The Bojanov-Naidenov problem for the functions with non-symmetric restrictions on the oldest derivative

Abstract

For given $r \in \NN$, $p, \alpha, \beta, \mu > 0$, we solve the
extremal problems
$$\int\nolimits_{a}^{b} x^q_{\pm}(t)dt \to \sup, \;\;\;\; q\ge p,\;$$
on the set of pair $(x, I)$ functions $x\in L^r_{\infty}$ and
intervals $I=[a,b] \subset \RR$ satisfying the inequalities $-\beta \le x^{(r)}(t) \le \alpha$ for almost everywhere $t \in \RR$ and
the both of conditions $L(x_{\pm})_p \le L(\varphi_{\lambda,r}^{\alpha, \beta})_{\pm})_p,$ and such that $\mu \left( {\rm supp}_{[a, b]} x_{+} \right) \le \mu$ or $\mu \left( {\rm supp}_{[a, b]} x_{-} \right) \le \mu$, where
$$L(x)_p:=\sup \left\{\left\|x \right\|_{L_p[a,b]}: \; a, b \in \RR,\; |x(t)|>0,\;t\in (a, b) \right\},$$
${\rm supp}_{[a, b]} x_{\pm}:= \{t\in [a, b]: x_{\pm}(t) > 0\}$
and $\varphi_{\lambda,r}^{\alpha, \beta}$ is the nonsymmetric
$(2\pi/\lambda)$-periodic spline of Euler of order $r$. In
particular, we solve the same problems for the intermediate
derivatives $x^{(k)}_{\pm}$, $k=1,...,r-1$, with $q \ge 1$.