On first order ordinary differential equations with infinitely many state dependent impulses (Q1374844)

The following system of differential equations with infinitely many state-dependent impulses is considered \[ x'(t)= f(x(t), t)+ \sum_{j\in J}\alpha_j(x(\tau_j)) \delta(t- \tau_j)\quad (t\in (0,T)),\tag{1} \] (2) \(x(0)= x_0\), where \(T>0\), \(x_0\in\mathbb{R}^n\), \(x: [0,T]\to \mathbb{R}^n\) is an unknown function, \(f:\mathbb{R}^n\times [0,T]\to \mathbb{R}^n\) is a given mapping, \(\delta\) is the Dirac delta function, \(J\) is a countable set of indices, \(\alpha_j: \mathbb{R}^n\to \mathbb{R}^n\) is a given mapping and \(\tau_j\) is a number from the interval \([0,T]\) for each \(j\in J\). It is assumed that the sum of the absolute values of all impulses is finite. Using the method of successive approximation it is shown that the problem (1)--(2) has a unique solution in the space of vector functions with bounded variation on the interval \([0,T]\).