In this paper we present some conditions which are sufficient for a mapping to have periodic points. Theorem. If
f
f
is a mapping of the space
X
X
into
X
X
and there exist subcontinua
H
H
and
K
K
of
X
X
such that (1) every subcontinuum of
K
K
has the fixed point property, (2)
f
[
K
]
f[K]
and every subcontinuum of
f
[
H
]
f[H]
are in class
W
W
, (3)
f
[
K
]
f[K]
contains
H
H
, (4)
f
[
H
]
f[H]
contains
H
∪
K
H \cup K
, and (5) if
n
n
is a positive integer such that
(
f
|
H
)
−
n
(
K
)
{(f|H)^{ - n}}(K)
intersects
K
K
, then
n
=
2
n = 2
, then
K
K
contains periodic points of
f
f
of every period greater than 1. Also included is a fixed point lemma: Lemma. Suppose
f
f
is a mapping of the space
X
X
into
X
X
and
K
K
is a subcontinuum of
X
X
such that
f
[
K
]
f[K]
contains
K
K
. If (1) every subcontinuum of
K
K
has the fixed point property, and (2) every subcontinuum of
f
[
K
]
f[K]
is in class
W
W
, then there is a point
x
x
of
K
K
such that
f
(
x
)
=
x
f(x) = x
. Further we show that: If
f
f
is a mapping of
[
0
,
1
]
[0,1]
into
[
0
,
1
]
[0,1]
and
f
f
has a periodic point which is not a power of 2, then
lim
{
[
0
,
1
]
,
f
}
\lim \{ [0,1],f\}
contains an indecomposable continuum. Moreover, for each positive integer
i
i
, there is a mapping of
[
0
,
1
]
[0,1]
into
[
0
,
1
]
[0,1]
with a periodic point of period
2
i
{2^i}
and having a hereditarily decomposable inverse limit.