On transcendental meromorphic solutions of certain types of differential equations
3
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this paper, for a transcendental meromorphic function $f$ and $a\in \mathbb{C}$, we have exhaustively studied the nature and form of solutions of a new type of non-linear differential equation of the following form which has never been investigated earlier: \[f^n+af^{n-2}f'+ P_d(z,f) = \sum_{i=1}^{k}p_i(z)e^{\alpha_i(z)}, \] where $P_d(z,f)$ is a differential polynomial of $f$, $p_i$'s and $\alpha_{i}$'s are non-vanishing rational functions and non-constant polynomials, respectively. When $a=0$, we have pointed out a major lacuna in a recent result of Xue \cite{xue_math slovaca20} and rectifying the result, presented the corrected form of the same equation at a large extent. In addition, our main result is also an improvement of a recent result of Chen-Lian \cite{chen lian_BKMS20} by rectifying a gap in the proof of the theorem of the same paper. The case $a\neq 0$ has also been manipulated to determine the form of the solutions. We also illustrate a handful number of examples for showing the accuracy of our results.
Abstract
In this paper, for a transcendental meromorphic function $f$ and $a\in \mathbb{C}$, we have exhaustively studied the nature and form of solutions of a new type of non-linear differential equation of the following form which has never been investigated earlier: \[f^n+af^{n-2}f'+ P_d(z,f) = \sum_{i=1}^{k}p_i(z)e^{\alpha_i(z)}, \] where $P_d(z,f)$ is a differential polynomial of $f$, $p_i$'s and $\alpha_{i}$'s are non-vanishing rational functions and non-constant polynomials, respectively. When $a=0$, we have pointed out a major lacuna in a recent result of Xue \cite{xue_math slovaca20} and rectifying the result, presented the corrected form of the same equation at a large extent. In addition, our main result is also an improvement of a recent result of Chen-Lian \cite{chen lian_BKMS20} by rectifying a gap in the proof of the theorem of the same paper. The case $a\neq 0$ has also been manipulated to determine the form of the solutions. We also illustrate a handful number of examples for showing the accuracy of our results.
참고문헌 (19)
- M. F. Chen, "Entire solutions of certain type of nonlinear differential equations and differential-difference equations", J. Comput. Anal. Appl, 24(1), pp.137~147 (2018)
- Jun-Fan Chen, "Expressions of meromorphic solutions of a certain type of nonlinear complex differential equations", 대한수학회보, 57(4), pp.1061~1073 (2020), 대한수학회[DOI]
- J. Clunie, "On integral and meromorphic functions", J. London Math. Soc, 37, pp.17~27 (1962)[DOI]
- W. K. Hayman, "Meromorphic Functions, Oxford Mathematical Monographs" (1964), Clarendon Press
- Pei-Chu Hu, "Existence of transcendental meromorphic solutions on some types of nonlinear differential equations", 대한수학회보, 57(4), pp.991~1002 (2020), 대한수학회[DOI]
- I. Laine, "Nevanlinna theory and complex differential equations", De Gruyter Studies in Mathematics, 15, pp.1515~ (1993), Walter de Gruyter & Co[DOI]
- Z. Latreuch, "On the existence of entire solutions of certain class of nonlinear difference equations", Mediterr. J. Math, 14(3), pp.16~ (2017)[DOI]
- P. Li, "Entire solutions of certain type of differential equations II", J. Math. Anal. Appl, 375(1), pp.310~319 (2011)[DOI]
- P. Li, "On the nonexistence of entire solutions of certain type of nonlinear differential equations", J. Math. Anal. Appl, 320(2), pp.827~835 (2006)[DOI]
- L. -W. Liao, "Non-linear differential equations and Hayman’s theorem on differential poly"nomials", Complex Var. Elliptic Equ, 60(6), pp.748~756 (2015)[DOI]
- L. -W. Liao, "On meromorphic solutions of certain type of non-linear differential equations", Ann. Acad. Sci. Fenn. Math, 38(2), pp.581~593 (2013)[DOI]
- L. -W. Liao, "On solutions to nonhomogeneous algebraic differential equations and their application", J. Aust. Math. Soc, 97(3), pp.391~403 (2014)[DOI]
- X. Q. Lu, "On meromorphic solutions of a certain type of nonlinear differential equations", Acta Math. Sin. (Engl. Ser. ), 33(12), pp.1597~1608 (2017)[DOI]
- F. Lu, "Growth and uniqueness related to complex differential and difference equations", Results Math, 74(1), pp.18~ (2019)[DOI]
- L. Mirsky, "An Introduction to Linear Algebra" (1955), Oxford, at the Clarendon Press
- J. X. Rong, "Three results on the nonlinear differential equations and differential-difference equations", Mathematics, 7(6), pp.539~ (2019)[DOI]
- B. Xue, "Entire solutions of certain type of non-linear differential equations", Math. Slovaca, 70(1), pp.87~94 (2020)[DOI]
- C.-C. Yang, "Uniqueness theory of meromorphic functions", Mathematics and its Applications, 557 (2003), Kluwer Academic Publishers Group
- J. J. Zhang, "Further results on transcendental meromorphic solutions of certain types of nonlinear differential equations", Acta Math. Sinica(Chin. Ser. ), 61(4), pp.529~540 (2018)