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

N. A. Chernyavskaya, J. Schiff, L. A. Shuster

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2 Scopus citations

Abstract

We consider a difference equation - h-2Δ (2)yn+qn(h)yn = fn(h), n ∈ Z = {0,±1, ±2,⋯}, (1) where h ∈ (0, h 0], h0 is a fixed positive number, Δ (2)yn = yn+1 - 2yn+yn-1, n∈Z; f={fn(h)}n∈Z∈Lp(h), p ∈ [1, ∞), Lp(h) = {f: ∥f∥Lp< ∞}, ∥f∥Lp(h)p = Σn∈Z, |fn|ph, and 0 ≤ qk(h) < ∞, Σk=-∞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 = {yn(h)} n∈Z ∈ Lp(h) (regardless of h), and y = (Gf)(h) =def {(Gf)n(h)}n∈Z, (Gf)n(h) = Σm∈ZGn,m(h)fm(h)h, n ∈ Z. (II) ∥y∥Lp(h) ≤ c(p)∥f∥Lp for any f ∈ Lp(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).

Original languageEnglish
Pages (from-to)245-260
Number of pages16
JournalJournal of Difference Equations and Applications
Volume11
Issue number3
DOIs
StatePublished - 1 Mar 2005

Keywords

  • Inversion problem
  • Properties of solutions
  • Sturm-Liouville difference equation

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