## International Journal of Differential Equations

### Analytic Solutions of an Iterative Functional Differential Equation near Resonance

#### Abstract

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

#### Article information

Source
Int. J. Differ. Equ., Volume 2009 (2009), Article ID 145213, 14 pages.

Dates
Accepted: 14 April 2009
First available in Project Euclid: 26 January 2017

https://projecteuclid.org/euclid.ijde/1485399813

Digital Object Identifier
doi:10.1155/2009/145213

Mathematical Reviews number (MathSciNet)
MR2525715

Zentralblatt MATH identifier
1207.34074

#### Citation

Liu, Tongbo; Li, Hong. Analytic Solutions of an Iterative Functional Differential Equation near Resonance. Int. J. Differ. Equ. 2009 (2009), Article ID 145213, 14 pages. doi:10.1155/2009/145213. https://projecteuclid.org/euclid.ijde/1485399813

#### References

• J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977.MR0508721
• R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, NY, USA, 1963.MR0147745
• A. D. Brjuno, “Analytic form of differential equations,” Transactions of the Moscow Mathematical Society, vol. 25, pp. 131–288, 1971.
• R. D. Driver, “Existence theory for a delay-differential system,” Contributions to Differential Equations, vol. 1, pp. 317–336, 1963.MR0150421
• E. Eder, “The functional-differential equation ${x}^{\prime }(t)=x(x(t))$,” Journal of Differential Equations, vol. 54, no. 3, pp. 390–400, 1984.MR760378
• V. R. Petuhov, “On a boundary value problem,” Trudy Seminara po Teorii Differencial'nyh Uravneniĭ s Otklonjajuščimsja Argumentom. Universitet Družby Narodov im. Patrisa Lumumby, vol. 3, pp. 252–255, 1965.MR0208109
• 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.MR1670209
• 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.MR1806761
• 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.MR1834580
• 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.MR2135843
• 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.MR2082681
• 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.MR2396300
• T. Carletti and S. Marmi, “Linearization of analytic and non-analytic germs of diffeomorphisms of $(\mathbb{C},0)$,” Bulletin de la Société Mathématique de France, vol. 128, no. 1, pp. 69–85, 2000.MR1765828
• A. M. Davie, “The critical function for the semistandard map,” Nonlinearity, vol. 7, no. 1, pp. 219–229, 1994.MR1260138