Boundary value problems for a second-order differential equation with selfadjoint operator coefficients (Q1357983)

This article deals with the following boundary value problem \[ \frac{d^2u}{dt^2}= 2A\frac{du}{dt}+Bu, \] \[ \begin{aligned} \alpha_{11}u(0)+ \alpha_{12}u_t(0)+ \beta_{11}u(T)+ \beta_{12}u_t(T) &= f_1,\\ \alpha_{21}u(0)+ \alpha_{22}u_t(0)+ \beta_{21}u(T)+ \beta_{22}u_t(T) &= f_2,\end{aligned} \] where \(A\) and \(B\) are selfadjoint and commuting linear operators with dense domains and discrete spectra in a Hilbert space \(H\), \(\alpha_{ij}\), \(\beta_{ij}\) reals which form a matrix \(2\times 4\) of range 2, \(f_1,f_2\) given functions defined on \([0,T]\) with values in \(H\), and \(u\) is an unknown function. The basic results are some natural energetic estimates for generalized solutions of the boundary value problem under consideration and, further, some necessary and sufficient conditions of unique solvability for almost all \(T\) of the problem. These results are obtained on the base of a convergence analysis of formal Fourier expansions of solutions and general number-theoretic reasoning.