We prove that for a given impulsive dynamical system there exists an isomorphism of the basic dynamical system such that in the new system equipped with the same impulse function each impulsive trajectory is global, i.e. the resulting dynamics is defined for all positive times. We also prove that for a given impulsive system it is possible to change the topology in the phase space so that we may consider the system as a semidynamical system (without impulses).

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.

In this paper we discuss the existence of oscillatory and nonoscillatory solutions of first order impulsive differential inclusions. We shall rely on a fixed point theorem of Bohnenblust-Karlin combined with lower and upper solutions method.

In this paper we discuss the existence of oscillatory and nonoscillatory solutions for first order impulsive dynamic inclusions on time scales. We shall rely of the nonlinear alternative of Leray-Schauder type combined with lower and upper solutions method.