A criterion for correct solvability of the Sturm-Liouville equation in the space 𝐿_{𝑝}(𝑅)

Abstract

Abstract. We consider an equation (1)

A criterion for compactness in Lp(ℝ) of the resolvent of the maximal general form Sturm–Liouville operator

“…We have the relations to concrete equations, one has to know the auxiliary functions h and d. In the r ≡ 1 case the requirement that B < ∞ can be replaced with a more simple one (see [5]). Usually, it is not possible to express these functions explicitly through the original coefficients r and q of (1.1).…”

Weighted estimates for solutions of a Sturm–Liouville equation in the space L1(ℝ)

“…Theorem 2.8 (Chernyavskaya and Shuster [6]). If m(a) > 0 for some a > 0 (see (2.1)), then the solution of (1.1), y ∈ L 1 , is of the form…”

“…Throughout the paper, c denotes absolute positive constants that are not essential for exposition and may differ even within a single chain of computations. Exact requirements for q, which guarantee (i)-(ii), are given in [6] (see also § 2 in this paper).…”

“…Chernyavskaya and Shuster[6]). Equation (1.1) is correctly solvable in L 1 if and only if there exists a > 0 such that m(a) > 0.…”

Continuous invertibility of minimal Sturm–Liouville operators in Lebesgue spaces

“…A linear operator T : D(T ) ⊂ X → Y will be called continuously invertible if it is bijective and T −1 is bounded. In [6], Chernyavskaya and Shuster investigated the continuous invertibility (which they refer to as 'correct solvability') of the Sturm-Liouville operator in the Banach space L p (R); in other words, they determined the necessary and sufficient conditions for the maximal operator T p,p in L p (R) defined by the symmetric differential expression M 1 [y] = −y + qy (where q is a non-negative function that is locally Lebesgue integrable) to satisfy the following properties: The following theorem is a principal result of [6]. Three corollaries of theorem 1.1 are:…”