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

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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.

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...

Analytic solutions of polynomial-like iterative functional equations with variable coefficients are discussed in the complex field ℂ by reducing to an auxiliary equation and by applying known results for systems of nonlinear functional equations of finite orders.

For nonlinear difference equations, it is difficult to obtain analytic solutions, especially when all the eigenvalues of the equation are of absolute value 1. We consider a second order nonlinear difference equation which can be transformed into the following simultaneous system of nonlinear difference equations: ⎧ x(t+1) = X(x(t),y(t)) ⎨ ⎩ y(t+1) = Y(x(t), y(t)) where $X(x,y)=\lambda \u2081x+\mu y+{\sum}_{i+j\ge 2}{c}_{ij}{x}^{i}{y}^{j}$, $Y(x,y)=\lambda \u2082y+{\sum}_{i+j\ge 2}{d}_{ij}{x}^{i}{y}^{j}$ satisfy some conditions. For these equations, we have obtained analytic solutions in the cases "|λ₁| ≠ 1 or |λ₂| ≠ 1" or "μ...

Flajolet and Richmond have invented a method to solve a large class of divide-and-conquer recursions. The essential part of it is the asymptotic analysis of a certain generating function for $z\to \infty $ by means of the Mellin transform. In this paper this type of analysis is performed for a reasonably large class of generating functions fulfilling a functional equation with polynomial coefficients. As an application, the average life time of a party of $N$ people is computed, where each person advances one...

A lot is known about the forward iterates of an analytic function which is bounded by 1 in modulus on the unit disk D. The Denjoy-Wolff Theorem describes their convergence properties and several authors, from the 1880's to the 1980's, have provided conjugations which yield very precise descriptions of the dynamics. Backward-iteration sequences are of a different nature because a point could have infinitely many preimages as well as none. However, if we insist in choosing preimages that are at a...

We study a class of nonlinear difference equations admitting a $1$-Gevrey formal power series solution which, in general, is not $1$- (or Borel-) summable. Using right inverses of an associated difference operator on Banach spaces of so-called quasi-functions, we prove that this formal solution can be lifted to an analytic solution in a suitable domain of the complex plane and show that this analytic solution is an accelero-sum of the formal power series.

In this paper we obtain that there are no transcendental entire solutions with finite order of some nonlinear difference equations of different forms.