Singular Dirichlet problem for ordinary differential equations with $\phi$-Laplacian

Abstract

We provide sufficient conditions for solvability of a singular Dirichlet boundary value problem with $$-Laplacian \[ \BOF\unknown. ((u^{\prime }))^{\prime } = f(t, u, u^{\prime }), u(0) = A, \ u(T) = B, \BOF\unknown. \] where $$ is an increasing homeomorphism, $(\mathbb{R})=\mathbb{R}$, $(0)=0$, $f$ satisfies the Carathéodory conditions on each set $[a, b]\times \mathbb{R}^{2}$ with $[a, b]\subset (0, T)$ and $f$ is not integrable on $[0, T]$ for some fixed values of its phase variables. We prove the existence of a solution which has continuous first derivative on $[0, T]$.