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Gevrey type solutions of nonlinear difference equations. (English) Zbl 1122.39018
The paper deals with the nonlinear functional equation \[ \varphi(z,y(z),y(z+1))=0,\tag{1} \] where \(\varphi\) is a \(\mathbb C^n\)-valued function, analytic in a domain of the form \(D\times U\times U\), \(D\) is an unbounded domain of \(\mathbb C\), \(U\) is a neighborhood of \(y_0\in \mathbb C^n\) and \(\varphi\) admits an asymptotic expansion \(\widehat \varphi(y_1,y_2)= \sum_{h=0}^{\infty}\varphi_h(y_1,y_2)z^{-h/p}\), with \(p\in \mathbb N\), as \(z\to \infty\), in \(D\). The author proves the existence of Gevrey type analitic solutions of equation (1), represented asymptotically by the formal solution \(\widehat f\in \mathbb C^n[[z^{-1/p}]]\) which has the constant term \(y_0\) and belongs to a generalized Gevrey class of divergent power series. This problem is studied in two different types of domains: “wide domains” which are bounded by curves with limiting directions \(\pi/2\) and \(-\pi/2\), and “narrow domains” which are bounded by curves with the same limiting direction viz. \(\pi/2\) (mod \(\pi\)) and contain an unbounded part of the imaginary axis.

MSC:
39B32 Functional equations for complex functions
30B10 Power series (including lacunary series) in one complex variable
39B52 Functional equations for functions with more general domains and/or ranges
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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