Solvability conditions for some difference operators
- PDF / 598,195 Bytes
- 13 Pages / 468 x 680 pts Page_size
- 100 Downloads / 221 Views
Infinite-dimensional difference operators are studied. Under the assumption that the coefficients of the operator have limits at infinity, limiting operators and associated polynomials are introduced. Under some specific conditions on the polynomials, the operator is Fredholm and has the zero index. Solvability conditions are obtained and the exponential behavior of solutions of the homogeneous equation at infinity is proved. 1. Introduction Infinite-dimensional difference operators may not satisfy the Fredholm property, and the Fredholm-type solvability conditions are not necessarily applicable to them. In other words, we do not know how to solve linear algebraic systems with infinite matrices. Various properties of linear and nonlinear infinite discrete systems are studied in [1, 2, 3, 4, 5, 6, 7, 8]. The goal of this paper is to establish the normal solvability for the difference operators of the form j
j
j
(Lu) j = a−m u j −m + · · · + a0 u j + · · · + am u j+m ,
j ∈ Z,
(1.1)
and to obtain the solvability conditions for the equation Lu = f , where m ≥ 0 is a given integer and f = { f j }∞ j =−∞ is an element of the Banach space
j =−∞ , u j ∈ R, sup u j < ∞ .
∞
E = u = uj
j ∈Z
(1.2)
The right-hand side in (1.1) does not necessarily contain an odd number of summands. We use this form of the operator to simplify the presentation. We will use here the approaches developed for elliptic problems in unbounded domains [9, 10] and adapt them for infinite-dimensional difference operators.
Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:1 (2005) 1–13 DOI: 10.1155/ADE.2005.1
2
Solvability conditions for some difference operators The operator L : E → E defined in (1.1) can be regarded as (Lu) j = A j U j , where
j
j
j
A j = a−m ,...,a0 ,...,am ,
U j = u j −m ,...,u j ,...,u j+m
(1.3)
are 2m + 1-vectors, A j is known, and U j is variable. We suppose that there exist the limits of the coefficients of the operator L as j → ±∞ j
a±l = lim al , j →±∞
l ∈ Z, −m ≤ l ≤ m,
(1.4)
and a±±m = 0. Denote by L± : E → E the limiting operators
L± u j = a±−m u j −m + · · · + a±0 u j + · · · + a±m u j+m ,
j ∈ Z.
(1.5)
Recall that a linear operator L : E → E is normally solvable if its image Im L is closed. If L is normally solvable with a finite-dimensional kernel and the codimension of its image is also finite, then L is called Fredholm operator. Denoting by α(L) and β(L) the dimension of ker L and the codimension of ImL, respectively, we can define the index κ(L) of the operator L as κ(L) = α(L) − β(L). It is known that the index does not change under deformation in the class of Fredholm operators. In Section 2 of this paper we introduce polynomials P + (σ) and P − (σ) associated with the limiting operators L+ and L− . We show that, if P + and P − do not have roots on the unit circle, then the limiting operators are invertible and the operator L is normally solvable with a finite-dimensional kernel. If moreover the polynomials have the same number of
Data Loading...
