Abstract
In this paper we study the existence and precise form of entire solutions of finite order for several system of Fermat type difference as well as differential difference equations. Our results are the generalization of the results due to Gao (Acta Math Sin (Chin Ser) 59:677–685, 2016) and Liu et al. (Arch Math 99:147–155, 2012) to a large extent. We exhibit some examples in support of our results.
Similar content being viewed by others
Data availability
Data sharing is not applicable to this article as no database were generated or analyzed during the current study.
References
Chiang, Y.M., and S.J. Feng. 2008. On the Nevanlinna characteristic of \(f(z+\eta )\) and difference equations in the complex plane. Ramanujan Journal 16(1): 105–129.
Gao, L.Y. 2016. Entire solutions of two types of systems of complex differential–difference equations. Acta Mathematica Sinica (Chinese Series) 59: 677–685.
Gross, F. 1966. On the equation \(f^n(z) + g^n(z) = 1\). Bulletin of the American Mathematical Society 72: 86–88.
Gross, F. 1966. On the equation \(f^n(z) + g^n(z) = h^n(z)\). The American Mathematical Monthly 73: 1093–1096.
Halburd, R.G., and R.J. Korhonen. 2006. Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. Journal of Mathematical Analysis and Applications 314: 477–487.
Halburd, R.G., and R.J. Korhonen. 2006. Nevanlinna theory for the difference operator. Annales Academiae Scientiarum Fennicae Mathematica 31: 463–478.
Hayman, W.K. 1964. Meromorphic functions. Oxford: The Clarendon Press.
Hu, P.C., P. Li, and C.C. Yang. 2003. Unicity of Meromorphic Mappings, Advances in Complex Analysis and Its Applications, vol. 1. Dordrecht: Kluwer Academic Publishers.
Iyer, G. 1939. On certain functional equations. Journal of the Indian Mathematical Society 3: 312–315.
Laine, I. 1993. Nevanlinna theory and complex differential equations. Berlin: Walter de Gruyter.
Liu, M.L., and L.Y. Gao. 2019. Transcendental solutions of systems of complex differential–difference equations. Science China Mathematics 49: 1–22.
Liu, K. 2009. Meromorphic functions sharing a set with applications to difference equations. Journal of Mathematical Analysis and Applications 359: 384–393.
Liu, K., and L.Z. Yang. 2012. On entire solutions of some differential–difference equations. Computational Methods and Function Theory 13: 433–447.
Liu, K., and T.B. Cao. 2013. Entire solutions of Fermat type q-difference–differential equations. Electronic Journal of Differential Equations 59: 1–10.
Liu, K., T.B. Cao, and H.Z. Cao. 2012. Entire solutions of Fermat-type differential–difference equations. Archiv der Mathematik 99: 147–155.
Liu, K., and X. Dong. 2015. Fermat-type differential and difference equations. Electronic Journal of Differential Equations 2015: 1–10.
Liu, K., and L.Z. Yang. 2016. A note on meromorphic solutions of Fermat types equations. Analele Stiintifice ale Universitatii Al I Cuza din Iasi - Matematica (N. S.) 1: 317–325.
Montel, P. 1927. Lecons sur les familles de nomales fonctions analytiques et leurs applications, 135–136. Paris: Gauthier-Viuars.
Tang, J.F., and L.W. Liao. 2007. The transcendental meromorphic solutions of a certain type of non-linear differential equations. Journal of Mathematical Analysis and Applications 334: 517–527.
Yang, C.C., and P. Li. 2004. On the transcendental solutions of a certain type of non-linear differential equations. Archiv der Mathematik 82: 442–448.
Yang, C.C., and H.X. Yi. 2003. Uniqueness theory of meromorphic functions. Dordrecht: Kluwer.
Acknowledgements
The authors would like to thank the referee(s) for the helpful suggestions and comments to improve the exposition of the paper.
Funding
There is no funding received from any organizations for this research work.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Additional information
Communicated by S. Ponnusamy.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Haldar, G. On entire solutions of system of Fermat type difference and differential–difference equations. J Anal (2024). https://doi.org/10.1007/s41478-023-00702-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s41478-023-00702-3