In this paper, we prove the following general result: Let
X
X
be a real Hilbert space and
J
:
X
→
R
J:X\to \textbf {R}
a
C
1
C^1
functional, with locally Lipschitzian derivative. Then, for each
x
0
∈
X
x_0\in X
with
J
′
(
x
0
)
≠
0
J’(x_0)\neq 0
, there exists
δ
>
0
\delta >0
such that, for every
r
∈
]
0
,
δ
[
r\in ]0,\delta [
, the restriction of
J
J
to the sphere
{
x
∈
X
:
‖
x
−
x
0
‖
=
r
}
\{x\in X : \|x-x_0\|=r\}
has a unique global minimum toward which every minimizing sequence strongly converges.