Let
(
X
,
τ
)
(X,\tau )
be a metrizable topological space,
P
(
τ
)
\mathcal {P}(\tau )
be the family of all metrics on X whose metric topologies are
τ
\tau
. Assume that the semigroup F of maps from X into itself, with composition as its semigroup operation, is equicontinuous under some
d
∈
P
(
τ
)
d \in \mathcal {P}(\tau )
; then we have the following results: I. There exists
d
′
∈
P
(
τ
)
d’ \in \mathcal {P}(\tau )
such that f is nonexpansive under
d
′
d’
for each
f
∈
F
f \in F
. II. If F is countable, commutative, and for each
f
∈
F
f \in F
, there is
x
f
∈
X
{x_f} \in X
such that the sequence
(
f
n
(
x
)
)
n
=
1
∞
({f^n}(x))_{n = 1}^\infty
converges to
x
f
,
∀
x
∈
X
{x_f},\forall x \in X
, then there exists
d
∈
P
(
τ
)
d \in \mathcal {P}(\tau )
such that f is contractive under
d
d
for each
f
∈
F
f \in F
. III. If there is
p
∈
X
p \in X
such that (1)
lim
n
→
∞
f
n
(
x
)
=
p
,
∀
x
∈
X
{\lim _{n \to \infty }}{f^n}(x) = p,\forall x \in X
and
∀
f
∈
F
\forall f \in F
, (2) there is a neighbourhood B of p such that
lim
m
→
∞
f
n
1
f
n
2
⋯
f
n
m
(
B
)
=
{
p
}
{\lim _{m \to \infty }}{f_{{n_1}}}{f_{{n_2}}} \cdots {f_{{n_m}}}(B) = \{ p\}
for any choice of
f
n
i
∈
F
,
i
=
1
,
…
,
m
{f_{{n_i}}} \in F,i = 1, \ldots ,m
, and the limit depends on m only, then for each
λ
\lambda
with
0
>
λ
>
1
0 > \lambda > 1
, there exists
d
′
∈
P
(
τ
)
d’ \in \mathcal {P}(\tau )
such that each f in F is a Banach contraction under
d
′
d’
with Lipschitz constant
λ
\lambda
.