A class of functional equations with nonlinear iterates is discussed on the unit circle . By lifting maps on and maps on the torus to Euclidean spaces and extending their restrictions to a compact interval or cube, we prove existence, uniqueness and stability for their continuous solutions.
The Levi-Civita functional equation (g,h ∈ G), for scalar functions on a topological semigroup G, has as the solutions the functions which have finite-dimensional orbits in the right regular representation of G, that is the matrix elements of G. In considerations of some extensions of the L-C equation one encounters with other geometric problems, for example: 1) which vectors x of the space X of a representation have orbits O(x) that are “close” to a fixed finite-dimensional subspace? 2) for...
We solve the mod G Cauchy functional equation f(x+y) = f(x) + f(y) (mod G), where G is a countable subgroup of ℝ and f:ℝ → ℝ is Borel measurable. We show that the only solutions are functions linear mod G.
In this paper, we consider Lipschitz conditions for tri-quadratic functional equations. We introduce a new notion similar to that of the left invariant mean and prove that a family of functions with this property can be approximated by tri-quadratic functions via a Lipschitz norm.