# Article

Full entry | PDF   (0.3 MB)
Keywords:
functional differential equation; functional nondifferential equation; asymptotic behaviour; transformation
Summary:
We discuss the asymptotic behaviour of all solutions of the functional differential equation $y^{\prime }(x)=\sum _{i=1}^ma_i(x)y(\tau _i(x))+b(x)y(x)\,,$ where $b(x)<0$. The asymptotic bounds are given in terms of a solution of the functional nondifferential equation $\sum _{i=1}^m|a_i(x)|\omega (\tau _i(x))+b(x)\omega (x)=0.$
References:
[1] Čermák, J.: On the asymptotic behaviour of solutions of certain functional differential equations. Math. Slovaca 48 (1998), 187–212. MR 1647674
[2] Čermák, J.: The asymptotic bounds of linear delay systems. J. Math. Anal. Appl. 225 (1998), 373–388. MR 1644331
[3] Dibl¡k, J.: Asymptotic equilibrium for a class of delay differential equations. Proc. of the Second International Conference on Difference Equations, S. Elaydi, I. Győri, G. Ladas (eds.), 1995, pp. 137–143. MR 1636319
[4] Hale, J. K., Verduyn Lunel, S. M.: Introduction to Functional Differential Equations, Springer-Verlag. New York, 1993. MR 1243878
[5] Heard, M. L.: A change of variables for functional differential equations. J. Differential Equations 18 (1975), 1–10. MR 0387766 | Zbl 0318.34069
[6] Kato, T., Mcleod, J. B.: The functional differential equation $y^{\prime }(x)=ay(\lambda x)+by(x)$. Bull. Amer. Math. Soc. 77 (1971), 891–397. MR 0283338
[7] Kuczma, M., Choczewski, B., Ger, R.: Iterative Functional Equations, Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1990. MR 1067720
[8] Neuman, F.: Simultaneous solutions of a system of Abel equations and differential equations with several deviations. Czechoslovak Math. J. 32 (107) (1982), 488–494. MR 0669790 | Zbl 0524.34070
[9] Neuman, F.: Transformations and canonical forms of functional-differential equations. Proc. Roy. Soc. Edinburgh 115A (1990), 349–357. MR 1069527
[10] Zdun, M.: On Simultaneous Abel equations. Aequationes Math. (1989), 163–177. MR 1018910 | Zbl 0686.39009

Partner of