411
1
n
n
a
f
f
¦
and
1
n
n
b
f
f
¦
;
(ii)
^ `
n
u
and
^ `
n
v
are bounded.
Then the iterative sequence
^ `
n
x
defined in
2.2
converges strongly to a fixed point of
T
if and only if
liminf
,
0
n
n
d x F T
of
.
Theorem 2.5
Let
X
,
C
,
T
and the iterative sequence
^ `
n
x
be as in Theorem 2.4 Suppose that conditions (i)
and (ii) in Theorem 2.4 hold. Assume further that
(1) the mapping
T
is asymptotically regular in
n
x
and
(2)
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 strongly to a fixed point of
T
.
Theorem 2.6
Let
X
,
C
,
T
and the iterative sequence
^ `
n
x
be as in Theorem 2.4 Suppose that conditions (i)
and (ii) in Theorem 2.4 hold . Assume further that mapping
T
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
,
n
n
n
x Tx f d x F T
t
for all
1
n
t
.
Then the sequence
^ `
n
x
converges to a fixed point of
T
.
References
Diaz, J.B. and Metcalf, F.T. (1969).
On the set of subsequential limit points of successive approximations
,
Trans. Amer. Math. Soc. 135, 459-485.
Dotson, W.G. (1972).
Fixed point of quasi-nonexpansive mappings
, J. Amer. Math. Soc. 167.
Petryshyn, W.V. and Williamson, T.E. (1972).
A necessary and sufficient condition for the convergence of a
sequence of iterates for quasi-nonexpansive mappings,
J. Amer. Math. Soc. 78, 1027-1031.
Tricomi, F. (1916).
Un teorema sulla convergenza delle successioni formate delle successivo iterate di una
funzione di una variable reale
, Giorn. Mat. Battaglini. 54, 1-9.
