ON ENTIRE SOLUTIONS OF SOME TYPE OF NONLINEAR DIFFERENCE EQUATIONS

Huifang LIU; Zhiqiang MAO

doi:10.1016/S0252-9602(18)30786-0

Acta mathematica scientia, Series B >

2018 , Vol. 38 >Issue 3: 819 - 828

DOI: https://doi.org/10.1016/S0252-9602(18)30786-0

Articles

ON ENTIRE SOLUTIONS OF SOME TYPE OF NONLINEAR DIFFERENCE EQUATIONS

Huifang LIU ,
Zhiqiang MAO

Expand

1. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China;
2. School of Mathematics and Computer, Jiangxi Science and Technology Normal University, Nanchang 330038, China

Received date: 2017-03-14

Revised date: 2017-10-20

Online published: 2018-06-25

Supported by

This work is supported by the National Natural Science Foundation of China (11661044).

Fold

Abstract

In this article, the existence of finite order entire solutions of nonlinear difference equations fⁿ + P_d(z, f)=p₁e^α₁z + p₂e^α₂z are studied, where n ≥ 2 is an integer, P_d(z, f) is a difference polynomial in f of degree d(≤ n-2), p₁, p₂ are small meromorphic functions of e^z, and α₁, α₂ are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.

Key words： Nevanlinna theory; difference polynomial; difference equation; entire solution

Cite this article

Huifang LIU , Zhiqiang MAO . ON ENTIRE SOLUTIONS OF SOME TYPE OF NONLINEAR DIFFERENCE EQUATIONS[J]. Acta mathematica scientia, Series B, 2018 , 38(3) : 819 -828 . DOI: 10.1016/S0252-9602(18)30786-0

References

[1] Hayman W. Meromorphic Functions. Oxford:Clarendon Press, 1964
[2] Laine I. Nevanlinna Theory and Complex Differential Equations. Berlin:Walter de Gruyter, 1993
[3] Yang C C, Li P. On the transcendental solutions of a certain type of nonlinear differential equations. Arch Math, 2004, 82:442-448
[4] Li P, Yang C C. On the nonexistence of entire solutions of a certain type of nonlinear differential equations. J Math Anal Appl, 2006, 320:827-835
[5] Li P. Entire solutions of certain type of differential equations. J Math Anal Appl, 2008, 344:253-259
[6] Laine I, Yang C C. Entire solutions of some non-linear differential equations. Bull De La Sociètè Des Sciences Et Des Lettres De Lódź, 2009, 59:19-23
[7] Liao L W, Yang C C, Zhang J J. On meromorphic solutions of certain type of non-linear differential equations. Ann Acad Sci Fenn Math, 2013, 38:581-593
[8] Yang C C, Laine I. On analogies between nonlinear difference and differential equations. Proc Japan Acad Ser A, 2010, 86:10-14
[9] Zhang J, Liao L W. On entire solutions of a certain type of nonlinear differential and difference equations. Taiwanese J Math, 2011, 15:2145-2157
[10] Wen Z T, Heittokangas J, Laine I. Exponential polynomials as solutions of certain nonlinear difference equations. Acta Math Sin, Engl Ser, 2012, 28:1295-1306
[11] Halburd R G, Korhonen R. Nevanlinna theory for the difference operator. Ann Acad Sci Fenn Math, 2006, 31:463-478
[12] Halburd R G, Korhonen R. Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J Math Anal Appl, 2006, 314:477-487
[13] Chiang Y M, Feng S J. On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane. Ramanujan J, 2008, 16:105-129
[14] Halburd R G, Korhonen R J, Tohge K. Holomorphic curves with shift-invariant hyperplane preimages. Trans Amer Math Soc, 2014, 366:4267-4298
[15] Yang C C, Yi H X. Uniqueness theory of meromorphic functions. New York:Kluwer Academic Publishers, 2003

Options

Abstract

Outlines

模态框（Modal）标题

Abstract

Cite this article

References