Let
X
X
be a real Banach space, let
K
K
be a closed convex subset of
X
X
, and let
T
T
, from
K
K
into
X
X
, be a pseudo-contractive mapping (i.e.
(
λ
−
1
)
(\lambda -1)
‖
u
−
v
‖
≤
‖
(
λ
I
−
T
)
(
u
)
−
(
λ
I
−
T
)
(
v
)
‖
\|u-v\|\le \|(\lambda I-T)(u)-(\lambda I-T)(v)\|
for all
u
,
v
∈
K
u,v\in K
and
λ
>
1
)
\lambda >1)
. Suppose the space
X
X
has a uniformly Gâteaux differentiable norm, such that every closed bounded convex subset of
K
K
enjoys the Fixed Point Property for nonexpansive self-mappings. Then the path
t
→
x
t
∈
K
t\to x_t\in K
,
t
∈
[
0
,
1
)
t\in [0,1)
, defined by the equation
x
t
=
t
T
x
t
+
(
1
−
t
)
x
0
x_t=tTx_t+(1-t)x_0
is continuous and strongly converges to a fixed point of
T
T
as
t
→
1
−
t\to 1^-
, provided that
T
T
satisfies the weakly inward condition.