Geodesic
On the theory of set-valued maps of bounded variation of one real variable
Sbornik. Mathematics, Tome 189 (1998) no. 5, pp. 797-819 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

(Set-valued) maps of bounded variation in the sense of Jordan defined on a subset of the real line and taking values in metric or normed linear spaces are studied. A structure theorem (more general than the Jordan decomposition) is proved for such maps; an analogue of Helly's selection principle is established. A compact set-valued map into a Banach space that is a map of bounded variation (or a Lipschitz or an absolutely continuous map) is shown to have a continuous selection of bounded variation (respectively, Lipschitz or absolutely continuous selection). 
@article{SM_1998_189_5_a7,
     author = {V. V. Chistyakov},
     title = {On the theory of set-valued maps of bounded variation of one real variable},
     journal = {Sbornik. Mathematics},
     pages = {797--819},
     year = {1998},
     volume = {189},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_5_a7/}
}
TY  - JOUR
AU  - V. V. Chistyakov
TI  - On the theory of set-valued maps of bounded variation of one real variable
JO  - Sbornik. Mathematics
PY  - 1998
SP  - 797
EP  - 819
VL  - 189
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/SM_1998_189_5_a7/
LA  - en
ID  - SM_1998_189_5_a7
ER  - 
%0 Journal Article
%A V. V. Chistyakov
%T On the theory of set-valued maps of bounded variation of one real variable
%J Sbornik. Mathematics
%D 1998
%P 797-819
%V 189
%N 5
%U http://geodesic.mathdoc.fr/item/SM_1998_189_5_a7/
%G en
%F SM_1998_189_5_a7
V. V. Chistyakov. On the theory of set-valued maps of bounded variation of one real variable. Sbornik. Mathematics, Tome 189 (1998) no. 5, pp. 797-819. http://geodesic.mathdoc.fr/item/SM_1998_189_5_a7/

[1] Natanson I. P., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974 | MR

[2] Shvarts L., Analiz, T. 1, Mir, M., 1972

[3] Wiener N., “The quadratic variation of a function and its Fourier coefficients”, Massachusetts J. Math. and Phys., 3 (1924), 72–94 | Zbl

[4] Chistyakov V. V., Galkin O. E., “On maps of bounded $p$-variation with $p>\nobreak 1$”, Positivity (to appear) | MR

[5] Young L. C., “Sur une généralisation de la notion de variation de puissance $p$-ième bornée au sens de N. Wiener, et sur la convergence des séries de Fourier”, C. R. Acad. Sci. Paris, 204:7 (1937), 470–472 | Zbl

[6] Chistyakov V. V., “On mappings of bounded variation”, J. Dynam. Control Systems, 3:2 (1997), 1–30 | DOI | MR

[7] Chistyakov V. V., Variatsiya, Izd-vo NNGU, Nizhnii Novgorod, 1992

[8] Castaing C., Valadier M., Convex Analysis and Measurable Multifunctions, Lecture Notes in Math., 580, Springer-Verlag, Berlin, 1977 | MR | Zbl

[9] Aubin J.-P., Cellina A., Differential Inclusions: Set-valued Maps and Viability Theory, Grundlehren Math. Wiss., 264, Springer-Verlag, Berlin, 1984 | MR | Zbl

[10] Hermes H., “On continuous and measurable selections and the existence of solutions of generalized differential equations”, Proc. Amer. Math. Soc., 29:3 (1971), 535–542 | DOI | MR

[11] Kikuchi N., Tonita Y., “On the absolute continuity of multifunctions and orientor fields”, Funkcial. Ekvac., 14:3 (1971), 161–170 | MR | Zbl

[12] Mordukhovich B. Sh., Metody approksimatsii v zadachakh optimizatsii i upravleniya, Nauka, M., 1988 | MR | Zbl

[13] Zhu Qiji, “Single valued representation of absolutely continuous set-valued mappings”, Kexue Tongbao, 31 (1986), 443–446 | MR | Zbl

[14] Nadler S. B., “Multi-valued contraction mappings”, Pacific J. Math., 30 (1969), 475–488 | MR | Zbl