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

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/

[1] J.-F. Bertazzon; V. Delecroix 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] 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 $C∞$-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 $f′(x)=af(2x)$ 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 $f′(x)=af(λx)$, $λ>1$, with $f(0)=0$, J. Math. Anal. Appl., Volume 332 (2007) no. 1, pp. 487-496

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