Combinatorics/Ordinary differential equations
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.

For each $λ∈N⁎$, we consider the integral equation:

where f is the concatenation of two continuous functions $fa,fb:[0,λ]→R$ along a word $u=u0u1⋯∈{a,b}N$ 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 $λ∈N⁎$ :

f est la concaténation de deux fonctions continues $fa,fb:[0,λ]→R$ le long d'un mot infini $u=u0u1⋯∈{a,b}N$ 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.

Accepted:
Published online:
DOI: 10.1016/j.crma.2018.01.013
Bertazzon, Jean-François 1

1 Lycée Notre-Dame-de-Sion, 231, rue Paradis, 13006 Marseille, France
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Bertazzon, Jean-François. 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. http://www.numdam.org/articles/10.1016/j.crma.2018.01.013/

[1] Bertazzon, J.-F.; Delecroix, V. Sommes de Birkhoff itérées sur des extensions finies d'odomètres. Construction de solutions auto-similaires à des équations différentielles avec délai, Bull. Soc. Math. Fr. (2018) (in press)

[2] Bogachev, L.; Derfel, G.; Molchanov, S.; Ockendon, J. 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

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

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