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Existence and growth of meromorphic solutions of some nonlinear qdifference equations
Advances in Difference Equations volume 2013, Article number: 33 (2013)
Abstract
In this paper, we investigate the existence of transcendental meromorphic solutions of order zero of some nonlinear qdifference equations and some more general equations. We also give some estimates on the growth of transcendental meromorphic solutions of these equations. Some examples are given to illustrate the sharpness of some of our results.
MSC:30D35, 39B32.
1 Introduction and main results
In this paper, a meromorphic function means being meromorphic in the whole complex plane. We also assume that the readers are familiar with the usual notations of Nevanlinna theory (see, e.g., [1–3]). Especially, we use notations $\sigma (f)$ and $\mu (f)$ for the order and the lower order of a meromorphic function f. We also denote by $S(r,f)$ any quantity satisfying $S(r,f)=o(T(r,f))$ for all r outside of a possible exceptional set of finite logarithmic measure. Moreover, the standard definitions of logarithmic measure and logarithmic density can be found in [4].
In last two decades, there has been a renewed interest in the complex analytic properties of complex differences and meromorphic solutions of complex difference equations owing to the introduction of Nevanlinna theory in this field. And the study of complex qdifferences and qdifference equations is an important component of the study of complex differences and difference equations.
The original study of complex nonlinear qdifference equations can be derived from the study of the nonautonomous Schröder equation
by Valiron [5], which is closely related to the equations in complex dynamic systems. Indeed, Ritt [6] is an earlier classical paper on the autonomous Schröder equation
And in the important collection [7] of research problems, Rubel posed the question: What can be said about the more general equation (1.1)?
Recently, a number of papers (including [8–23]) focus on complex differences and complex qdifferences. These papers also investigate the existence and the growth of meromorphic solutions of complex difference equations and complex qdifference equations.
In particular, YangLaine [21] pointed out some similarities between results on the existence and uniqueness of finite order entire solutions of the nonlinear differential equations and differentialdifference equations. They obtained the following results.
Theorem A Let $p(z)$, $q(z)$ be polynomials. Then a nonlinear difference equation
has no transcendental entire solutions of finite order.
Theorem B Let $n\ge 4$ be an integer, $M(z,f)$ be a linear differentialdifference polynomial of f, not vanishing identically, and h be a meromorphic function of finite order. Then the differentialdifference equation
possesses at most one admissible transcendental entire solution of finite order such that all coefficients of $M(z,f)$ are small functions of f. If such a solution f exists, then f is of the same order as h.
Some generalizations of Theorems A and B can be found in PengChen [19], and we omit those results here.
In this paper, we consider a similar problem on the existence and the growth of transcendental meromorphic solutions of complex qdifference equations (resp. differentialqdifference equations) instead of complex difference equation (1.2) (resp. differentialdifference equation (1.3)). And the involved equations are more general than (1.1). Moreover, the fact that all meromorphic solutions of the Riccati qdifference equation and linear qdifference equation, both with rational coefficients, are of order zero, shows that it is of great importance to investigate meromorphic solutions of order zero of qdifference equations.
Theorem 1.1 Let $s(z)$ (≢0), $t(z)$ be rational functions, $q\in \mathbb{C}\mathrm{\setminus}\{0,1\}$, and n, m be positive integers such that $n\ne m$.

(i)
If $n>m$, then the nonlinear qdifference equation
$${f}^{n}(z)+s(z)f(qz)f\left({q}^{2}z\right)\cdots f\left({q}^{m}z\right)=t(z)$$(1.4)
has no transcendental meromorphic solutions of order zero. Furthermore, when $q>1$, if there exists a transcendental meromorphic solution f of positive order of (1.4), then

(ii)
If $n<m$, then nonlinear qdifference equation (1.4) has no transcendental entire solutions of order zero.
Corollary 1.1 Let $s(z)$ (≢0), $t(z)$ be rational functions, $q\in \mathbb{C}\mathrm{\setminus}\{0,1\}$, and $n\ge 2$ be an integer. Then the nonlinear qdifference equation
has no transcendental meromorphic solutions of order zero. Furthermore, when $q>1$, if there exists a transcendental meromorphic solution f of positive order of (1.5), then
Remark 1.1 Wittich [24] and Ishizaki [18] had earlier treated equation (1.5) of the case $n=1$, which is the first order linear qdifference equation.
Remark 1.2 Equation (1.4) may have meromorphic solutions of order zero, when $n=m$. For example, the function
is a transcendental meromorphic function of order zero (see Ramis [20]), and it satisfies the nonlinear qdifference equation
where $n=m=2$.
Remark 1.3 Equations (1.4) and (1.5) may have meromorphic solutions of positive orders, when $q>1$. For example, the function
satisfies the nonlinear qdifference equation
where $n=20>2=m$, $q=2$ and $\sigma (f)=\mu (f)=2>{log}_{4}10=\frac{lognlogm}{mlogq}$. And the function
satisfies the nonlinear qdifference equation
where $n=2$, $q=2$ and $\sigma (f)=\mu (f)=1=\frac{logn}{logq}$. The above two examples show that the estimates on the growth of meromorphic solutions of equations (1.4) and (1.5) are sharp.
In the following, we consider the existence of entire solutions of order zero of a type of differentialqdifference equation, which includes equations (1.4) and (1.5) as its special cases. We define a differentialqdifference polynomial in f, which is a finite sum of products of f, derivatives of f and of their qshifts, with all meromorphic coefficients of these monomials of growth $S(r,f)$. Concretely, we denote a differentialqdifference polynomial in f by
where J is a finite set of indices, ${b}_{\lambda}(z)$, $\lambda \in J$ are meromorphic functions of growth $S(r,f)$, and $q\in \mathbb{C}\mathrm{\setminus}\{0,1\}$. And we denote the degree of ${U}_{q}(z,f)$ by
In particular, if each monomial of ${U}_{q}(z,f)$ is of the same degree, then we call ${U}_{q}(z,f)$ a homogeneous differentialqdifference polynomial in f.
Theorem 1.2 Let n, m be integers such that $n>2m>0$, ${U}_{q}(z,f)$ (≢0) be a homogeneous differentialqdifference polynomial in f of degree m, with all meromorphic coefficients of growth $S(r,f)$, and $t(z)$ be a rational function. Then the differentialqdifference equation
has no transcendental entire solutions of order zero.
Corollary 1.2 Let $n\ge 3$ be an integer, ${U}_{q}(z,f)$ (≢0) be a linear differentialqdifference polynomial in f, with all meromorphic coefficients of growth $S(r,f)$, and $t(z)$ be a rational function. Then differentialqdifference equation (1.6) has no transcendental entire solutions of order zero.
Corollary 1.3 Let ${s}_{j}(z)$, $j=1,\dots ,m+1$ be rational functions, not all vanishing identically, $q\in \mathbb{C}\mathrm{\setminus}\{0,1\}$, and let n (≥3), m be positive integers. Then the nonlinear qdifference equation
has no transcendental entire solutions of order zero. Furthermore, if $n>m$, then nonlinear qdifference equation (1.7) has no transcendental meromorphic solutions of order zero, and any transcendental meromorphic solution f of positive order of (1.7) satisfies $\sigma (f)\ge \mu (f)\ge \frac{lognlogm}{mlogq}$.
Remark 1.4 The results concerning the existence of meromorphic solutions of order zero in Theorem 1.1(i) and Corollary 1.1 are not only special cases of Theorem 1.2 and Corollary 1.2 respectively, but also more precise.
Remark 1.5 Clearly, equations (1.4)(1.7) can have rational solutions.
2 Preliminary lemmas
Lemma 2.1 (See [22])
Let f be a nonconstant zeroorder meromorphic function and $q\in \mathbb{C}\mathrm{\setminus}\{0\}$. Then
on a set of lower logarithmic density 1.
The following two lemmas can be seen as special cases of [[23], Theorem 3]. In fact, the present versions we give here are more precise than the original one. To facilitate the readers, we give the corresponding proofs here.
Lemma 2.2 Suppose that f is a transcendental meromorphic solution of the equation
where $q\in \mathbb{C}$, $q>1$ and all meromorphic coefficients are of growth $S(r,f)$. If $d=max\{p,t\}>m$, then for sufficiently large r,
where K (>0) is a constant. Thus, the lower order of f satisfies $\mu (f)\ge \frac{logdlogm}{mlogq}$.
Proof We will use the observation (see [[10], p.249]) as
Noting that $q>1$, by (2.1), (2.2) and ValironMohon’ko theorem (see [[2], Theorem 2.2.5 and Corollary 2.2.7]), we have that for any given ε ($0<\epsilon <\frac{dm}{d+m}$),
outside of a possible exceptional set of finite logarithmic measure. We apply [[2], Lemma 1.1.1] to deal with the exceptional set here. It follows by (2.3) that for any given $\alpha >1$, there exists an ${r}_{0}>0$ such that
holds for all $r\ge {r}_{0}$. Hence,
Inductively, for any $k\in \mathbb{N}$, we have by (2.4) that
For sufficiently large s, there exists a $k\in \mathbb{N}$ such that
We have by (2.5)(2.6) that
Letting $\epsilon \to 0$ and $\alpha \to 1$, we have by (2.7) that
where $K={(\frac{d}{m})}^{\frac{log({q}^{m}{r}_{0})}{mlogq}}T({r}_{0},f)$ (>0) is a constant. Thus, we get $\mu (f)\ge \frac{logdlogm}{mlogq}$. □
Lemma 2.3 Suppose that f is a transcendental meromorphic solution of the equation
where $q\in \mathbb{C}$, $q>1$ and all meromorphic coefficients are of growth $S(r,f)$. If $d=max\{p,t\}>m$, then for sufficiently large r,
where K (>0) is a constant. Thus, the lower order of f satisfies $\mu (f)\ge \frac{logdlogm}{mlogq}$.
Proof Note that (2.3) holds for both (2.1) and (2.8), then the proof of Lemma 2.3 is similar to that of Lemma 2.2. □
Lemma 2.4 (See [9])
Let f be a nonconstant zeroorder meromorphic function and $q\in \mathbb{C}\mathrm{\setminus}\{0\}$. Then
on a set of logarithmic density 1.
Lemma 2.5 (See [14])
Suppose that f is a transcendental meromorphic solution of an equation of the form (1.1) with $q>1$ and meromorphic coefficients of growth $S(r,f)$. Then we have that
where $d={deg}_{f}R$.
3 Proofs of Theorem 1.1 and Corollary 1.1
3.1 Proof of Theorem 1.1

(i)
Suppose that $f(z)$ is a transcendental meromorphic solution of order zero of (1.4). It follows by (1.4) and Lemma 2.1 that
$$\begin{array}{rcl}nT(r,f)& =& T(r,{f}^{n})=T(r,t(z)s(z)f(qz)f\left({q}^{2}z\right)\cdots f\left({q}^{m}z\right))\\ \le & \sum _{j=1}^{m}T(r,f\left({q}^{j}z\right))+T(r,t)+T(r,s)\le m(1+o(1))T(r,f)+O(logr)\\ =& m(1+o(1))T(r,f)\end{array}$$(3.1)
on a set of lower logarithmic density 1. It is clear that (3.1) is a contradiction since $n>m$. Thus, (1.4) has no transcendental meromorphic solutions of order zero if $n>m$.
Furthermore, when $q>1$, we can transform (1.4) into
By (3.2) and Lemma 2.2, we have

(ii)
Suppose that $f(z)$ is a transcendental entire solution of order zero of (1.4). By (3.2) and Lemmas 2.1, 2.4, we have
$$\begin{array}{rcl}mT(r,f)& =& mm(r,f)=m(r,{f}^{m})\\ \le & m(r,\frac{{f}^{m}}{f(qz)f({q}^{2}z)\cdots f({q}^{m}z)})+m(r,f(qz)f\left({q}^{2}z\right)\cdots f\left({q}^{m}z\right))\\ \le & \sum _{j=1}^{m}m(r,\frac{f(z)}{f({q}^{j}z)})+m(r,\frac{t{f}^{n}}{s})\\ \le & \sum _{j=1}^{m}o\left(T(r,f\left({q}^{j}z\right))\right)+m(r,{f}^{n})+O(logr)\\ =& n(1+o(1))T(r,f)\end{array}$$(3.3)
on a set of logarithmic density 1. It is clear that (3.3) is a contradiction since $n<m$. Thus, (1.4) has no transcendental meromorphic solutions of order zero if $n<m$.
3.2 Proof of Corollary 1.1
Since $n\ge 2>1=m$, we immediately know by Theorem 1.1(i) that (1.5) has no transcendental meromorphic solutions of order zero. And when $q>1$, if there is a transcendental meromorphic solution f of positive order of (1.5), then we have
On the other hand, we have by (1.5) and Lemma 2.5 that $\sigma (f)=\frac{logn}{logq}$. Thus, f has a regular order
4 Proofs of Theorem 1.2 and Corollaries 1.2, 1.3
4.1 Proof of Theorem 1.2
Suppose that $f(z)$ is a transcendental entire solution of order zero of (1.6). It follows by (1.6) that
Noting that ${U}_{q}(z,f)$ is a homogeneous differentialqdifference polynomial in f of degree m, by the logarithmic derivative lemma, Lemma 2.4 and (4.1), we have
on a set of logarithmic density 1. It is clear that (4.2) is a contradiction since $n>2m$. Thus, (1.6) has no transcendental entire solution of order zero if $n>2m$.
4.2 Proof of Corollary 1.2
Since ${U}_{q}(z,f)$ is a linear differentialqdifference polynomial in f, the degree of ${U}_{q}(z,f)$ is $m=1$. Thus, by $n\ge 3>2=2m$, we immediately know by Theorem 1.2 that (1.6) has no transcendental entire solutions of order zero.
4.3 Proof of Corollary 1.3
The first part result of Corollary 1.3 is a special case of Corollary 1.2. The second part result of Corollary 1.3 can be proved similarly to Theorem 1.1(i) by replacing Lemma 2.2 with Lemma 2.3.
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Acknowledgements
The authors thank the referees and the editors for their valuable comments to improve the readability of our paper. And this work was supported by the National Natural Science Foundation of China (11126145 and 11171119) and the Natural Science Foundation of Jiangxi Province in China (20114BAB211003 and 20122BAB211005).
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Zheng, XM., Tu, J. Existence and growth of meromorphic solutions of some nonlinear qdifference equations. Adv Differ Equ 2013, 33 (2013). https://doi.org/10.1186/16871847201333
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Keywords
 qdifference equation
 differentialqdifference equation
 meromorphic solution
 order zero