400
We give some sufficient and necessary conditions so that the sequence
^ `
n
x
defined above converges to some
common fixed point of
1
T
and
2
T
Keywords
: Quasi-nonexpansive mapping, Banach space, Common fixed point
1. Introduction
Let
X
be a real Banach space,
C
a closed subset of
X
and
T
a mapping of
C
into
C
such that
T
has a nonempty set of fixed point
F T
in
C
and
,
Tx p x p
d
for all
x C
,
.
p F T
We shall refer to
T
satisfying the above conditions as
quasi-nonexpansive
. It is
introduced by Tricomi (1916) for real functions and further studied by Diaz and Metcalf (1969) and Dotson (1970)
for mapping in Banach spaces.
In 1972, Petryshyn and Williamson 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 subsets of the Banach spaces and with
nonempty sets 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 paper, we construct a new iterative procedure to approximate common fixed points of quasi-
nonexpansive mappings and prove some convergence theorems.
Definition 1.1. (Fixed Point).
A
fixed point
of a mapping
:
T X X
o
of a set
X
into itself is an
x X
which
is mapped onto itself (is “kept fixed” by
T
), that is,
Tx x
, the image
Tx
coincides with
x
. The set of all fixed
points of
T
is denoted by
F T
, that is,
^
`
|
.
F T x X x Tx
Definition 1.2. (Quasi-nonexpansive mapping).
Let
X
be a real Banach space,
C
a closed subset of
X
and
T
a continuous mapping of
C
into
C
such that
T
has a nonempty set of fixed points
F T
in
.
C
T
is called
a
quasi-nonexpansive mapping
of
C
into
C
if
Tx p x p
d
,
for all
x
in
C
and
p
in
F T
.
Definition 1.3. ( Little-o notation).
Given two functions
f
and
,
g
the statement
f o g
is equivalent to the
statement
( )
lim 0.
( )
x
f x
g x
of
This statement is voiced
f
is
little-o
of
g
or simply
f
is
little-o
.
g
