma 2.2 Let { z n } and { w n } be bounded sequences in a Banach space E and let { α n } be

a sequence in [0, 1] with 0 < liminf n α n ≤lim supn α n < 1 Suppose that z n+1 = α n w n+ (1−

α n)z n for all n ∈ N and

lim sup

n →∞

w n+1 − w n −z n+1 − z n  ≤0. (2.19)

Then lim n
w n − z n
= 0.

Proof We put a =lim infn α n > 0, M =2 sup{
z n
+
w n
:n ∈ N} < ∞, and d =

lim supn
w n − z n
< ∞ We assumed > 0 Then fix k ∈ Nwith (1 +ka)d > M ByLemma2.1, we have

lim inf

n →∞ w n+k − z n −1 +α n+α n+1+···+α n+k −1

d =0. (2.20)Thus, there exists a subsequence{ n i }of a sequence{ n }inNsuch that

We prove the following lemmas, which are connected with real numbers

Lemma 2.3 Let { α n } be a real sequence with lim n(α n+1 − α n)= 0 Then every t ∈ R with

lim infn α n < t < limsup n α n is a cluster point of { α n }

Proof We assume that there exists t ∈(lim infn α n, lim supn α n) such thatt is not a cluster

point of{ α n } Then there existε > 0 and n1∈ Nsuch that

We put

n5=max

n : n < n4,α n ≤ t − ε ≥ n3. (2.25)Then we have

α n5≤ t − ε < t + ε ≤ α n5+1 (2.26)and hence

ε ≤2ε ≤ α n5+1− α n5=α n5+1− α n

This is a contradiction Therefore we obtain the desired result

Lemma 2.4 For α,β ∈(0, 1/2) and n ∈ N ,

We know the following.

Lemma 2.5 Let C be a subset of a Banach space E and let { V n } be a sequence of pansive mappings on C with a common fixed point w ∈ C Let x1∈ C and define a sequence

nonex-{ x n } in C by x n+1 = V n x n for n ∈ N Then {
x n − w
} is a nonincreasing sequence inR.

Proof We have
x n+1 − w
=
V n x n − V n w
≤
x n − w
for alln ∈ N

3 Three nonexpansive mappings

In this section, we prove a convergence theorem for three nonexpansive mappings Thepurpose for this is that we give the idea of our results

Lemma 3.1 Let C be a closed convex subset of a Banach space E Let T1and T2be pansive mappings on C with T1◦ T2= T2◦ T1 Let { t n } be a sequence in (0, 1) converging to

nonex-0 and let { z n } be a sequence in C such that { z n } converges strongly to some w ∈ C and

Then w is a common fixed point of T1and T2.

Proof It is obvious that

T1◦ T2w = T2◦ T1w = T2w. (3.5)

We assume thatw is not a fixed point of T2 Put

Lemma 3.2 Let C be a closed convex subset of a Banach space E Let T1, T2, and T3 be commuting nonexpansive mappings on C Let { t n } be a sequence in (0, 1/2) converging to 0 and let { z n } be a sequence in C such that { z n } converges strongly to some w ∈ C and

Then w is a common fixed point of T1, T2, and T3.

Proof We note that { T1z n },{ T2z n }and{ T3z n }are bounded sequences inC because { z n }

This is a contradiction Hence,w is a common fixed point of T1,T2, andT3

Theorem 3.3 Let C be a compact convex subset of a Banach space E Let T1, T2, and T3be commuting nonexpansive mappings on C Fix λ ∈ (0, 1) Let { α n } be a sequence in [0, 1/2] satisfying

for alln ∈ N Since

Therefore we have



1− t − t2 

T1z t+tT2z t+t2T3z t = z t (3.27)

for allt ∈ Rwith 0< t < α Since C is compact, there exists a real sequence { t n }in (0,α)

such that limn t n =0, and{ z t n }converges strongly to some pointw ∈ C ByLemma 3.2,

we obtain that suchw is a common fixed point of T1,T2, andT3 We note thatw is a

cluster point of{ x n }because so arez t n for alln ∈ N Hence, lim infn
x n − w
=0 Wealso have that{
x n − w
}is nonincreasing byLemma 2.5 Thus, limn
x n − w
=0 This

We give an example concerning{ α n }

Example 3.4 Define a sequence { β n }in [−1/2,1/2] by

forn ∈ N Then{ α n }satisfies the assumption ofTheorem 3.3

Remark 3.5 The sequence { α n }is as follows:

In this section, we prove our main results

Lemma 4.1 Let C be a closed convex subset of a Banach space E Let  ∈ N with  ≥ 2 and let

T1,T2, ,T  be commuting nonexpansive mappings on C Let { t n } be a sequence in (0, 1/2) converging to 0 and let { z n } be a sequence in C such that { z n } converges strongly to some

Then w is a common fixed point of T1,T2, ,T 

Proof We will prove this lemma by induction We have already proved the conclusion

in the case of =2, 3 Fix ∈ Nwith ≥4 We assume that the conclusion holds forevery integer less than and greater than 1 We note that { T1z n },{ T2z n }, , { T  z n }arebounded sequences inC because { z n }is bounded We have

lim sup

n →∞

1− −2

k =1t k n

This is a contradiction Hence,w is a common fixed point of T1,T2, ,T  By induction,

Lemma 4.2 Let C be a bounded closed convex subset of a Banach space E Let { T n:n ∈ N}

be an infinite family of commuting nonexpansive mappings on C Let { t n } be a sequence in

(0, 1/2) converging to 0 and let { z n } be a sequence in C such that { z n } converges strongly to some w ∈ C and

Proof Fix  ∈ Nwith ≥2 We putM =2 sup{
x
:x ∈ C } < ∞ We have

Theorem 4.3 Let C be a compact convex subset of a Banach space E Let { T n:n ∈ N} be

an infinite family of commuting nonexpansive mappings on C Fix λ ∈ (0, 1) Let { α n } be a sequence in [0, 1/2] satisfying

for n ∈ N Then { x n } converges strongly to a common fixed point of { T n:n ∈ N}

Remark 4.4 We know that∞

n =1F(T n)=∅by DeMarr’s result in [5] We define 0

for alln ∈ N Since

≤x n+1 − x n+ 8Mα n+1 − α n+ 2M 1

2n

(4.15)forn ∈ N, we have

for allt ∈ Rwith 0< t < α Since C is compact, there exists a real sequence { t n }in (0,α)

such that limn t n =0 and{ z t n }converges strongly to some pointw ∈ C ByLemma 4.2,

we obtain that suchw is a common fixed point of { T n:n ∈ N} We note thatw is a cluster

point of{ x n }because so arez t n for alln ∈ N Hence, lim infn
x n − w
=0 We also havethat{
x n − w
}is nonincreasing byLemma 2.5 Thus, limn
x n − w
=0 This completes

Similarly, we can prove the following

Theorem 4.5 Let C be a compact convex subset of a Banach space E Let { T n:n ∈ N} be

an infinite family of commuting nonexpansive mappings on C Fix λ ∈ (0, 1) Let { α n } be a sequence in [0, 1/2] satisfying

for n ∈ N Then { x n } converges strongly to a common fixed point of { T n:n ∈ N}

As direct consequences, we obtain the following

Theorem 4.6 Let C be a compact convex subset of a Banach space E Let S and T be expansive mappings on C with ST = TS Let { α n } be a sequence in [0, 1] satisfying

for n ∈ N Then { x n } converges strongly to a common fixed point of S and T.

Remark 4.7 This theorem is simpler thanTheorem 1.2

Theorem 4.8 Let C be a compact convex subset of a Banach space E Let  ∈ N with  ≥2

and let { T1,T2, ,T  } be a finite family of commuting nonexpansive mappings on C Let

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3 Three nonexpansive mappings< /b>

In this section, we prove a convergence theorem for three nonexpansive mappings Thepurpose for this