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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:
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