Discontinuous Solutions to Ordinary Nonlinear Differential Equations in the Space of Functions of Bounded Variation

Abstract

In this chapter we continue studying differential equations with generalized right—hand sides, which was started in Section 2.6, but here we deal with distributions to be distributional derivatives of functions of bounded variation. We concern a definition of solutions to ordinary differential equations in the space of functions of bounded variation. We define discontinuous solutions by means of closing the absolutely continuous solutions set. It is shown that solutions defined in such a way satisfy some integral inclusion. The case in which such an inclusion turns into an integral equation is considered. A Cauchy formula for the discontinuous solutions to bilinear systems is obtained. Discontinuous solutions to neutral type nonlinear differential equations are discussed. In particular, we obtain a generalization of Gronwall—Bellman’s lemma for the space of functions of bounded variation.