*(English)*Zbl 0849.34055

This paper deals with systems of differential delay equations with a cyclic feedback structure of the form ${\dot{x}}^{0}\left(t\right)={f}^{0}(t,{x}^{0}\left(t\right),{x}^{1}\left(t\right))$, ${\dot{x}}^{i}\left(t\right)={f}^{i}(t,{x}^{i-1}\left(t\right),{x}^{i}\left(t\right),{x}^{i+1}\left(t\right))$ for $1\le i\le N-1$ and ${\dot{x}}^{N}\left(t\right)={f}^{N}(t,{x}^{N-1}\left(t\right)$, ${x}^{N}\left(t\right)$, ${x}^{0}(t-1))$ which is a nearest neighbor system: the derivative of each ${x}^{i}$ depends on ${x}^{i}$ and on its neighbors ${x}^{i\pm 1}$. The functions ${f}^{0}$, ${f}^{1},\cdots ,{f}^{N}$ satisfy some monotonicity conditions. The purpose of the paper is twofold. First the authors define and investigate a Lyapunov-like integer-valued function ${V}^{\delta}$, where $\delta =\pm 1$, as follows: Let $\mathbb{K}$ be the closed interval $[-1,0]$ joined with the positive integers and let $\phi :\mathbb{K}\to \mathbb{R}$ be a continuous function not identically zero. Let $sc\left(\phi \right)$ denote the number of sign changes of $\phi (\xb7)$ if this is finite, otherwise it is $+\infty $. Then define ${V}^{+}\left(\phi \right)=sc\left(\phi \right)$, if $sc\left(\phi \right)$ is even or $+\infty $, ${V}^{+}\left(\phi \right)=sc\left(\phi \right)+1$, if $sc\left(\phi \right)$ is odd and ${V}^{-}\left(\phi \right)=sc\left(\phi \right)$, if $sc\left(\phi \right)$ is odd or $+\infty $, ${V}^{-}\left(\phi \right)=sc\left(\phi \right)+1$, if $sc\left(\phi \right)$ is even. If $({x}^{0},{x}^{1},\cdots ,{x}^{N})$ is a solution and $t$ is fixed, define ${x}_{t}\left(\vartheta \right)={x}^{0}(t+\vartheta )$, if $-1\le \vartheta \le 0$ and ${x}_{t}\left(\vartheta \right)={x}^{\vartheta}\left(t\right)$, if $\vartheta =1,2,\cdots ,N$. After these denotations some sufficient conditions are provided such that ${V}^{\pm}\left({x}_{t}\right)$ is nonincreasing in $t$ (Theorem 2.10) or finite (Theorem 2.4).

The second purpose of the paper is to apply the above general results to nonautonomous linear nearest neighbor systems in order to provide information about the relation between the Floquet multipliers and the values of ${V}^{\pm}\left({x}_{t}\right)$ when the ${x}_{t}$ belong to the Floquet subspaces. Some nice results are given for the autonomous linear case.

##### MSC:

34K99 | Functional-differential equations |

34K20 | Stability theory of functional-differential equations |

34C25 | Periodic solutions of ODE |