Let
\(\Omega := ( a,b ) \subset \mathbb {R}\),
\(m\in L^{1} ( \Omega ) \) and
\(\phi :\mathbb {R\rightarrow R}\) be an odd increasing homeomorphism. We consider the existence of
positive solutions for problems of the form
$$\begin{aligned} \left\{ \begin{array} c]{ll} -\phi ( u^{\prime } ) ^{\prime }=m ( x ) f ( u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on } \partial \Omega , \end{array} \right. \end{aligned}$$
where
\(f: 0,\infty ) \rightarrow 0,\infty ) \) is a continuous function which is, roughly speaking, superlinear with respect to
\(\phi \). Our approach combines the Guo-Krasnoselski? fixed-point theorem with some estimates on related nonlinear problems. We mention that our results are new even in the case
\(m\ge 0\).