The Bojanov – Naidenov problem for functions with nonsymmetric restrictions on the highest derivative
For given $r \in \bfN , p, \alpha , \beta , \mu > 0$, we solve the extreme problems $$\int^b_ax^q_{\pm} (t)dt \rightarrow \mathrm{s}\mathrm{u}\mathrm{p}, q \geq p,$$ in the set of pairs $(x, I)$ of functions $x \in L^r_{\infty}$ and intervals $I = [a, b] \subset R$ satisfying the inequ...
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| Дата: | 2019 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2019
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/1445 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| Резюме: | For given $r \in \bfN , p, \alpha , \beta , \mu > 0$, we solve the extreme problems
$$\int^b_ax^q_{\pm} (t)dt \rightarrow \mathrm{s}\mathrm{u}\mathrm{p}, q \geq p,$$
in the set of pairs $(x, I)$ of functions $x \in L^r_{\infty}$ and intervals $I = [a, b] \subset R$ satisfying the inequalities $\beta \leq x(r)(t) \leq \alpha$ for almost all $t \in R$ , the conditions $L(x_{\pm})p \leq L\bigl(( \varphi^{\alpha ,\beta}_{\lambda ,r}) \bigr)_p$, and the corresponding condition $\mu\Bigl(\mathrm{s}\mathrm{u}\mathrm{p} \mathrm{p}_{[a,b]}x_{+}\Bigr) \leq \mu$ or
$\mu \Bigl( \mathrm{s}\mathrm{u}\mathrm{p} \mathrm{p}_{[a,b]}x \Bigr) \leq \mu$, where
$$L(x)p := \mathrm{s}\mathrm{u}\mathrm{p}
\Bigl\{ \| x\| L_{p[a,b]} : a, b \in R , | x(t)| > 0, t \in (a, b)\Bigr\},$$
$\mathrm{s}\mathrm{u}\mathrm{p} \mathrm{p}_{[a,b]}x_{\pm} := \{ t \in [a, b] : x_{\pm} (t) > 0\} , \varphi^{\alpha ,\beta}_{\lambda ,r}$ is the nonsymmetric $(2\pi /\lambda)$-periodic Euler spline of order $r$. As a
consequence, we solve the same problems for the intermediate derivatives $x(k)_{\pm} , k = 1,..., r_1,$ with $q \geq 1$. |
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