Solutions of equivariance for iterative differential equations. (English) Zbl 1068.34069

The authors study an iterative differential equation of the form \(x'(t)= G[x^{n_1}(t),\dots, x^{n_k}(t)]\) on an internal \(I\subset\mathbb{R}\), where \(G: I^k\to\mathbb{R}\) is continuous, \(n_j\in\mathbb{N}\) for \(j= 1,\dots, k\), and \(x^n\) denotes the \(n\)th iteration of \(x: I\to I\). If \(\Gamma\) is a Lie group acting on \(\mathbb{R}\), a solution \(x\) of the above equation is said to be \(\Gamma\)-equivariant if \(x(\gamma t)= \gamma x(t)\), for \(\gamma\in\Gamma\) and \(t\in I\). Under the assumption that \(\Gamma\) is topologically finitely generated, they find sufficient conditions on \(G\) under which there exist \(\Gamma\)-equivariant solutions.


34K17 Transformation and reduction of functional-differential equations and systems, normal forms
34A26 Geometric methods in ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
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[1] Hale, J.K., Theory of functional differential equations, (1977), Springer New York · Zbl 0425.34048
[2] Cooke, K.L., Functional differential systems: some models and perturbation problems, () · Zbl 0189.40301
[3] Stephan, B.H., On the existence of periodic solutions of z^{1}(t) = −az(t − r + μk(t,z(t))) · Zbl 0184.12101
[4] Eder, E., The functional differential equation x′(t) = x(x(t)), J. differential equations, 54, 390-400, (1984) · Zbl 0497.34050
[5] Feckn, E., On certain type of functional differential equations, Math. slovaca, 43, 39-43, (1993)
[6] Wang, K., On the equation x′(t) = f(x(x(t))), Funkcial. ekvac., 33, 405-425, (1990) · Zbl 0714.34026
[7] Si, J.; Cheng, S.S., Smooth solutions of a nonhomogeneous iterative functional differential equation, (), 821-831 · Zbl 0912.34057
[8] Si, J.; Wang, X., Smooth solutions of a nonhomogeneous iterative functional differential equation with variable coefficients, J. math. anal. appl., 226, 377-392, (1998) · Zbl 0917.34055
[9] Si, J.; Li, W.R.; Cheng, S.S., Analytic solutions of an iterative functional differential equation, Computers math. applic., 33, 6, 47-51, (1997) · Zbl 0872.34042
[10] Hsing, D.P.K., Existence and uniqueness theorem for the one-dimensional backwards two-body problem of electrodynamics, Phys. rev. D., 16, 4, 974-982, (1977)
[11] Staněk, S., On global properties of solutions of functional differential equation x′(t) = x(x(t)) + x(t), Dynam. systems appl., 4, 263-278, (1995) · Zbl 0830.34064
[12] Golubitsky, M.; Schaeffer, D.G., Singularities and groups in bifurcation theory, volume 1, () · Zbl 0607.35004
[13] Golubitsky, M.; Stewart, I.N.; Schaeffer, D.G., Singularities and groups in bifurcation theory, volume 2, ()
[14] Zhang, W., A criterion of sequential compactness of smooth functions (in Chinese), Math. in practice & theory, 27, 372-375, (1997)
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