Bounded variation evolution equations

AbstractA “fundamental theory” is presented for the equation x(t) = ∫0t q(x(s), s) ds where the integral is Stieltjes and x is of bounded variation with values in Rn. This includes the ordinary differential equation (o.d.e.) case with impulses. The principal conclusion is that the corresponding conditions for Carathéodory's o.d.e. problem carry over almost unchanged to the more general case. Areas treated include existence (local and global), uniqueness, dependence, integral funnels, stability, and Picard iterates