On Non-Invertible Boundary Value Problems

For non-invertible boundary value problems, i.e. where the boundary conditions as such do not determine the solution uniquely, the usual concepts of condition numbers and stability do not apply. Such problems typically arise when the interval is semi-infinite. If one assumes that the desired solution is bounded the boundary conditions are sufficient to give a unique solution (as an element of the bounded solutions manifold). Another type of problems are eigenvalue problems, where both the dynamics and the boundary conditions are homogeneous. We shall introduce sub-condition numbers that indicate the sensitivity of the problem with respect to perturbations of a relevant subproblem. We also discuss a numerical method that computes such sub-condition number to demonstrate its applicability. Finally we give a number of numerical examples to illustrate both the theory and the computational method.