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

Warning

This publication doesn't include Faculty of Sports Studies. It includes Faculty of Science. Official publication website can be found on muni.cz.
Authors

KALAS Josef

Year of publication 2005
Type Article in Periodical
Magazine / Source Nonlinear Analysis, Theory, Methods & Applications
MU Faculty or unit

Faculty of Science

Citation
Field General mathematics
Keywords Delayed differential equation; Asymptotic behaviour; Boundedness of solutions; Two-dimensional systems; Lyapunov method; Wazewski topological principle
Description The asymptotic behaviour of the solutions of a real two-dimensional system x'=A(t)x(t)+B(t)x(t-r)+h(t,x(t),x(t-r)), where r>0 is a constant delay, is studied under the assumption of instability. Here A, B and h are matrix functions and a vector function, respectively. 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 considered real system to one equation with complex-valued coefficients. Asymptotic properties of this equation are studied by means of a suitable LyapunovKrasovskii functional and by virtue of the Wazewski topological principle. The results supplement those of Kalas and Baráková [J. Math. Anal. Appl. 269(1)(2002) 278300], where the stability and asymptotic behaviour were investigated for the stable case.
Related projects:

You are running an old browser version. We recommend updating your browser to its latest version.

More info