An elliptic functional-differential equation with contractions of the arguments

From the text (translated from the Russian): "Functional-differential equations with contraction of the argument in the one-dimensional case were considered by many authors, including T. Kato and J. B. McLeod [Bull. Amer. Math. Soc. {bf 77} (1971), 891--937; [msn] MR0283338 (44 #570) [/msn]] in connection with applications in technology. Boundary value problems for elliptic differential-difference equations and the closely related nonlocal boundary value problems for elliptic differential equations were studied by A. L. Skubachevskiĭ [Mat. Sb. (N.S.) {bf 129(171)} (1986), no.~2, 279--302; [msn] MR0832122 (87h:35089) [/msn]; {it Elliptic functional-differential equations and applications}, Birkhäuser, Basel, 1997; [msn] MR1437607 (98c:35164) [/msn]]. A natural development of the above-mentioned papers is the investigation of boundary value problems for elliptic functional-differential equations containing dilations and contractions of the arguments of the unknown function [L. E. Rossovskiĭ, Tr. Mosk. Mat. Obs. {bf 62} (2001), 199--228; [msn] MR1907255 (2003e:35309) [/msn]]. Note that in a domain containing the origin, which is a fixed point of the contraction operator, such a problem can have not only a unique smooth solution but also an infinite number of nonsmooth generalized solutions. It is convenient to analyze the behavior of solutions in a neighborhood of the fixed point by choosing a suitable weight. In this paper, we study the solvability of an equation with contraction of the arguments in weighted spaces."