Majoration of the dimension of the space of concatenated solutions to a specific pantograph equation

Jean-François Bertazzon

doi:10.1016/j.crma.2018.01.013

Combinatorics/Ordinary differential equations

Majoration of the dimension of the space of concatenated solutions to a specific pantograph equation

Presented by: the Editorial Board

Jean-François Bertazzon ¹

¹Lycée Notre-Dame-de-Sion, 231, rue Paradis, 13006 Marseille, France

Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 235-242

Abstract (Original language)
Résumé

For each $λ \in N^{⁎}$ , we consider the integral equation:

\int_{λ y}^{λ x} f (t) d t = f (x) - f (y) for every (x, y) \in R_{+}^{2},

where f is the concatenation of two continuous functions

f_{a}, f_{b} : [0, λ] \to R

along a word

u = u_{0} u_{1} \dots \in {a, b}^{N}

such that

u = σ (u)

, where σ is a λ-uniform substitution satisfying some combinatorial conditions.

There exists some non-trivial solutions ([1]). We show in this work that the dimension of the set of solutions is at most two.

Nous considérons les équations intégrales de la forme suivante pour $λ \in N^{⁎}$ :

\int_{λ y}^{λ x} f (t) d t = f (x) - f (y) for every (x, y) \in R_{+}^{2},

où f est la concaténation de deux fonctions continues

f_{a}, f_{b} : [0, λ] \to R

le long d'un mot infini

u = u_{0} u_{1} \dots \in {a, b}^{N}

tel que

u = σ (u)

, où σ est une substitution λ-uniforme vérifiant certaines propriétés combinatoires.

Il existe des solutions non triviales à ces équations ([1]). Nous montrons dans ce travail que l'espace des solutions est de dimension au plus 2.

Received: 2017-09-27
Accepted: 2018-01-22
Published online: 2018-02-01

DOI: 10.1016/j.crma.2018.01.013

Author's affiliations:

Jean-François Bertazzon ¹

¹ Lycée Notre-Dame-de-Sion, 231, rue Paradis, 13006 Marseille, France

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Jean-François Bertazzon. Majoration of the dimension of the space of concatenated solutions to a specific pantograph equation. Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 235-242. doi: 10.1016/j.crma.2018.01.013

References
Cited by

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