The Bojanov-Naidenov problem for the functions with non-symmetric restrictions on the oldest derivative
For given $r \in \NN$, $p, \alpha, \beta, \mu > 0$, we solve theextremal problems$$\int\nolimits_{a}^{b} x^q_{\pm}(t)dt \to \sup, \;\;\;\; q\ge p,\;$$on the set of pair $(x, I)$ functions $x\in L^r_{\infty}$ andintervals $I=[a,b] \subset \RR$ satisfying the inequalities $ -\beta\le x^{(r)...
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| Date: | 2021 |
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Institute of Mathematics, NAS of Ukraine
2021
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/254 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860506984160165888 |
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| author | Кофанов, Володимир Олександрович |
| author_facet | Кофанов, Володимир Олександрович |
| author_sort | Кофанов, Володимир Олександрович |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2021-01-29T15:36:34Z |
| description | For given $r \in \NN$, $p, \alpha, \beta, \mu > 0$, we solve theextremal problems$$\int\nolimits_{a}^{b} x^q_{\pm}(t)dt \to \sup, \;\;\;\; q\ge p,\;$$on the set of pair $(x, I)$ functions $x\in L^r_{\infty}$ andintervals $I=[a,b] \subset \RR$ satisfying the inequalities $ -\beta\le x^{(r)}(t) \le \alpha $ for almost everywhere $t \in \RR $ andthe both of conditions $ L(x_{\pm})_p \leL(\varphi_{\lambda,r}^{\alpha, \beta})_{\pm})_p, $ and such that $\mu \left( {\rm supp}_{[a, b]} x_{+} \right) \le \mu$ or $ \mu\left( {\rm supp}_{[a, b]} x_{-} \right) \le \mu$, where$$ L(x)_p:=\sup\left\{\left\|x \right\|_{L_p[a,b]}: \; a, b \in \RR,\;|x(t)|>0,\;t\in (a, b) \right\},$$$ {\rm supp}_{[a, b]} x_{\pm}:= \{t\in [a, b]: x_{\pm}(t) > 0\} $and $\varphi_{\lambda,r}^{\alpha, \beta}$ is the nonsymmetric$(2\pi/\lambda)$-periodic spline of Euler of order $r$. Inparticular, we solve the same problems for the intermediatederivatives $x^{(k)}_{\pm}$, $k=1,...,r-1$, with $q \ge 1$.
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| last_indexed | 2026-03-24T02:02:06Z |
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| spelling | umjimathkievua-article-2542021-01-29T15:36:34Z The Bojanov-Naidenov problem for the functions with non-symmetric restrictions on the oldest derivative Задача Боянова-Найденова для функций с несимметричными ограничениями на старшую производную Задача Боянова-Найдьонова для функцій з несиметричними обмеженнями на старшу похідну Кофанов, Володимир Олександрович For given $r \in \NN$, $p, \alpha, \beta, \mu > 0$, we solve theextremal problems$$\int\nolimits_{a}^{b} x^q_{\pm}(t)dt \to \sup, \;\;\;\; q\ge p,\;$$on the set of pair $(x, I)$ functions $x\in L^r_{\infty}$ andintervals $I=[a,b] \subset \RR$ satisfying the inequalities $ -\beta\le x^{(r)}(t) \le \alpha $ for almost everywhere $t \in \RR $ andthe both of conditions $ L(x_{\pm})_p \leL(\varphi_{\lambda,r}^{\alpha, \beta})_{\pm})_p, $ and such that $\mu \left( {\rm supp}_{[a, b]} x_{+} \right) \le \mu$ or $ \mu\left( {\rm supp}_{[a, b]} x_{-} \right) \le \mu$, where$$ L(x)_p:=\sup\left\{\left\|x \right\|_{L_p[a,b]}: \; a, b \in \RR,\;|x(t)|>0,\;t\in (a, b) \right\},$$$ {\rm supp}_{[a, b]} x_{\pm}:= \{t\in [a, b]: x_{\pm}(t) > 0\} $and $\varphi_{\lambda,r}^{\alpha, \beta}$ is the nonsymmetric$(2\pi/\lambda)$-periodic spline of Euler of order $r$. Inparticular, we solve the same problems for the intermediatederivatives $x^{(k)}_{\pm}$, $k=1,...,r-1$, with $q \ge 1$.   Для заданных $r \in \NN$, $p, \alpha, \beta, \mu > 0$, решеныэкстремальные задачи$$\int\nolimits_{a}^{b} x^q_{\pm}(t)dt \to \sup, \;\;\;\; q\ge p,\;$$   на классе пар $(x, I)$ функций $x\in L^r_{\infty}$ и отрезков$I=[a,b] \subset \RR$, для которых выполнены неравенства  $ -\beta \lex^{(r)}(t) \le \alpha $ почти для всех  $t \in \RR $, оба условия $L(x_{\pm})_p \le L(\varphi_{\lambda,r}^{\alpha, \beta})_{\pm})_p $, и соответствующее требование $ \mu \left( {\rm supp}_{[a, b]} x_{+} \right)\le \mu$ или \mu \left( {\rm supp}_{[a, b]} x_{-} \right) \le\mu$, где$$ L(x)_p:=\sup\left\{\left\|x \right\|_{L_p[a,b]}: \; a, b \in \RR,\;|x(t)|>0,\;t\in (a, b) \right\},$$$ {\rm supp}_{[a, b]} x_{\pm}:= \{t\in [a, b]: x_{\pm}(t) > 0\} $,$\varphi_{\lambda,r}^{\alpha, \beta}-$ несимметричный$(2\pi/\lambda)$-периодический  сплайн Эйлера порядка $r$. Как следствиерешены те же задачи для промежуточных производных$x^{(k)}_{\pm}$, $k=1,...,r-1$, при $q \ge 1$.   Для заданих $r \in \NN$, $p, \alpha, \beta, \mu > 0$, розв'язанiекстремальнi проблеми$$\int\nolimits_{a}^{b} x^q_{\pm}(t)dt \to \sup, \;\;\;\; q\ge p,\;$$на класi пар $(x, I)$ функцiй $x\in L^r_{\infty}$ та вiдрiзкiв$I=[a,b] \subset \RR$, для яких виконанi нерiвностi $ -\beta \lex^{(r)}(t) \le \alpha $ майже для всiх $t \in \RR $, обидвi умови $L(x_{\pm})_p \le L(\varphi_{\lambda,r}^{\alpha, \beta})_{\pm})_p $,та вiдповiдна вимога $ \mu \left( {\rm supp}_{[a, b]} x_{+} \right)\le \mu$ або $ \mu \left( {\rm supp}_{[a, b]} x_{-} \right) \le\mu$, де$$ L(x)_p:=\sup\left\{\left\|x \right\|_{L_p[a,b]}: \; a, b \in \RR,\;|x(t)|>0,\;t\in (a, b) \right\},$$$ {\rm supp}_{[a, b]} x_{\pm}:= \{t\in [a, b]: x_{\pm}(t) > 0\} $,$\varphi_{\lambda,r}^{\alpha, \beta}-$ несиметричний$(2\pi/\lambda)$-перiодичний сплайн Ейлера порядку $r$. Як наслiдок,розв'язанi тi ж самi екстремальнi проблеми для промiжних похiдних$x^{(k)}_{\pm}$, $k=1,...,r-1$, при $q \ge 1$.   Institute of Mathematics, NAS of Ukraine 2021-01-29 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/254 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 3 (2019); 368-381 Український математичний журнал; Том 71 № 3 (2019); 368-381 1027-3190 Copyright (c) 20219 Володимир Олександрович Кофанов |
| spellingShingle | Кофанов, Володимир Олександрович The Bojanov-Naidenov problem for the functions with non-symmetric restrictions on the oldest derivative |
| title | The Bojanov-Naidenov problem for the functions with non-symmetric restrictions on the oldest derivative |
| title_alt | Задача Боянова-Найденова для функций с несимметричными ограничениями на старшую производную Задача Боянова-Найдьонова для функцій з несиметричними обмеженнями на старшу похідну |
| title_full | The Bojanov-Naidenov problem for the functions with non-symmetric restrictions on the oldest derivative |
| title_fullStr | The Bojanov-Naidenov problem for the functions with non-symmetric restrictions on the oldest derivative |
| title_full_unstemmed | The Bojanov-Naidenov problem for the functions with non-symmetric restrictions on the oldest derivative |
| title_short | The Bojanov-Naidenov problem for the functions with non-symmetric restrictions on the oldest derivative |
| title_sort | bojanov-naidenov problem for the functions with non-symmetric restrictions on the oldest derivative |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/254 |
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