Sufficient Conditions of Solvability of a Functional Differential Equation with Orthotropic Contractions in Weighted Spaces

In this paper the solvability of a functional-differential equation is studied in the scale of Kondrat’ev weighted spaces. The equation is considered in the real plane, it has constant coefficients and transformations of arguments of required function, and this transformation consists in the contraction of one argument and the expansion of another. These transformations are called by orthotropic contractions here. The considered problem boils down to an invertibility of a difference operator on the real line with variable smooth coefficients stabilized in the infinity. Sufficient conditions of the invertibility of the difference operator and the initial functional differential operator were obtained in algebraic form. It is well-known that the properties of functional differential equations are largely defined by the structure of point orbits under the action of a group generated by transformations attended in the equation. The orbits of isotropic contractions are situated on the rays passing through the origin and condense near the origin”- a fixed point of the operator. In case of contraction of one argument and expansion of another the orbits are situated on curves having a form of hyperbolas. Herewith the origin is a fixed point of the operator as before. Therefore it is natural to assume that problems with orthotropic contractions differ in their properties and analysis methods from problems with isotropic contractions.

functional-differential equations, Kondrat’ev weighted spaces, weighted shift operator, orthotropic contractions, difference equations