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 - 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 fixed positive number, Δ ⁽²⁾y_n = y_n+1 - 2y_n+y_n-1, n∈Z; f={f_n(h)}_n∈Z∈L_p(h), p ∈ [1, ∞), L_p(h) = {f: ∥f∥L_p< ∞}, ∥f∥_Lp(h)^p = Σ_n∈Z, |f_n|^ph, and 0 ≤ _qk(h) < ∞, Σ_k=-∞ⁿqn(h) > 0, n ∈ Z. We obtain necessary and sufficient conditions under which assertions (I) and (II) hold together:. (I) for a given p ∞ [1, ∞), for any f ∈ L _p(h), equation (1) has a unique solution y = {y_n(h)} _n∈Z ∈ L_p(h) (regardless of h), and y = (Gf)(h) =^def {(Gf)_n(h)}_n∈Z, (Gf)_n(h) = Σ_m∈ZG_n,m(h)f_m(h)h, n ∈ Z. (II) ∥y∥_Lp(h) ≤ c(p)∥f∥L_p for any f ∈ L_p(h). Here c(p) is an absolute positive constant, {G _n,m(h)}_n,m∈Z is the difference Green function corresponding to equation (1).