Asymptotic behaviour of a two‐dimensional differential system with non‐constant delay

Abstract

Key words Delayed differential system, asymptotic behaviour, stability, boundedness of solutions MSC (2000) Primary: 34K12; Secondary: 34K20The asymptotic behaviour and stability properties are studied for a real two-dimensional systemand h are matrix functions and a vector function, respectively. The method of investigation is based on the transformation of the considered real system to one equation with complex-valued coefficients. Stability and asymptotic properties of this equation are studied by means of a suitable Lyapunov-Krasovskii functional. The results generalize the great part of the results of J. Kalas and L. Baráková [J. Math. Anal. Appl. 269, No. 1, 278-300 (2002)] for two-dimensional systems with a constant delay.

Asymptotic behaviour of a two-dimensional differential system with a nonconstant delay under the conditions of instability

“…The results were extended and new corollaries were presented for systems with a finite number of constant delays in [12], [11] (stable case only). The results concerning asymptotic properties of solutions for the stable case of (1.1) with (generally unbounded) nonconstant delay can be found in [4] and in [10]. In the present paper we shall give results for the unstable case for (1.1).…”

Eighty fifth anniversary of birthday and scientific legacy of Professor Miloš Ráb

“…The asymptotic properties of the two-dimensional system with constant delay in the stable case were studied by J. Kalas and L. Baráová in [28], the asymptotic behavior of solutions under the conditions of instability was investigated by J. Kalas in [29], [30]. The equations with a finite system of constant delays were inspected by J. Rebenda in [49], [50].…”