Nina A. Chernyavskaya

Conditions for Correct Solvability of a Simplest Singular Boundary Value Problem

We consider the singular boundary value problem −r(x)y(x) + q(x)y(x) = f(x), x ∈ R lim |x|→∞ y(x) = 0, where f ∈ Lp(R), p ∈ [1,∞] (L∞(R) := C(R)), r is a continuous positive function for x ∈ R, q ∈ Lloc 1 (R), q ≥ 0. A solution of this problem is, by definition, any absolutely continuous function y satisfying the limit condition and almost everywhere the differential equation. This problem is called correctly solvable in a given space Lp(R) if for any function f ∈ Lp(R) it has a unique solution y ∈ Lp(R) and if the following inequality holds with an absolute constant cp ∈ (0,∞) : ‖y‖Lp(R) ≤ cp‖f‖Lp(R), f ∈ Lp(R). We find minimal requirements for r and q under which the above problem is correctly solvable in Lp(R).

Necessary and sufficient conditions for solvability of the Hartman–Wintner problem for difference equations

The equation is viewed as a perturbation of the equation which does not oscillate at infinity. The sequences are assumed real, r n >0 for all n ≥ 0, the sequences may be complex-valued. We study the Hartman–Wintner problem on asymptotic ‘integration’ of (1) for large n in terms of solutions of (2) and the perturbation .

Regularity of the inversion problem for the Sturm-Liouville difference equation III. A criterion for regularity of the inversion problem

We consider a difference equation where h 0 is a fixed positive number, We obtain necessary and sufficient conditions under which assertions (I) and (II) hold together: 1. (I) for a given for any equation (1) has a unique solution (regardless of h), and 2. (II) for any Here c(p) is an absolute positive constant, is the difference Green function corresponding to equation (1).

Regularity of the inversion problem for the Sturm-Liouville difference equation IV. Stability conditions for a three-point difference scheme with non-negative coefficients

Consider a three-point difference scheme where is a given positive number, Assume that the sequence satisfies the a priori condition We obtain criteria for the stability of scheme (1) in

An embedding theorem

We consider a weighted space W (2) 1 (R, q) of Sobolev type: W (2) 1 (R, q) = { y ∈ AC loc (R) : ‖y ‖L1(R) + ‖qy‖L1(R) < ∞ } , where 0 ≤ q ∈ Lloc 1 (R) and ‖y‖ W (2) 1 (R,q) = ‖y‖L1(R) + ‖qy‖L1(R). We obtain a precise condition which guarantees the embedding W (2) 1 (R, q) ↪→ Lp(R), p ∈ [1,∞).