Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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THREE RESULTS ON TRANSCENDENTAL MEROMORPHIC SOLUTIONS OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS

  • Li, Nan (School of Mathematics Qilu Normal University) ;
  • Yang, Lianzhong (School of Mathematics Shandong University)
  • Received : 2020.03.22
  • Accepted : 2020.09.15
  • Published : 2021.07.31
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Abstract

In this paper, we study the transcendental meromorphic solutions for the nonlinear differential equations: fn + P(f) = R(z)eα(z) and fn + P*(f) = p1(z)eα1(z) + p2(z)eα2(z) in the complex plane, where P(f) and P*(f) are differential polynomials in f of degree n - 1 with coefficients being small functions and rational functions respectively, R is a non-vanishing small function of f, α is a nonconstant entire function, p1, p2 are non-vanishing rational functions, and α1, α2 are nonconstant polynomials. Particularly, we consider the solutions of the second equation when p1, p2 are nonzero constants, and deg α1 = deg α2 = 1. Our results are improvements and complements of Liao ([9]), and Rong-Xu ([11]), etc., which partially answer a question proposed by Li ([7]).

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Acknowledgement

The authors would like to thank the referee for his/her thorough review with constructive suggestions and valuable comments on the paper, especially the construction of h(z) in formula (54) which plays an important role in the proof of Theorem 1.3 Case 1.

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