Geodesic
A generalization of the Helly theorem for functions with values in a~uniform space
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2010), pp. 41-54.

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In this paper we consider sequences of functions that are defined on a subset of the real line with values in a uniform Hausdorff space. For such sequences we obtain a sufficient condition for the existence of pointwise convergent subsequences. We prove that this generalization of the Helly theorem includes many results of the recent research. In addition, we prove that the sufficient condition is also necessary for uniformly convergent sequences of functions. We also obtain a representation for regular functions whose values belong to the uniform space.
Keywords: selection principle, pointwise convergence, proper functions with respect to a dense set, uniform space. 
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Yu. V. Tret'yachenko. A generalization of the Helly theorem for functions with values in a~uniform space. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2010), pp. 41-54. http://geodesic.mathdoc.fr/item/IVM_2010_5_a5/

[1] Helly E., “Über lineare Functionaloperationen”, Wien. Ber., 121 (1912), 265–297 | Zbl

[2] Natanson I. P., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974, 480 pp. | MR

[3] Chistyakov V. V., “Selections of bounded variation”, J. Appl. Anal., 10:1 (2004), 1–82 | MR

[4] Musielak J., Orlicz W., “On generalized variations. I”, Stud. Math., 18:1 (1959), 11–41 | MR | Zbl

[5] Waterman D., “On $\Lambda$-bounded variation”, Stud. Math., 57:1 (1976), 33–45 | MR | Zbl

[6] Gniłka S., “On the generalized Helly's theorem”, Funct. Approx. Comment. Math., 4 (1976), 109–112 | MR | Zbl

[7] Schramm M., “Functions of $\Phi$-bounded variation and Riemann–Stieltjes integration”, Trans. Amer. Math. Soc., 287:1 (1985), 49–63 | DOI | MR | Zbl

[8] Belov S. A., Chistyakov V. V., “A selection principle for mappings of bounded variation”, J. Math. Anal. Appl., 249:2 (2000), 351–366 | DOI | MR | Zbl

[9] Chistyakov V. V., “On mappings of bounded variation”, J. Dynam. Control Syst., 3:2 (1997), 261–289 | DOI | MR | Zbl

[10] Chistyakov V. V., “K teorii mnogoznachnykh otobrazhenii ogranichennoi variatsii odnoi veschestvennoi peremennoi”, Matem. sb., 189:5 (1998), 153–176 | MR | Zbl

[11] Chistyakov V. V., “Mappings of bounded variation with values in a metric space: generalizations”, J. Math. Sci., 100:6 (2000), 2700–2715 | DOI | MR

[12] Chistyakov V. V., “Generalized variation of mappings with applications to composition operators and multifunctions”, Positivity, 5:4 (2001), 323–358 | DOI | MR | Zbl

[13] Chistyakov V. V., “Metric space-valued mappings of bounded variation”, J. Math. Sci., 111:2 (2002), 3387–3429 | DOI | MR | Zbl

[14] Chistyakov V. V., “Mappings of generalized variation and composition operators”, J. Math. Sci., 110:2 (2002), 2455–2466 | DOI | MR | Zbl

[15] Chistyakov V. V., “O mnogoznachnykh otobrazheniyakh ogranichennoi obobschennoi variatsii”, Matem. zametki, 71:4 (2002), 611–632 | MR | Zbl

[16] Barbu V., Precupanu Th., Convexity and optimization in Banach spaces, Reidel, Dordrecht, 1986, 301 pp. | MR | Zbl

[17] Chistyakov V. V., “Printsip vybora dlya funktsii so znacheniyami v ravnomernom prostranstve”, Dokl. RAN, 409:5 (2006), 591–593 | MR | Zbl

[18] Chistyakov V. V., “Potochechnyi printsip vybora dlya funktsii odnoi peremennoi so znacheniyami v ravnomernom prostranstve”, Matem. tr., 9:1 (2006), 176–204 | MR

[19] Tretyachenko Yu. V., “Uslovie suschestvovaniya potochechno skhodyascheisya podposledovatelnosti”, Teoriya funktsii, ee prilozheniya i smezhnye voprosy, Tez. dokl., Tr. Matem. tsentra im. N. I. Lobachevskogo, 35, Kazan, 2007, 246–247

[20] Tretyachenko Yu. V., “Ob odnom printsipe vybora dlya potochechno ogranichennykh posledovatelnostei funktsii”, Lobachevskie chteniya – 2007, Tez. dokl., Tr. Matem. tsentra im. N. I. Lobachevskogo, 36, Kazan, 2007, 219–221

[21] Kelli Dzh., Obschaya topologiya, Nauka, M., 1981, 432 pp. | MR

[22] Dudley R. M., Norvaiša R., Differentiability of six operators on nonsmooth functions and $p$-variation, Springer-Verlag, Berlin, 1999, 277 pp. | MR | Zbl

[23] Jeffery R. L., “Generalized integrals with respect to functions of bounded variation”, Canad. J. Math., 10:4 (1958), 617–628 | MR

[24] Chanturiya Z. A., “Modul izmeneniya funktsii i ego primeneniya v teorii ryadov Fure”, DAN SSSR, 214:1 (1974), 63–66 | Zbl

[25] Chistyakov V. V., “A selection principle for functions of a real variable”, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, 53:1 (2005), 25–43 | MR | Zbl

[26] Chistyakov V. V., “The optimal form of selection principles for functions of a real variable”, J. Math. Anal. Appl., 310:2 (2005), 609–625 | MR | Zbl

[27] Tolstonogov A. A., “O nekotorykh svoistvakh prostranstva pravilnykh funktsii”, Matem. zametki, 35:6 (1984), 803–812 | MR | Zbl

[28] Tretyachenko Yu. V., Chistyakov V. V., “Printsip vybora dlya potochechno ogranichennykh posledovatelnostei funktsii”, Matem. zametki, 84:3 (2008), 428–439 | MR | Zbl

[29] Chistyakov V. V., Maniscalco C., Tretyachenko Y. V., “Variants of a selection principle for sequences of regulated and non-regulated functions”, Topics in Classical Analysis and Applications in Honor of Professor Daniel Waterman, World Scientific, Singapore, 2008, 45–72 | Zbl