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

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 Lyapunov-Krasovskii functional and by virtue of the Ważewski topological principle. The results supplement those of Kalas and Baráková [J. Math. Anal. Appl. 269(1) (2002) 278-300], where the stability and asymptotic behaviour were investigated for the stable case.