Ważewski’s method for systems of dynamic equations on time scales

How to find initial data generating bounded solutions of discrete equations

“…Due to (11) and Lemma 3.1, we can find an interval I s−1 ⊆Î s−1 such that g s−1 (I s−1 ) = I s . Recursively, we find I s−2 , .…”

“…The aim is to find a solution u = u(n) of equation (1) such that u(n) ∈ ω(n) for every n ∈ Z ∞ a . Similar problems are studied in many papers, e.g., [2], [4]- [11]. In those papers, various conditions concerning the set ω and the right-hand side of equation (1) are considered.…”

Ważewski type theorem for non-autonomous systems of equations with a disconnected set of egress points

“…More precisely, in previous papers, see, e.g., [6][7][8]10], devoted to n-dimensional systems, one considered the time section t (set with fixed t ∈ T) of a simple hyper-prism shape. In the present paper t (for every t ∈ T) is an arbitrary closed bounded set homeomorphic to the disk.…”

“…[4][5][6]. Finally and recently, it was used for dynamic equations on time scales, see e.g., [1,[7][8][9][10]12].…”

New sufficient conditions for the asymptotic stability of discrete time-delay systems

“…During the last decades, the stability analysis of switched linear continuous/discrete time-delay systems has attracted a lot of attention [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The main approach for stability analysis relies on the use of Lyapunov-Krasovskii functionals and linear matrix inequlity (LMI) approach for constructing a common Lyapunov function [19][20][21][22][23][24]. Although many important results have been obtained for switched linear continuoustime systems, there are few results concerning the stability of switched linear discrete systems with time-varying delays.…”