For given r ∈ N, p, 𝛼, β,μ > 0, we solve the extreme problems
in the set of pairs (x, I) of functions \( x\in {L}_{\infty}^r \) and intervals I = [a, b] ⊂R satisfying the inequalities −β ≤ x(r)(t) ≤ 𝛼 for almost all t ∈ R, the conditions\( L{\left({x}_{\pm}\right)}_p\le L{\left({\left({\varphi}_{\leftthreetimes, r}^{\alpha, \beta}\right)}_{\pm}\right)}_p \), and the corresponding condition μ(supp[a, b]x+) ≤ μ or μ(supp[a, b]x−) ≤ μ, where L(x)p; = sup {‖x‖Lp[a, b]; a, b ∈ R, |x(t)| > 0, t ∈ (a, b)}, supp[a, b]x ± ≔ {t ∈ [a, b] : x±(t) > 0},, and \( {\varphi}_{\leftthreetimes, r}^{\alpha, \beta } \) is an asymmetric (2π/⋋) -periodic Euler spline of order r. As a consequence, we solve the same extreme problems for the intermediate derivatives \( {x}_{\pm}^{(k)} \), k = 1, . . . , r − 1, with q ≥ 1.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 3, pp. 368–381, March, 2019.
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Kofanov, V.A. Bojanov–Naidenov Problem for Functions with Asymmetric Restrictions for the Higher Derivative. Ukr Math J 71, 419–434 (2019). https://doi.org/10.1007/s11253-019-01655-2
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DOI: https://doi.org/10.1007/s11253-019-01655-2