Unique Solutions of Discontinuous O.D.E’s in Banach Spaces

shows that a Carathéodory solution may not exist, even locally in time. Various assumptions on the discontinuous vector field f , which guarantee the existence or the uniqueness of local solutions, have been studied in the literature [1,2,4,5,8]. In essence, for the well-posedness of the Cauchy problem one needs that the vector field f should point transversally to the surfaces where discontinuities occur. As shown in [1], this transversality condition is best formulated as an assumption of bounded directional variation. The main result in [1] shows that the Cauchy problem (1.1) has a unique solution provided that there exists a cone Γ ⊂ IR with the two properties: