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

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