Previous |  Up |  Next


functional differential equations; ordinary differential equations; global transformations; functional equations in a single variable; functional equations in several variables
The paper describes the general form of functional-differential equations of the first order with $m (m\ge 1)$ delays which allows nontrivial global transformations consisting of a change of the independent variable and of a nonvanishing factor. A functional equation \[ f(t, uv, u_{1}v_{1}, \ldots , u_{m}v_{m}) = f(x, v, v_{1}, \ldots , v_{m})g(t, x, u, u_{1}, \ldots , u_{m})u + h(t, x, u, u_{1}, \ldots , u_{m})v \] for $u\ne 0$ is solved on $\mathbb R$ and a method of proof by J. Aczél is applied.
[n1] Aczél, J.: Über Zusammenhänge zwischen Differential- und Funktionalgleichungen. Jahresber. Deutsch. Math.-Verein. 71 (1969), 55–57. MR 0256014
[n3] Aczél, J.: Lectures on Functional Equations and Their Applications. Academic Press, New York, 1966. MR 0208210
[n4] Čermák, J.: Continuous transformations of differential equations with delays. Georgian Math. J. 2 (1995), 1–8. DOI 10.1007/BF02257729 | MR 1310496
[n5] Ladde, G.S., Lakshmikantham, V., Zhang, B.G.: Oscillation Theory of Differential Equations with Deviating Arguments. Marcel Dekker, New York and Basel, 1987. MR 1017244
[n6] Neuman, F.: On transformations of differential equations and systems with deviating argument. Czechoslovak Math. J. 31 (106) (1981), 87–90. MR 0604115 | Zbl 0463.34051
[n7] Neuman, F.: Simultaneous solutions of a system of Abel equations and differential equations with several delays. Czechoslovak Math. J. 32 (107) (1982), 488–494. MR 0669790
[n8] Neuman, F.: Transformations and canonical forms of functional-differential equations. Proc. Roy. Soc. Edinburgh 115 A (1990), 349–357. MR 1069527
[n9] Neuman, F.: Global Properties of Linear Ordinary Differential Equations. Mathematics and Its Applications (East European Series) 52, Kluwer Acad. Publ., Dordrecht-Boston-London, 1991. MR 1192133 | Zbl 0784.34009
[n10] Neuman, F.: On equivalence of linear functional-differential equations. Results Math. 26 (1994), 354–359. DOI 10.1007/BF03323059 | MR 1300618 | Zbl 0829.34054
[n11] Tryhuk, V.: The most general transformations of homogeneous linear differential retarded equations of the first order. Arch. Math. (Brno) 16 (1980), 225–230. MR 0594470
[n12] Tryhuk, V.: The most general transformation of homogeneous linear differential retarded equations of the $n$-th order. Math. Slovaca 33 (1983), 15–21. MR 0689272
[n13] Zdun, M.: Note on commutable functions. Aequationes Math. 36 (1988), 153–164. DOI 10.1007/BF01836087 | MR 0972282 | Zbl 0662.39004
[n14] Zdun, M.: On simultaneous Abel equations. Aequationes Math. 38 (1989), 163–177. DOI 10.1007/BF01840002 | MR 1018910 | Zbl 0686.39009
Partner of
EuDML logo