ABSTRACT. We consider functional-differential equations with the Dirichlet conditions and with contraction and dilatation of the arguments. Necessary and sufficient conditions are obtained under which a Gs type inequality holds. These results allow us to verify coerciveness by using a special "symbol" of the equation considered.The coerciveness problem for quadratic forms generated by differential operators was studied by M. Figueiredo [4], and J. Necas in the book [5]. In particular, it was proved that the strong ellipticity of a system of differential operators with Dirichlet conditions on the boundary is a necessary and sufficient condition for this system to satisfy a Gs type inequality. A. Skubachevskii [6] obtained necessary as well as sufficient conditions in algebraic form for differencedifferential operators to satisfy a Gs type inequality. He also proved that these conditions coincide for almost all domains Q c R".The present paper treats functional-differential equations with Dirichlet conditions and with contracted and expanded arguments. We present necessary and sufficient conditions for a Gs type inequality. With the equations under consideration we associate an algebraic expression (the symbol of the equation) whose positiveness is the desired condition. Theorems 1 and 2 contain the main results. Let us begin with some preliminary considerations.w Let S =-1 be the unit sphere in R n. To each function g G C (S "-1) we assign a function G G L~(R ") by setting c(=)=g (= e R").
(i)The multiplication by G(x) is a bounded operator in L~(R ") k. It will be also denoted by G. Thus,Obviously, the mapping g ~-. G is a homomorphism of the algebra C(S n-1 ) into the algebra of bounded operators B(L2(R")). Let us show that this is an isometry. On the one hand, we have
The "commensurability" of transformations has been a crucial assumption in the study of solvability and regularity of solutions for elliptic functional differential equations in domains, while equations with incommensurable transformations are much less studied. In the paper, we consider an equation containing multiplicatively incommensurable contractions of the arguments of the unknown function in the principal part. Algebraic conditions for unique solvability of the Dirichlet problem will be obtained as well as conditions ensuring the existence of an infinite-dimensional null-space. The equation considered is an elliptic analog of the generalized pantograph equation studied by many authors. As a complementary conclusion, we observe that the spectral properties of functional operators with contractions are unstable with respect to small perturbations of scaling parameters.
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