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

Tom 72 / 1999

Annales Polonici Mathematici 72 (1999), 115-130 DOI: 10.4064/ap-72-2-115-130

Streszczenie

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.

• Josef Diblík

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