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Meromorphic Solutions of Nonlinear Differential-Difference Equations Involving Periodic Functions

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Abstract

We investigate the following two types of nonlinear differential-difference equations

$$ L(z,f)+H(z,f)=\sum _{k=1}^r\alpha _k(z)e^{\beta _k z}; \ \ \ \ $$
$$L(z,f)+H(z,f)=\sum _{k=1}^rF_k(z), \ \ \ \ \ \ \ \ \ $$

where \(\alpha _1, \ldots , \alpha _r\) are meromorphic functions of order \(<1,\) and \(F_1,\ldots , F_r\) are periodic transcendental entire functions, and L, H are defined by \(L(z,f)=\sum _{k=1}^pa_k(z)f^{(m_k)}(z+\tau _k)\not \equiv 0,\) \(H(z,f)=\sum _{k=1}^qb_k(z)\big [f^{(n_k)}(z+\zeta _k)\big ]^{s_k} \ \ \) with small meromorphic coefficients \(a_i, b_j.\) By introducing a new method, we obtain the exact forms of the solutions of these two equations under certain growth conditions.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Shuang-Shuang Yang & Liang-Wen Liao

  2. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, People’s Republic of China

    Xian-Jing Dong

Authors
  1. Shuang-Shuang Yang
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  2. Xian-Jing Dong
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  3. Liang-Wen Liao
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Correspondence to Liang-Wen Liao.

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Communicated by Rosihan M. Ali.

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This research is supported by the National Natural Science Foundation of China (Grant No: 12171232).

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Yang, SS., Dong, XJ. & Liao, LW. Meromorphic Solutions of Nonlinear Differential-Difference Equations Involving Periodic Functions. Bull. Malays. Math. Sci. Soc. 47, 91 (2024). https://doi.org/10.1007/s40840-024-01681-9

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  • DOI: https://doi.org/10.1007/s40840-024-01681-9

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