# Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay

## Josef Rebenda

Address: Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic

E-mail: rebenda@math.muni.cz

Abstract: 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.

AMSclassification: primary 34K20; secondary 34K25, 34K12.

Keywords: stability, asymptotic behaviour, differential system, nonconstant delay, Lyapunov method.