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

Josef Diblík¹

¹

Annales Polonici Mathematici, Tome 72 (1999) no. 2, pp. 115-130.

Résumé

The linear homogeneous differential equation with variable delays $ ẏ(t) = ∑_{j=1}^n α_j(t)[y(t) - y(t-τ_j(t))]$ is considered, where $α_j ∈ C(I,ℝ͞͞⁺)$, I = [t₀,∞), ℝ⁺ = (0,∞), $∑_{j=1}^n α _j(t) > 0$ on I, $τ_j ∈ C(I,ℝ⁺),$ the functions $t - τ_j(t)$, j=1,...,n, are increasing and the delays $τ_j$ 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.

Détail

DOI : 10.4064/ap-72-2-115-130

Keywords: topological principle of Ważewski (Rybakowski's approach), asymptotic convergence of solutions, linear homogeneous delay differential equation