On multiplicatively-additive iteration groups
The main motivation of the paper is an economical model describing the so-called impatience. Impatience is a special order on a sequences space of preferences of consumption outcomes. The main problem of the paper is to find the continuous solutions of the functional equation \[ F\bigl((t,a),F\bigl((s, b\bigr), x)\bigr)=F\bigl((ts,tb+a),x\bigr), \qquad t,s > 0,\quad a,b\in\mathbb{R},\quad x\in I, \] where $I$ is a nonempty open interval. It turns out, that every sequence of consumption outcomes can be represented by an increasing homeomorphism with one fixed point and the problem of the existence of impatience order can be reduced to the determination of suitable properties of semigroups of increasing homeomorphisms of $\mathbb{R}$ satisfying some extra conditions. \par The author gives a construction of large families of homeomorphisms semigroups possessing at most one fixed point satisfying the extra conditions. These semigroups are conjugate to the family of all affine functions. Their construction relates to the determination of two parameter families of homeomorphisms satisfying the functional equation above.
scientific article; zbMATH DE number 7035780