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Keywords:
stability; asymptotic behaviour; differential system; nonconstant delay; Lyapunov method
stability; asymptotic behaviour; differential system; nonconstant delay; Lyapunov method
Summary:
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.
References:
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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 2391945 | Zbl 1157.34356
[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
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