410
0
0
.
4
n
x p
H
2.7
Combining
2.4
,
2.6
and
2.7
, for any positive integer
m
, we have
0
0
0
0
0
0
0
0
0
0
0
1
1
2
2
4 2
,
n m n
n m
n
n m
n
i
n
i n
n m
n
i
i n
x
x
x
p x p
x p
d x p
x p
d
H
H
H
d
d
§ ·
¨ ¸
© ¹
¦
¦
which implies that
^ `
n
x
is a Cauchy sequences in
X
. But
X
is a Banach space, so there must exist
p X
such
that
n
x p
o
. Since
C
is closed and
^ `
n
x
is a sequence in
C
converging to
p
, we have that
p C
. Also, by
Lemma 1.4, we have that
1
F T
and
2
F T
are closed. Thus
1
2
F T F T
is closed. From the continuity of
1
2
,
d x F T F T
with
n
x p
o
as
n
of
, we have
1
2
1
2
,
,
.
n
d x F T F T d p F T F T
o
From
2.5
, we have
1
2
,
0
n
d x F T F T
o
. So that
1
2
,
0
d p F T F T
.
Since
1
2
F T F T
is closed,
1
2
p F T F T
. Therefore
^ `
n
x
converges strongly to a common fixed
point of
1
T
and
2
T
, as desired.
Theorem 2.2
Let
X
,
C
,
1, 2
i
T i
and the iterative sequence
^ `
n
x
be as in Theorem 2.1 Suppose that
conditions (i) and (ii) in Theorem 2.1 hold and
(1) the mapping
1, 2
i
T i
is asymptotically regular in
n
x
and
(2)
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
.
Theorem 2.3
Let
X
,
C
,
1, 2
i
T i
and the iterative sequence
^ `
n
x
be as in Theorem 2.1 Suppose that
conditions (i) and (ii) in Theorem 2.1 hold . Assume further that the mapping
1, 2
i
T i
is asymptotically
regular in
n
x
, and there exists an increasing function
:
f R R
o
with
0
f r
!
for all
0
r
!
and 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.
Theorem 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 with nonempty fixed point set
F T
. Let
^ `
n
a
,
^ `
n
b
,
^ `
n
D
and
^ `
n
E
be sequences in
> @
0,1
and
^ `
n
u
and
^ `
n
v
be sequences in
C
. Assume that
(i)
^
`
n
n
a
E
and
^
`
n
n
b
D
are sequences in
> @
0,1
and
