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 -h^-2Δ ⁽²⁾y_n + q_n(h)y_n = f_n(h), n ∈ Z = {0, ± 1, ± 2,...} (1) where h ∈ (0, h ₀], h₀ is a given positive number, Δ ⁽²⁾y_n = y_n+1 - 2y_n + y _n-1, f(h) =def {f_n(h)}_n∈Z ∈ L _p(h),p ∈ [1, ∞), L_p(h) = { f(h): ∥ f (h) ∥ L_p(h) < ∞}, ∥f(h)∥|^pL _p(h) =∑n∈|f _n(h)|^p h. Assume that the sequence q(h) =def {q_n(h)}_n∈Z satisfies the a priori condition 0 ≤ q_n(h) < ∞ ∀n ∈ Z, ∀h ∈ (0, h₀]. We obtain criteria for the stability of scheme (1) in L_p(h), p ∈ [1, ∞).