Mathematica Bohemica, Vol. 148, No. 1, pp. 73-94, 2023


On the meromorphic solutions of a certain type of nonlinear difference-differential equation

Sujoy Majumder, Lata Mahato

Received December 6, 2020.   Published online March 30, 2022.

Abstract:  The main objective of this paper is to give the specific forms of the meromorphic solutions of the nonlinear difference-differential equation $f^n(z)+P_d(z,f)=p_1(z){\rm e}^{\alpha_1(z)}+p_2(z){\rm e}^{\alpha_2(z)}$, where $P_d(z,f)$ is a difference-differential polynomial in $f(z)$ of degree $d\leq n-1$ with small functions of $f(z)$ as its coefficients, $p_1$, $p_2$ are nonzero rational functions and $\alpha_1$, $\alpha_2$ are non-constant polynomials. More precisely, we find out the conditions for ensuring the existence of meromorphic solutions of the above equation.
Keywords:  nonlinear differential equation; differential polynomial; Nevanlinna's value distribution theory
Classification MSC:  34M05, 30D35, 33E30, 30D30


References:
[1] R. Halburd, R. Korhonen, K. Tohge: Holomorphic curves with shift-invariant hyperplane preimages. Trans. Am. Math. Soc. 366 (2014), 4267-4298. DOI 10.1090/S0002-9947-2014-05949-7 | MR 3206459 | Zbl 1298.32012
[2] W. K. Hayman: Meromorphic Functions. Oxford Mathematical Monographs. Clarendon Press, Oxford (1964). MR 0164038 | Zbl 0115.06203
[3] J. Heittokangas, R. Korhonen, I. Laine: On meromorphic solutions of certain nonlinear differential equations. Bull. Aust. Math. Soc. 66 (2002), 331-343. DOI 10.1017/S000497270004017X | MR 1932356 | Zbl 1047.34101
[4] I. Laine: Nevanlinna Theory and Complex Differential Equations. de Gruyter Studies in Mathematics 15. Walter de Gruyter, Berlin (1993). DOI 10.1515/9783110863147 | MR 1207139 | Zbl 0784.30002
[5] P. Li: Entire solutions of certain type of differential equations. II. J. Math. Anal. Appl. 375 (2011), 310-319. DOI 10.1016/j.jmaa.2010.09.026 | MR 2735715 | Zbl 1206.30046
[6] P. Li, C.-C. Yang: On the nonexistence of entire solutions of certain type of nonlinear differential equations. J. Math. Anal. Appl. 320 (2006), 827-835. DOI 10.1016/j.jmaa.2005.07.066 | MR 2225998 | Zbl 1100.34066
[7] L.-W. Liao, C.-C. Yang, J.-J. Zhang: On meromorphic solutions of certain type of nonlinear differential equations. Ann. Acad. Sci. Fenn., Math. 38 (2013), 581-593. DOI 10.5186/aasfm.2013.3840 | MR 3113096 | Zbl 1303.30029
[8] W. Lü, L. Wu, D. Wang, C.-C. Yang: The existence of solutions of certain type of nonlinear difference-differential equations. Open Math. 16 (2018), 806-815. DOI 10.1515/math-2018-0071 | MR 3830193 | Zbl 1412.34239
[9] C.-C. Yang: On deficiencies of differential polynomials. II. Math. Z. 125 (1972), 107-112. DOI 10.1007/BF01110921 | MR 0294642 | Zbl 0217.38402
[0] C.-C. Yang, P. Li: On the transcendental solutions of a certain type of nonlinear differential equations. Arch. Math. 82 (2004), 442-448. DOI 10.1007/s00013-003-4796-8 | MR 2061450 | Zbl 1052.34083
[11] C.-C. Yang, H.-X. Yi: Uniqueness Theory of Meromorphic Functions. Mathematics and its Applications (Dordrecht) 557. Kluwer Academic, Dordrecht (2003). DOI 10.1007/978-94-017-3626-8 | MR 2105668 | Zbl 1070.30011

Affiliations:   Sujoy Majumder, Department of Mathematics, Raiganj University, Raiganj, West Bengal-733134, India, e-mail: sm05math@gmail.com, Lata Mahato, Department of Mathematics, Mahadevananda Mahavidyalaya, SN Banerjee Rd., Monirampore, Kolkata, West Bengal-700120, India, e-mail: himangshulata@gmail.com


 
PDF available at: