Stability and asymptotic properties of a system of functional differential equations with nonconstant delays

The asymptotic properties of a real two-dimensional differential system  x′(t)=A(t)x(t)+∑k=1mBk(t)x(θk(t))+h(t,x(t),x(θ1(t)),…,x(θm(t))) with unbounded nonconstant delays t-θk(t)⩾0 satisfying limt→∞θk(t)=∞ are studied. Here A,Bk and h are supposed to be matrix functions and a vector function. The conditions for the stability and asymptotic stability of solutions and the conditions under which all solutions tend to zero are given. The methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of a Lyapunov-Krasovskii functional. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one or more constant delays or one nonconstant delay were studied.