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.