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An example of nonlinear \(q\)-difference equation. (English) Zbl 1087.39023
Summary: We study the formal solutions of the nonlinear \(q\)-difference equation \[ x\sigma_qf-f=b(f,x) \] where \(\sigma_qf(x)= f(qx)\), with a real number \(q>1\) and \(b(f,x)\) belongs to \(\mathbb{C}\{f,x\}\) with the conditions \(b(0,0)=0\) and \((\partial_fb)(0,0)=0\). We prove that a solution of this equation can be conjugated to the solution \(ue_q(x)=u q^{-\log_qx(\log_qx-1)/2}(\sigma_qu=u)\) of the associated homogeneous equation, with the help of a formal substitution automorphism \(\Theta\in\mathbb{C}[[x,e_q,u,\partial_u]]\).
Following the methods developed by J. Ecalle [Ann. Inst. Fourier 42, No. 1–2, 73–164 (1992; Zbl 0940.32013)], we first express this conjugating operator \(\Theta\) as a mould-comould expansion. The mould \(W^\bullet\) can be computed and each of its components is a formal series in \(x\).
When \(b(0,x)=0\), these components happen to be convergent and we prove that the conjugating operator is also convergent in a well-adapted topology.
In the generic case, the components of the mould are no more convergent. Nevertheless, these components are \(q\)-multisummable. This is not sufficient to define a good resummation process for the conjugating operator. Some unsolved problems call for new results in the \(q\)-resummation theory. Besides this, it also seems that the arborification of moulds yields some simplifications of the encountered problems.

MSC:
39A13 Difference equations, scaling (\(q\)-differences)
40H05 Functional analytic methods in summability
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References:
[1] Ecalle ( J. ). - Singularités non abordables par la géométrie , Annales de l’Institut Fourier 42 , p. 73 - 164 ( 1992 ). Numdam | MR 1162558 | Zbl 0940.32013 · Zbl 0940.32013
[2] Zhang ( C. ). - Développements asymptotiques q-Gevrey et séries Gqsommables , Annales de l’institut Fourier 49 , p. 227 - 261 ( 1999 ). Numdam | MR 1688144 | Zbl 0974.39009 · Zbl 0974.39009
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