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 :$$x{\sigma }_{q}y=y\text{ou}x{\sigma }_{q}y=y+x$$

Nous donnons une version $q$-analogue de l’asymptotique Gevrey et de la sommabilité de Borel, dues respectivement à G. Watson et E. Borel et systématiquement développées depuis une quinzaine d’années par J.-P. Ramis, Y. Sibuya, etc. Le but de ces auteurs était l’étude des équations différentielles dans le champ complexe. De même notre but est l’étude des équations aux $q$-différences dans le champ complexe, dans la ligne de G.D. Birkhoff et W.J. Trjitzinsky.Plus précisément, nous introduisons une nouvelle...

We investigate the existence and uniqueness of entire solutions of order zero of the nonlinear q-difference equation of the form fⁿ(z) + L(z) = p(z), where p(z) is a polynomial and L(z) is a linear differential-q-difference polynomial of f with small growth coefficients. We also study the zeros distribution of some special type of q-difference polynomials.

In this paper, we study a new class of three-point boundary value problems of nonlinear second-order q-difference inclusions. Our problems contain different numbers of q in derivatives and integrals. By using fixed point theorems, some new existence results are obtained in the cases when the right-hand side has convex as well as noncovex values.

On commence par présenter une méthode de résolution d’une famille de systèmes fuchsiens d’opérateurs de pseudo-dérivations associées à une famille à deux paramètres d’homographies, qui unifie et généralise les cas connus des systèmes différentiels, aux différences ou aux $q$-différences. Nous traitons ensuite dans cette famille des problèmes de confluence que l’on peut voir comme des problèmes de continuité en ces deux paramètres.

Choose $q\in ℂ$ with $0\<\left|q\right|\<1$. The main theme of this paper is the study of linear $q$-difference equations over the field $K$ of germs of meromorphic functions at $0$. A systematic treatment of classification and moduli is developed. It turns out that a difference module $M$ over $K$ induces in a functorial way a vector bundle $v\left(M\right)$ on the Tate curve $E_q:={ℂ}^{*}/q^{\mathbb{Z}}$ that was known for modules with ”integer slopes“, [Saul, 2]). As a corollary one rediscovers Atiyah’s classification $\left(\right[\mathrm{A}t\left]\right)$ of the indecomposable vector bundles on the complex Tate...

In this paper, we investigate the problem of global output-feedback regulation for a class of switched nonlinear systems with unknown linear growth condition and uncertain output function. Based on the backstepping method, an adaptive output-feedback controller is designed to guarantee that the state of the switched nonlinear system can be globally regulated to the origin while maintaining global boundedness of the resulting closed-loop switched system under arbitrary switchings. A numerical example...

In this paper, the problem of global finite-time stabilization via output-feedback is investigated for a class of stochastic nonlinear cascaded systems (SNCSs). First, based on the adding a power integrator technique and the homogeneous domination approach, a global output-feedback finite-time control law is constructed for the driving subsystem. Then, based on homogeneous systems theory, it is shown that under some mild conditions the global finite- time stability in probability of the driving...