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

• Volume: 72, Issue: 2, page 115-130
• ISSN: 0066-2216

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## Abstract

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The linear homogeneous differential equation with variable delays $ẏ\left(t\right)={\sum }_{j=1}^{n}{\alpha }_{j}\left(t\right)\left[y\left(t\right)-y\left(t-{\tau }_{j}\left(t\right)\right)\right]$ is considered, where ${\alpha }_{j}\in C\left(I,ℝ͞͞⁺\right)$, I = [t₀,∞), ℝ⁺ = (0,∞), ${\sum }_{j=1}^{n}{\alpha }_{j}\left(t\right)>0$ on I, ${\tau }_{j}\in C\left(I,ℝ⁺\right),$ the functions $t-{\tau }_{j}\left(t\right)$, j=1,...,n, are increasing and the delays ${\tau }_{j}$ 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.

## How to cite

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Diblík, Josef. "A criterion for convergence of solutions of homogeneous delay linear differential equations." Annales Polonici Mathematici 72.2 (1999): 115-130. <http://eudml.org/doc/262578>.

@article{Diblík1999,
abstract = {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.},
author = {Diblík, Josef},
journal = {Annales Polonici Mathematici},
keywords = {topological principle of Ważewski (Rybakowski's approach); asymptotic convergence of solutions; linear homogeneous delay differential equation; topological principle of Rybakowski},
language = {eng},
number = {2},
pages = {115-130},
title = {A criterion for convergence of solutions of homogeneous delay linear differential equations},
url = {http://eudml.org/doc/262578},
volume = {72},
year = {1999},
}

TY - JOUR
AU - Diblík, Josef
TI - A criterion for convergence of solutions of homogeneous delay linear differential equations
JO - Annales Polonici Mathematici
PY - 1999
VL - 72
IS - 2
SP - 115
EP - 130
AB - 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.
LA - eng
KW - topological principle of Ważewski (Rybakowski's approach); asymptotic convergence of solutions; linear homogeneous delay differential equation; topological principle of Rybakowski
UR - http://eudml.org/doc/262578
ER -

## References

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