The paper presents the existence result for positive solutions of the differential equation $(g(x))''=f(t,x,(g(x))')$ satisfying the nonlocal boundary conditions $x(0)=x(T)$, $\min\{ x(t): t \in J\}=0$. Here the positive function $f$ satisfies local Carathéodory conditions on $[0,T] \times (0,\infty) \times (\R {\setminus} \{0\})$ and $f$ may be singular at the value 0 of both its phase variables. Existence results are proved by Leray-Schauder degree theory and Vitali's convergence theorem.