Asymptotic properties of an unstable two-dimensional differential system with delay

Title:

Asymptotic properties of an unstable two-dimensional differential system with delay

Creator:

Kalas, Josef

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:d5457858-7b5a-4228-8ab9-90f7c9f4e2ea
uuid:d5457858-7b5a-4228-8ab9-90f7c9f4e2ea
doi:10.21136/MB.2006.134142

Subject:

delayed differential equation, asymptotic behaviour, boundedness of solutions, two-dimensional systems, Lyapunov method, and Ważewski topological principle

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

The asymptotic behaviour of the solutions is studied for a real unstable twodimensional system x ' (t) = A(t)x(t) + B(t)x(t − r) + h(t, x(t), x(t − r)), where r > 0 is a constant delay. It is supposed that A, B and h are matrix functions and a vector function, respectively. Our results complement those of Kalas [Nonlinear Anal. 62(2) (2005), 207–224], where the conditions for the existence of bounded solutions or solutions tending to the origin as t → ∞ are given. The method of investigation is based on the transformation of the real system considered to one equation with complex-valued coefficients. Asymptotic properties of this equation are studied by means of a suitable Lyapunov-Krasovskii functional and by virtue of the Wazewski topological principle. Stability and asymptotic behaviour of the solutions for the stable case of the equation considered were studied in Kalas and Baráková [J. Math. Anal. Appl. 269(1) (2002), 278–300].

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

305-319

Source:

Mathematica bohemica | 2006 Volume:131 | Number:3

Harvested from:

CDK

Metadata only:

false