Shifts and periodicity in algebraic analysis

Abstract

Abstract Shifts and periodicity for functional-differential equations and their generalizations have been studied by the author in various aspects (cf. for instance, [17]–[20] and following papers). Here we would like to give a comprehensive survey of some of these results (without proofs) in order to recall the most important properties of the considered shifts. In particular, it is shown that the so-called true shifts in complete linear metric spaces are hypercyclic and that a necessary and sufficient condition for true shifts in commutative algebras to be multiplicative is that the generating operator D satisfies the Leibniz condition. A consequence of this fact is that in commutative Leibniz algebras with logarithms the operator D is uniquely determined by an isomorphism acting on $$ \frac{d} {{dt}} $$. There are also studied generalized periodic and exponential-periodic solutions of linear and some nonlinear equations with shifts and generalizations of the classical Birkhoff theorem and Floquet theorem. These results are obtained by means of tools given by Algebraic Analysis (cf. the author [17]). A generalization of binomial formula of Umbral Calculus is shown in Section 7 (cf. Roman and Rota [36]). Section 11 contains a perturbation theorem for linear differential-difference equations with non-commensurable deviations and some its consequences.