In this article, stability and asymptotic properties of solutions of a real two-dimensional system $x^{\prime }(t) = \mathbf{A} (t) x(t) + \mathbf{B} (t) x (\tau (t)) + \mathbf{h} (t, x(t), x(\tau (t)))$ are studied, where $\mathbf{A}$, $\mathbf{B}$ are matrix functions, $\mathbf{h}$ is a vector function and $\tau (t) \le t$ is a nonconstant delay which is absolutely continuous and satisfies $\lim \limits _{t \rightarrow \infty } \tau (t) = \infty $. Generalization of results on stability of a two-dimensional differential system with one constant delay is obtained using the methods of complexification and Lyapunov-Krasovskii functional and some new corollaries and examples are presented.
In this article, stability and asymptotic properties of solutions of a real two-dimensional system $x^{\prime }(t) = \mathbf{A} (t) x(t) + \mathbf{B} (t) x (\tau (t)) + \mathbf{h} (t, x(t), x(\tau (t)))$ are studied, where $\mathbf{A}$, $\mathbf{B}$ are matrix functions, $\mathbf{h}$ is a vector function and $\tau (t) \le t$ is a nonconstant delay which is absolutely continuous and satisfies $\lim \limits _{t \rightarrow \infty } \tau (t) = \infty $. Generalization of results on stability of a two-dimensional differential system with one constant delay is obtained using the methods of complexification and Lyapunov-Krasovskii functional and some new corollaries and examples are presented.
@article{ARM_2009_45_3_a6,
author = {Rebenda, Josef},
title = {Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay},
journal = {Archivum mathematicum},
pages = {223--236},
year = {2009},
volume = {45},
number = {3},
mrnumber = {2591678},
zbl = {1212.34235},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ARM_2009_45_3_a6/}
}
TY - JOUR
AU - Rebenda, Josef
TI - Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay
JO - Archivum mathematicum
PY - 2009
SP - 223
EP - 236
VL - 45
IS - 3
UR - http://geodesic.mathdoc.fr/item/ARM_2009_45_3_a6/
LA - en
ID - ARM_2009_45_3_a6
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%0 Journal Article
%A Rebenda, Josef
%T Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay
%J Archivum mathematicum
%D 2009
%P 223-236
%V 45
%N 3
%U http://geodesic.mathdoc.fr/item/ARM_2009_45_3_a6/
%G en
%F ARM_2009_45_3_a6
Rebenda, Josef. Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay. Archivum mathematicum, Tome 45 (2009) no. 3, pp. 223-236. http://geodesic.mathdoc.fr/item/ARM_2009_45_3_a6/
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[4] Rebenda, J.: Asymptotic properties of solutions of real two-dimensional differential systems with a finite number of constant delays. Mem. Differential Equations Math. Phys. 41 (2007), 97–114. | MR | Zbl
[5] Rebenda, J.: Stability of the trivial solution of real two-dimensional differential system with nonconstant delay. In 6. matematický workshop - sborník, FAST VUT Brno 2007, 2007, 49–50 (abstract). Fulltext available at http://math.fce.vutbr.cz/~pribyl/workshop_2007/prispevky/Rebenda.pdf
[6] Rebenda, J.: Asymptotic behaviour of real two-dimensional differential system with a finite number of constant delays. Demonstratio Math. 41 (4) (2008), 845–857. | MR | Zbl
