Admissible pair of spaces for not correctly solvable linear differential equations

2014/09/27 by N. Chernyavskaya, N. A. Chernyavskaya, Lela S. Dorel +6
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Numerical methods for differential equations #Polynomial and algebraic computation #math.CA #msc:34A30 #msc:34B40

paper · pdf · doi:10.48550/arxiv.1409.7823

arxiv created 2014/09/27 · arxiv updated 2014/09/30

Abstract

We consider the differential equation -y'(x)+q(x)y(x)=f(x), x ∈ \mathbb R, where f ∈ L_p(\mathbb R), p∈ [1,∞), and 0≤ q ∈ L₁^{\rm loc}(\mathbb R), ∫_-∞⁰q(t) dt=∫₀^∞q(t) dt=∞, q₀(a)=inf_{x∈ \mathbb R}∫_x-a^x+aq(t) dt=0 \rm for ~ any a∈ (0,∞). Under these conditions, the equation (\rm \refab) is not correctly solvable in L_p(\mathbb R) for any p ∈ [1, ∞) . Let q^*(x) be the Otelbaev-type average of the function q(t), t∈ ℝ, at the point t=x; θ(x) be a continuous positive function for x ∈ \mathbb R, and L_p,θ(\mathbb R) = \f∈ L_p^{\rm loc}(\mathbb R): ∫_-∞^∞|θ(x)f(x)|^p dx<∞ \, ‖f‖_L_p,θ(\mathbb R)=(∫_-∞^∞|θ(x)f(x)|^p dx)^1/p We show that if there exists a constant c∈ [1, ∞), such that the inequality c^-1q^*(x)≤ θ(x)≤ cq^*(x) holds for all x ∈ ℝ, then under some additional conditions for q the pair of spaces \L_{p, θ}(\mathbb R); L_p(\mathbb R)\ is admissible for the equation (\rm \refab).

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