Beyond-all-orders boundary-value sensitivity

A homogeneous mode can be order one inside an interval and exponentially small at an endpoint. Adjusting an endpoint value beyond all algebraic orders can then change the interior solution by order one. In the nested layers at an essential singular endpoint example, the mode $\exp (-x/ϵ-2{ϵ}^2/x)$ has this behavior: prescribed exact endpoint values give a unique solution for each fixed positive $ϵ$, but leading asymptotic endpoint data do not select a unique bounded family.