Functional-Differential Equations with Dilation and Symmetry

We examine the Dirichlet problem in a bounded plane domain for a strongly elliptic functional-differential equation of the second order containing the argument transformations $ x\mapsto px $ ( $ p>0 $ ) and $ x\mapsto-x $ in higher-order derivatives. The study of solvability of the problem relies on a Gårding-type inequality for which some necessary and sufficient conditions are obtained in algebraic form.