Solvability of quasi-linear multi-point boundary value problem at resonance

In this paper, we consider the following second order quasi-linear differential equation:
$$ (\Phi_p(x'))'+f(t,x)=0,\qquad0<t<1, $$
where $\Phi_p(s)=|s|^{p-2}s$, $p\geq2$, subject to certain boundary conditions. The criteria of solvability of these boundary value problems are given by employing the recent generalization of coincidence degree method. We also give an example to illustrate our conclusions.