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Differentiable solutions for a class of functional equations

V. Murugan, P. V. Subrahmanyam (2007)

Annales Polonici Mathematici

We give a set of sufficient conditions for the existence of differentiable solutions for a functional equation involving a series of iterates, using a method different from that of Baker and Zhang [Ann. Polon. Math. 73 (2000)].

General construction of non-dense disjoint iteration groups on the circle

Krzysztof Ciepliński (2005)

Czechoslovak Mathematical Journal

Let ℱ = { F v 𝕊 1 → 𝕊 1 , v ∈ V } be a disjoint iteration group on the unit circle 𝕊 1 , that is a family of homeomorphisms such that F v 1 ∘ F v 2 = F v 1 + v 2 for v 1 , v 2 ∈ V and each F v either is the identity mapping or has no fixed point ( ( V , + ) is a 2 -divisible nontrivial Abelian group). Denote by L ℱ the set of all cluster points of { F v ( z ) , v ∈ V } for z ∈ 𝕊 1 . In this paper we give a general construction of disjoint iteration groups for which ∅ ≠ L ℱ ≠ 𝕊 1 .

Geometric characteristics for convergence and asymptotics of successive approximations of equations with smooth operators

Boris Godunov, Petr Zabreĭko (1995)

Studia Mathematica

We discuss the problem of characterizing the possible asymptotic behaviour of the iterates of a sufficiently smooth nonlinear operator acting in a Banach space in small neighbourhoods of a fixed point. It turns out that under natural conditions, for the most part of initial approximations these iterates tend to "lie down" along a finite-dimensional subspace generated by the leading (peripherical) eigensubspaces of the Fréchet derivative at the fixed point and moreover the asymptotic behaviour of...

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