On a Class of Elliptic Functional–Differential Equations with Orthotropic Contractions–Expansions

The paper is devoted to the study of an elliptic-type functional differential equation containing the contraction transformation of the arguments of the unknown function in its leading part, and the contractions are different for different arguments. Some necessary and sufficient conditions for the validity of a Gårding-type inequality, which is an analog of the strong ellipticity condition, are presented in an explicit form. The Fredholm solvability and the structure of the spectrum of the first boundary value problem in the Sobolev spaces are studied. Sufficient conditions for the solvability of the equation in the Kondrat'ev weighted spaces on the plane are given. In the course of the proof, sufficient conditions for the invertibility of a finite-difference operator with variable coefficients on a line are obtained. Some concrete examples illustrating the results thus obtained are presented.