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DOI: https://doi.org/10.35634/vm220301

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.

Received: 04.02.2022
Accepted: 19.07.2022

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Citation: V. N. Baranov, V. I. Rodionov, “On nonlinear metric spaces of functions of bounded variation”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 32:3 (2022), 341–360

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This publication is cited in the following articles:

1. M. B. Zvereva, “Model deformatsii sistemy stiltesovskikh strun s nelineinym usloviem”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 32:4 (2022), 528–545      
2. V. Ya. Derr, “O nekotorykh svoistvakh *-integrala”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 33:1 (2023), 66–89      
3. V. N. Baranov, V. I. Rodionov, A. G. Rodionova, “O banakhovykh prostranstvakh pravilnykh funktsii mnogikh peremennykh. Analog integrala Rimana”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 33:3 (2023), 387–401    

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles