On the transcendental entire and meromorphic solutions of certain non-linear generalized delay-differential equations
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Abhijit Banerjee
Abstract
The prime intention of this paper is to study the conditions under which certain non-linear generalized delay-differential equations possess a solution. In this respect, by extending and improving recent results of [5, 15], we characterize the nature of solutions. By providing relevant examples in a remark, we also show that for the uniqueness of solutions the conditions on the Borel exceptional value can not be removed. Finally, to determine explicitly all forms of the solutions of a traditional generalized delay-differential equation, we deal with the situation under the aegis of generalized c-delay-differential equations. We exhibit some examples to show that the conclusions of the theorems actually occur.
Funding statement: The first author is thankful to DST-PURSE–II Programme for financial assistance. The second author is thankful to University Grant Commission (UGC), Govt. of India, for financial support under UGC reference number 1174/(CSIR-UGC NET DEC. 2017) dated 21/01/2019.
Acknowledgements
The authors wish to thank the referee for his/her valuable suggestions towards the improvement of the paper.
References
[1] A. Banerjee and S. Bhattacharyya, Uniqueness of meromorphic functions with their reduced linear c-shift operators sharing two or more values or sets, Adv. Difference Equ. 2019 (2019), Paper No. 509. 10.1186/s13662-019-2425-5Search in Google Scholar
[2] C. A. Berenstein and R. Gay, Complex Variables, Grad. Texts in Math. 125, Springer, New York, 1991. 10.1007/978-1-4612-3024-3Search in Google Scholar
[3] M. F. Chen and Z. S. Gao, Entire solutions of certain type of nonlinear differential equations and differential-difference equations, J. Comput. Anal. Appl. 24 (2018), no. 1, 137–147. Search in Google Scholar
[4] M.-F. Chen, Z.-S. Gao and J.-L. Zhang, Entire solutions of certain type of non-linear difference equations, Comput. Methods Funct. Theory 19 (2019), no. 1, 17–36. 10.1007/s40315-018-0250-6Search in Google Scholar
[5] X. Dong and L. Liao, Meromorphic solutions of certain type of differential-difference equations, J. Math. Anal. Appl. 470 (2019), no. 1, 327–339. 10.1016/j.jmaa.2018.10.005Search in Google Scholar
[6] W. K. Hayman, Meromorphic Functions, Oxford Math. Monogr., Clarendon, Oxford, 1964. Search in Google Scholar
[7] I. Laine, Nevanlinna Theory and Complex Differential Equations, De Gruyter Stud. Math. 15, Walter de Gruyter, Berlin, 1993. 10.1515/9783110863147Search in Google Scholar
[8] Z. Latreuch, On the existence of entire solutions of certain class of nonlinear difference equations, Mediterr. J. Math. 14 (2017), no. 3, Paper No. 115. 10.1007/s00009-017-0914-xSearch in Google Scholar
[9] P. Li, Entire solutions of certain type of differential equations II, J. Math. Anal. Appl. 375 (2011), no. 1, 310–319. 10.1016/j.jmaa.2010.09.026Search in Google Scholar
[10] P. Li and C.-C. Yang, On the nonexistence of entire solutions of certain type of nonlinear differential equations, J. Math. Anal. Appl. 320 (2006), no. 2, 827–835. 10.1016/j.jmaa.2005.07.066Search in Google Scholar
[11] L.-W. Liao, C.-C. Yang and J.-J. Zhang, On meromorphic solutions of certain type of non-linear differential equations, Ann. Acad. Sci. Fenn. Math. 38 (2013), no. 2, 581–593. 10.5186/aasfm.2013.3840Search in Google Scholar
[12] W. Lü, L. Wu, D. Wang and C. Yang, The existence of solutions to certain type of nonlinear difference-differential equations, Open Math. 16 (2018), no. 1, 806–815. 10.1515/math-2018-0071Search in Google Scholar
[13] X. Q. Lu, L. W. Liao and J. Wang, On meromorphic solutions of a certain type of nonlinear differential equations, Acta Math. Sin. (Engl. Ser.) 33 (2017), no. 12, 1597–1608. 10.1007/s10114-017-6484-9Search in Google Scholar
[14] J. Rong and J. Xu, Three results on the non-linear differential equations and differential-difference equations, Mathematics 7 (2019), no. 6, 539–549. 10.3390/math7060539Search in Google Scholar
[15] Q. Wang and Q. Wang, Study on the existence of solutions to two specific types of differential-difference equations, Turkish J. Math. 43 (2019), no. 2, 941–954. 10.3906/mat-1811-48Search in Google Scholar
[16] Q. Wang, G. Zhan and P. Hu, Growth on meromorphic solutions of differential-difference equations, Bull. Malays. Math. Sci. Soc. 43 (2020), no. 2, 1503–1515. 10.1007/s40840-019-00753-5Search in Google Scholar
[17] J. Xu and J. Rong, Exponential polynomials and nonlinear differential-difference equations, J. Funct. Spaces 2020 (2020), Article ID 6901270. 10.1155/2020/6901270Search in Google Scholar
[18] C.-C. Yang, On entire solutions of a certain type of nonlinear differential equation, Bull. Austral. Math. Soc. 64 (2001), no. 3, 377–380. 10.1017/S0004972700019845Search in Google Scholar
[19] C.-C. Yang and I. Laine, On analogies between nonlinear difference and differential equations, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 1, 10–14. 10.3792/pjaa.86.10Search in Google Scholar
[20] C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, Math. Appl. 557, Kluwer Academic, Dordrecht, 2003. 10.1007/978-94-017-3626-8Search in Google Scholar
[21] F. Zhang, N. Liu, W. Lü and C. Yang, Entire solutions of certain class of differential-difference equations, Adv. Difference Equ. 150 (2015), 1–9. 10.1186/s13662-015-0488-5Search in Google Scholar
[22] J. Zhang and L.-W. Liao, On entire solutions of a certain type of nonlinear differential and difference equations, Taiwanese J. Math. 15 (2011), no. 5, 2145–2157. 10.11650/twjm/1500406427Search in Google Scholar
[23] R.-R. Zhang and Z.-B. Huang, On meromorphic solutions of non-linear difference equations, Comput. Methods Funct. Theory 18 (2018), no. 3, 389–408. 10.1007/s40315-017-0223-1Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On λ-Changhee–Hermite polynomials
- Growth properties of meromorphic solutions of some higher-order linear differential-difference equations
- On the transcendental entire and meromorphic solutions of certain non-linear generalized delay-differential equations
- Connections between normalized Wright functions with families of analytic functions with negative coefficients
- Weak solutions for fractional p(x,·)-Laplacian Dirichlet problems with weight
Articles in the same Issue
- Frontmatter
- On λ-Changhee–Hermite polynomials
- Growth properties of meromorphic solutions of some higher-order linear differential-difference equations
- On the transcendental entire and meromorphic solutions of certain non-linear generalized delay-differential equations
- Connections between normalized Wright functions with families of analytic functions with negative coefficients
- Weak solutions for fractional p(x,·)-Laplacian Dirichlet problems with weight
