  Let $\Omega:=\left( a,b\right) \subset\mathbb{R}$, $m\in L^{1}\left(
\Omega\right) $ and $\lambda>0$ be a real parameter. Let $\mathcal{L}$ be the
differential operator given by $\mathcal{L}u:=-\phi\left( u^{\prime}\right)
^{\prime}+r\left( x\right) \phi\left( u\right) $, where $\phi
:\mathbb{R\rightarrow R}$ is an odd increasing homeomorphism and $0\leq r\in
L^{1}\left( \Omega\right) $. We study the existence of positive solutions for
problems of the form $\mathcal{L}u=\lambda m\left( x\right) f\left( u\right)$
in $\Omega,$ $u=0$ on $\partial\Omega$, where $f:\left[ 0,\infty\right)
\rightarrow\left[ 0,\infty\right) $ is a continuos function which is, roughly
speaking, sublinear with respect to $\phi$. Our approach combines the sub and
supersolution method with some estimates on related nonlinear problems. We
point out that our results are new even in the cases $r\equiv0$ and/or
$m\geq0$.
