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A Galois D -groupoid for q -difference equations

Anne Granier (2011)

Annales de l’institut Fourier

We first recall Malgrange’s definition of D -groupoid and we define a Galois D -groupoid for q -difference equations. Then, we compute explicitly the Galois D -groupoid of a constant linear q -difference system, and show that it corresponds to the q -difference Galois group. Finally, we establish a conjugation between the Galois D -groupoids of two equivalent constant linear q -difference systems, and define a local Galois D -groupoid for Fuchsian linear q -difference systems by giving its realizations.

A role of the coefficient of the differential term in qualitative theory of half-linear equations

Pavel Řehák (2010)

Mathematica Bohemica

The aim of this contribution is to study the role of the coefficient r in the qualitative theory of the equation ( r ( t ) Φ ( y Δ ) ) Δ + p ( t ) Φ ( y σ ) = 0 , where Φ ( u ) = | u | α - 1 sgn u with α > 1 . We discuss sign and smoothness conditions posed on r , (non)availability of some transformations, and mainly we show how the behavior of r , along with the behavior of the graininess of the time scale, affect some comparison results and (non)oscillation criteria. At the same time we provide a survey of recent results acquired by sophisticated modifications of the Riccati...

An example of local analytic q-difference equation : Analytic classification

Frédéric Menous (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

Using the techniques developed by Jean Ecalle for the study of nonlinear differential equations, we prove that the q -difference equation x σ q y = y + b ( y , x ) with ( σ q f ) ( x ) = f ( q x ) ( q > 1 ) and b ( 0 , 0 ) = y b ( 0 , 0 ) = 0 is analytically conjugated to one of the following equations : x σ q y = y ou x σ q y = y + x

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