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.

The aim of this contribution is to study the role of the coefficient $r$ in the qualitative theory of the equation ${\left(r\left(t\right)\Phi \left(y^{\Delta }\right)\right)}^{\Delta }+p\left(t\right)\Phi \left(y^{\sigma }\right)=0$, where $\Phi \left(u\right)={\left|u\right|}^{\alpha -1}\mathrm{sgn}u$ with $\alpha >1$. 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...

Using the techniques developed by Jean Ecalle for the study of nonlinear differential equations, we prove that the $q$-difference equation$$x{\sigma }_{q}y=y+b(y,x)$$with $\left({\sigma }_{q}f\right)\left(x\right)=f\left(qx\right)$ ($q\>1$) and $b(0,0)={\partial }_{y}b(0,0)=0$ is analytically conjugated to one of the following equations :$$x{\sigma }_{q}y=y\text{ou}x{\sigma }_{q}y=y+x$$