A criterion for convergence of solutions of homogeneous delay linear differential equations

The linear homogeneous differential equation with variable delays ${\displaystyle ẏ\left(t\right)={\sum }_{j=1}^n{\alpha }_j\left(t\right)[y\left(t\right)-y(t-{\tau }_j\left(t\right))]}$ is considered, where ${\displaystyle {\alpha }_j\in C(I,ℝ͞͞⁺)}$, I = [t₀,∞), ℝ⁺ = (0,∞), ${\displaystyle {\sum }_{j=1}^n{\alpha }_j\left(t\right)>0}$ on I, ${\displaystyle {\tau }_j\in C(I,ℝ⁺),}$ the functions ${\displaystyle t-{\tau }_j\left(t\right)}$, j=1,...,n, are increasing and the delays ${\displaystyle {\tau }_j}$ are bounded. A criterion and some sufficient conditions for convergence of all solutions of this equation are proved. The related problem of nonconvergence is also discussed. Some comparisons to known results are given.