Estimates for Green′s Function of the Sturm-Liouville Operator

Methods of analysis of the condition for correct solvability in L p (ℝ) of general Sturm-Liouville equations

“…(We say an estimate of a function f (x) for x ∈ (a, b) through a function g(x) is sharp by order if c −1 |g(x)| |f (x)| c|g(x)|, x ∈ (a, b), c = const. Note that in this paper some results from [4], [5] are strengthened. Moreover, in the special case r ≡ 1, 0 q ∈ L loc (R), another more effective form for the solution of the considered problem is obtained (for details see [1], [3])).…”

AN ASYMPTOTIC MAJORANT FOR SOLUTIONS OF STURM–LIOUVILLE EQUATIONS IN $L_P(\mathbb{R})$

“…This paper continues the authors' work in [2,3,5]. We consider the equation By a solution of equation (1.1), we mean any function y such that y, y ∈ AC loc (R) and equality (1.1) hold almost everywhere in R. We also assume that (1.1) is correctly solvable in L p (R).…”

Conditions for Solvability of the Hartman–wintner Problem in Terms of Coefficients

“…From Lemma 1.1 it follows (see [6]) that v 1 (x) is a principal solution of (1. We now can formulate our main problem.…”

“…(2.9) Lemma 2.6 (see [6]). An FSS {u 1 (x), v 1 (x)} of (1.2) (see Lemma 1.3) satisfies the following inequalities for every x ∈ R:…”

“…These requirements were introduced without necessary comments because they appeared for purely technical and not conceptual reasons. To be more precise, under assumptions (1.13), (1.14) we know sharp-by-order two-sided estimates for the function ρ 1 (x) = u 1 (x)v 1 (x), x ∈ R, in terms of some auxiliary functions in r(x) and q 1 (x) (see [6] and § 2 below). Note that to apply Theorem 1.1 (and any other statement from [4] concerning the solvability of problem (1.6)-(1.8)), one has to know the function ρ 1 (x) (estimates are not sufficient).…”