A selection principle for mappings of bounded variation

E. Helly's selection principle states that an infinite bounded family of re al functions on the closed interval, which is bounded in variation, contain s a pointwise convergent sequence whose limit is a function of bounded vari ation. We extend this theorem to metric space valued mappings of bounded va riation. Then we apply the extended Helly selection principle to obtain the existence of regular selections of (non-convex) set-valued mappings: any s et-valued mapping from an interval of the real line into nonempty compact s ubsets of a metric space, which is of bounded variation with respect to the Hausdorff metric, admits a selection of bounded variation. Also, we show t hat a compact-valued set-valued mapping which is Lipschitzian, absolutely c ontinuous, or of bounded Riesz Phi-variation admits a selection which is Li pschitzian, absolutely continuous, or of bounded Riesz Phi-variation, respe ctively. (C) 2000 Academic Press.