The paper investigates the analytic properties and asymptotic behaviors of solutions to the non-homogeneous functional differential equation $y^{\prime }(x)=ay(qx)+by(x)+\frac{1}{1+x}$, where $q$ is a constant satisfying $0<q<1$, and $a\ne 0$, $b\ne 0$ are complex numbers, by following methods developed by J. P. Ramis for singular differential equations and $q$-difference equations. First, we study the existence and analytic properties of solutions expressed as series expansions around both zero and infinity. Next, by considering the equation as a perturbation of a differential equation, we derive a solution represented as a sum of integrals containing an infinite number of singularities. Finally, we establish a connection formula between the integral-sum solution and the series expansion at zero, which is crucial for determining the asymptotic behavior of the series solution as $x\rightarrow \infty $, given the initial conditions.
Dans cet article, nous examinons les propriétés analytiques et les comportements asymptotiques des solutions de l’équation différentielle fonctionnelle non homogène $y^{\prime }(x)=ay(qx)+by(x)+\frac{1}{1+x}$, où $q$ est une constante satisfaisant $0<q<1$, et $a\ne 0$, $b\ne 0$ sont des nombres complexes, en suivant les méthodes développées par J. P. Ramis pour les équations différentielles autour des singularités et les équations aux $q$-différences. Nous étudions tout d’abord l’existence et les propriétés analytiques des solutions exprimées sous forme de séries autour de zéro et à l’infini. Ensuite, en considérant l’équation comme une perturbation d’une équation différentielle, nous obtenons une solution représentée par une somme d’intégrales comportant un nombre infini de singularités. Enfin, nous établissons la formule de connexion entre la fonction somme-intégrale et la solution en série à zéro, qui joue un rôle crucial dans la détermination des comportements asymptotiques à l’infini de ladite solution en série, une condition initiale étant donnée.
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Keywords: Functional differential equation, non-homogeneous term, analytic properties, perturbation of equation, asymptotic expansion
Mots-clés : Équation différentielle fonctionnelle, terme non homogène, propriétés analytiques, perturbation de l’équation, expansion asymptotique
Huan Dai 1 , 2
CC-BY 4.0
@article{CRMATH_2025__363_G10_959_0,
author = {Huan Dai},
title = {Analytic and asymptotic properties of solutions to a non-homogeneous functional differential equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {959--975},
year = {2025},
publisher = {Acad\'emie des sciences, Paris},
volume = {363},
doi = {10.5802/crmath.781},
language = {en},
}
TY - JOUR AU - Huan Dai TI - Analytic and asymptotic properties of solutions to a non-homogeneous functional differential equation JO - Comptes Rendus. Mathématique PY - 2025 SP - 959 EP - 975 VL - 363 PB - Académie des sciences, Paris DO - 10.5802/crmath.781 LA - en ID - CRMATH_2025__363_G10_959_0 ER -
Huan Dai. Analytic and asymptotic properties of solutions to a non-homogeneous functional differential equation. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 959-975. doi: 10.5802/crmath.781
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