402
2. Main Results
Let
X
be a real Banach space, and let
C
be a nonempty closed convex subset of
.
X
For
1, 2
i
let
:
i
T C C
o
be a quasi-nonexpansive mapping such that
1
2
( )
( )
F T F T
I
z
in
.
C
Let
^ `
n
D
and
^ `
n
E
be
sequences in
[0,1)
and
^ `
n
u
and
^ `
n
v
be sequences in
.
C
We are interested in sequences in the following
process. For
1
x C
and
1,
n
t
define the sequences
^ `
n
x
and
^ `
n
y
by
2
1
n
n n
n n
n
y
T x
x v
E
E
1
1
1
n
n n
n n
n
x
T y
y u
D
D
(2.1)
We have the following theorems.
Theorem 2.1.
Let
, ,
i
X C T
1, 2 ,
i
^ `
,
n
D
^ `
,
n
E
^ `
n
x
and
^ `
n
y
be defined as above. Let
1
x C
be such
that the iterative sequences
^ `
n
x
and
^ `
n
y
defined by
(2.1).
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 sequence
^ `
n
x
converges strongly to a common fixed point
of
1
T
and
2
T
if and only if
1
2
liminf
,
0.
n
n
d x F T F T
of
(2.2)
Proof.
For the necessary, we assume that
^ `
n
x
converges strongly to a common fixed point of
1
T
and
2
T
, i.e.,
there exists
1
2
p F T F T
such that
lim
0.
n
n
x p
of
From this, we have
liminf
0.
n
n
x p
of
We see that
1
2
1
2
,
inf
,
n
n
n
q F T F T
d x F T F T
x q x p
d
for all
.
n
Taking limit infimum as
n
o f
and using the sandwich theorem, we obtain that
1
2
liminf
,
0,
n
n
d x F T F T
of
as desired.
For the sufficiency, we let
1
2
.
p F T F T
Since
:
i
T C C
o
is a quasi-nonexpansive mapping
for
1, 2
i
and by the triangle inequality, we get
n
y p
2
1
n n
n n
n
T x
x v p
E
E
2
2
1
n n
n n
n
n
n
T x
x v p T p p
E
E
E
E
2
1
n
n
n
n
n
T x p
x p v
E
E
2
1
n
n
n n
n
T x p
x p v
E
E
d
,
n
n
x p v
d
(2.3)
for all
.
n
By assumption, we have
1
n
n
u o
D
cc
so by definition of the little-o notation, we get
lim 0.
1
n
n
n
u
D
of
cc
