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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