A class of functional equations with nonlinear iterates is discussed on the unit circle ${𝕋}^1$. By lifting maps on ${𝕋}^1$ and maps on the torus ${𝕋}^n$ 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 $X(x,y)=\lambda ₁x+\mu y+{\sum }_{i+j\ge 2}{c}_{ij}{x}^i{y}^j$, $Y(x,y)=\lambda ₂y+{\sum }_{i+j\ge 2}{d}_{ij}{x}^i{y}^j$ 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...