Generalized and classical solutions to the second and third boundary value problem for difference-differential equations

In this paper, the author considers the equation -(R_0u)"(x)+(R_1u)'(x)+(R_2u)(x)=f(x),quad xin (0, d), with boundary value conditions lim_{xto 0^+}(-(R_0u)'(x)+sigma_1u(x))=0,quad lim_{xto d^-}((R_0u)'(x)+sigma_2u(x)=0, and under the assumption that u(x)=0, xin Bbb{R}sbs (0, d) , where sigma_1, sigma_2ge 0, and R_i, {i=0, 1, 2}, are difference operators defined by the formulas R_iu(x)=sumlimits_{j=-m}^{m}b_{ij}(x)u(x+j),quad i=0,1,2. Here m is an integer and b_{ij} in C^{infty}(Bbb{R}) are complex-valued functions. par Necessary and sufficient conditions are obtained for the existence of a classical solution for an arbitrary continuous function f if the boundary value problem under consideration has a generalized solution. It is proved that such conditions are that certain coefficients of the difference operators on the orbits generated by the shifts be equal to zero. It is also shown that in contrast to Dirichlet boundary value problems, the necessary condition for the existence of a classical solution does not coincide with the sufficient condition. Two examples are given to illustrate the results.