  We deal with positive solutions for the Neumann boundary value problem
associated with the scalar second order ODE $$ u" + q(t)g(u) = 0, \quad t \in
[0, T], $$ where $g: [0, +\infty[\, \to \mathbb{R}$ is positive on $\,]0,
+\infty[\,$ and $q(t)$ is an indefinite weight. Complementary to previous
investigations in the case $\int_0^T q(t) < 0$, we provide existence results
for a suitable class of weights having (small) positive mean, when $g'(x) < 0$
at infinity. Our proof relies on a shooting argument for a suitable equivalent
planar system of the type $$ x' = y, \qquad y' = h(x)y^2 + q(t), $$ with $h(x)$
a continuous function defined on the whole real line.
