The Neumann problem for a second-order singular system (Q1033585)

Consider the boundary value problem \[ \begin{gathered} u''+ t^{-1} ku' = -\varphi_0(t, v),\;v''+ t^{-1} kv'= \psi_0(t, u),\\ u'(0)= 0,\quad u'(\tau)= a+ A_{11}u(\tau)+ A_{12} v(\tau),\\ v'(0)= 0,\quad v'(\tau)= b- A_{21} u(\tau)= A_{22} v(\tau),\end{gathered}\tag{\(*\)} \] where \(u,v,a,b\in\mathbb{R}^n\), \(\varphi_0,\psi_0\in C(I\times \mathbb{R}^n)\), \(I= [0,\tau]\), \(A_{ij}\) are \(n\times n\)-matrices, \(k>-1\). The author derives conditions such that \((*)\) has for any \(a\) and \(b\) a solution.