404
2
,
4 2
H
H
H
§ ·
¨ ¸
© ¹
which means that
^ `
n
x
is a Cauchy sequence in
.
X
But
X
is a Banach space, so there exists
x X
such that
.
n
x x
o
Since
C
is closed and
^ `
n
x
is a sequence in
,
C
we get
.
x C
Since
1
2
F T F T
I
z
in
C
and
,
n
x x
o
it follows by Lemma 1.7 and (2.8) that
1
2
1
2
0 lim ,
,
.
n
n
d x F T F T d x F T F T
of
Thus
1
2
,
0.
d x F T F T
Since
1
2
F T F T
is closed, it follows by Lemma 1.5 that
1
2
.
x F T F T
Therefore
^ `
n
x
converges to a common fixed point of
1
T
and
2
,
T
as desired.
Corollary 2.2.
Let
, ,
i
X C T
1, 2
i
and the iterative sequence
^ `
n
x
be as in Theorem 2.1. Suppose that
( )
i
The mapping
i
T
1, 2
i
is asymptotically regular in
;
n
x
( )
ii
liminf
0
n
i n
n
x T x
of
implies that
1
2
liminf
,
0.
n
n
d x F T F T
of
Then the sequence
^ `
n
x
converges strongly to a common fixed point of
1
T
and
2
T
Corollary 2.3.
Let
, ,
i
X C T
1, 2
i
and the iterative sequence
^ `
n
x
be as in Theorem 2.1. Assume further
that the mapping
i
T
1, 2
i
is asymptotically regular in
n
x
and there exists an increasing function
:
f
R
+
o
R
+
with
0
f r
!
for all
0
r
!
such that for
1, 2,
i
we have
1
2
,
,
n
i n
n
x T x f d x F T F T
t
for all
1
n
t
. Then the sequence
^ `
n
x
converges to a common fixed point of
1
T
and
2
.
T
If
1
2
T T T
we have the following result.
Corollary 2.4.
Let
X
be a real Banach space and let
C
be a nonempty closed convex subset of
.
X
Let
:
T C C
o
be a quasi-nonexpansive mapping such that the fixed point set
F T
I
z
in
.
C
Let
^ `
n
D
and
^ `
n
E
be sequence in
[0,1)
and
^ `
n
u
and
^ `
n
v
be sequences in
.
C
Suppose there exists an element
1
x C
for
which the iterative sequences
^ `
n
x
and
^ `
n
y
defined in
(2.1)
are in
.
C
Assume that
( )
i
n
n
n
u u u
c
c
for
1,
n
t
1
,
n
n
v
f
f
¦
1
n
n
u
f
c
f
¦
and
1 ;
n
n
u o
D
cc
( )
ii
1
1
.
n
n
D
f
f
¦
Then the iterative
^ `
n
x
converges strongly to a fixed point of
T
if and only if
liminf
,
0.
n
n
d x F T
of
Corollary 2.5.
Let
, ,
X C T
and the iterative sequence
^ `
n
x
be as in Corollary 2.4. Suppose that
( )
i
The mapping
T
is asymptotically regular in
n
x
and
( )
ii
liminf
0
n
n
n
x Tx
of
implies that
liminf
,
0.
n
n
d x F T
of
Then the sequence
^ `
n
x
converges to a common fixed point of
.
T
The following is the theorem that tells some sufficient conditions.
Corollary 2.6.
Let
, ,
X C T
and the iterative sequence
^ `
n
x
be as in Corollary 2.4. Assume further that the
mapping
T
is asymptotically regular in
^ `
n
x
and there exists an increasing function
:
f
R
+
o
R
+
with
0
f r
!
for all
0
r
!
such that
