Geodesic
On the meromorphic solutions of a certain type of nonlinear difference-differential equation
Mathematica Bohemica, Tome 148 (2023) no. 1, pp. 73-94
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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.
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.
DOI : 10.21136/MB.2022.0186-20
Classification : 30D30, 30D35, 33E30, 34M05
Keywords: nonlinear differential equation; differential polynomial; Nevanlinna's value distribution theory 
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Majumder, Sujoy; Mahato, Lata. On the meromorphic solutions of a certain type of nonlinear difference-differential equation. Mathematica Bohemica, Tome 148 (2023) no. 1, pp. 73-94. doi: 10.21136/MB.2022.0186-20

[1] Halburd, R., Korhonen, R., Tohge, K.: Holomorphic curves with shift-invariant hyperplane preimages. Trans. Am. Math. Soc. 366 (2014), 4267-4298. | DOI | MR | JFM

[2] Hayman, W. K.: Meromorphic Functions. Oxford Mathematical Monographs. Clarendon Press, Oxford (1964). | MR | JFM

[3] Heittokangas, J., Korhonen, R., Laine, I.: On meromorphic solutions of certain nonlinear differential equations. Bull. Aust. Math. Soc. 66 (2002), 331-343. | DOI | MR | JFM

[4] Laine, I.: Nevanlinna Theory and Complex Differential Equations. de Gruyter Studies in Mathematics 15. Walter de Gruyter, Berlin (1993). | DOI | MR | JFM

[5] Li, P.: Entire solutions of certain type of differential equations. II. J. Math. Anal. Appl. 375 (2011), 310-319. | DOI | MR | JFM

[6] Li, P., Yang, C.-C.: On the nonexistence of entire solutions of certain type of nonlinear differential equations. J. Math. Anal. Appl. 320 (2006), 827-835. | DOI | MR | JFM

[7] Liao, L.-W., Yang, C.-C., Zhang, J.-J.: On meromorphic solutions of certain type of nonlinear differential equations. Ann. Acad. Sci. Fenn., Math. 38 (2013), 581-593. | DOI | MR | JFM

[8] Lü, W., Wu, L., Wang, D., Yang, C.-C.: The existence of solutions of certain type of nonlinear difference-differential equations. Open Math. 16 (2018), 806-815. | DOI | MR | JFM

[9] Yang, C.-C.: On deficiencies of differential polynomials. II. Math. Z. 125 (1972), 107-112. | DOI | MR | JFM

[0] Yang, C.-C., Li, P.: On the transcendental solutions of a certain type of nonlinear differential equations. Arch. Math. 82 (2004), 442-448. | DOI | MR | JFM

[11] Yang, C.-C., Yi, H.-X.: Uniqueness Theory of Meromorphic Functions. Mathematics and its Applications (Dordrecht) 557. Kluwer Academic, Dordrecht (2003). | DOI | MR | JFM

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