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

Diblík, Josef

Annales Polonici Mathematici, Tome 72 (1999), p. 115-130 / Harvested from The Polish Digital Mathematics Library

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

Publié le : 1999-01-01

Zbl 0953.34065

EUDML-ID : urn:eudml:doc:262578

Bibliographie

[00000] [1] O. Arino, I. Győri and M. Pituk, Asymptotically diagonal delay differential systems, J. Math. Anal. Appl. (in the press). | Zbl 0876.34078

[00001] [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. | Zbl 0529.34065

[00002] [3] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963. | Zbl 0105.06402

[00003] [4] K. Borsuk, Theory of Retracts, PWN, Warszawa, 1967. | Zbl 0153.52905

[00004] [5] J. Čermák, On the asymptotic behaviour of solutions of certain functional differential equations, Math. Slovaca 48 (1998), 187-212. | Zbl 0942.34060

[00005] [6] J. Čermák, The asymptotic bounds of solutions of linear delay systems, J. Math. Anal. Appl. 225 (1998), 373-388. | Zbl 0913.34063

[00006] [7] J. Diblík, Asymptotic representation of solutions of equation ẏ(t) = β(t)[y(t)-y(t-τ(t))], ibid. 217 (1998), 200-215. | Zbl 0892.34067

[00007] [8] I. Győri and M. Pituk, Comparison theorems and asymptotic equilibrium for delay differential and difference equations, Dynam. Systems Appl. 5 (1996), 277-302. | Zbl 0859.34053

[00008] [9] I. Győri and M. Pituk, L²-perturbation of a linear delay differential equation, J. Math. Anal. Appl. 195 (1995), 415-427. | Zbl 0853.34070

[00009] [10] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, 1993.

[00010] [11] T. Krisztin, Asymptotic estimation for functional differential equations via Lyapunov functions, J. Math. Anal. Appl. 109 (1985), 509-521. | Zbl 0586.34061

[00011] [12] T. Krisztin, On the rate of convergence of solutions of functional differential equations, Funkcial. Ekvac. 29 (1986), 1-10. | Zbl 0601.34046

[00012] [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. | Zbl 0693.45012

[00013] [14] F. Neuman, On equivalence of linear functional-differential equations, Results in Math. 26 (1994), 354-359. | Zbl 0829.34054

[00014] [15] F. Neuman, On transformations of differential equations and systems with deviating argument, Czechoslovak Math. J. 31 (1981), 87-90. | Zbl 0463.34051

[00015] [16] K. P. Rybakowski, Ważewski's principle for retarded functional differential equations, J. Differential Equations 36 (1980), 117-138. | Zbl 0407.34056

[00016] [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. | Zbl 0032.35001

[00017] [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).