On the solvability of initial-value problems for nonlinear implicit difference equations
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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 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 {vi }r1 and {v j }m r+1 be any bases of KerQ and N, respectively. ˜ = diag(Or ,Im−r ), where Or Denote by V = (v1 ,...,vm ) a column matrix and denote Q and Im−r stand for r × r zero matrix and (m − r) × (m − r) identity matrix, respectively. ˜ −1 , and this decomposition depends on the choice of Then V is nonsingular, Q = V QV m the bases {vi }1 , that is, on V . Now, suppose Nα and Nβ are two subspaces of the same dimension m − r (1 r m − 1) in Rm . Then any projections Qα and Qβ onto Nα and Nβ can be decomposed Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:3 (2004) 195–200 2000 Mathematics Subject Classification: 34A09, 39A10 URL: http://dx.doi.org/10.1155/S1687183904402015
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IVPs for nonlinear implicit difference equations
˜ α−1 and Qβ = Vβ QV ˜ β−1 , respectively. Define an operator connecting two as Qα = Vα QV ˜ β−1 . Clearly, subspaces Nα and Nβ (connecting operator, for short) Qαβ := Vα QV Qαβ = Qα Qαβ = Qαβ Qβ = Qα Vα Vβ−1 = Vα Vβ−1 Qβ , Qαβ Qβα = Qα ,
Qβα Qαβ = Qβ .
(2.1)
We consider a system of LIDEs An xn+1 + Bn xn = qn
(n 0),
(2.2)
where An ,Bn ∈ Rm×m , qn ∈ Rm are given and rank An ≡ r (1 r m − 1) for all n 0. Let Qn be any projection onto Ker An ,Pn = I − Qn and consider decompositions Qn = ˜ n−1 (n 0). For definiteness, we put A−1 := A0 , Q−1 := Q0 , P−1 := P0 , and V−1 := Vn QV ˜ n−1 are determined for all n 0. V0 . Thus, the connecting operators Qn−1,n := Vn−1 QV Recall that a linear DAE A(t)x + B(t)x = q(t), t
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