Published online by Cambridge University Press: 06 August 2013
In this paper, we prove that if $X$ is an infinite-dimensional real Hilbert space and
$J: X\rightarrow \mathbb{R} $ is a sequentially weakly lower semicontinuous
${C}^{1} $ functional whose Gâteaux derivative is non-expansive, then there exists a closed ball
$B$ in
$X$ such that
$(\mathrm{id} + {J}^{\prime } )(B)$ intersects every convex and dense subset of
$X$.