aequationes mathematicae

, Volume 67, Issue 1, pp 117–131

On regular solutions of some simple iterative functional equations

Research paper

DOI: 10.1007/s00010-003-2712-8

Cite this article as:
Brillouët-Belluot, N. Aequ. Math. (2004) 67: 117. doi:10.1007/s00010-003-2712-8
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Summary.

We consider the iterative functional equation

\( \Phi(\alpha\, x) = L[\Phi(x)] + F(x) \quad (x \in E) \qquad(2) \)

where \( \Phi : E \rightarrow G \) is the unknown function, E and G are complex normed linear spaces, \( F : E \rightarrow G \) is a given function, \( \alpha \) is a root of unity of order n, I is the identical mapping in G and \( L : G \rightarrow G \) is a given linear mapping satisfying \( L^{n} = I \) .

By using simple direct methods, we obtain the general solution of (2), its continuous solutions, and, in the case \( E = \mathbb{C}^q \) , its differentiable solutions. The results presented here complete a previous paper dealing with the case \( L(x) = \beta\, x\, (x \in G) \) where \( \beta \) is some complex number.

Mathematics Subject Classification (2000).

Primary 39B12 39B52. 

Keywords.

Linear functional equation Iterative functional equation. 

Copyright information

© Birkhäuser-Verlag 2004

Authors and Affiliations

  1. 1.Département d’Informatique et de Mathématiques École Centrale de NantesNantesFrance

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