On the stability of a set of systems of impulsive equations (Q656253)

The author considers a nonlinear impulsive differential equation \[ D_HX= F(t,X),\quad t\neq t_k,\tag{1} \] \[ X(t^+_k)= I_k(X(t_k)),\quad t= t_k,\tag{2} \] where \(X\in{\mathcal K}_C(\mathbb{R}^n)\), \(F\in PC(\mathbb{R}_+\times{\mathcal K}_C(\mathbb{R}^n,{\mathcal K}_C(\mathbb{R}^n))\), \(I:{\mathcal K}_C(\mathbb{R}^n)\to{\mathcal K}_C(\mathbb{R}^n)\) for any \(k= 1,2,\dots\) and \(t_k< t_{k+1}\), \(t_k\to\infty\) \((k\to\infty)\), and \(X(t;t_0, X_0)= X_k(t; t_k,X^+_k)\), \(t_{k-1}< t\leq t_{k+1}\) solves (1), (2) on \(t_{k-1}< t\leq t_{k+1}\). Along with the set of impulsive differential equations (1), (2) the author considers the impulsive scalar equation \[ {dw\over dt}= g(t,w),\quad t\neq t_k,\tag{3} \] \[ w(t^+_k)= \phi_k(w(t_k)),\quad t= t_k.\tag{4} \] The maximal solution of (3), (4) is a function \(r(t; t_0,w_0)\) defined as \(r(t; t_0,w_0)= r_k(t; t_k,t^+_k)\) on \(t_k< t\leq t_{k+1}\), and \(w(t; t_0,w_0)\leq r(t; t_0,w_0)\), \(t= t_k\) for all \(t\in\mathbb{R}\) and all solutions \(w(t;t_0, w_0)\) of (3) (4). A connection between the solutions of (1), (2) and the maximal solutions of (3), (4) is proved.