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STRONG CONVERGENCE THEOREMS FOR
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 nonexpan-
sive mappings in general Banach spaces. Motivated by Ishikawa’s result, we prove strong
convergence theorems for infinite families of nonexpansive mappings.
1. Introduction
Throughout this paper, we denote by N the set of positive integers and by R the set of real
numbers. For an arbitrary set A, we also denote by A the cardinal number of A.
Let C be a closed convex subset of a Banach space E.LetT be a nonexpansive mapping
on C, that is,
Tx−Ty≤x − y (1.1)
for all x, y ∈ C. We denote by F(T) the set of fixed points of T.WeknowF(T) = ∅ in the
case that E is uniformly convex and C is bounded; see Browder [2], G
¨
ohde [9], and Kirk
[13]. Common fixed point theorems for families of nonexpansive mappings are proved
in [2, 4, 5], and other references.
Many convergence theorems for nonexpansive mappings and families of n onexpansive
mappings 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]. Let C beacompactconvexsubsetofaBanachspaceE.Let
{T
1
,T
2
, ,T
k
}

be a finite family of commuting nonexpansive mappings on C.Let{β
i
}
k
i=1
be a finite sequence
in (0,1) and put S
i
x = β
i
T
i
x +(1−β
i
)x for x ∈C and i = 1,2, ,k.Letx
1
∈ C and define
asequence{x
n
} in C by
x
n+1
=

n

n
k−1
=1


S
k
n
k−1

n
k−2
=1

S
k−1
···

S
3
n
2

n
1
=1

S
2
n
1

n
0
=1

S
1

···

x
1
(1.2)
for n ∈ N. Then {x
n
} converges strongly to a common fixed point of {T
1
,T
2
, ,T
k
}.
Copyright © 2005 Hindawi Publishing Corporation
Fixed Point Theory and Applications 2005:1 (2005) 103–123
DOI: 10.1155/FPTA.2005.103
104 Convergence to common fixed point
The author thinks this theorem is one of the most interesting convergence theorems
for families of nonexpansive mappings. In the case of k = 4, this iteration scheme is as
follows:
x
2
= S
4
S
3

S
2
S
1
x
1
,
x
3
= S
4
S
3
S
2
S
1
S
1
S
2
S
1
S
3
S
2
S
1
x

2
,
x
4
= S
4
S
3
S
2
S
1
S
1
S
1
S
2
S
1
S
1
S
2
S
1
S
3
S
2

S
1
S
1
S
2
S
1
S
3
S
2
S
1
x
3
,
x
5
= S
4
S
3
S
2
S
1
S
1
S

1
S
1
S
2
S
1
S
1
S
1
S
2
S
1
S
1
S
2
S
1
S
3
S
2
S
1
S
1
S

1
S
2
S
1
S
1
S
2
S
1
S
3
S
2
S
1
S
1
S
2
S
1
S
3
S
2
S
1
x

4
,
x
6
= S
4
S
3
S
2
S
1
S
1
S
1
S
1
S
1
S
2
S
1
S
1
S
1
S
1

S
2
S
1
S
1
S
1
S
2
S
1
S
1
S
2
S
1
S
3
S
2
S
1
S
1
S
1
S
1

S
2
S
1
S
1
S
1
S
2
S
1
S
1
S
2
S
1
S
3
S
2
S
1
S
1
S
1
S
2

S
1
S
1
S
2
S
1
S
3
S
2
S
1
S
1
S
2
S
1
S
3
S
2
S
1
x
5
,
x

7
= S
4
S
3
S
2
S
1
S
1
S
1
S
1
S
1
S
1
S
2
S
1
S
1
S
1
S
1
S

1
S
2
S
1
S
1
S
1
S
1
S
2
S
1
S
1
S
1
S
2
S
1
S
1
S
2
S
1
S

3
S
2
S
1
S
1
S
1
S
1
S
1
S
2
S
1
S
1
S
1
S
1
S
2
S
1
S
1
S

1
S
2
S
1
S
1
S
2
S
1
S
3
S
2
S
1
S
1
S
1
S
1
S
2
S
1
S
1
S

1
S
2
S
1
S
1
S
2
S
1
S
3
S
2
S
1
S
1
S
1
S
2
S
1
S
1
S
2
S

1
S
3
S
2
S
1
S
1
S
2
S
1
S
3
S
2
S
1
x
6
.
(1.3)
We remar k that S
i
S
j
= S
j
S

i
does not hold in general.
Very recently, in 2002, the following theorem was proved in [19].
Theorem 1.2 [19]. Let C be a compact convex subset of a Banach space E and let S and T
be nonexpansive mappings on C with ST
= TS.Letx
1
∈ C and de fine a sequence {x
n
} in C
by
x
n+1
=
α
n
n
2
n

i=1
n

j=1
S
i
T
j
x
n

+

1 −α
n

x
n
(1.4)
for n ∈ N,where{α
n
} is a sequence in [0,1] such that 0 < liminf
n
α
n
≤ limsup
n
α
n
< 1.
Then {x
n
} converges strongly to a common fixed point z
0
of S and T.
This theorem is simpler than Theorem 1.1. However, this is not a convergence theorem
for infinite families of nonexpansive mappings.
Under the assumption of the strict convexity of the Banach space, convergence theo-
rems for infinite families of nonexpansive mappings were proved. In 1972, Linhart [15]
proved the following; see also [20].
Theorem 1.3 [15]. Let C beacompactconvexsubsetofastrictlyconvexBanachspaceE.

Let
{T
n
: n ∈N} be an infinite family of commuting nonexpansive mappings on C.Let{β
n
}
beasequencein(0,1).PutS
i
x =β
i
T
i
x +(1−β
i
)x for i ∈ N and x ∈ C.Let f be a mapping
on N satisfying ( f
−1
(i)) =∞for all i ∈N. Define a seque nce {x
n
} in C by x
1
∈ C and
x
n+1
= S
f (n)
◦S
f (n−1)
◦···◦S
f (1)

x
1
(1.5)
for n ∈ N. Then {x
n
} converges strongly to a common fixed point of {T
n
: n ∈N}.
Tomonari Suzuki 105
The following mapping f on N satisfies the assumption in Theorem 1.3:ifn ∈N sat-
isfies
k−1

j=1
j<n≤
k

j=1
j (1.6)
for some k
∈ N,thenput
f (n) =n −
k−1

j=1
j. (1.7)
That is,
f (1)
= 1,
f (2) =1, f (3) =2,

f (4) =1, f (5) =2, f (6) =3,
f (7) =1, f (8) =2, f (9) =3, f (10) =4,
f (11) =1, f (12) =2, f (13) =3, f (14) =4, f (15) =5,
f (16) =1, f (17) =2,
(1.8)
It is a natural problem whether or not there exists an iteration to find a common fixed
point for infinite families of commuting nonexpansive mappings without assuming the
strict convexity of the Banach space. This problem has not been solved for twenty-five
years. In this paper, we give such iteration. That is, our answer of this problem is positive.
2. Lemmas
In this section, we prove some lemmas. The following lemma is connected with Kras-
nosel’ski
˘
ı and Mann’s type sequences [14, 16]. This is a generalization of [19, Lemma 1].
See also [8, 20].
Lemma 2.1. Let
{z
n
} and {w
n
} be sequences in a Banach space E and let {α
n
} beasequence
in [0,1] with limsup
n
α
n
< 1.Put
d =limsup
n→∞



w
n
−z
n


or d =liminf
n→∞


w
n
−z
n


. (2.1)
Suppose that z
n+1
= α
n
w
n
+(1−α
n
)z
n
for all n ∈N,

limsup
n→∞



w
n+1
−w
n


−


z
n+1
−z
n



≤ 0, (2.2)
and d<∞. Then
liminf
n→∞




w

n+k
−z
n


−

1+α
n
+ α
n+1
+ ···+ α
n+k−1

d


=
0 (2.3)
hold for all k
∈ N.
106 Convergence to common fixed point
Proof. Since


w
n+1
−z
n+1



−


w
n
−z
n


≤


w
n+1
−w
n


+


w
n
−z
n+1


−



w
n
−z
n


=


w
n+1
−w
n


−


z
n+1
−z
n


,
(2.4)
we have
limsup
n→∞




w
n+ j
−z
n+ j


−


w
n
−z
n



=
limsup
n→∞
j−1

i=0



w
n+i+1

−z
n+i+1


−


w
n+i
−z
n+i



≤ limsup
n→∞
j−1

i=0



w
n+i+1
−w
n+i


−



z
n+i+1
−z
n+i



≤
j−1

i=0
limsup
n→∞



w
n+i+1
−w
n+i


−


z
n+i+1
−z
n+i




≤ 0
(2.5)
for j ∈ N.Puta = (1 −limsup
n
α
n
)/2. We note that 0 <a<1. Fix k, ∈ N and ε>0.
Then there exists m

≥  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/2foralln ≥ m

and j = 1,2, ,k. In the case of d = limsup
n
w
n
−
z
n
,wechoosem ≥m

satisfying


w
m+k
−z
m+k


≥ d −
ε
2
(2.6)
and w
n
−z
n
≤d + ε for all n ≥ m. We note that



w
m+ j
−z
m+ j


≥


w
m+k
−z
m+k


−
ε
2
≥ d −ε (2.7)
for j =0,1, ,k −1. In the case of d =liminf
n
w
n
−z
n
,wechoosem ≥m

satisfying



w
m
−z
m


≤ d +
ε
2
(2.8)
and w
n
−z
n
≥d −ε for all n ≥ m. We note that


w
m+ j
−z
m+ j


≤


w
m

−z
m


+
ε
2
≤ d + ε (2.9)
for j = 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
d −ε ≤


w
m+ j
−z
m+ j


≤ d + ε (2.10)

for j =0,1, ,k. We next show


w
m+k
−z
m+ j


≥

1+α
m+ j
+ α
m+ j+1
+ ···+ α
m+k−1

d −
(k − j)(2k +1)
a
k−j
ε (2.11)
Tomonari Suzuki 107
for j =0,1, ,k −1. Since
d −ε ≤


w
m+k

−z
m+k


=


w
m+k
−α
m+k−1
w
m+k−1
−

1 −α
m+k−1

z
m+k−1


≤ α
m+k−1


w
m+k
−w
m+k−1



+

1 −α
m+k−1



w
m+k
−z
m+k−1


≤ α
m+k−1


z
m+k
−z
m+k−1


+ ε +

1 −α
m+k−1




w
m+k
−z
m+k−1


= α
2
m+k−1


w
m+k−1
−z
m+k−1


+ ε +

1 −α
m+k−1



w
m+k
−z
m+k−1



≤ α
2
m+k−1
d +2ε +

1 −α
m+k−1



w
m+k
−z
m+k−1


,
(2.12)
we obtain


w
m+k
−z
m+k−1


≥


1 −α
2
m+k−1

d −3ε
1 −α
m+k−1
≥

1+α
m+k−1

d −
2k +1
a
ε.
(2.13)
Hence (2.11)holdsfor j = k −1. We assume (2.11)holdsforsomej ∈{1,2, ,k −1}.
Then since

1+
k−1

i=j
α
m+i

d −
(k − j)(2k +1)

a
k−j
ε
≤


w
m+k
−z
m+ j


=


w
m+k
−α
m+ j−1
w
m+ j−1
−

1 −α
m+ j−1

z
m+ j−1



≤ α
m+ j−1


w
m+k
−w
m+ j−1


+

1 −α
m+ j−1



w
m+k
−z
m+ j−1


≤ α
m+ j−1
k−1

i=j−1



w
m+i+1
−w
m+i


+

1 −α
m+ j−1



w
m+k
−z
m+ j−1


≤ α
m+ j−1
k−1

i=j−1



z
m+i+1
−z

m+i


+ ε

+

1 −α
m+ j−1



w
m+k
−z
m+ j−1


≤ α
m+ j−1
k−1

i=j−1


z
m+i+1
−z
m+i



+ kε +

1 −α
m+ j−1



w
m+k
−z
m+ j−1


=
α
m+ j−1
k−1

i=j−1
α
m+i


w
m+i
−z
m+i



+ kε +

1 −α
m+ j−1



w
m+k
−z
m+ j−1


≤ α
m+ j−1
k−1

i=j−1
α
m+i
(d + ε)+kε +

1 −α
m+ j−1



w
m+k
−z

m+ j−1


≤ α
m+ j−1
k−1

i=j−1
α
m+i
d +2kε+

1 −α
m+ j−1



w
m+k
−z
m+ j−1


,
(2.14)
108 Convergence to common fixed point
we obtain


w

m+k
−z
m+ j−1


≥
1+

k−1
i=j
α
m+i
−α
m+ j−1

k−1
i=j−1
α
m+i
1 −α
m+ j−1
d −
(k − j)(2k +1)/a
k−j
+2k
1 −α
m+ j−1
ε
≥


1+
k−1

i=j−1
α
m+i

d −
(k − j + 1)(2k +1)
a
k−j+1
ε.
(2.15)
Hence (2.11)holdsfor j := j −1. Therefore (2.11)holdsforallj = 0, 1, , k −1. Spe-
cially, we have


w
m+k
−z
m


≥

1+α
m
+ α
m+1
+ ···+ α

m+k−1

d −
k(2k +1)
a
k
ε. (2.16)
On the other hand, we have


w
m+k
−z
m


≤


w
m+k
−z
m+k


+
k−1

i=0



z
m+i+1
−z
m+i


=


w
m+k
−z
m+k


+
k−1

i=0
α
m+i


w
m+i
−z
m+i



≤ d + ε +
k−1

i=0
α
m+i
(d + ε)
≤ d +
k−1

i=0
α
m+i
d +(k +1)ε.
(2.17)
From (2.16)and(2.17), we obtain




w
m+k
−z
m


−

1+α
m

+ α
m+1
+ ···+ α
m+k−1

d


≤
k(2k +1)
a
k
ε. (2.18)
Since 
∈ N and ε>0 are arbitrary, we obtain the desired result. 
By using Lemma 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
asequencein[0, 1] with 0 < liminf
n
α
n
≤ lim sup
n

α
n
< 1.Supposethatz
n+1
= α
n
w
n
+(1−
α
n
)z
n
for all n ∈N and
limsup
n→∞



w
n+1
−w
n


−


z
n+1

−z
n



≤ 0. (2.19)
Then lim
n
w
n
−z
n
=0.
Tomonari Suzuki 109
Proof. We put a = liminf
n
α
n
> 0, M = 2sup{z
n
 + w
n
 : n ∈ N} < ∞,andd =
limsup
n
w
n
−z
n
 < ∞.Weassumed>0. Then fix k ∈ N with (1 + ka)d>M.ByLemma

2.1,wehave
liminf
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} in N such that

lim
i→∞



w
n
i
+k
−z
n
i


−

1+α
n
i
+ α
n
i
+1
+ ···+ α
n
i
+k−1

d


=
0, (2.21)
the limit of {w
n
i
+k
−z
n
i
}exists, and the limits of {α
n
i
+ j
}exist for all j ∈{0,1, ,k −1}.
Put β
j
= lim
i
α
n
i
+ j
for j ∈{0,1,···,k −1}. It is obvious that β
j
≥ a for all j ∈{0,1, ,
k
−1}.Wehave
M<(1 + ka)d
≤


1+β
0
+ β
1
+ ···+ β
k−1

d
= lim
i→∞

1+α
n
i
+ α
n
i
+1
+ ···+ α
n
i
+k−1

d
= lim
i→∞


w
n

i
+k
−z
n
i


≤
limsup
n→∞


w
n+k
−z
n


≤ M.
(2.22)
This is a contradiction. Therefore d
= 0. 
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.Theneveryt ∈ R with
liminf
n
α
n
<t<limsup
n
α
n
is a cluster point of {α
n
}.
Proof. We assume that there exists t ∈ (liminf
n
α
n
,limsup
n
α
n
)suchthatt is not a cluster
point of {α
n
}. Then there exist ε>0andn
1
∈ N such that
liminf
n→∞
α

n
<t−ε<t<t+ ε<limsup
n→∞
α
n
,
α
n
∈ (−∞, t −ε] ∪[t + ε,∞),
(2.23)
for all n ≥n
1
.Wechoosen
2
≥ n
1
such that |α
n+1
−α
n
| <εfor all n ≥ n
2
. Then there exist
n
3
,n
4
∈ N such that n
4
≥ n

3
≥ n
2
,
α
n
3
∈ (−∞, t −ε], α
n
4
∈ [t + ε,∞). (2.24)
110 Convergence to common fixed point
We put
n
5
= max

n : n<n
4
, α
n
≤ t −ε

≥ n
3
. (2.25)
Then we have
α
n
5

≤ t −ε<t+ ε ≤ α
n
5
+1
(2.26)
and hence
ε ≤2ε ≤α
n
5
+1
−α
n
5
=


α
n
5
+1
−α
n
5


<ε. (2.27)
This is a contradiction. Therefore we obtain the desired result. 
Lemma 2.4. For α,β ∈ (0,1/2) and n ∈N,



α
n
−β
n


≤|α −β|,
∞

k=1


α
k
−β
k


≤ 4|α −β|
(2.28)
hold.
Proof. We assume that n ≥2 because the conclusion is obvious in the case of n =1. Since
α
n
−β
n
= (α −β)
n−1

k=0

α
n−1−k
β
k
, (2.29)
we have


α
n
−β
n


=|
α −β|
n−1

k=0
α
n−1−k
β
k
≤|α −β|
n−1

k=0
1
2
n−1

=|α −β|
n
2
n−1
≤|α −β|.
(2.30)
We also have
∞

k=1


α
k
−β
k


=





∞

k=1

α
k

−β
k






=




α
1 −α
−
β
1 −β




=




α −β
(1 −α)(1 −β)





≤ 4|α −β|.
(2.31)
This completes the proof. 
Tomonari Suzuki 111
We know the following.
Lemma 2.5. Let C be a subset of a B anach space E and let {V
n
} beasequenceofnonex-
pansive mappings on C w ith a common fixed point w ∈C.Letx
1
∈ C and define a sequence
{x
n
} in C by x
n+1
= V
n
x
n
for n ∈ N. Then {x
n
−w} is a nonincreasing sequence in R.
Proof. We have x
n+1
−w=V
n
x

n
−V
n
w≤x
n
−w for all n ∈N. 
3. Three nonexpansive mappings
In this section, we prove a convergence theorem for three nonexpansive mappings. The
purpose 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.LetT
1
and T
2
be nonex-
pansive mappings on C with T
1
◦T
2
= T
2
◦T
1
.Let{t
n
} beasequencein(0,1) converging to
0 and let {z
n
} beasequenceinC such that {z
n
} converges strongly to some w ∈ C and

lim
n→∞



1 −t
n

T
1
z
n
+ t
n
T
2
z
n
−z
n


t
n
= 0. (3.1)
Then w is a common fixed point of T
1
and T
2
.

Proof. It is obvious that
sup
m,n∈N


T
1
z
m
−T
1
z
n


≤ sup
m,n∈N


z
m
−z
n


. (3.2)
So {T
1
z
n

} is bounded because {z
n
} is bounded. Similarly, we have that {T
2
z
n
} is also
bounded. Since
lim
n→∞



1 −t
n

T
1
z
n
+ t
n
T
2
z
n
−z
n



=
0, (3.3)
we have


T
1
w −w


≤ limsup
n→∞



T
1
w −T
1
z
n


+


T
1
z
n

−

1 −t
n

T
1
z
n
−t
n
T
2
z
n


+



1 −t
n

T
1
z
n
+ t
n

T
2
z
n
−z
n


+


z
n
−w



≤ limsup
n→∞

2


w −z
n


+ t
n



T
1
z
n
−T
2
z
n


+



1 −t
n

T
1
z
n
+ t
n
T
2
z
n
−z
n




=
0
(3.4)
and hence w is a fixed point of T
1
. We note that
T
1
◦T
2
w =T
2
◦T
1
w =T
2
w. (3.5)
112 Convergence to common fixed point
We assume that w is not a fixed point of T
2
.Put
ε =


T
2
w −w



3
> 0. (3.6)
Then there exists m ∈ N such that


z
m
−w


<ε,



1 −t
m

T
1
z
m
+ t
m
T
2
z
m
−z

m


t
m
<ε. (3.7)
Since
3ε =


T
2
w −w


≤


T
2
w −z
m


+


z
m
−w



<


T
2
w −z
m


+ ε,
(3.8)
we have
2ε<


T
2
w −z
m


. (3.9)
So, we obtain


T
2
w −z

m


≤


T
2
w −

1 −t
m

T
1
z
m
−t
m
T
2
z
m


+



1 −t

m

T
1
z
m
+ t
m
T
2
z
m
−z
m


≤

1 −t
m



T
2
w −T
1
z
m



+ t
m


T
2
w −T
2
z
m


+



1 −t
m

T
1
z
m
+ t
m
T
2
z
m

−z
m


=

1 −t
m



T
1
◦T
2
w −T
1
z
m


+ t
m


T
2
w −T
2
z

m


+



1 −t
m

T
1
z
m
+ t
m
T
2
z
m
−z
m


≤

1 −t
m




T
2
w −z
m


+ t
m


w −z
m


+



1 −t
m

T
1
z
m
+ t
m
T
2

z
m
−z
m


<

1 −t
m



T
2
w −z
m


+2t
m
ε
<

1 −t
m



T

2
w −z
m


+ t
m


T
2
w −z
m


=


T
2
w −z
m


.
(3.10)
This is a contradiction. Hence, w is a common fixed point of T
1
and T
2

. 
Tomonari Suzuki 113
Lemma 3.2. Let C be a closed convex subset of a Banach space E.LetT
1
, T
2
,andT
3
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
lim
n→∞



1 −t
n
−t
2
n

T
1

z
n
+ t
n
T
2
z
n
+ t
2
n
T
3
z
n
−z
n


t
2
n
= 0. (3.11)
Then w is a common fixed point of T
1
, T
2
,andT
3
.

Proof. We note that {T
1
z
n
}, {T
2
z
n
} and {T
3
z
n
} are bounded sequences in C because {z
n
}
is bounded. We have
limsup
n→∞



1 −t
n

T
1
z
n
+ t
n

T
2
z
n
−z
n


t
n
≤ limsup
n→∞



1 −t
n
−t
2
n

T
1
z
n
+ t
n
T
2
z

n
+ t
2
n
T
3
z
n
−z
n


+ t
2
n


T
1
z
n
−T
3
z
n


t
n
= lim

n→∞

t
n



1 −t
n
−t
2
n

T
1
z
n
+ t
n
T
2
z
n
+ t
2
n
T
3
z
n

−z
n


t
2
n
+ t
n


T
1
z
n
−T
3
z
n



= 0.
(3.12)
So, by Lemma 3.1,wehavethatw is a common fixed point of T
1
and T
2
. We note that
T

1
◦T
3
w =T
3
◦T
1
w =T
3
w, T
2
◦T
3
w =T
3
◦T
2
w =T
3
w. (3.13)
We assume that w is not a fixed point of T
3
.Put
ε =


T
3
w −w



3
> 0. (3.14)
Then there exists m
∈ N such that


z
m
−w


<ε,



1 −t
m
−t
2
m

T
1
z
m
+ t
m
T
2

z
m
+ t
2
m
T
3
z
m
−z
m


t
2
m
<ε. (3.15)
Since
3ε =


T
3
w −w


≤


T

3
w −z
m


+


z
m
−w


<


T
3
w −z
m


+ ε,
(3.16)
we have
2ε<


T
3

w −z
m


. (3.17)
114 Convergence to common fixed point
So, we obtain


T
3
w −z
m


≤


T
3
w −

1 −t
m
−t
2
m

T
1

z
m
−t
m
T
2
z
m
−t
2
m
T
3
z
m


+



1 −t
m
−t
2
m

T
1
z

m
+ t
m
T
2
z
m
+ t
2
m
T
3
z
m
−z
m


≤

1 −t
m
−t
2
m



T
3

w −T
1
z
m


+ t
m


T
3
w −T
2
z
m


+ t
2
m


T
3
w −T
3
z
m



+



1 −t
m
−t
2
m

T
1
z
m
+ t
m
T
2
z
m
+ t
2
m
T
3
z
m
−z
m



=

1 −t
m
−t
2
m



T
1
◦T
3
w −T
1
z
m


+ t
m


T
2
◦T
3

w −T
2
z
m


+ t
2
m


T
3
w −T
3
z
m


+



1 −t
m
−t
2
m

T

1
z
m
+ t
m
T
2
z
m
+ t
2
m
T
3
z
m
−z
m


≤

1 −t
m
−t
2
m




T
3
w −z
m


+ t
m


T
3
w −z
m


+ t
2
m


w −z
m


+



1 −t

m
−t
2
m

T
1
z
m
+ t
m
T
2
z
m
+ t
2
m
T
3
z
m
−z
m


<

1 −t
2

m



T
3
w −z
m


+2t
2
m
ε
<

1 −t
2
m



T
3
w −z
m


+ t
2

m


T
3
w −z
m


=


T
3
w −z
m


.
(3.18)
This is a contradiction. Hence, w is a common fixed point of T
1
, T
2
,andT
3
. 
Theorem 3.3. Let C beacompactconvexsubsetofaBanachspaceE.LetT
1
, T

2
,andT
3
be
commuting nonexpansive mappings on C.Fixλ ∈(0,1).Let{α
n
} be a sequence in [0,1/2]
satisfying
liminf
n→∞
α
n
= 0, limsup
n→∞
α
n
> 0, lim
n→∞

α
n+1
−α
n

=
0. (3.19)
Define a sequence {x
n
} in C by x
1

∈ C and
x
n+1
= λ

1 −α
n
−α
2
n

T
1
x
n
+ λα
n
T
2
x
n
+ λα
2
n
T
3
x
n
+(1−λ)x
n

(3.20)
for n ∈ N. Then {x
n
} converges strongly to a common fixed point of T
1
, T
2
,andT
3
.
Proof. Put
α = limsup
n→∞
α
n
> 0, M =sup
x∈C
x < ∞,
y
n
=

1 −α
n
−α
2
n

T
1

x
n
+ α
n
T
2
x
n
+ α
2
n
T
3
x
n
,
(3.21)
for n ∈ N. We note that
x
n+1
= λy
n
+(1−λ)x
n
(3.22)
Tomonari Suzuki 115
for all n ∈N.Since


y

n+1
− y
n


=



1 −α
n+1
−α
2
n+1

T
1
x
n+1
+ α
n+1
T
2
x
n+1
+ α
2
n+1
T
3

x
n+1
−

1 −α
n
−α
2
n

T
1
x
n
−α
n
T
2
x
n
−α
2
n
T
3
x
n


≤


1 −α
n+1
−α
2
n+1



T
1
x
n+1
−T
1
x
n


+ α
n+1


T
2
x
n+1
−T
2
x

n


+ α
2
n+1


T
3
x
n+1
−T
3
x
n


+


α
n+1
+ α
2
n+1
−α
n
−α
2

n




−T
1
x
n


+


α
n+1
−α
n




T
2
x
n


+



α
2
n+1
−α
2
n




T
3
x
n


≤

1 −α
n+1
−α
2
n+1



x
n+1
−x

n


+ α
n+1


x
n+1
−x
n


+ α
2
n+1


x
n+1
−x
n


+


α
n+1
+ α

2
n+1
−α
n
−α
2
n


M +


α
n+1
−α
n


M +


α
2
n+1
−α
2
n


M

≤


x
n+1
−x
n


+4


α
n+1
−α
n


M
(3.23)
for n ∈ N,wehave
limsup
n→∞



y
n+1
− y
n



−


x
n+1
−x
n



≤ 0. (3.24)
So, by Lemma 2.2,wehavelim
n
x
n
− y
n
=0. Fix t ∈R with 0 <t<α.ThenbyLemma
2.3, there exists a subsequence
{α
n
k
} of {α
n
} converging to t.SinceC is compact, there
exists a subsequence {x
n
k

j
} of {x
n
k
} converging to some point z
t
∈ C.Wehave



1 −t −t
2

T
1
z
t
+ tT
2
z
t
+ t
2
T
3
z
t
−z
t



≤



1 −t −t
2

T
1
z
t
+ tT
2
z
t
+ t
2
T
3
z
t
− y
n


+


y

n
−x
n


+


x
n
−z
t


=



1 −t −t
2

T
1
z
t
+ tT
2
z
t
+ t

2
T
3
z
t
−

1 −α
n
−α
2
n

T
1
x
n
−α
n
T
2
x
n
−α
2
n
T
3
x
n



+


y
n
−x
n


+


x
n
−z
t


≤

1 −t −t
2



T
1
z

t
−T
1
x
n


+ t


T
2
z
t
−T
2
x
n


+ t
2


T
3
z
t
−T
3

x
n


+


t + t
2
−α
n
−α
2
n




−T
1
x
n


+


t −α
n





T
2
x
n


+


t
2
−α
2
n




T
3
x
n


+



y
n
−x
n


+


x
n
−z
t


≤

1 −t −t
2



z
t
−x
n


+ t



z
t
−x
n


+ t
2


z
t
−x
n


+


t + t
2
−α
n
−α
2
n


M +



t −α
n


M +


t
2
−α
2
n


M
+


y
n
−x
n


+


x

n
−z
t


≤ 2


z
t
−x
n


+4


t −α
n


M +


y
n
−x
n



(3.25)
for n ∈ N, and hence



1 −t −t
2

T
1
z
t
+ tT
2
z
t
+ t
2
T
3
z
t
−z
t


≤ lim
j→∞

2



z
t
−x
n
k
j


+4


t −α
n
k
j


M +


y
n
k
j
−x
n
k
j




= 0.
(3.26)
116 Convergence to common fixed point
Therefore we have

1 −t −t
2

T
1
z
t
+ tT
2
z
t
+ t
2
T
3
z
t
= z
t
(3.27)
for all t ∈R with 0 <t<α.SinceC is compact, there exists a real sequence {t
n

} in (0,α)
such that lim
n
t
n
= 0, and {z
t
n
} converges strongly to some point w ∈ C.ByLemma 3.2,
we obtain that such w is a common fixed point of T
1
, T
2
,andT
3
. We note that w is a
cluster point of
{x
n
} because so are z
t
n
for all n ∈ N. Hence, lim inf
n
x
n
−w=0. We
also have that {x
n
−w} is nonincreasing by Lemma 2.5.Thus,lim

n
x
n
−w=0. This
completes the proof. 
We give an example concerning {α
n
}.
Example 3.4. Define a sequence {β
n
} in [−1/2,1/2] by
β
n
=


















1
2k
if 2
k−1

j=1
j<n≤2
k−1

j=1
j + k for some k ∈N,
−
1
2k
if 2
k−1

j=1
j + k<n≤2
k

j=1
j for some k ∈ N.
(3.28)
Define a sequence {α
n
} in [0,1/2] by
α
n

=
n

k=1
β
k
(3.29)
for n ∈ N.Then{α
n
} satisfies the assumption of Theorem 3.3.
Remark 3.5. The sequence {α
n
} is as follows:
α
1
= 1/2, α
2
= 0, α
3
= 1/4, α
4
= 2/4, α
5
= 1/4,
α
6
= 0, α
7
= 1/6, α
8

= 2/6, α
9
= 3/6, α
10
= 2/6,
α
11
= 1/6, α
12
= 0, α
13
= 1/8, α
14
= 2/8, α
15
= 3/8,
α
16
= 4/8, α
17
= 3/8, α
18
= 2/8, α
19
= 1/8, α
20
= 0,
α
21
= 1/10, α

22
= 2/10, α
23
= 3/10, α
24
= 4/10, α
25
= 5/10,
α
26
= 4/10, α
27
= 3/10, α
28
= 2/10,
(3.30)
4. Main results
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
T
1
,T
2
, ,T

be commuting nonexpansive mappings on C.Let{t
n
} beasequencein(0,1/2)
converging to 0 and let {z
n

} be a sequence in C such that {z
n
} converges strongly to some
Tomonari Suzuki 117
w ∈C and
lim
n→∞



1 −

−1
k=1
t
k
n

T
1
z
n
+


k=2
t
k−1
n
T

k
z
n
−z
n


t
−1
n
= 0. (4.1)
Then w is a common fixed point of T
1
,T
2
, ,T

.
Proof. We will prove this lemma by induction. We have already proved the conclusion
in the case of  = 2, 3. Fix  ∈ N with  ≥ 4. We assume that the conclusion holds for
every integer less than  and greater than 1. We note that {T
1
z
n
},{T
2
z
n
}, ,{T


z
n
} are
bounded sequences in C because {z
n
} is bounded. We have
limsup
n→∞



1 −

−2
k=1
t
k
n

T
1
z
n
+

−1
k=2
t
k−1
n

T
k
z
n
−z
n


t
−2
n
≤ limsup
n→∞



1 −

−1
k=1
t
k
n

T
1
z
n
+



k=2
t
k−1
n
T
k
z
n
−z
n


+ t
−1
n


T
1
z
n
−T

z
n


t
−2

n
= lim
n→∞

t
n



1 −

−1
k=1
t
k
n

T
1
z
n
+


k=2
t
k−1
n
T
k

z
n
−z
n


t
−1
n
+ t
n


T
1
z
n
−T

z
n



= 0.
(4.2)
So, by the assumption of induction, we have that w is a common fixed point of T
1
,T
2

, ,
T
−1
. We note that
T
k
◦T

w =T

◦T
k
w =T

w (4.3)
for all k ∈ N with 1 ≤ k<. We assume that w is not a fixed point of T

.Put
ε =


T

w −w


3
> 0. (4.4)
Then there exists m ∈ N such that



z
m
−w


<ε,



1 −

−1
k=1
t
k
m

T
1
z
m
+


k=2
t
k−1
m
T

k
z
m
−z
m


t
−1
m
<ε. (4.5)
Since
3ε =


T

w −w


≤


T

w −z
m


+



z
m
−w


<


T

w −z
m


+ ε,
(4.6)
118 Convergence to common fixed point
we have
2ε<


T

w −z
m


. (4.7)

So, we obtain


T

w −z
m


≤





T

w −

1 −
−1

k=1
t
k
m

T
1
z

m
−


k=2
t
k−1
m
T
k
z
m





+






1 −
−1

k=1
t
k

m

T
1
z
m
+


k=2
t
k−1
m
T
k
z
m
−z
m





<

1 −
−1

k=1

t
k
m



T

w −T
1
z
m


+


k=2
t
k−1
m


T

w −T
k
z
m



+ t
−1
m
ε
=

1 −
−1

k=1
t
k
m



T
1
◦T

w −T
1
z
m


+
−1


k=2
t
k−1
m


T
k
◦T

w −T
k
z
m


+ t
−1
m


T

w −T

z
m


+ t

−1
m
ε
≤

1 −
−1

k=1
t
k
m



T

w −z
m


+
−1

k=2
t
k−1
m



T

w −z
m


+ t
−1
m


w −z
m


+ t
−1
m
ε
<

1 −t
−1
m



T

w −z

m


+2t
−1
m
ε
<

1 −t
−1
m



T

w −z
m


+ t
−1
m


T

w −z
m



=


T

w −z
m


.
(4.8)
This is a contradiction. Hence, w is a common fixed point of T
1
,T
2
, ,T

.Byinduction,
we obtain the desired result. 
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
} beasequencein
(0,1/2) converging to 0 and let {z
n
} beasequenceinC such that {z

n
} converges strongly to
some w ∈ C and

1 −
∞

k=1
t
k
n

T
1
z
n
+
∞

k=2
t
k−1
n
T
k
z
n
= z
n
(4.9)

for all n ∈N. Then w is a common fixed point of {T
n
: n ∈N}.
Tomonari Suzuki 119
Proof. Fix  ∈ N with  ≥2. We put M = 2sup{x : x ∈C}< ∞.Wehave
limsup
n→∞



1 −

−1
k=1
t
k
n

T
1
z
n
+


k=2
t
k−1
n
T

k
z
n
−z
n


t
−1
n
≤ limsup
n→∞




1 −

∞
k=1
t
k
n

T
1
z
n
+


∞
k=2
t
k−1
n
T
k
z
n
−z
n


t
−1
n
+

∞
k=+1
t
k−1
n


T
1
z
n
−T

k
z
n


t
−1
n

=
limsup
n→∞

∞
k=+1
t
k−1
n


T
1
z
n
−T
k
z
n



t
−1
n
≤ limsup
n→∞
∞

k=+1
t
k−
n
M
= lim
n→∞
t
n
1 −t
n
M
= 0.
(4.10)
So, by Lemma 4.1,wehavethatw is a common fixed point of T
1
,T
2
, ,T

.Since ∈N is
arbitrary, we obtain that w is a common fixed point of
{T

n
: n ∈ N}. This completes the
proof. 
Theorem 4.3. Let C beacompactconvexsubsetofaBanachspaceE.Let{T
n
: n ∈ N} be
an infinite family of commuting nonexpansive mappings on C.Fixλ ∈ (0,1).Let{α
n
} be a
sequence in [0,1/2] satisfying
liminf
n→∞
α
n
= 0, limsup
n→∞
α
n
> 0, lim
n→∞

α
n+1
−α
n

=
0. (4.11)
Define a sequence {x
n

} in C by x
1
∈ C and
x
n+1
= λ

1 −
n−1

k=1
α
k
n

T
1
x
n
+ λ

n

k=2
α
k−1
n
T
k
x

n

+(1−λ)x
n
(4.12)
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=1
F(T
n
) = ∅ by DeMarr’ s result in [5]. We define

0
k=1
α
k
1
=
0and

1
k=2
α
k−1

1
T
k
x
1
= 0.
Proof. Put
α
= lim sup
n→∞
α
n
> 0, M =sup
x∈C
x < ∞,
y
n
=

1 −
n−1

k=1
α
k
n

T
1
x

n
+
n

k=2
α
k−1
n
T
k
x
n
(4.13)
for n ∈ N. We note that
x
n+1
= λy
n
+(1−λ)x
n
(4.14)
120 Convergence to common fixed point
for all n ∈N.Since


y
n+1
− y
n



=






1 −
n

k=1
α
k
n+1

T
1
x
n+1
+
n+1

k=2
α
k−1
n+1
T
k
x

n+1
−

1 −
n−1

k=1
α
k
n

T
1
x
n
−
n

k=2
α
k−1
n
T
k
x
n






≤

1 −
n

k=1
α
k
n+1



T
1
x
n+1
−T
1
x
n


+
n+1

k=2
α
k−1
n+1



T
k
x
n+1
−T
k
x
n


+





n

k=1
α
k
n+1
−
n

k=1
α
k

n







−T
1
x
n


+
n+1

k=2


α
k−1
n+1
−α
k−1
n





T
k
x
n


+


α
n
n




−T
1
x
n


+


α
n
n





T
n+1
x
n


≤

1 −
n

k=1
α
k
n+1



x
n+1
−x
n


+
n+1

k=2

α
k−1
n+1


x
n+1
−x
n


+2M
n

k=1


α
k
n+1
−α
k
n


+2M


α
n

n


≤


x
n+1
−x
n


+8M


α
n+1
−α
n


+2M
1
2
n
(4.15)
for n ∈ N,wehave
limsup
n→∞




y
n+1
− y
n


−


x
n+1
−x
n



≤ 0. (4.16)
So, by Lemma 2.2,wehavelim
n
x
n
− y
n
=0. Fix t ∈R with 0 <t<α.ThenbyLemma
2.3, there exists a subsequence {α
n
k
} of {α

n
} converging to t.SinceC is compact, there
exists a subsequence {x
n
k
j
} of {x
n
k
} converging strongly to some point z
t
∈ C.Wehave






1 −
∞

=1
t


T
1
z
t
+

∞

=2
t
−1
T

z
t
−z
t





≤






1 −
∞

=1
t



T
1
z
t
+
∞

=2
t
−1
T

z
t
− y
n





+


y
n
−x
n



+


x
n
−z
t


=






1 −
∞

=1
t


T
1
z
t
+
∞


=2
t
−1
T

z
t
−

1 −
n−1

=1
α

n

T
1
x
n
−
n

=2
α
−1
n
T


x
n





+


y
n
−x
n


+


x
n
−z
t


≤

1 −
∞


=1
t




T
1
z
t
−T
1
x
n


+
∞

=2
t
−1


T

z
t
−T


x
n


+





n−1

=1
t

−
n−1

=1
α

n







−T

1
x
n


+
n

=2


t
−1
−α
−1
n




T

x
n


+
∞

=n

t



−T
1
x
n


+
∞

=n+1
t
−1


T

x
n


+


y
n
−x

n


+


x
n
−z
t


Tomonari Suzuki 121
≤

1 −
∞

=1
t




z
t
−x
n



+
∞

=2
t
−1


z
t
−x
n


+





n−1

=1
t

−
n−1

=1
α


n





M
+
n

=2


t
−1
−α
−1
n


M +
∞

=n
t

M +
∞


=n+1
t
−1
M +


y
n
−x
n


+


x
n
−z
t


≤ 2


z
t
−x
n



+8


t −α
n


M +
2t
n
1 −t
M +


y
n
−x
n


(4.17)
for n ∈ N, and hence






1 −
∞


=1
t


T
1
z
t
+
∞

=2
t
−1
T

z
t
−z
t





≤ lim sup
j→∞

2



z
t
−x
n
k
j


+8


t −α
n
k
j


M +
2t
n
k
j
1 −t
M +


y
n

k
j
−x
n
k
j



=0.
(4.18)
Therefore we have

1 −
∞

=1
t


T
1
z
t
+
∞

=2
t
−1

T

z
t
= z
t
(4.19)
for all t
∈ R with 0 <t<α.SinceC is compact, there exists a real sequence {t
n
} in (0,α)
such that lim
n
t
n
= 0and{z
t
n
} converges strongly to some point w ∈ C.ByLemma 4.2,
we obtain that such w is a common fixed point of {T
n
: n ∈N}. We note that w is a cluster
point of {x
n
} becausesoarez
t
n
for all n ∈ N. Hence, liminf
n
x

n
−w=0. We also have
that {x
n
−w}is nonincreasing by Lemma 2.5.Thus,lim
n
x
n
−w=0. This completes
the proof. 
Similarly, we can prove the following.
Theorem 4.5. Let C beacompactconvexsubsetofaBanachspaceE.Let{T
n
: n ∈ N} be
an infinite family of commuting nonexpansive mappings on C.Fixλ ∈ (0,1).Let{α
n
} be a
sequence in [0,1/2] satisfying
liminf
n→∞
α
n
= 0, limsup
n→∞
α
n
> 0, lim
n→∞

α

n+1
−α
n

=
0. (4.20)
Define a sequence {x
n
} in C by x
1
∈ C and
x
n+1
= λ

1 −
∞

k=1
α
k
n

T
1
x
n
+ λ

∞


k=2
α
k−1
n
T
k
x
n

+(1−λ)x
n
(4.21)
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.
122 Convergence to common fixed point
Theorem 4.6. Let C beacompactconvexsubsetofaBanachspaceE.LetS and T be non-
expansive mappings on C with ST = TS.Let{α
n
} beasequencein[0,1] satisfy ing
liminf
n→∞
α
n
= 0, limsup
n→∞

α
n
> 0, lim
n→∞

α
n+1
−α
n

=
0. (4.22)
Define a sequence {x
n
} in C by x
1
∈ C and
x
n+1
=
1 −α
n
2
Sx
n
+
α
n
2
Tx

n
+
1
2
x
n
(4.23)
for n ∈ N. Then {x
n
} converges strongly to a common fixed point of S and T.
Remark 4.7. This theorem is simpler than Theorem 1.2.
Theorem 4.8. Let C beacompactconvexsubsetofaBanachspaceE.Let ∈N with  ≥2
and let {T
1
,T
2
, ,T

} be a finite family of commuting nonexpansive mappings on C.Let
{α
n
} beasequencein[0,1/2] satisfying
liminf
n→∞
α
n
= 0, limsup
n→∞
α
n

> 0, lim
n→∞

α
n+1
−α
n

= 0. (4.24)
Define a sequence {x
n
} in C by x
1
∈ C and
x
n+1
=
1
2

1 −
−1

k=1
α
k
n

T
1

x
n
+
1
2



k=2
α
k−1
n
T
k
x
n

+
1
2
x
n
(4.25)
for n ∈ N. Then {x
n
} converges strongly to a common fixed point of {T
1
,T
2
, ,T


}.
Remark 4.9. This theorem is simpler than Theorem 1.1.
Acknowledgment
The author wishes to express his thanks to Professor W. A. Kirk for giving the historical
comment.
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eor
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eme de type ergodique pour les cont ractions non lin
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Tomonari Suzuki: Department of Mathematics, Kyushu Institute of Technology, 1-1 Sensuicho,
Tobataku, Kitakyushu 804-8550, Japan
E-mail address: