In 1995, in the monograph by A. M. Samoilenko and N. A. Perestyuk [Impulsive differential equations. Transl. from the Russian by Yury Chapovsky. Singapore: World Scientific (1995; Zbl 0837.34003)], necessary and sufficient conditions for the exponential stability have been established for the first order impulsive differential equations
\[ x'(t)=Ax(t);\quad t\neq t_i,\tag{1a} \]
\[ x(t^+_i)-x(t^-_i)=Bx(t^-_i)\tag{1b} \]
for \(i=1,2,\ldots\). They used the basic properties of the Cauchy matrix to prove their results. For example, the solution of (1a)–(1b) can be explained by the equation
\[ X(t,t_0)=e^{A(t-t_i)}\prod_{t<t_j<t_j}(I+B)e^{A(t_j-t_j-1)},\qquad t_i<t\leq t_{i+1}. \]
In the present paper, the authors modify the approach given in [loc cit.] to the impulsive equations, they obtain some stability results.
The first result concerns the asymptotic stability of equations of the form:
\[ \begin{aligned} &x'(t)=Ax(t);\qquad t\in[s_i,t_{i+1}],\\ &x(t^+_i)=Bx(t^-_i),\\ &x(t)=Bx(t^-_i);\qquad t\in(t_i,s_i],\\ &x(s^+_i)=x(s^-_i) \end{aligned} \]
for \(i=1,2,\ldots\), the second concerns the asymptotic stability of the equations of the form:
\[ \begin{aligned} &z'(t)=Az(t)+P(t)z(t);\qquad t\in[s_i,t_{i+1}],\\ &z(t^+_i)=Bz(t^-_i)+I_iz(t^-_i),\\ &z(t)=Bz(t^-_i)+I_iz(t^-_i);\qquad t\in(t_i,s_i],\\ &z(s^+_i)=z(s^-_i)\end{aligned} \]
for \(i=1,2,\ldots\). Finally, they investigate the existence, uniqueness of solutions and the Ulam-Hyers-Rassias stability of the more general impulsive equations of the form: \[ \begin{aligned} &y'(t)=Ay(t)+g(t,y(t));\qquad t\in[s_i,t_{i+1}],\\ &y(t^+_i)=By(t^-_i)+b_i,\\ &y(t)=By(t^-_i)+b_i;\qquad t\in(t_i,s_i],\\ &y(s^+_i)=y(s^-_i)\end{aligned} \]
for \(i=1,2,\ldots\).