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## Annales Polonici Mathematici

1999 | 72 | 2 | 115-130
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
115-130
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-03-23
poprawiono
1999-03-25
Twórcy
autor
• Department of Mathematics Faculty of Electrical Engineering and Computer Science Technical University of Brno Technická 8 616 00 Brno, Czech Republic
Bibliografia
• [1] O. Arino, I. Győri and M. Pituk, Asymptotically diagonal delay differential systems, J. Math. Anal. Appl. (in the press).
• [2] F. V. Atkinson and J. R. Haddock, Criteria for asymptotic constancy of solutions of functional differential equations, J. Math. Anal. Appl. 91 (1983), 410-423.
• [3] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963.
• [4] K. Borsuk, Theory of Retracts, PWN, Warszawa, 1967.
• [5] J. Čermák, On the asymptotic behaviour of solutions of certain functional differential equations, Math. Slovaca 48 (1998), 187-212.
• [6] J. Čermák, The asymptotic bounds of solutions of linear delay systems, J. Math. Anal. Appl. 225 (1998), 373-388.
• [7] J. Diblík, Asymptotic representation of solutions of equation ẏ(t) = β(t)[y(t)-y(t-τ(t))], ibid. 217 (1998), 200-215.
• [8] I. Győri and M. Pituk, Comparison theorems and asymptotic equilibrium for delay differential and difference equations, Dynam. Systems Appl. 5 (1996), 277-302.
• [9] I. Győri and M. Pituk, L²-perturbation of a linear delay differential equation, J. Math. Anal. Appl. 195 (1995), 415-427.
• [10] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, 1993.
• [11] T. Krisztin, Asymptotic estimation for functional differential equations via Lyapunov functions, J. Math. Anal. Appl. 109 (1985), 509-521.
• [12] T. Krisztin, On the rate of convergence of solutions of functional differential equations, Funkcial. Ekvac. 29 (1986), 1-10.
• [13] T. Krisztin, A note on the convergence of the solutions of a linear functional differential equation, J. Math. Anal. Appl. 145 (1990), 17-25.
• [14] F. Neuman, On equivalence of linear functional-differential equations, Results in Math. 26 (1994), 354-359.
• [15] F. Neuman, On transformations of differential equations and systems with deviating argument, Czechoslovak Math. J. 31 (1981), 87-90.
• [16] K. P. Rybakowski, Ważewski's principle for retarded functional differential equations, J. Differential Equations 36 (1980), 117-138.
• [17] T. Ważewski, Sur un principe topologique de l'examen de l'allure asymptotique des intégrales des équations différentielles ordinaires, Ann. Soc. Polon. Math. 20 (1947), 279-313.
• [18] S. N. Zhang, Asymptotic behaviour and structure of solutions for equation ẋ(t) = p(t)[x(t) - x(t-1)], J. Anhui Univ. (Natural Science Edition) 2 (1981), 11-21 (in Chinese).
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Bibliografia
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