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

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

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

Abstract
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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.01.013/

References
Cited by

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[2] L. Bogachev; G. Derfel; S. Molchanov; J. Ockendon On bounded solutions of the balanced generalized pantograph equation, Topics in Stochastic Analysis and Nonparametric Estimation, The IMA Volumes in Mathematics and its Applications, vol. 145, 2008, pp. 24-49

[3] J. Fabius A probabilistic example of a nowhere analytic $C^{\infty}$ -function, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 5 (1966) no. 2, pp. 173-174

[4] A. Saadatmandi; M. Dehghan Variational iteration method for solving a generalized pantograph equation, Comput. Math. Appl., Volume 58 (2002) no. 11, pp. 2190-2196

[5] T. Yoneda On the functional-differential equation of advanced type $f^{'} (x) = a f (2 x)$ with $f (0) = 0$ , J. Math. Anal. Appl., Volume 37 (2006) no. 1, pp. 320-330

[6] T. Yoneda On the functional-differential equation of advanced type $f^{'} (x) = a f (λ x)$ , $λ > 1$ , with $f (0) = 0$ , J. Math. Anal. Appl., Volume 332 (2007) no. 1, pp. 487-496

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