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Summability of formal solutions of a class of nonlinear difference equations. (English) Zbl 0973.39003
The author shows that the solutions of the equation \[ \Phi(z,y(z),y(z+1))=0 \] can be characterized by their asymptotic behaviour in slightly large domains. Here \(\Phi\) is an \(n\)-dimensional vector-function, analytical in a domain of the form \(D\times U\times U\), where \(D\) is some unbounded domain of \(C\) and \(U\) is a neighbourhood of a given point in \(C^n\). The notion of weak Borel-sum is used. It is proven that, under additional hypotheses on \(\Phi\), the above equation has a unique formal solution \[ \sum_{n=0}^{\infty}a_n z^{-n}. \]

MSC:
39A10 Additive difference equations
39A11 Stability of difference equations (MSC2000)
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