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October 2021 On the existence of meromorphic solutions of certain nonlinear difference equations
Xiao-Min Li, Chen-Shuang Hao, Hong-Xun Yi
Rocky Mountain J. Math. 51(5): 1723-1748 (October 2021). DOI: 10.1216/rmj.2021.51.1723

Abstract

By Cartan’s version of Nevanlinna’s theory, we prove the following result: Let m and n be two positive integers satisfying nm+2 and m1, let p be a polynomial such that p0, let η be a finite complex number such that η0 and Δηf0, let α1,α2,,αm be m distinct finite nonzero complex numbers, and let Hj be either an exponential polynomial of degree less than q or an ordinary polynomial in z such that Hj0 for 1jm. Suppose that f is a nonconstant meromorphic solution of the difference equation

fn(z)+p(z)Δηf(z)=H1(z)eα1zq+H2(z)eα2zq++Hm(z)eαmzq.

Then m2 and f is reduced to a transcendental entire function such that its order ρ(f)= or its order satisfies ρ(f)=q with m=2, while f can be either expressed as

f(z)=A1(z)eα1zq, with A1(z)=H1fpΔηf and nα1α2=0,

or

f(z)=A2(z)eα2zq, with A2(z)=H2fpΔηf and nα2α1=0,

where A1 and A2 are entire function such that their characteristic functions satisfy

T(r,Aj)=o(T(r,f)),1j2,

as rE and r. Here, E(0,+) is a subset of finite linear measure.

An example is provided to show that the assumption nm+2, in a sense, is the best possible.

This result improves and extends the corresponding results from Yang–Laine (2010), Liu–Lü–Shen–Yang (2014), and Latreuch (2017).

Citation

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Xiao-Min Li. Chen-Shuang Hao. Hong-Xun Yi. "On the existence of meromorphic solutions of certain nonlinear difference equations." Rocky Mountain J. Math. 51 (5) 1723 - 1748, October 2021. https://doi.org/10.1216/rmj.2021.51.1723

Information

Received: 25 July 2019; Revised: 5 February 2021; Accepted: 18 March 2021; Published: October 2021
First available in Project Euclid: 17 February 2022

Digital Object Identifier: 10.1216/rmj.2021.51.1723

Subjects:
Primary: 30D35
Secondary: 39B32

Keywords: Cartan’s version of Nevanlinna’s theory , difference Nevanlinna’s theory , meromorphic functions , Nevanlinna’s theory , nonlinear difference equations

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium

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Vol.51 • No. 5 • October 2021
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