SOLVABILITY CONDITIONS FOR SOME DIFFERENCE OPERATORS N. C. APREUTESEI AND V. A. VOLPERT Received 24 June 2004 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. | SOLVABILITY CONDITIONS FOR SOME DIFFERENCE OPERATORS N. C. APREUTESEI AND V. A. VOLPERT Received 24 June 2004 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 Lu j a-muj-m a0uj amu j m j _ Z and to obtain the solvability conditions for the equation Lu f where m 0 is a given integer and f fj J-D is an element of the Banach space -Jit _ J I Ỉ _ if z Lp oil TA ill rv L. u u u ị _ UR su. d u A DO . 1 -K j Í j_Z J The right-hand side in 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 2 Solvability conditions for some difference operators The operator L E E defined in can be regarded as Lu j AjUj where Aj a-m . a0 . a Uj uj-m . Uj . Uj m are 2m 1-vectors Aj is known and Uj is variable. We suppose that there exist the limits of the coefficients of the operator L as j TO a
