Generalized and classical solutions of boundary value problems for differential-difference equations (Q1945180)

This paper deals with the boundary value problem \[ -\frac{d^2}{dt^2}R_0u(t)+\frac{d}{dt}R_1u(t)+R_2u(t)=f(t),\quad t\in (0, d), \] with the homogeneous boundary condition \[ u(t)=0\text{ for } t\in {\mathbb R}\backslash (0, d), \] where \(R_i\) are difference operators defined by the formula \[ R_iu(t)=\sum\limits_{j=-m}^{m}b_{ij}u(t+j),\, i=0, 1, 2. \] Here, \(m\) is a positive integer, \(b_{ij}\) are real numbers, and \(b_{00}>0\). Sufficient conditions are given under which the boundary value condition has a classical solution for any continuous functions \(f\). Assuming that the problem in question has a weak solution, it turns out that a classical solution of the problem exists if and only if the derivatives of \(u\) have no argument shifts.