Asymptotic behaviour of a two-dimensional differential system with delay under the conditions of instability
| Authors | |
|---|---|
| Year of publication | 2005 |
| Type | Article in Periodical |
| Magazine / Source | Nonlinear Analysis, Theory, Methods & Applications |
| MU Faculty or unit | |
| 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. |
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