# zbMATH — the first resource for mathematics

On mappings of bounded variation. (English) Zbl 0940.26009
Summary: We present properties of mappings of bounded variation defined on a subset of the real line with values in metric and normed spaces and show that major aspects of the theory of real-valued functions of bounded variation remain valid in this case. In particular, we prove the structure theorem and obtain the continuity properties of these mappings as well as jump formulas for the variation. We establish the existence of Lipschitz continuous geodesic paths and prove an analog of the well-known Helly selection principle. For normed space-valued smooth mappings we obtain the usual integral formula for the variation without the completeness assumption on the space of values. As an application of our theory we show that compact set-valued mappings (= multifunctions) of bounded variation admit regular selections of bounded variation.

##### MSC:
 26A45 Functions of bounded variation, generalizations 26E25 Set-valued functions 49J53 Set-valued and variational analysis 54C60 Set-valued maps in general topology 54C65 Selections in general topology
Full Text:
##### References:
 [1] J.-P. Aubin and A. Cellina, Differential inclusions: Set-valued maps and viability theory.Springer-Verlag, Berlin, Heidelberg, New York, 1984. · Zbl 0538.34007 [2] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces.Noordhoff Intern. Publ. Leyden, The Netherlands, 1976. [3] V. Barbu and Th. Precupanu, Convexity and optimization in Banach spaces.Sijthoff and Noordhoff Intern. Publ., The Netherlands, 1978. · Zbl 0379.49010 [4] N. Bourbaki, Éléments de mathématique. Livre IV. Fonctions d’une variable reélle (Théorie élémentaire).Hermann, Paris, 1962. [5] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions.Lect. Notes Math. 580 (1977). · Zbl 0346.46038 [6] V. V. Chistyakov, Variation (Lecture Notes). (Russian)Univ. of Nizhny Novgorod, Nizhny Novgorod, 1992. [7] J. Dieudonné, Foundations of modern analysis.Academic Press, New York and London, 1960. [8] H. Federer, Geometric measure theory.Springer-Verlag, Berlin-Heidelberg-New York, 1969. · Zbl 0176.00801 [9] G. B. Folland, Real analysis: Modern techniques and their applications.Wiley-Interscience, New York, 1984. · Zbl 0549.28001 [10] H. Hermes, On continuous and measurable selections and the existence of solutions of generalized differential equations.Proc. Am. Math. Soc.,29 (1971), No. 3, 535–542. · Zbl 0214.09802 [11] N. Kikuchi and Y. Tonita, On the absolute continuity of multifunctions and orientor fields.Funkc. Ekvac. 14 (1971), No. 3, 161–170. · Zbl 0248.49023 [12] Y. Komura, Nonlinear semigroups in Hilbert spaces.J. Math. Soc. Japan 19 (1967), 493–507. · Zbl 0163.38302 [13] E. B. Lee and L. Markus, Foundations of optimal control theory.John Wiley and Sons, New York, London, Sydney, 1967. · Zbl 0159.13201 [14] E. A. Michael, Continuous slections. I.Ann. Math. 63 (1956), No. 2, 361–382. · Zbl 0071.15902 [15] –, Continuous selections. II.Ann. Math. 64 (1956) 562–580. · Zbl 0073.17702 [16] –, Continuous selections. III.Ann. Math. 65 (1957), 375–390. · Zbl 0088.15003 [17] B. Sh. Mordukhovich, Approximations methods in the problems of optimization and control. (Russian),Nauka, Moscow, 1988. [18] S. B. Nadler, Multivalued contraction mappings.Pacif. J. Math. 30 (1969), 475–488. · Zbl 0187.45002 [19] I. P. Natanson, Theory of functions of a real variable.Ungar, New York, 1961. [20] L. Schwartz, Analyse mathématique. Vol. 1.Hermann, Paris, 1967.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.