401
Definition 1.4. (Asymptotically regular).
Let
X
be a real Banach space,
C
a closed subset of
,
X
and
T
a
quasi-nonexpansive mapping of
C
into
C
with nonempty fixed point set
.
F T
T
is said to be
asymptotically
regular
in
n
x
if
liminf
0.
n
n
n
x Tx
of
Lamma 1.5.
(Kreyszig, 1978)
If
C
is a nonempty closed subset of a normed space
,
X
x X
and
,
0,
d x C
then
.
x C
Lamma 1.6.
(Ayaragarnchanakul, 2008)
Let
X
be a real Banach space,
C
a closed subset of
,
X
T
is a quasi-
nonexpansive mapping of
C
into
C
and
( )
F T
I
z
in
.
C
Then
F T
is a closed subset of
.
C
Note that for a quasi-nonexpansive mapping
:
i
T C C
o
1, 2
i
with the common fixed point set
1
2
( )
( ) ,
F T F T
I
z
this shows that
1
F T
I
z
and
2
.
F T
I
z
From above we get
1
F T
and
2
F T
is
closed. Thus
1
2
F T F T
is closed.
Lamma 1.7.
Let
C
be a nonempty closed subset of Banach space
X
and
:
T C C
o
be a quasi-nonexpansive
mapping with the fixed point set
.
F T
I
z
If
,
n
x x
o
then
lim ,
,
.
n
n
d x F T d x F T
of
Proof
.
Let
.
n
x x
o
We will prove that
lim ,
,
.
n
n
d x F T d x F T
of
By the triangle inequality, for each
n
N
, we obtain
,
,
, .
n
n
d x F T d x F T d x x
d
From this, for each
n
N
, we get
,
,
, .
n
n
d x F T d x F T d x x
d
(1.1)
Similarly, for each
n
N
, we can obtain that
,
,
,
.
n
n
d x x d x F T d x F T
d
(1.2)
From (1.1) and (1.2) , we get
,
,
, .
n
n
d x F T d x F T d x x
d
(1.3)
Since
,
n
x x
o
lim ,
0.
n
n
d x x
of
From this, (1.3) and the sandwich theorem we get
lim ,
,
0.
n
n
d x F T d x F T
of
Hence
lim ,
,
,
n
n
d x F T d x F T
of
as desired.
Note that, for a quasi-nonexpansive mapping
:
i
T C C
o
1, 2
i
with the common fixed point set
1
2
( )
( )
F T F T
I
z
and
.
n
x x
o
From above we get
1
2
1
2
lim ,
,
.
n
n
d x F T F T d x F T F T
of
Lamma 1.8.
(Ayaragarnchanakul, 2008)
Let
^ `
,
n
a
^ `
n
b
and
^ `
n
G
be sequences of nonnegative real numbers
satisfying the inequality
1
1
,
n
n n
n
a
a b
G
d
n
N
.
If
1
n
n
G
f
f
¦
and
1
,
n
n
b
f
f
¦
then
(1)
lim
n
n
a
of
f
exists.
(2) If
^ `
n
a
has a subsequence converging to zero, then
lim 0.
n
n
a
of
