Positive solutions for nonlinear problems involving the one-dimensional {\phi}-Laplacian

Let $\mathrm{\Omega }:=(a,b)\subset \mathbb{R}$, $m\in L^{1}\left( \Omega\right) $and$\lambda>0$bearealparameter\mathrm{.}Let$$\mathcal{L}$$ be the differential operator given by $$\mathcal{L}$u:=-\phi\left( u^{\prime}\right) ^{\prime}+r\left( x\right) \phi\left( u\right) $,where$\phi :$\mathbb{R}\to \mathbb{R}$$isanoddincreasinghomeomorphismand$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 $\mathrm{\Omega },$ $u=0$ on $\mathrm{\partial }\mathrm{\Omega }$, where $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ is a continuos function which is, roughly speaking, sublinear with respect to $\varphi$. 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 $r\equiv 0$ and/or $m\ge 0$.