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 shown-that 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.

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A study is made of several nonlinear boundary-value problems of singular perturbation type for which a straightforward application of boundary-layer theory leads to spurious solutions. It is shown that these problems can be treated successfully by a slight modification of the method of matched asymptotic expansions. The analysis leads to several novel features which are not present in routine singular perturbation problems.