Ulam's type stability of impulsive ordinary differential equations (Q441976)

The authors consider the following impulsive differential equations NEWLINE\[NEWLINEx'(t)= f(t,x(t)),\quad t\in J':= J\setminus\{t_1,\dotsc, t_m\},\quad J:= [0,T],\;T> 0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\Delta x(t_k)= I_k(x(t^-_k)),\quad k= 1,2,\dotsc, m,NEWLINE\]NEWLINE where \(f: J\times\mathbb{R}\to \mathbb{R}\) is continuous, \(I_k: \mathbb{R}\to \mathbb{R}\), \(T< \infty\), and NEWLINE\[NEWLINEx(t^+_k)= \lim_{\varepsilon\to 0^+} x(t_k+\varepsilon)\quad\text{and} \quad x(t^-_k)= \lim_{\varepsilon\to 0^-} x(t_k+ \varepsilon).NEWLINE\]NEWLINE The authors introduce four types of Ulam's stability and find sufficient conditions for them.