On the solvability of initial-value problems for nonlinear implicit difference equations

Abstract

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.

On stability and Bohl exponent of linear singular systems of difference equations with variable coefficients

“…The main aim of this paper is to extend some well-known stability theorems from ordinary difference equations to singular difference equations. These results complement those in [8,9,13]. They can also be considered as the discrete-time analogues of some recent results for DAEs, see [15][16][17][18][19].…”

“…While the theory of DAEs, the continuous-time counterpart of (1.1), has been almost well established, qualitative results in the theory of singular difference equations, particularly those for non-autonomous systems, are very few. Though the first result on LSDEs with variable coefficients was given a long time ago in [3], interesting results on the existence and the stability of solutions have been published only recently, see [7][8][9][10][11][12][13]. Most of the stability and robust stability results for singular difference equations are obtained for autonomous systems, e.g.…”

Floquet theorem for linear implicit nonautonomous difference systems

“…We define the so-called connecting operators (see [2]) as follows: Q n−1,n := V n−1Q V −1 n and Q n,n−1 := V nQ V −1 n−1 . Clearly, Q n−1,n = Q n−1 Q n−1,n = Q n−1,n Q n ; Q n−1,n Q n,n−1 = Q n−1 and Q n,n−1 Q n−1,n = Q n .…”

On discrete analogues of nonlinear implicit differential equations

“…It can be verified (cf. [2]) that the matrices G n (y,x) are nonsingular if and only if S n (y,x) ∩ ᏺ n−1 = {0} ∀y, x ∈ R m ; ∀n ≥ 0, (2.4) where, as in the DAE case, S n (y,x) denotes the set ξ ∈ R m : ∂ f n ∂x (y,x)ξ ∈ Im ∂ f n ∂y (y,x) . (2.5) P. K. Anh and L. C. Loi 3 Since condition (2.4) does not depend on the choice of connecting operators, the correctness of the index-1 notion for nonlinear IDEs is guaranteed.…”