General-kind differential-difference elliptic equations in half-plane: asymptotic properties of solutions

The contemporary theory of nonlocal problems (actively developed by various researchers nowadays) and the general theory of functional-differential equations, i. e., equations containing other operators (apart from differential ones) acting on the unknown function, are closely related between each other. Actually, it is reasonably to talk about a unified research area of mathematical physics, specified by the following circumstance: unlike the classical theory of equations of mathematical physics, the equation (boundary-value condition) links the values of derivatives of the unknown function at different points. This qualitative difference generates various novelties both in research results and in available research methods. Also, it opens application areas that are not available for the classical theory.
An important part of the above theory is the theory of partial differential-difference equations. To investigate them, one can apply operational methods because translation operators are Fourier multipliers. For the parabolic case, it is implemented by the author in J. Math. Sci. (New York), 216 (2016), No 3, 345–496. The elliptic case (for unbounded domains) is still an almost open research area.
In the present talk, the Dirichlet problem (with bounded continuous boundary-value functions) in the half-plane is investigated for the second-order differential-difference equation containing, apart from differential operators, translation operators with respect to the variable parallel to the boundary line (the so-called nonlocal variable). Those nonlocal terns are assumed to be restricted by the only condition: the real part of the symbol of the operator acting with respect to the nonlocal variable is bounded from below by a positive constant.
We investigate the solution of the specified problem, which is smooth outside the boundary line, and prove its asymptotical closeness to the unique bounded solution of the same Dirichlet problem for the partial differential equation obtained from the original differential-difference one by the following way: all translations are assigned to be equal to zero.