Singular perturbations of multi-point eigenvalue problems (Q760573)

We consider multi-point eigenvalue problems of the form \[ (1)\quad \epsilon^ P \sum^{m}_{j=n+1} \epsilon^{(j-n-1)P} b_ j(x,\lambda,\epsilon) y^{(j)} = \sum^{n}_{j=0} b_ j(x,\lambda,\epsilon)y^{(j)}, \] \[ (2)\quad \sum^{m-1}_{j=0} \sum^{k}_{i=1} c_{vij}(\lambda,\epsilon) y^{(j)}(x_ i),\quad 1\leq v\leq n \] where \(0\leq x\leq 1\), \(2\leq n\leq m\)-1, \(0=x_ 1<x_ 2<...<x_ k=1\), \(\epsilon\) is a positive parameter and \(\lambda\) is the eigenvalue parameter. Further we assume that the coefficients \(b_ j\) and \(c_{vij}\) have asymptotic expansions with respect to \(\epsilon\) as \(\epsilon\to 0\) and that \(b_ m(x,\lambda,0)\neq 0\neq b_ n(x,\lambda,0)\) for all \((x,\lambda)\in [0,1]\times {\mathbb{C}}\). We give conditions which ensure that (in the nonexceptional case) eigenvalues and eigenfunctions of problem (1),(2) exist for \(\epsilon\) sufficiently small and that these eigenvalues have limits as \(\epsilon\) tends to zero. Under suitable conditions the corresponding eigenfunctions can be so normalized that they also have limits and the limiting functions are eigenfunctions of the reduced problem. In the last section we state analogous results for the exceptional case with splitting multi-point conditions.