Some results on the boundedness of difference equations with unbounded delay (Q2735413)

The author obtains a new Razumikhin-type boundedness theorem for the following difference equation with unbounded delay NEWLINE\[NEWLINEx(n+ 1)= G\bigl(n,x(\tau); \tau=l,l+1, \dots,n\bigr) \equiv G\bigl(n,x(\cdot) \bigr),\;n\geq n_0,\tag{1}NEWLINE\]NEWLINE where \(x\in\mathbb{R}^k,l\) is a integer or \(-\infty,n_0\) and \(n\) are integers, and \(G(n,\psi(\cdot)) \in\mathbb{R}^k\) holds for \(n\geq l\) and any function \(\psi:\{l,l+1, \dots,n\}\to \mathbb{R}^k\).NEWLINENEWLINENEWLINELet \(u,v,w:\mathbb{R}_+: =[0, \infty) \to \mathbb{R}_+,u(0)= \nu(0)=0\), \(u(s)\), \(w(s)\to+ \infty\) as \(s\to+ \infty \), with \(u,v\) strictly increasing and \(w(s)>0\) for \(s>0\). Let further \(\alpha_0>0\) and \(M_0\geq 0\) be constants. The main result embodied in Theorem 1 can be restated as follows:NEWLINENEWLINENEWLINETheorem 1. If there exists a Lyapunov-type function \(V:\{l,l+1,l+2, \dots\}\times \mathbb{R}^k\to \mathbb{R}^k\) that satisfies the conditions: (i) \(u(|x|)\leq V(n,x)\leq v(|x|)\); (ii) For every \(\beta(\geq \alpha_0)\) and \(\sigma>M_0\) there exist a number \(\eta=\eta (\beta, \sigma) >0\) and a positive integer \(q=q(\beta, \sigma)\) such that, when \(n\geq l\), \(\alpha_0\leq |x(n+1)|\leq \beta\) and \(\sup\{|x(\tau) |: l \leq\tau\leq n\} \leq\beta\), the relation NEWLINE\[NEWLINEV\bigl( \tau,x(\tau) \bigr)\leq V \bigl(n+1,x (n+1)\bigr)+ \eta\text{ holds for }\max\{l,n-q\} \leq\tau\leq n.NEWLINE\]NEWLINE Then all solutions of Eq. (1) are uniformly bounded and uniformly ultimate bounded.NEWLINENEWLINENEWLINEThe last result generalizes the Theorem 2 and Theorem 3 of \textit{S. Zhang} [Nonlinear Anal., Theory Methods Appl. 22, No. 10, 1209-1219 (1994; Zbl 0805.39003)]. An application example of Theorem 1 is also given for which the mentioned Zhang's theorems are not applicable.