# Article

 Title: Asymptotic behaviour of solutions of some linear delay differential equations (English) Author: Čermák, Jan Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 Volume: 125 Issue: 3 Year: 2000 Pages: 355-364 Summary lang: English . Category: math . Summary: In this paper we investigate the asymptotic properties of all solutions of the delay differential equation y'(x)=a(x)y(\tau(x))+b(x)y(x),\qquad x\in I=[x_0,\infty). We set up conditions under which every solution of this equation can be represented in terms of a solution of the differential equation z'(x)=b(x)z(x),\qquad x\in I and a solution of the functional equation |a(x)|\varphi(\tau(x))=|b(x)|\varphi(x),\qquad x\in I. (English) Keyword: asymptotic behaviour Keyword: differential equation Keyword: delayed argument Keyword: functional equation MSC: 34K15 MSC: 34K25 MSC: 39B05 MSC: 39B22 MSC: 39B99 idZBL: Zbl 0972.34066 idMR: MR1790126 . Date available: 2009-09-24T21:44:21Z Last updated: 2015-09-15 Stable URL: http://hdl.handle.net/10338.dmlcz/126125 . Reference: [1] F. V. Atkinson J. R. Haddock: Criteria for asymptotic constancy of solutions of functional differential equations.J. Math. Anal. Appl. 91 (1983), 410-423. MR 0690880, 10.1016/0022-247X(83)90161-0 Reference: [2] N. G. de Bruijn: The asymptotically periodic behavior of the solutions of some linear functional equations.Amer. J. Math. 71 (1949), 313-330. Zbl 0033.27002, MR 0029065, 10.2307/2372246 Reference: [3] J. Čermák: On the asymptotic behaviour of solutions of some functional-differential equations.Math. Slovaca 48 (1998), 187-212. MR 1647674 Reference: [4] J. Čermák: The asymptotic bounds of solutions of linear delay systems.J. Math. Anal. Appl. 225 (1998), 373-388. MR 1644331, 10.1006/jmaa.1998.6018 Reference: [5] J. Diblík: Asymptotic representation of solutions of equation $\dot{y} (t) = \beta (t)[y(t) - y(t - \tau (t))].J. Math. Anal Appl 217 (1998), 200-215. MR 1492085, 10.1006/jmaa.1997.5709 Reference: [6] I. Győri M. Pituk: Comparison theorems and asymptotic equilibrium for delay differential and difference equations.Dynam. Systems Appl. 5 (1996), 277-302. MR 1396192 Reference: [7] J. K. Hale S. M. Verduyn Lunel: Functional Differential Equations.Springer-Verlag, New York, 1993. Reference: [8] M. L. Heard: A change of variables for functional differential equations.J. Differential Equations 18 (1975), 1-10. Zbl 0318.34069, MR 0387766, 10.1016/0022-0396(75)90076-5 Reference: [9] T. Kato J. B. McLeod: The functional differential equation$y'(x) = a y(\lambda x) + b y(x).Bull. Amer. Math. Soc. 77 (1971), 891-937. MR 0283338 Reference: [10] M. Kuczma B. Choczewski R. Ger: Iterative Functional Equations.Encyclopedia of Mathematics and Its Applications, Cambridge Univ. Press, Cambridge, England, 1990. MR 1067720 Reference: [11] F. Neuman: On transformations of differential equations and systems with deviating argument.Czechoslovak Math, J. 31 (1981), 87-90. Zbl 0463.34051, MR 0604115 Reference: [12] S. N. Zhang: Asymptotic behaviour and structure of solutions for equation $\dot{x} (t) = p(t)[x(t) - x(t - 1)]$.J. Anhui Normal Univ. Nat. Sci. 2 (1981), 11-21. (In Chinese.) .

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