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Holomorphic Solutions of a Functional Equation Related to Nonlinear Difference Systems

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Abstract

Here we consider the following functional equation,

$$\Psi(X(x,\Psi(x)))=Y(x, \Psi(x)),$$

where X(x, y) and Y(x, y) are holomorphic functions in |x| < δ 1, |y| < δ 1. When we consider a nonlinear simultaneous system of two variables difference equations, we can reduce it to a single difference equation of first order by a solution Ψ of the above functional equation. We obtain a matrix by the linear terms of functions X and Y. When the all eigenvalues of the matrix are equal to 1, it is difficult to have a solution of the above functional equation. In the present paper, we derive a formal solution of the above functional equation under the condition. Further we prove the existence of a solution which is holomorphic and have an asymptotically expansion of the formal solution. Moreover, we will show an example of nonlinear difference system such that our results are applicable.

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Authors and Affiliations

  1. Department of Mathematics, College of Liberal Arts, J. F. Oberlin University, 3758 Tokiwa-cho, Machida-City, Tokyo, 194-0294, Japan

    Mami Suzuki

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  1. Mami Suzuki
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Correspondence to Mami Suzuki.

Additional information

To my supervisor, the late Professor Niro Yanagihara

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Suzuki, M. Holomorphic Solutions of a Functional Equation Related to Nonlinear Difference Systems. Results. Math. 58, 17–35 (2010). https://doi.org/10.1007/s00025-010-0044-2

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  • DOI: https://doi.org/10.1007/s00025-010-0044-2

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