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

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 $$$\|x^{(r)}\|_{\infty} \leqslant 1$$$, $$$\|x\|_{p,\delta} \leqslant \| \varphi_{\lambda,r} \|_{p,\delta}$$$, $$$\delta \in (0,\pi / \lambda)$$$.