``Truncation'' of countable systems of differential equations with impulse effect (Q909083)

The paper deals with the countable system of differential equations with impulse effect \[ (1)\quad dx_ k/dt=\epsilon X_ k(t,x_ 1,x_ 2,...);\quad \Delta x_ k|_{t_ i}=\epsilon I_{ik}(x_ 1,x_ 2,....),\quad i,k=1,2,..., \] where roughly speaking \(X_ k(t,x_ 1,x_ 2,..)\) and \(I_{ik}(x_ 1,x_ 2,...)\) are continuous, satisfy the generalized Cauchy-Lipschitz conditions and \((X_ k(t,0,0,...)|_ 0\) are majorized on [0,T] by a piecewise continuous function B(t) and \(| I_{ik}(0,0,...)|_ 0\leq L_ i<\infty.\) \(| \cdot |_ 0\) is the norm in Banach space of piecewise continuous functions defined on [0,T]. It is assumed that \(\{t_ i\}^{\infty}_{i=1}\) has no finite limit points. The author considers the shortened system \[ (2)\quad \frac{dx_ k}{dt}=\epsilon X_ k(t,x_ 1,...,x_ n,0,...),\Delta x_ k|_{t_ i}=\epsilon I_{ik}(x_ 1,x_ 2,...,x_ n,0,0,...),\quad k=1,2,...,n \] and proves that the solutions \(u_{sn}(t,\epsilon)\), \(s=1,2,..\). of (2) converge to the solutions \(x_ s(t,\epsilon)\) of the system (1). In the second part the author considers the averaged shortened system corresponding to the system (2) and establishes the relation between the solutions of this systems and the solutions of the system (1).