tailieunhanh - ON THE SOLVABILITY OF INITIAL-VALUE PROBLEMS FOR NONLINEAR IMPLICIT DIFFERENCE EQUATIONS PHAM KY ANH

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. | 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 nonau-tonomous implicit difference equations which is similar to that of linear differential-algebraic 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 differential-algebraic 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 problems IVPs as well as multipoint boundary-value problems MBVPs for index 1 LIDEs has been studied. In this paper we propose a natural definition of index for LIDEs so that it can be extended to a class of nonlinear IDEs. The paper is organized as follows. Section 2 is concerned with index 1 LIDEs and their reduction to ordinary difference equations. In Section 3 we study the index concept and the solvability of IVPs for nonlinear IDEs. The result of this paper can be considered as a discrete version of the corresponding result of 4 . 2. Index 1 linear implicit difference equations Let Q be an arbitrary projection onto a given subspace N of dimension m - r 1 r m - 1 in Rm. Further let v ir and vj im 1 be any bases of KerQ and N respectively. Denote by V v1 . vm a column matrix and denote Q diag Or Im-r where Or and Im-r stand for r X r zero matrix and m - r X m - r identity matrix respectively. Then V is nonsingular Q VQV-1 and this decomposition depends on the choice of the bases vịim that is on V. Now