Bounded solutions of discrete equations (Q1914356)

It is well known that, for ordinary differential equations, sufficient conditions for boundedness of solutions can be obtained from considering Lyapunov functions in a neighborhood of infinity [see, e.g. \textit{T. Yoshizawa}, Stability theory by Lyapunov's second method, Tokyo (1966; Zbl 0144.10802)]. The author presents three analogous theorems for discrete-time systems described by difference equations of the type \(x_{i + 1} = f_i (x_i, x_{i - 1}, \dots, x_{i - k})\), \(i = i_0\), \(i_0 + 1, \dots\), where \(k\) is a fixed integer, \(i_0 \in T = \{0,1, \dots\}\), and \(f : T \times \mathbb{R}^{k + 1} \to \mathbb{R}\). As an example he considers a second-order relay impulse system \(x_{i + 1} = ax_i + bx_{i - 1} - c \text{sign} x_i - d \text{sign} x_{i - 1}\) and shows that its solutions are uniformly bounded if \(c + d \leq 0\).