For a (metric) continuum
Z
Z
, let
2
Z
{2^Z}
(resp.,
C
(
Z
)
C(Z)
) denote the space of all nonempty compacta (resp., continua) in
Z
Z
with the Hausdorff metric. We prove: (1) If
f
f
is a monotone map of a continuum
X
X
onto a Peano continuum
Y
Y
, then, for any maps
g
:
2
X
→
2
Y
g:{2^X} \to {2^Y}
and
h
:
C
(
X
)
→
C
(
Y
)
h:C(X) \to C(Y)
, there is
A
∈
2
X
A \in {2^X}
and
B
∈
C
(
X
)
B \in C(X)
such that
f
(
A
)
=
g
(
A
)
f(A) = g(A)
and
f
(
B
)
=
h
(
B
)
f(B) = h(B)
. We use (1) to prove: (2) If
X
X
is an inverse limit of dendrites with quasi-monotone bonding maps, then
2
X
{2^X}
and
C
(
X
)
C(X)
have the fixed point property. Thus, we have a proof that for certain indecomposable continua
X
,
2
X
X,{2^X}
has the fixed point property.