### A characterization of damped and undamped harmonic oscillations by a superposition property II.

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The functional equation (F(x)-F(y))/(x-y) = (G(x)+G(y))(H(x)+H(y)) where F,G,H are unknown functions is considered. Some motivations, coming from the equality problem for means, are presented.

A class of functional equations with nonlinear iterates is discussed on the unit circle ${\mathbb{T}}^{1}$. By lifting maps on ${\mathbb{T}}^{1}$ and maps on the torus ${\mathbb{T}}^{n}$ to Euclidean spaces and extending their restrictions to a compact interval or cube, we prove existence, uniqueness and stability for their continuous solutions.

A competition model is described by a nonlinear first-order differential equation (of Riccati type). Its solution is then used to construct a functional equation in two variables (admitting essentially the same solution) and several iterative functional equations; their continuous solutions are presented in various forms (closed form, power series, integral representation, asymptotic expansion, continued fraction). A constant C = 0.917... (inherent in the model) is shown to be a transcendental number....

In this paper we characterize those bounded linear transformations $Tf$ carrying ${L}^{1}\left({\mathbb{R}}^{1}\right)$ into the space of bounded continuous functions on ${\mathbb{R}}^{1}$, for which the convolution identity $T(f*g)=Tf\xb7Tg$ holds. It is shown that such a transformation is just the Fourier transform combined with an appropriate change of variable.

The Levi-Civita functional equation $f\left(gh\right)={\sum}_{k=1}^{n}{u}_{k}\left(g\right){v}_{k}\left(h\right)$ (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 $g\mapsto {T}_{g}$ 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.