Abstract:
The article is devoted to the study of one elliptic-type functional differential equation that contains in its upper part the contraction transformation of unknown function arguments herewith contractions are different for every argument, i.e. contractions are orthotropic. Some necessary and sufficient conditions of strong ellipticity in the terms of the Garding-type inequality fulfilment were presented in explicit form. Thus, a new class of equations satisfying the Kato square root problem was obtained. The first boundary valued problem for the strongly elliptic functional differential equation with contractions was considered in the domain containing the origin—the fixed point of contraction transformation, and star-shaped regarding it. The Fredholm solvability and spectrum structure in the Sobolev spaces were studied. Further the sufficient conditions for solvability of the equation considered in the Kondratiev weighted spaces on the plane were obtained. It is remarkable that the conditions depend on the weight parameter. 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 obtained results were presented. Acknowledgements The work is supported by Russian Foundation of Basic Research, project No. 20-01-00288.
