Solution of Bojanov-Naidenov problem with constraints for the norm $$$\|x\|_{p,\delta} = \sup \bigl\{ \| x \|_{L_p[a;b]} \colon a,b\in \mathbb{R}, b-a\leqslant \delta \bigr\}$$$

V.A. Kofanov (Oles Honchar Dnipropetrovsk National University), http://orcid.org/0000-0003-0392-2257

Abstract


For given $$$r\in \mathbb{N}$$$; $$$p,\lambda > 0$$$ and fixed interval $$$[a;b] \subset \mathbb{R}$$$ we solve the extremal problems 1) $$$\int\limits_a^b |x(t)|^q dt \rightarrow \sup$$$, $$$q > p$$$, 2) $$$\int\limits_a^b |x^{(k)}(t)|^q dt \rightarrow \sup$$$, $$$q \geqslant 1$$$, $$$k\in \mathbb{N}$$$, $$$k < r$$$, on the set of functions $$$f\in L^r_{\infty}$$$ such that $$$\|x^{(r)}\|_{\infty} \leqslant 1$$$, $$$\|x\|_{p,\delta} \leqslant \| \varphi_{\lambda,r} \|_{p,\delta}$$$, $$$\delta \in (0,\pi / \lambda)$$$.

Keywords


inequalities of various metrics; Kolmogorov-type inequalities; Bojanov-Naidenov problem

References


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ISSN (Online): 2518-7996
ISSN (Print): 2312-9557