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Analytic solutions of an iterative functional differential equation near resonance. (English) Zbl 1207.34074

Summary: We investigate the existence of analytic solutions of a class of second-order differential equations involving iterates of the unknown function
\[ x''(z) + cx'(z) = x(az + bx(z)) \]
in the complex field \(\mathbb C\). By reducing the equation with the Schröder transformation to another functional differential equation without iteration of the unknown function
\[ \lambda^2g''(\lambda z)g'(z)-\lambda g'(\lambda z)g''(z)+c(g'(z))^2(\lambda g'(\lambda z)-ag'(z))=(g'(z))^3(\lambda^2z)-ag(\lambda z)), \]
we get its local invertible analytic solutions.

MSC:

34K05 General theory of functional-differential equations
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References:

[1] J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977. · Zbl 0461.05016 · doi:10.1016/0021-8693(77)90310-6
[2] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, NY, USA, 1963. · Zbl 0226.76030 · doi:10.1073/pnas.50.2.222
[3] A. D. Brjuno, “Analytic form of differential equations,” Transactions of the Moscow Mathematical Society, vol. 25, pp. 131-288, 1971.
[4] R. D. Driver, “Existence theory for a delay-differential system,” Contributions to Differential Equations, vol. 1, pp. 317-336, 1963. · Zbl 0126.10102
[5] E. Eder, “The functional-differential equation x\(^{\prime}\)(t)=x(x(t)),” Journal of Differential Equations, vol. 54, no. 3, pp. 390-400, 1984. · Zbl 0497.34050 · doi:10.1016/0022-0396(84)90150-5
[6] V. R. Petuhov, “On a boundary value problem,” Trudy Seminara po Teorii Differencial/nyh Uravneniĭ s Otklonjaju\vs\vcimsja Argumentom. Universitet Dru\vzby Narodov im. Patrisa Lumumby, vol. 3, pp. 252-255, 1965.
[7] J.-G. Si and X.-P. Wang, “Analytic solutions of a second-order functional-differential equation with a state derivative dependent delay,” Colloquium Mathematicum, vol. 79, no. 2, pp. 273-281, 1999. · Zbl 0927.34077
[8] J.-G. Si and X.-P. Wang, “Analytic solutions of a second-order iterative functional differential equation,” Journal of Computational and Applied Mathematics, vol. 126, no. 1-2, pp. 277-285, 2000. · Zbl 0983.34056 · doi:10.1016/S0377-0427(99)00359-3
[9] J.-G. Si and X.-P. Wang, “Analytic solutions of a second-order functional differential equation with a state dependent delay,” Results in Mathematics, vol. 39, no. 3-4, pp. 345-352, 2001. · Zbl 1017.34074 · doi:10.1007/BF03322694
[10] J.-G. Si and W. Zhang, “Analytic solutions of a second-order nonautonomous iterative functional differential equation,” Journal of Mathematical Analysis and Applications, vol. 306, no. 2, pp. 398-412, 2005. · Zbl 1083.34060 · doi:10.1016/j.jmaa.2005.01.005
[11] J.-G. Si and W. Zhang, “Analytic solutions of a q-difference equation and applications to iterative equations,” Journal of Difference Equations and Applications, vol. 10, no. 11, pp. 955-962, 2004. · Zbl 1067.39035 · doi:10.1080/10236190412331272607
[12] T. Liu and H. Li, “Local analytic solution of a second-order functional differential equation with a state derivative dependent delay,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 158-166, 2008. · Zbl 1143.34042 · doi:10.1016/j.amc.2007.07.057
[13] T. Carletti and S. Marmi, “Linearization of analytic and non-analytic germs of diffeomorphisms of (\Bbb C,0),” Bulletin de la Société Mathématique de France, vol. 128, no. 1, pp. 69-85, 2000. · Zbl 0997.37017
[14] A. M. Davie, “The critical function for the semistandard map,” Nonlinearity, vol. 7, no. 1, pp. 219-229, 1994. · Zbl 0997.37500 · doi:10.1088/0951-7715/7/1/009
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