JOllrnal
of Mathematical
Rcsearch
& Exposition
Vo
l.
l5,
No
.4,
519-523
,
Nov.
1995
On
Subsequential
Limit
Points
of
A
Sequence
of
Iteraters
*
Liu
Zeqing
(Dept of Math.
,
Li
a.Ol山
19
Nonnal Ulliv., Daliall
116029
)
Abstract
In
this papcr
wc
show thrcc rcslllts conccrning thc strllcturc of the sct of
subscqucntiallimit points of
a seqllcnce of iterates of a
cOlltillllO
Il
S sclf-map of a mctric
space. Our results ullify alld extelld
correspOll
正
ling
reslllts of Diaz alld Metcalf, Maiti
a.
lld B
a.
bll, and Park.
Keywords
sllbseqllel1 tial lirnit point,
CO
l1
tim
lO
ll
点
map
，
mctric space, ,
COllllcctcd
sc
t.
Classification
AMS(1991) 54H25/CCL
189.1
1.
Introduction
Diaz
and
Metca
u1
1
]
obtained
the
structure
of
the
set
of
subsequential
limit
points
of
a sequence
of
iterates
of
a
continuous
self-map
of
a
metric
space
(X
,
d).
Later
,
Maiti
and
Babu[2]
estab
lÍ
shed
a
similar
result
on
maps
which
are
contractive
over two consecutive
elements
of
an
orbit.
Park[3]
extended
a few
results
of
[1
,
2].
Replacing
the
distance
function
by
a
continuous
function
on
X x X ,
Maiti
and
Babu
gave a
number
of
variations
and
generalizations
of
the
results
in
[1].
Motivated
by
the
results
of
P
a:町
d
记』
Babu[
间
4]
，
we
establ
且
ish
three
resul
扰
ts
on
the
structure
of
the
set
of
subsequen
刊创
tiallim
咀
it
points
of
a
s
肥
eq
伊
uenc
臼
e
of
i
让
te
盯
ra
川
te
臼
s
of
a
c
∞
ontinuous
sel
匹
f-ma
叩
p
of
a
metric
space.
Our
results
unify
and
extend
corresponding
restùts
of
[1
-4].
Three
illustrative
examples
are
given
in
support
of
our
results.
Let
f
be
a self-map
of
a
metric
space
(X
,
d)
and
F(J)
denote
the
set
offixed
points
of
f.
The
orbit
of
z
已
X
generated
by
f is
denoted
by
O(z
,1)
and
its
closure
by
O(z
,
I)
.L(z)
denotes
the
set
of
subsequential
limit
points
of
the
sequence
{1
H
Z
}
旱。.
Let
Ro
be
the
subspace
[0
，∞)
of
the
realline
with
usual
topology
,
and
h a
function
from
X x X
into
Ro.
For z
ε
X
and
A ,B c X , define
d(x
，
A)
二
inf{
d( z ,
Y)IY
ξ
A}
，
d(A
，
B)
二
inf{
d(x
,
y)lx
ε
A
and
y
ε
B}
and
h(z
,
A)
二
inf{
h(
x ,
y)ly
ε
A}.
2.
Lemmas
In
order
to
prove
our
main
theorems
, we first show
the
following
Lemmas.
Lenuna
1
Let
f
be
a sel
f-
map
of
a
metric
space
(X
, d).
Assume
tlwt
tllere
exists
z E X
*Rcccived
Dcc.25
,
1993.
Projcct S
lI
pprotcd
by
thc Natnr
ll.
l
Scicncc
FOllndation
of
Liaoning Provincc.
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