407
Introduction
Let
X
be a real Banach space,
C
a closed subset of
X
, and
:
T C X
o
such that
T
has a nonempty
set of fixed points
F T C
.
T
is called
quasi-nonexpansive
if
( )
T x p x p
d
for all
x
in
C
and
p
in
F T
. It is introduced by Tricomi (1916) for real functions and further studied by Diaz
and Metcalf (1969) .
In 1967, Diaz and Metcalf provided definition of quasi-nonexpansive self-mapping as follow:
Definition.
A self-mapping
T
of a subset
C
of a normed linear space is said to be
quasi-nonexpansive
provided
T
has at least one fixed point in
C
, and if
p C
is any fixed point of
T
then
( )
T x p x p
d
holds for all
x C
.
In 1972, Petryshyn and Williamson had presented two new theorems which provided necessary and
sufficient conditions for the convergence of the successive approximation method and of the convex combination
iteration method for quasi-nonexpansive mapping defined on suitable subset of Banach space and with nonempty
set of fixed points. They also indicated briefly how these theorems were used to deduce a number of known, as
well as some new, convergence results for various special classes of mappings of nonexpansive, P-compact, and
1-set-contractive type which recently have been extensively studied by a number of authors.
In this study, we create new iterative process with errors for quasi-nonexpansive mappings in Banach
space and prove some convergence theorems.
Definition 1.1
Let
X
be a real Banach space,
C
a closed subset of
X
, and
:
T C C
o
such that
T
has a
nonempty set of fixed points
F T C
.
T
is called
quasi-nonexpansive
if
( )
T x p x p
d
for all
x
in
C
and
p
in
F T
.
Definition 1.2
Let
X
be a real Banach space. Let
C
be a nonempty subset of
X
. A mapping
:
T C C
o
is said
to be
asymptotically regular
in
n
x
, if
liminf
0
n
n
n
x Tx
of
.
Lemma 1.3
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
for all
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)
lim 0
n
n
a
of
if
^ `
n
a
has a subsequence converging to zero.
Lemma 1.4
Let
C
be a nonempty closed subset of a Banach space
X
and
:
T C C
o
be a quasi-nonexpansive
mapping with the fixed point set
F T
z
. Then
F T
is a closed subset in
C
.
