On Continuous and Measurable Selections and the Existence of Solutions of Generalized Differential Equations

Let <!-- MATH $\mathcal{C}({B^n})$ --> denote the space of nonempty compact subsets of some bounded set in Euclidean n dimensional space , topologized with the Hausdorff metric topology. The existence of a solution to the initial value problem for the generalized differential equation <!-- MATH $dx(t)/dt \in R(x(t))$ --> is shown under the assumption that <!-- MATH $R:{E^n} \to \mathcal{C}({B^n})$ --> has bounded variation in some neighborhood of the initial value, and under a less restrictive condition on the variation of R. Included are continuous and Lipschitz continuous selection results for mappings <!-- MATH $Q:{E^1} \to \mathcal{C}({B^n})$ --> which are, respectively, of bounded variation and Lipschitz continuous.