On Perturbative Hardy-Type Inequalities
DOI:
https://doi.org/10.15407/mag19.01.128Ключові слова:
нерiвнiсть Хардi, головнi i неголовнi розв’язки, теорiя коливань, оператори Штурма–ЛiувiлляАнотація
Для даного трьохкоефiцiєнтного диференцiального виразу Штурма–Лiувiлля $\tau_0 = r_0^{-1}[-(d/dx)p_0(d/dx)+q_0]$ та його збурення $\tau_{q_1}=\tau_0+r_0q^{-1}$ на iнтервалi $(a,b)\subset\mathbb{R}$, ми використовуємо iснування строго додатного розв’язку $u_0(\lambda_0,\cdot)>0$ на $(a,b)$ для $\tau_0u_0=\lambda_0u_0$ для того, щоб одержати для $\tau_{q_1}$ нерiвнiсть у квадратичнiй формi, яка природно узагальнює добре вiдому нерiвнiсть Хардi i зводиться до неї в окремому випадку $p_0=r_0=u_0(0,\cdot)=1$, $q_0=\lambda_0=0$, $a\in \mathbb{R}, b=\infty.$
Mathematical Subject Classification 2020: 34A40, 34B24, 34C10, 47E05,
26D20, 34L05
Посилання
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