 Research
 Open Access
Entire solutions of nonlinear differentialdifference equations
 Cuiping Li^{1},
 Feng Lü^{1} and
 Junfeng Xu^{2}Email author
 Received: 26 January 2016
 Accepted: 29 April 2016
 Published: 12 May 2016
Abstract
In this paper, we describe the properties of entire solutions of a nonlinear differentialdifference equation and a Fermat type equation, and improve several previous theorems greatly. In addition, we also deduce a uniqueness result for an entire function f(z) that shares a set with its shift \(f(z+c)\), which is a generalization of a result of Liu.
Keywords
 Difference equation
 Meromorphic function
 Logarithmic order
 Nevanlinna theory
 Difference polynomials
Mathematics Subject Classification
 30D35
 39B12
Introduction and main result
The complex oscillation theory of meromorphic solutions of differential equations is an important topic in complex analysis. Some results can be found in Yi and Yang (2003), where Nevanlinna theory is an effective research tool. Recently, many results on meromorphic solutions of difference equations have been rapidly obtained. In this note, we are interested in the properties of entire solutions of difference and differentialdifference equations.
Before proceeding, we spare the reader for a moment and assume some familiarity with the basics of Nevanlinna theory of meromorphic functions in \({\mathbb {C}}\) such as the first and second main theorems, and the usual notations such as the characteristic function T(r, f), the proximity function m(r, f) and the counting function N(r, f). S(r, f) denotes any quantity satisfying \(S(r, f) = o (T (r, f))\) as \(r\rightarrow \infty\), except possibly on a set of finite logarithmic measure—not necessarily the same at each occurrence. Let a, f be meromorphic functions on \({\mathbb {C}}\). a is said to be a small function of f whenever \(T(r, a)=S(r, f)\). S(f) denotes the family of all the small functions of f. \(\lambda (f)\) denotes the exponent of convergence of zeros of f, \(\sigma (f)\) denotes the order of f. A differential polynomial of f means that it is a polynomial in f and its derivatives with coefficients that are small functions of f. A differentialdifference polynomial of f means that it is a polynomial in f, its derivatives and its shifts \(f(z+c)\) with coefficients that are small functions of f.
Recently, there has been a renewed interest in studying meromorphic solutions of differentialdifference equations, see Peng and Chen (2013), Yang and Laine (2010) and Zhang and Liao (2011). Xu et al. (2015) considered a general differentialdifference equation to obtain the following theorem.
Theorem A
 (1)
\(\lambda (f)<\sigma (f)=\infty\), \(\sigma _2(f)<\infty\);
 (2)
\(\lambda _2(f)<\sigma _2(f)<\infty\).
Then f can not be an entire solution of (1).
After studying Theorem A, we ask whether the conclusion still holds if the condition \(\sigma _2(f)<\infty\) is omitted in (1). In the paper, we consider the problem and give an affirmative answer.
Theorem 1
Liu (2009) used the idea of shared set (see Lü and Xu 2008) and studied the uniqueness problem of entire function f(z) shares a set with its difference shift \(f(z+c)\) as follows.
Theorem B
 (1)
\(f(z)\equiv f(z+c)\)
 (2)
\(f(z)+f(z+c)\equiv 0\)
 (3)
\(f(z)=\frac{1}{2}(h_1(z)+h_2(z))\), where \(\frac{h_1(z+c)}{h_1(z)}=e^{\gamma }\), \(\frac{h_2(z+c)}{h_2(z)}=e^{\gamma }\), \(h_1(z)h_2(z)=a^2(z)(1e^{2\gamma })\) and \(\gamma\) is a polynomial.
Note that the form of conclusion (3) is not similar to (1) and (2). So, it is necessary to further study the problem. In the paper, we consider Theorem B again. Due to the different method of proof we employ, we obtain the following result.
Theorem 2
 (I)
\(f(z)\equiv f(z+2c)\);
 (II)
\(f(z)+f(z+2c)\equiv 0\).
Examples

(a) Let \(f(z)=e^z\) and \(c=2\pi i\). Then for any \(a(z)\in S(f)\), we notice that f(z) and \(f(z+c)\) share {\(a(z),a(z)\)} and we can easily see that \(f(z)=f(z+2c)\). This example satisfies (I) of Theorem 2.

(b) Let \(f(z)=\cos \,z\) and \(c=\frac{\pi }{2}\). Then for any \(a=\frac{\sqrt{2}}{2}\), we notice that f(z) and \(f(z+c)\) share {\(a(z),a(z)\)}. Furthermore, we can easily obtain \(f(z)+f(z+2c)=0\). This example satisfies case (II) of Theorem 2.
Tang and Liao (2007) considered the entire solutions of a differential equation. Liu and Cao (2013) considered a qdifference analogue of the above differential equation. Liu and Yang (2013) further generalized the result of Tang and Liao (2007) from differential equations to difference equations. They deduced the entire solutions of generalization of Fermat type equation and obtain below result.
Theorem C
At the end of the paper, by considering a different proof of Theorem C, we generalize Theorem C from polynomial P to small function P as follows.
Theorem 3
Under the conditions of Theorem C and suppose that P(z) is nonzero small entire function of f, then the conclusions of Theorem C still hold.
Some lemmas
In this section, we state some results that we employ in our proofs.
Lemma 1
Lemma 2
(Yang and Laine 2010, Theorem 2.3) Let f be a transcendental entire function, Q(z) is the canonical product of f constructed by the zeros of f. Then \(\sigma (Q)=\lambda (Q)=\lambda (f)\).
The Hadamard theorem of entire functions of infinite order with \(\sigma _2(f)<\infty\) has been proved in Jank and Volkmann (1985). In the following proof, we need to remove the condition \(\sigma _2(f)<\infty\). Similar to the proof of the Hadamard theorem, we prove the following result.
Lemma 3
Proof
Since Q(z) is the canonical product of f constructed by the zeros of f, then \(\lambda (f)=\lambda (Q)\). By Lemma 2, we have \(\sigma (Q)=\lambda (Q)=\lambda (f)<\infty\).
Note that \(\sigma (Q)<\sigma (f)=\infty\), we have \(\sigma (e^g)=\max \{\sigma (Q), \sigma (f)\}=\sigma (f)=\infty\). \(\square\)
Proof of Theorem 1
Suppose that f is an entire solution of Eq. (1) and satisfying \(\lambda (f)<\sigma (f)\). By Theorem A, it is suffice to prove Theorem 1 for the case \(\sigma _2(f)=\infty\).
Thus, we finish the proof of Theorem 1.
Proof of Theorem 2
From (9)–(12), it follows that F(z), \(F(z)a^{2}(z)(1e^{\alpha (z)})\) and \(F(z)a^{2}(z)(1e^{\alpha (zc)})\) just have multiple zeros.
Suppose that the three functions 0, \(a^{2}(z)(1e^{\alpha (z)})\) and \(a^{2}(z)(1e^{\alpha (zc)})\) are distinct from each other.
 (i)
If \(a^{2}(z)(1e^{\alpha (z)})=0\), then \(e^{\alpha (z)}=1\), which implies \(f(z) \equiv f(z+c)\) or \(f(z)+f(z+c) \equiv 0\). Furthermore, it leads to the case (I).
 (ii)
If \(a^{2}(z)(1e^{\alpha (zc)})=0\), then \(e^{\alpha (zc)}=1\), which implies \(e^{\alpha (z)}=1\), we get the same conclusion of (i).
 (iii)If \(a^{2}(z)(1e^{\alpha (z)})=a^{2}(z)(1e^{\alpha (zc)})\), thenwhich implies that \(1=e^{\alpha (z)+\alpha (z+c)}\). Then, a calculation leads to \(\alpha\) is a constant and \(e^{2\alpha }=1\). So, \(e^\alpha =\pm 1\).$$\begin{aligned} 1e^{\alpha (z+c)}=1e^{\alpha (z)}, \end{aligned}$$
If \(e^{\alpha }=1\), then we get the same conclusion of (i) and (ii).
If \(e^{\alpha }=1\), thenFurthermore,$$\begin{aligned} f^{2}(z+c)a^{2}(z)=f^{2}(z)+a^{2}(z). \end{aligned}$$which implies \(f^{2}(z)=f^{2}(z+2c)\). We obtain \(f(z)\equiv f(z+2c)\) or \(f(z)+f(z+2c)\equiv 0\), which is (I) or (II).$$\begin{aligned} \begin{aligned} f^{2}(z+2c)a^{2}(z)&=f^{2}(z+2c)a^{2}(z+c)=f^{2}(z+c)+a^{2}(z+c)\\&=f^{2}(z+c)+a^{2}(z)=f^2(z)a^{2}(z), \end{aligned} \end{aligned}$$Thus, we finish the proof of Theorem 2.
Proof of Theorem 3
From (13)–(15), we obtain G, \(G(z)Q(z)\), \(G(z)\frac{Q(zc)}{P^{2}(zc)}\) just have multiple zeros.
Because Q(z) is a nonzero polynomial, we have \(P^2(z)\equiv 1\) and Q(z) reduces to a constant. Furthermore, by Liu et al. (2012, Theorem 1.1), we obtain the desired result.
Thus, we finish the proof of Theorem 3.
Conclusions
This paper provides three results. Firstly, we consider the existence of the solutions of a nonlinear differentialdifference equation under a general condition. Secondly, we prove a uniqueness theorem of entire function f(z) shares a set with its difference shift \(f(z+c)\). At last, we obtain the entire function solutions of a general Fermat type equation. The above three results were obtained by the different proofs, which can be used later.
Declarations
Authors' contributions
CPL, FL and JFX completed the main part of this article. All authors read and approved the final manuscript.
Acknowledgements
The research was supported by the Natural Science Foundation of Shandong Province Youth Fund Project (ZR2012AQ021), the Fundamental Research Funds for the Central Universities (15CX08011A,15CX05063A), the training plan for the Outstanding Young Teachers in Higher Education of Guangdong (Nos. Yq2013159, SYq2014002) and NSF of Guangdong Province (Nos. 2016A030313002, 2015A030313644).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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