Báo Cáo Báo cáo hóa học ON THE SOLVABILITY OF INITIAL-VALUE PROBLEMS FOR NONLINEAR IMPLICIT DIFFERENCE EQUAT

ON THE SOLVABILITY OF INITIAL-VALUE PROBLEMS FOR NONLINEAR IMPLICIT DIFFERENCE EQUATIONS PHAM KY ANH AND HA THI NGOC YEN Received 18 February 2004 Our aim is twofold. First, we propose a natural definition of index for linear nonautonomous implicit difference equations, which is similar to that of linear differentialalgebraic equations. Then we extend this index notion to a class of nonlinear implicit difference equations and prove some existence theorems for their initial-value problems. 1. Introduction Implicit difference equations (IDEs) arise in various applications, such as the Leontief dynamic model of a multisector economy, the Leslie population growth model, and so forth. On the other hand, IDEs may be regarded as discrete analogues of differentialalgebraic equations (DAEs) which have already attracted much attention of researchers. Recently [1, 3], a notion of index 1 linear implicit difference equations (LIDEs) has been introduced and the solvability of initial-value