## Abstract

We consider a difference equation - h^{-2}Δ ^{(2)}y_{n}+q_{n}(h)y_{n} = f_{n}(h), n ∈ Z = {0,±1, ±2,⋯}, (1) where h ∈ (0, h _{0}], h_{0} 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}|^{p}h, and 0 ≤ _{qk}(h) < ∞, Σ_{k=-∞}^{n}qn(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∈Z}G_{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).

Original language | English |
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Pages (from-to) | 245-260 |

Number of pages | 16 |

Journal | Journal of Difference Equations and Applications |

Volume | 11 |

Issue number | 3 |

DOIs | |

State | Published - 1 Mar 2005 |

## Keywords

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