The Bojanov – Naidenov problem for functions with nonsymmetric restrictions on the highest derivative
Abstract
For given r∈\bfN,p,α,β,μ>0, we solve the extreme problems ∫baxq±(t)dt→sup,q≥p, in the set of pairs (x,I) of functions x∈Lr∞ and intervals I=[a,b]⊂R satisfying the inequalities β≤x(r)(t)≤α for almost all t∈R , the conditions L(x±)p≤L((φα,βλ,r))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},φα,βλ,r is the nonsymmetric (2π/λ)-periodic Euler spline of order r. As a consequence, we solve the same problems for the intermediate derivatives x(k)±,k=1,...,r1, with q≥1.
Published
25.03.2019
How to Cite
Kofanov, V. A. “The Bojanov – Naidenov Problem for Functions With Nonsymmetric restrictions
on the Highest Derivative”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, no. 3, Mar. 2019, pp. 368-81, https://umj.imath.kiev.ua/index.php/umj/article/view/1445.
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Section
Research articles