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An ultrametric version of the Maillet-Malgrange theorem for nonlinear \(q\)-difference equations. (English) Zbl 1152.33011
An analog of the Maillet-Malgrange theorem for ultrametric nonlinear \(q\)-difference equation is established in this paper, under the assumption \(\mod (q)\) not equal to unity. The result so obtained, generalizes to nonlinear \(q\)-difference equation, a theorem of J.-P. Bézivin and A. Boutabaa [Collect. Math. 43, No. 2, 125–140 (1992; Zbl 0778.39009)]. In Section 2, examples of the result are given, Section 3 gives the proof of the main result. The author claims to have obtained a much stronger version of Theorem 7 (page 2811, this paper). The references are exhaustive and some of them may encourage researchers to study Gevrey theory for linear \(q\)-difference equations.

MSC:
33E99 Other special functions
39A13 Difference equations, scaling (\(q\)-differences)
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