On Linear Implicit Non-autonomous Systems of Difference Equations

Abstract

This paper deals with the solvability of initial-value problems (IVPs) and multipoint boundaryvalue problems (MPBVPs) for linear implicit non-autonomous systems of difference equations.

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

“…Corollary 2.2 confirms that it suffices to restrict our consideration to orthogonal projections onto KerA n , as was done in [3]. However, in the mentioned paper, a singularvalue decomposition (SVD) of A n is employed for constructing an orthogonal projection Q n onto KerA n and it seems not to be convenient for a further extension of the index notion to nonlinear cases.…”

“…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 one-step-map for switched singular systems in discrete-time

“…Although the following results are partially obtained by similar arguments as in [2], [3], [14], we will give their proofs here to make our presentation self-contained. Furthermore, these properties also play a crucial rule for the treatment of the inhomogeneous case later.…”

“…Puttingx(k) := (v(k) , w(k) ) , where v(k) ∈ R r , w(k) ∈ R n−r , we can reduce system (14) to system v(k + 1) =Ā 1 σ(k),σ(k−1) v(k) , which yields the claimed condition by applying the coordinate transformation V σ(−1) .…”

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

“…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.…”