New existence results are presented for the two point singular “resonant” boundary value problem $\frac{1}{p}{\left(p{y}^{\text{'}}\right)}^{\text{'}}+ry+{\lambda }_{m}qy=f(t,y,p{y}^{\text{'}})$ a.eȯn $[0,1]$ with $y$ satisfying Sturm Liouville or Periodic boundary conditions. Here ${\lambda }_m$ is the ${(m+1)}^{st}$ eigenvalue of $\frac{1}{pq}[{\left(p{u}^{\text{'}}\right)}^{\text{'}}+rpu]+\lambda u=0$ a.eȯn $[0,1]$ with $u$ satisfying Sturm Liouville or Periodic boundary data.

It is known that the vector stop operator with a convex closed characteristic $Z$ of class $C^1$ is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping $n$ is Lipschitz continuous on the boundary $\partial Z$ of $Z$. We prove that in the regular case, this condition is also necessary.

Existence results are established for the resonant problem $y^{\text{'}\text{'}}+{\lambda }_{m}ay=f(t,y)$ a.e. on $[0,1]$ with $y$ satisfying Dirichlet boundary conditions. The problem is singular since $f$ is a Carathéodory function, $a\in L_{\mathrm{l}oc}^1(0,1)$ with $a>0$ a.e. on $[0,1]$ and ${\int }_0^{1}x(1-x)a\left(x\right)\mathrm{d}x<\infty$.