Singular Perturbation Analysis of Boundary Value Problems for Differential-Difference Equations. V. Small Shifts with Layer Behavior

In this paper, an investigation is initiated of boundary-value problems for singularly perturbed linear second-order differential-difference equations with small shifts, i.e., where the secondorder derivative is multiplied by a small parameter and the shift depends on the small parameter. Similar boundary-value problems are associated with expected first-exit times of the membrane potential in models for neurons. In particular, this paper focuses on problems with solutions that exhibit layer behavior at one or both of the boundaries. The analyses of the layer equations using Laplace transforms lead to novel results. It is shownthat the layer behavior can change its character and even be destroyed as the shifts increase but remain small. In the companion paper [SIAM J. Appl. Math., 54 (1994), pp. 273-283], similar boundary-value problems with solutions that exhibit rapid oscillations are studied.