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 λN, we consider the integral equation:

λyλxf(t)dt=f(x)f(y) for every (x,y)R+2,
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 λN :

λyλxf(t)dt=f(x)f(y) for every (x,y)R+2,
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.

Received:
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

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

[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

[5] Yoneda, T. 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] Yoneda, T. 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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