On continuous and measurable selections and the existence of solutions of generalized differential equations

Abstract:

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