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Keywords:
asymptotic behaviour; differential equation; delayed argument; functional equation
asymptotic behaviour; differential equation; delayed argument; functional equation
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.
References:
[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. DOI 10.1016/0022-247X(83)90161-0 | MR 0690880
[2] N. G. de Bruijn: The asymptotically periodic behavior of the solutions of some linear functional equations. Amer. J. Math. 71 (1949), 313-330. DOI 10.2307/2372246 | MR 0029065 | Zbl 0033.27002
[3] J. Čermák: On the asymptotic behaviour of solutions of some functional-differential equations. Math. Slovaca 48 (1998), 187-212. MR 1647674
[4] J. Čermák: The asymptotic bounds of solutions of linear delay systems. J. Math. Anal. Appl. 225 (1998), 373-388. DOI 10.1006/jmaa.1998.6018 | MR 1644331
[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. DOI 10.1006/jmaa.1997.5709 | MR 1492085
[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
[7] J. K. Hale S. M. Verduyn Lunel: Functional Differential Equations. Springer-Verlag, New York, 1993.
[8] M. L. Heard: A change of variables for functional differential equations. J. Differential Equations 18 (1975), 1-10. DOI 10.1016/0022-0396(75)90076-5 | MR 0387766 | Zbl 0318.34069
[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
[10] M. Kuczma B. Choczewski R. Ger: Iterative Functional Equations. Encyclopedia of Mathematics and Its Applications, Cambridge Univ. Press, Cambridge, England, 1990. MR 1067720
[11] F. Neuman: On transformations of differential equations and systems with deviating argument. Czechoslovak Math, J. 31 (1981), 87-90. MR 0604115 | Zbl 0463.34051
[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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