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A general class of iterative equations on the unit circle

Marek Cezary Zdun, Wei Nian Zhang (2007)

Czechoslovak Mathematical Journal

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.

Addition theorems and related geometric problems of group representation theory

Ekaterina Shulman (2013)

Banach Center Publications

The Levi-Civita functional equation f ( g h ) = k = 1 n u k ( g ) v k ( h ) (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 T g have orbits O(x) that are “close” to a fixed finite-dimensional subspace? 2) for...

Analytic solutions of a nonlinear two variables difference system whose eigenvalues are both 1

Mami Suzuki (2011)

Annales Polonici Mathematici

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 ) = λ x + μ y + i + j 2 c i j x i y j , Y ( x , y ) = λ y + i + j 2 d i j 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...

Asymptotic analysis of a class of functional equations and applications

P. J. Grabner, H. Prodinger, R. F. Tichy (1993)

Journal de théorie des nombres de Bordeaux

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 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...

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