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Russia Izhevsk
Year
2022
Volume
32
Issue
3
Pages
341-360
>>
Section Mathematics
Title On nonlinear metric spaces of functions of bounded variation
Author(-s) Baranov V.N.a, Rodionov V.I.a
Affiliations Udmurt State Universitya
Abstract In the first part of the paper, the nonlinear metric space $\langle\overline{\rm G}^\infty[a,b],d\rangle$ is defined and studied. It consists of functions defined on the interval $[a,b]$ and taking the values in the extended numeric axis $\overline{\mathbb R}$. For any $x\in\overline{\rm G}^\infty[a,b]$ and $t\in(a,b)$ there are limit numbers $x(t-0),x(t+0) \in\overline{\mathbb R}$ (and numbers $x(a+0),x(b-0)\in\overline{\mathbb R}$). The completeness of the space is proved. It is the closure of the space of step functions in the metric $d$. In the second part of the work, the nonlinear space ${\rm RL}[a,b]$ is defined and studied. Every piecewise smooth function defined on $[a,b]$ is contained in ${\rm RL}[a,b]$. Every function $x\in{\rm RL}[a,b]$ has bounded variation. All one-sided derivatives (with values in the metric space $\langle\overline{\mathbb R},\varrho\rangle$) are defined for it. The function of left-hand derivatives is continuous on the left, and the function of right-hand derivatives is continuous on the right. Both functions extended to the entire interval $[a,b]$ belong to the space $\overline{\rm G}^\infty[a,b]$. In the final part of the paper, two subspaces of the space ${\rm RL}[a,b]$ are defined and studied. In subspaces, promising formulations for the simplest variational problems are stated and discussed.
Keywords non-linear analysis, non-smooth analysis, bounded variation, one-sided derivative
UDC 517.988, 517.518.24
MSC 49J52, 26A45
DOI 10.35634/vm220301
Received 4 February 2022
Language Russian
Citation Baranov V.N., Rodionov V.I. On nonlinear metric spaces of functions of bounded variation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 3, pp. 341-360.
References
  1. Rodionov V.I. On the space of regular smooth functions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2011, issue 1, pp. 87-98 (in Russian). https://doi.org/10.20537/vm110109
  2. Schwartz L. Analyse mathématique. I, Paris: Hermann, 1967.
  3. Dieudonné J. Foundations of modern analysis, New York-London: Academic Press, 1960. https://zbmath.org/?q=an:0100.04201
  4. Tvrdý M. Regulated functions and the Perron-Stieltjes integral, Časopis pro Pěstování Matematiky, 1989, vol. 114, issue 2, pp. 187-209. https://doi.org/10.21136/CPM.1989.108713
  5. Rodionov V.I. On family of subspaces of the space of regulated functions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2009, issue 4, pp. 7-24 (in Russian). https://doi.org/10.20537/vm090402
  6. Hönig Ch.S. Volterra-Stieltjes integral equations. Functional analytic methods, linear constraints, Amsterdam: North-Holland, 1975. https://zbmath.org/?q=an:0307.45002
  7. Cichoń M., Cichoń K., Satco B. Measure differential inclusions through selection principles in the space of regulated functions, Mediterranean Journal of Mathematics, 2018, vol. 15, no. 4, article number: 148. https://doi.org/10.1007/s00009-018-1192-y
  8. Hanung U.M., Tvrdý M. On the relationships between Stieltjes type integrals of Young, Dushnik and Kurzweil, Mathematica Bohemica, 2019, vol. 144, no. 4, pp. 357-372. https://doi.org/10.21136/MB.2019.0015-19
  9. Federson M., Mesquita J.G., Slavík A. Basic results for functional differential and dynamic equations involving impulses, Mathematische Nachrichten, 2013, vol. 286, nos. 2-3, pp. 181-204. https://doi.org/10.1002/mana.201200006
  10. Monteiro G.A., Slavík A. Extremal solutions of measure differential equations, Journal of Mathematical Analysis and Applications, 2016, vol. 444, no. 1, pp. 568-597. https://doi.org/10.1016/j.jmaa.2016.06.035
  11. Monteiro G.A., Hanung U.M., Tvrdý M. Bounded convergence theorem for abstract Kurzweil-Stieltjes integral, Monatshefte für Mathematik, 2016, vol. 180, no. 3, pp. 409-434. https://doi.org/10.1007/s00605-015-0774-z
  12. Rodionov V.I. On a family of analogs of the Perron-Stieltjes integral, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2011, issue 3, pp. 95-106 (in Russian). https://doi.org/10.20537/vm110309
  13. Derr V.Ya. On the extension of a Rieman-Stieltjes integral, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 2, pp. 135-152 (in Russian). https://doi.org/10.20537/vm190201
  14. Derr V.Ya., Kim I.G. The spaces of regulated functions and differential equations with generalized functions in coefficients, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 1, pp. 3-18 (in Russian). https://doi.org/10.20537/vm140101
  15. Rodionov V.I. Analogue of the Cauchy matrix for system of quasi-integral equations with constant coefficients, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 2, pp. 44-62 (in Russian). https://doi.org/10.20537/vm120205
  16. Ioffe A.D., Tikhomirov V.M. Extension of variational problems, Trudy Moskovskogo Matematicheskogo Obshchestva, 1968, vol. 18, pp. 187-246 (in Russian). http://mi.mathnet.ru/eng/mmo209
  17. Miller B.M., Rubinovich E.Ya. Discontinuous solutions in the optimal control problems and their representation by singular space-time transformations, Automation and Remote Control, 2013, vol. 74, no. 12, pp. 1969-2006. https://doi.org/10.1134/S0005117913120047
  18. Voitushenko E.S. Weakly nonlinear impulsive problems for degenerate differential systems, Journal of Mathematical Sciences, 2017, vol. 220, no. 4, pp. 394-401. https://doi.org/10.1007/s10958-016-3191-5
  19. Maksimov V.P. The structure of the Cauchy operator to a linear continuous-discrete functional differential system with aftereffect and some properties of its components, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 1, pp. 40-51. https://doi.org/10.20537/vm190104
  20. Maksimov V.P. On a class of optimal control problems for functional differential systems, Proceedings of the Steklov Institute of Mathematics, 2019, vol. 305, suppl. 1, pp. S114-S124. https://doi.org/10.1134/S0081543819040126
  21. Maksimov V.P. Continuous-discrete dynamic models, Ufa Mathematical Journal, 2021, vol. 13, no. 3, pp. 95-103. https://doi.org/10.13108/2021-13-3-95
  22. Shilov G.E. Matematicheskii analiz. Funktsii odnogo peremennogo. Chasti 1-2 (Mathematical analysis. Functions of one variable. Parts 1-2), Moscow: Nauka, 1969.
  23. Kolmogorov A.N., Fomin S.V. Elementy teorii funktsii i funktsional'nogo analiza (Elements of the theory of functions and functional analysis), Moscow: Nauka, 1981.
  24. Rudin W. Functional analysis, New York: McGraw-Hill, 1973.
  25. Alekseev V.M., Tikhomirov V.M., Fomin S.V. Optimal'noe upravlenie (Optimal control), Moscow: Nauka, 1979.
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