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Differential equations of the following forms are considered: and where is a positive selfadjoint operator in a Hilbert space . The question of whether the solutions of such equations have boundary values at the end points of the interval on which they are considered is investigated, as well as the problem of recovering a solution from its boundary values. A characterization of the boundary values is given in terms of the behavior of the solution near the end points and . A number of examples are cited in which is realized as a differential operator in various function spaces. When applied to these concrete situations, the abstract theorems yield the existence and characteristics of the boundary values for certain classes of elliptic and parabolic equations; in particular, the well-known results of F. Riesz, Kothe and Komatsu are obtained and sharpened in this way. The approach is based on the spectral theory of selfadjoint operators.Bibliography: 18 titles.