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Analytic Solutions of a Second-Order Functional Differential Equation

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Abstract

In this paper, we study the existence of analytic solutions of a second-order differential equation

$$\begin{aligned} \alpha z+\beta x'(z)+\gamma x''(z)=x(az+bx''(z)), \end{aligned}$$

in the complex field \(\mathbb C,\) where \(\alpha , \beta , \gamma , a, b\) are complex numbers. We discuss not only the constant \(\lambda \) at resonance, i.e. at a root of the unity, but also those \(\lambda \) near resonance (near a root of the unity) under the Brjuno condition.

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Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11326120), Foundation of Chongqing Municipal Education Commission (Grant No. KJ1400528), Program of Chongqing Innovation Team Project in University (Grant No. KJTD201308) and the Natural Science Foundation of Chongqing Normal University (Grant No. 12XLZ04).

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Authors and Affiliations

  1. School of Mathematics, Chongqing Normal University, Chongqing, 401331, People’s Republic of China

    Houyu Zhao

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  1. Houyu Zhao
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Correspondence to Houyu Zhao.

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Communicated by Shangjiang Guo.

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Zhao, H. Analytic Solutions of a Second-Order Functional Differential Equation. Bull. Malays. Math. Sci. Soc. 38, 719–731 (2015). https://doi.org/10.1007/s40840-014-0046-4

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  • DOI: https://doi.org/10.1007/s40840-014-0046-4

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