Functions of Bounded Variation as Solutions of Differential Systems

1. This paper investigates certain linear differential systems whose solutions are required to be merely functions of bounded variation. Subsequently the equivalence of such problems with appropriate integral equations with Lebesgue integrals and, also, their equivalence to certain integral equations with Stieltjes integrals is established. This study is a generalization of differential systems with interface conditions [ó]. For such problems the usual condition that the solution be absolutely continuous is too stringent; in fact the interface conditions usually require a solution to have a jump discontinuity at each interface. Matrix notation is used extensively. The elements of a matrix A are denoted by A a. Any analytic properties such as continuity, bounded variation, or differentiability, postulated for a matrix are understood to be assumed for each element separately. The inverse of a matrix A, if it has one, is denoted by A~l. Let 9„(7) denote the class of all nonsingular nXn matrices G which are absolutely continuous on the real interval I, and let (Rn(I) denote the class of all nXn matrices R which have the following properties.