The Bojanov – Naidenov problem for functions with nonsymmetric restrictions on the highest derivative

Abstract

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 $q \geq 1$.