In this paper, we investigate oscillation results for the solutions of impulsive conformable fractional differential equations of the form tkDαpttkDαxt+rtxt+qtxt=0,t≥t0,t≠tk,xtk+=akx(tk−),tkDαxtk+=bktk−1Dαx(tk−),k=1,2,…. $$\left\{\begin{array}{c}{t}_k{D}^{\alpha }\left(p\left(t\right)\left[t_k{D}^{\alpha }x\left(t\right)+r\left(t\right)x\left(t\right)\right]\right)+q\left(t\right)x\left(t\right)=0,t\ge t_0,t\ne t_k,\hfill \\ x\left(t_k^{+}\right)=a_{k}x\left(t_k^{-}\right),t_k{D}^{\alpha }x\left(t_k^{+}\right)=b_{k{t}_{k-1}}{D}^{\alpha }x\left(t_k^{-}\right),k=1,2,....\hfill \end{array}\right.$$ Some new oscillation results are obtained by using the equivalence transformation and the associated Riccati techniques.

On étudie les systèmes différentiels singulièrement perturbés de dimension 3 du type$$\{\begin{array}{ccc}\dot{x}\hfill & =& f(x,y,z,\epsilon ),\hfill \\ \dot{y}\hfill & =& g(x,y,z,\epsilon ),\hfill \\ \epsilon \dot{z}\hfill & =& h(x,y,z,\epsilon ),\hfill \end{array}$$où $f$, $g$, $h$ sont analytiques quelconques. Les travaux antérieurs étudiaient les points réguliers où la surface lente $h=0$ est transverse au champ rapide vertical. C’est le domaine d’application du théorème de Tikhonov. Dans d’autres travaux antérieurs, on étudiait les singularités de certains types : plis et fronces de la surface lente, ainsi que certaines singularités plus compliquées, analogues aux points tournants...

Sufficient conditions are given which guarantee that the linear transformation converting a given linear Hamiltonian system into another system of the same form transforms principal (antiprincipal) solutions into principal (antiprincipal) solutions.

We study the rank–2 distributions satisfying so-called Goursat condition (GC); that is to say, codimension–2 differential systems forming with their derived systems a flag. Firstly, we restate in a clear way the main result of[7] giving preliminary local forms of such systems. Secondly – and this is the main part of the paper – in dimension 7 and 8 we explain which constants in those local forms can be made 0, normalizing the remaining ones to 1. All constructed equivalences are explicit. ...