# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions. (English) Zbl 0849.34055

This paper deals with systems of differential delay equations with a cyclic feedback structure of the form ${\stackrel{˙}{x}}^{0}\left(t\right)={f}^{0}\left(t,{x}^{0}\left(t\right),{x}^{1}\left(t\right)\right)$, ${\stackrel{˙}{x}}^{i}\left(t\right)={f}^{i}\left(t,{x}^{i-1}\left(t\right),{x}^{i}\left(t\right),{x}^{i+1}\left(t\right)\right)$ for $1\le i\le N-1$ and ${\stackrel{˙}{x}}^{N}\left(t\right)={f}^{N}\left(t,{x}^{N-1}\left(t\right)$, ${x}^{N}\left(t\right)$, ${x}^{0}\left(t-1\right)\right)$ which is a nearest neighbor system: the derivative of each ${x}^{i}$ depends on ${x}^{i}$ and on its neighbors ${x}^{i±1}$. 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 $\delta =±1$, as follows: Let $𝕂$ be the closed interval $\left[-1,0\right]$ joined with the positive integers and let $\varphi :𝕂\to ℝ$ be a continuous function not identically zero. Let $sc\left(\varphi \right)$ denote the number of sign changes of $\varphi \left(·\right)$ if this is finite, otherwise it is $+\infty$. Then define ${V}^{+}\left(\varphi \right)=sc\left(\varphi \right)$, if $sc\left(\varphi \right)$ is even or $+\infty$, ${V}^{+}\left(\varphi \right)=sc\left(\varphi \right)+1$, if $sc\left(\varphi \right)$ is odd and ${V}^{-}\left(\varphi \right)=sc\left(\varphi \right)$, if $sc\left(\varphi \right)$ is odd or $+\infty$, ${V}^{-}\left(\varphi \right)=sc\left(\varphi \right)+1$, if $sc\left(\varphi \right)$ is even. If $\left({x}^{0},{x}^{1},\cdots ,{x}^{N}\right)$ is a solution and $t$ is fixed, define ${x}_{t}\left(\vartheta \right)={x}^{0}\left(t+\vartheta \right)$, 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 ${V}^{±}\left({x}_{t}\right)$ 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 ${V}^{±}\left({x}_{t}\right)$ 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