Dorota Krassowska

A Pair of Functional Inequalities of Iterative Type Related to a Cauchy Functional Equation

It is shown that, under some general algebraic conditions on fixed real numbers a, b, α, β, every continuous at a point solution f of the system of functional inequalities f(x + a) ≤ f(x) + α, f(x + b) ≤ f(x) + β (x ∈ ℝ) must be a polynomial of order 1. Analogous results for three remaining counterparts of this simultaneous system are presented. An application to characterization of L p -norm is given.

On principal iteration semigroups in the case of multiplier zero

We collect and generalize various known definitions of principal iteration semigroups in the case of multiplier zero and establish connections among them. The common characteristic property of each definition is conjugating of an iteration semigroup to different normal forms. The conjugating functions are expressed by suitable formulas and satisfy either Böttcher’s or Schröder’s functional equation.