MR2177123
El-Rabih, Abir
Abir EL-RABIH
Existence of Local Analytic Solutions for Systems of Difference Equations with Small Step Size
Funkcialaj Ekvacioj. Serio Internacia
48
2005
313--330
http://fe.math.kobe-u.ac.jp/FE/FullPapers/48-2/48_313.pdf
http://www.ams.org/mathscinet-getitem?mr=MR2177123
We study the existence of analytic solutions of systems of difference equations where we have vector valued functions $y$ of $\epsilon$ and $(\epsilon+x)$ equals to vector valued analytic functions $F$ of $\epsilon$, $x$ and $y$ in a neighborhood of $(0,x^*,y^*)$ with $y^*=F(0,x^*,y^*)$. Under the assumption that the Jacobian of $F$ with respect to $y$ at $(0,x^*,y^*)$ minus the idendity is invertible, we first show the existence of a unique formal solution that is Gevrey-1. We also show, by applying a fixed point theorem, the existence of analytic solutions having a Gevrey-1 asymptotic expansion in small $\epsilon$-sectors. This requires the construction of some bounded right inverse operators on a certain Banach space.
Difference equation, Difference operator, Inverse operator, Fixed point theorem, Gevrey asymptotic, Formal solution, Quasi-solution, Nagumo norm, Borel-Laplace transform.
39A, 47B39.
48-313
2005
Existence of Local Analytic Solutions for Systems of Difference Equations with Small Step Size
EL-RABIH, Abir
Abir EL-RABIH
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