Positive solutions for concave-convex type problems for the one-dimensional $\phi$-Laplacian

Let $\mathrm{\Omega }=(a,b)\subset \mathbb{R}$, $0\le m,n\in L^1(\mathrm{\Omega })$, $\lambda ,\mu >0$ be real parameters, and $\phi :\mathbb{R}\to \mathbb{R}$ be an odd increasing homeomorphism. In this paper we consider the existence of positive solutions for problems of the form $$\{\begin{array}{ll}-\phi {\left(u^{\mathrm{\prime }}\right)}^{\mathrm{\prime }}=\lambda m(x)f(u)+\mu n(x)g(u)& \text{ in }\mathrm{\Omega },\\ u=0& \text{ on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $f,g:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ are continuous functions which are, roughly speaking, sublinear and superlinear with respect to $\phi$, respectively. Our assumptions on $\phi$, $m$ and $n$ are substantially weaker than the ones imposed in previous works. The approach used here combines the Guo-Krasnoselskiĭ\ fixed-point theorem and the sub-supersolutions method with some estimates on related nonlinear problems.