Introduction. In a recent publication1 a matrix type of boundary value problem was introduced in order to simplify the description of nuclear reactions. It appeared that this type of boundary value problem could find applications in other branches of mathematical physics, and the purpose of the present note is to illustrate them. When we deal with vibrations of continuous media, with problems o...

smooth coefficients aα(x), L+ be a formally adjoint differential operation. Let L0, L0 be minimal operators (i.e., for example, D(L0) is the clozure of C∞ 0 (Ω) in the norm of the graph ‖u‖L = ‖u‖L2(Ω)+‖Lu‖L2(Ω)), and L, L+ be maximal expansions of L,L+ in the space L2(Ω) respectively (i.e. L = (L0 ) ∗, L+ = (L0)∗), L̃ = L|D(L̃) where D(L̃) is the clozure of C∞(Ω̄) in the norm of the graph ‖u‖L and...

where the initial time, t0, is a given real number, the initial position, ~ ξ0 ∈ IR, is a given vector and ~ F : IR × IR → IR is a given function. We shall assume throughout these notes that ~ F is C. By definition, a solution to the initial value problem (1) on the interval I (which may be open, closed or half–open, but which, of course, contains t0) is a differentiable function ~x(t) which obeys