Consider the equation -(r(x))y′(x))′ + q(x)y(x) = f(x), x ∈ ℝ, (∗) where f ∈ Lp(ℝ), p ∈ (1, ∞) and r > 0, q ≥ 0, 1/r ∈ L1loc(ℝ), q ∈L1loc(ℝ) lim |d|→∞∫x-dx dt/r(t) ∫x-dx q(t) dt = ∞ ∀x ∈ ℝ. By a solution of (∗), we mean any function y absolutely continuous together with (ry′) and satisfying (∗) almost everywhere on ℝ. In addition, we assume that (∗) is correctly solvable in the space Lp (ℝ), i.e. (1) for any function f ∈ Lploc(ℝ), there exists a unique solution y ∈ Lp(ℝ) of (∗); (2) there exists an absolute constant c1(p) > 0 such that the solution y ∈ Lp(ℝ) of (∗) satisfies the inequality ∥y∥Lp(ℝ) ≤ c1(p)∥f∥Lp(ℝ) ∀f ∈ Lp(ℝ). (∗∗) We study the following problem on the strengthening estimate (∗∗). Let a non-negative function θ ∈ L1loc(ℝ) be given. We have to find minimal additional restrictions for θ under which the following inequality holds: ∥θy∥Lp(ℝ) ≤ c2(p)∥f∥Lp(ℝ) ∀f ∈ Lp(ℝ). Here, y is a solution of (∗) from the class Lp(ℝ), and c2(p) is an absolute positive constant.
- Everitt-Giertz problem
- linear differential equations
- second-order equation
ASJC Scopus subject areas
- Mathematics (all)