Boundary value problems associated with multiple roots of the characteristic equation (Q1088039)

We consider boundary value problems of the form \[ (1)\quad L(x,\lambda;y)=\sum^{n}_{k=0}\lambda^{H(n-k)}a_{n- k}(x,\lambda)y^{(k)}=0\quad (2)\quad U_{\nu}(y)=U_{\nu 0}(y)+U_{\nu 1}(y)=0, \] \(1\leq \nu \leq n\), where \(\lambda\) is the spectral parameter, \(x\in [0,1]\), \(H\in {\mathbb{N}}\) is chosen as small as possible, \(a_{n-k}(x,\lambda)=\sum^{H(n-k)}_{\nu =0}\alpha_{n- k,\nu}(x)\lambda^{-\nu},\) \(0\leq k\leq n\), \(a_ 0(x,\lambda)\equiv 1\) and where the characteristic equation \(\sum^{n}_{k=0}\alpha_{n- k,0}(x)\rho^ k=0\) has multiple roots. For regular problems (1), (2) we prove estimates for the Green's function and an expansion theorem for functions satisfying the boundary conditions (2).