On an adequate description of adjoint operator

We study multipoint boundary value problems for quasidifferential equations, under certain (broad) assumptions on the coefficients of the equation so that there exists the formally adjoint (in the sense of Lagrange) quasidifferential equation. The operator corresponding to the original boundary value problem is densely defined in a reflexive Banachian space and has closed image in its adjoint; the operator corresponding to the adjoint problem has exactly the same properties. We note that the adjoint boundary value problem is not classical: its solution satisfies the quasidifferential equation only in the open intervals between points in which boundary conditions are specified. These considerations lead us to the notion of the generalized boundary value problem. In particular, we introduce the notion of the generalized Valle-Pousin problem (GVPP), where the number of boundary conditions may exceed the order of the equation by allowing higher quasiderivatives of the solution to be discontinuous at the interior points in which boundary conditions are specified. We also show that the boundary value problem adjoint to GVPP is itself a GVPP.