\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 175, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/175\hfil Some properties of meromorphic solutions]
{Some properties of meromorphic solutions for $q$-difference equations}

\author[H. Y. Xu, S. Y. Liu,  X. M. Zheng\hfil EJDE-2017/175\hfilneg]
{Hong Yan Xu, San Yang Liu,  Xiu Min Zheng}

\address{Hong Yan Xu (corresponding author) \newline
School of mathematics and statistics,
Xidian University,
Xi'an, Shaanxi 710126, China. \newline
Department of Informatics and Engineering,
Jingdezhen Ceramic Institute (Xiang Hu Xiao Qu),
Jingdezhen, Jiangxi 333403, China}
\email{xhyhhh@126.com}

\address{San Yang Liu \newline
School of mathematics and statistics,
Xidian University,
Xi'an, Shaanxi 710126, China}
\email{liusanyang@126.com}

\address{Xiu Min Zheng \newline
Department of Mathematics,
Jiangxi Normal University,
Nanchan, Jiangxi 330022, China}
\email{zhengxiumin2008@sina.com}

\thanks{Submitted February 14, 2017. Published July 10, 2017.}
\subjclass[2010]{39A50, 30D35}
\keywords{Meromorphic function; $q$-difference equation; zero order}

\begin{abstract}
 The main purpose of this article is to investigate some properties
 on the meromorphic solutions of some types of $q$-difference equations,
 which can be seen the $q$-difference analogues of Painev\'e equations.
 We obtain estimates of the exponent of convergence of poles of
 $\Delta_qf(z):=f(qz)-f(z)$, which extends some earlier results by Chen et al.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction and statement of main results}

Throughout this paper, the term ``meromorphic'' will  mean
meromorphic in the complex plane $\mathbb{C}$. Also, we shall
assume that readers are familiar
with the fundamental results and the standard notation of the
Nevanlinna value distribution theory of meromorphic functions such
as  $m(r,f)$,  $N(r,f)$, $T(r,f)$, etc.  (see Hayman \cite{8}, Yang \cite{19} and
Yi and Yang \cite{20}). We  use $\sigma(f),\lambda(f)$ and
$\lambda(1/f)$ to denote the order, the exponent of
convergence of zeros and the exponent of convergence of poles
 of $f(z)$  respectively, and we also use $S(r,f)$ denotes any quantity
satisfying $S(r,f)=o(T(r,f))$ for all $r$ on a set $F$ of logarithmic density 1,
the logarithmic density of a set $F$ is defined by
$$
\limsup_{r\to\infty}\frac{1}{\log r}\int_{[1,r]\cap
F}\frac{1}{t}dt.
$$
Throughout this article, where the set $F$ of logarithmic density will be
not necessarily the same at each occurrence.

A century ago,  Painlev\'e and his colleagues \cite{pain} considered the class
$$
w''(z) = F(z;w;w'),
$$
where $F$ is rational in $w$ and $w'$ and (locally) analytic in $z$.
They singled out a list of 50 equations,
six of which could not be integrated in terms of known functions.
These equations are now known as the Painlev\'e equations.
The first two of these equations are $P_I$ and $P_{II}$:
$$
w'' = 6w^2 + z, \quad w''= 2w^2 + zw + \alpha,
$$
where $\alpha$ is a constant. However, after that, essentially nothing happened
until about 1980, and just after that differential
Painlev\'e equations became an important research subject.

 In the 1990s, the discrete Painlev\'e equations have become important
research problems (see \cite{fokas, grammat}). For example,  the
following equations
$$
y_{n+1}+y_{n-1}=\frac{an+b}{y_n}+c,\quad
y_{n+1}+y_{n-1}=\frac{an+b}{y_n}+\frac{c}{y_n^2},
$$
are some known  as the special discretization of discrete $P_I$,  and the equation
$$y_{n+1}+y_{n-1}=\frac{(an+b)y_n+c}{1-y_n^2},$$
is known as the special discretization of the discrete $P_{II}$, where $a,b,c$
are constants, $n\in N$.

Recently, a number of papers (see \cite{3,jh,14}) focused on complex
difference equations and difference analogues of Nevanlinn's theory.
Around 2006s,  Halburd and Korhonen \cite{5,hklondon,7} used Nevanlinna
value distribution theory to single out the difference Painlev\'e $I$ and $II$ equations from the following form
\begin{equation} \label{e1}
  w(z+1)+w(z-1)=R(z,w),
\end{equation}
where $R(z,w)$ is rational in $w$ and meromorphic in $z$.
They obtained that if \eqref{e1} has an admissible meromorphic
solution of finite order, then either $w$ satisfies a difference Riccati
equation, or \eqref{e1} can be transformed by  a linear change in $w$ to some
difference equations, which include difference Painlev\'e $I$ equations
\begin{gather}
w(z+1)+w(z-1)=\frac{az+b}{w(z)}+c, \label{e2} \\
w(z+1)+w(z-1)=\frac{(az+b)}{w(z)}+\frac{c}{w(z)^2}, \label{e3}
\end{gather}
and  difference Painlev\'e $II$ equation
\begin{equation} \label{e4}
w(z+1)+w(z-1)=\frac{(az+b)w(z)+c}{1-w(z)^2}.
\end{equation}

Chen et al \cite{cs11,ch10,pc} studied some properties of finite order
transcendental meromorphic
solutions of \eqref{e2}--\eqref{e4}, and obtained a lot of interesting results.
In 2007, Barnett, Halburd, Korhonen and Morgan
\cite{1} firstly established an analogue of the Logarithmic
Derivative Lemma on $q$-difference operators.
Closely related to difference expressions are $q$-difference expressions,
 where the usual shift $f(z + c)$ of a meromorphic function
will be replaced by the $q$-difference $f(qz)$,
$q \in \mathbb{C} \backslash \{0, 1\}$. By this way, there
were lots of results about difference operators, difference equations,
$q$-difference operators, $q$-difference equations, and so on
(see \cite{gw,gund,li,liao1,ly,wangj,zhangkorhonen,zhch,zhch2}).

In 2015, Qi and Yang \cite{qiyang} investigated the  equations
\begin{gather}
  f(qz)+f(\frac{z}{q})=\frac{az+b}{f(z)}+c, \label{e5}\\
  f(qz)+f(\frac{z}{q})=\frac{(az+b)f(z)+c}{1-f(z)^2}, \label{e6}
\end{gather}
which can be seen $q$-difference analogues of \eqref{e2} and \eqref{e4},
and obtained some theorems as follows.

\begin{theorem}[{\cite[Theorem 1.1]{qiyang}}] \label{thmA}
Let $f(z)$ be a transcendental meromorphic solution with zero order of
\eqref{e5}, and $a, b, c$ be three constants such that $a, b$ cannot vanish
simultaneously. Then
\begin{itemize}
\item[(i)] $f(z)$ has infinitely many poles.

\item[(ii)] If $a \neq 0$ and any $d\in \mathbb{C}$, then
$f(z)-d$ has infinitely many zeros.

\item[(iii)] If $a = 0$ and $f(z)$ takes a finite value $A$ finitely often,
then $A$ is a solution of $2z^2 -cz -b = 0$.
\end{itemize}
\end{theorem}

  \begin{theorem}[{\cite[Theorem 1.3]{qiyang}}] \label{thmB}
Let $a,b, c$ be constants with $ac \neq 0$, and let $f(z)$ be a transcendental
meromorphic solution with zero order of equation \eqref{e6}.
Then $f(z)$ has infinitely
many poles and $f(z)-d$ has infinitely many zeros, where $d\in\mathbb{C}$.
\end{theorem}


In this article, we further investigated some properties of transcendental
 meromorphic solutions of the equations \eqref{e5}, \eqref{e6} and
\begin{equation} \label{e7}
 f(qz)+f(\frac{z}{q})=\frac{az+b}{f(z)}+\frac{c}{f(z)^2},
\end{equation}
and obtained the following theorems, which extends the previous results
given by Qi and Yang \cite{qiyang}.

\begin{theorem} \label{thm1.1}
  Let $a, b, c$ be constants with $|a| + |b|\neq 0$. Suppose that $f (z)$
is a zero order transcendental meromorphic solution of \eqref{e5}. Then
\begin{itemize}
\item[(i)] if $a \neq 0$, $p(z)$ is a polynomial of degree $k\geq0$ and
$|q|\neq 1$, then $f (z)-p(z)$ has infinitely many zeros and
$\lambda(f-p) =\sigma(f)$;

if $a= 0$, then Borel exceptional values of $f (z)$ can only come from the set
$E= \{z |2z^2-cz-b =0\}$;

\item[(ii)] $\lambda(\frac{1}{f})=\lambda(\frac{1}{\Delta_qf})
 =\sigma(\Delta_qf) = \sigma(f)$.
\end{itemize}
\end{theorem}

\begin{theorem} \label{thm1.2}
  Let $a, b, c$ be constants with $|a|+|b|+ |c|\neq0$. Suppose that $f (z)$ is a zero
order transcendental meromorphic solution of \eqref{e6}. Then
\begin{itemize}

\item[(i)] if $a\neq0$, $p(z)$ is a polynomial of degree $k\geq0$ and $|q|\neq 1$, then $f (z)-p(z)$ has infinitely many zeros
and $\lambda(f-p) = \sigma(f )$;
if $a = 0$, then Borel exceptional values of $f (z)$ can only come from the set
$E = \{z | 2z^3+ (b-2)z + c =0\}$;
if $c\neq0$, then $\lambda(f ) = \sigma(f )$;

\item[(ii)] $\lambda(\frac{1}{f})=\lambda(\frac{1}{\Delta_qf})
= \sigma(\Delta_qf ) = \sigma(f )$.
\end{itemize}
\end{theorem}

\begin{theorem} \label{thm1.3}
 Let $a, b, c$ be constants with $|a| + |b| + |c|\neq0$. Suppose that $f (z)$
 is a zero order transcendental meromorphic solution of \eqref{e7}. Then
\begin{itemize}
\item[(i)] if $a=0$, then Borel exceptional values of $f (z)$ can only come
from the set $E = \{z |2z^3-bz- c =0\}$;

\item[(ii)] $\lambda(\frac{1}{f})=\lambda(\frac{1}{\Delta_qf})
=\sigma(\Delta_qf ) =\sigma(f )$.
\end{itemize}
\end{theorem}


\section{Some Lemmas}

The following result can be called an analogue of $q$-difference Clunie lemma,
recently proved by Barnett et al.\ \cite[Theorem 2.1]{1}.
Here a $q$-difference polynomial of $f$ for $q \in \mathbb{C} \backslash \{0, 1\}$
 is a polynomial in $f(z)$ and finitely many of its $q$-shifts
$f(qz), \ldots, f(q^nz)$ with meromorphic coefficients in
the sense that their Nevanlinna characteristic functions are $o(T(r, f))$
on a set of logarithmic density 1.

\begin{lemma}[{\cite[Theorem 2.5]{laineyang}}]\label{lem2.1}
  Let $f$ be a transcendental meromorphic solution of order zero of a
$q$-difference equation of the form
$$
U_q(z,f)P_q(z,f)=Q_q(z,f),
$$
where $U_q(z,f),P_q(z, f)$ and $Q_q(z, f$) are
$q$-difference polynomials such that the total degree $\deg U_q(z, f)=n$
in $f(z)$ and its $q$-shifts, whereas $\deg Q_q(z, f)\leq n$.
Moreover, we assume that
$U_q(z, f)$ contains just one term of maximal total degree in $f(z)$
and its $q$-shifts. Then
$$
m(r,P_q(z,f))=o(T(r,f)),
$$
on a set of logarithmic density 1.
\end{lemma}


\begin{lemma}[{\cite[Theorem 2.5]{1}}] \label{lem2.2}
 Let $f$ be a nonconstant zero-order meromorphic solution
of $P_q(z,f)=0$, where $P_q(z, f)$ is a $q$-difference polynomial
in $f(z)$. If $P_q(z, a)\not\equiv0$ for slowly moving target $a(z)$, then
$$
m(r,\frac{1}{f-a})=o(T(r,f)),
$$
on a set of logarithmic density 1.
\end{lemma}

\begin{lemma}[{\cite[Theorem 1.1 and 1.3]{zhangkorhonen}}] \label{lem2.3}
 Let $f(z)$ be a nonconstant zero-order \\ meromorphic function and
$q \in \mathbb{C}\setminus \{0\}$. Then
$$
T(r,f(qz))=(1+o(1))T(r,f(z)),\quad N(r,f(qz))=(1+o(1))N(r,f(z)),
$$
on a set of lower logarithmic density 1.
\end{lemma}

\begin{lemma}[Valiron-Mohon'ko \cite{12}] \label{lem2.4}
Let $f(z)$ be a meromorphic function. Then for all irreducible
rational functions in $f$,
  $$
  R(z,f(z))=\frac{\sum_{i=0}^ma_i(z)f(z)^i}{\sum_{j=0}^nb_j(z)f(z)^j},
  $$
with meromorphic coefficients $a_i(z),b_j(z)$, the characteristic
function of $R(z,f(z))$ satisfies that
  $$
  T(r,R(z,f(z)))=dT(r,f)+O(\Psi(r)),
$$
  where $d=\max\{m,n\}$ and
  $\Psi(r)=\max_{i,j}\{T(r,a_i),T(r,b_j)\}$.
\end{lemma}


\section{Proof of Theorem \ref{thm1.1}}

Suppose that $f(z)$ is a zero order transcendental meromorphic solution
of \eqref{e5}.

(i) $a\neq0$.  Let $p(z)$ is a polynomial of degree $k$ and $p(z)=a_kz^k+\dots$.
Let $g(z)=f(z)-p(z)$. Substituting $f(z)=g(z)+p(z)$ into
equation \eqref{e5}, we have
$$
  g(qz)+p(qz)+g(\frac{z}{q})+p(\frac{z}{q})=\frac{az+b}{g(z)+p(z)}+c.
$$
It follows that
\begin{equation} \label{e8}
\begin{aligned}
  P_q(z,g):=&[g(qz)+p(qz)+g(\frac{z}{q})+p(\frac{z}{q})]
[g(z)+p(z)]\\
&-(az+b)-c[g(z)+p(z)]=0.
\end{aligned}
\end{equation}
Then, we have
\begin{equation} \label{e9}
P_q(z,0)=[p(qz)+p(\frac{z}{q})]p(z)-(az+b)-cp(z).
\end{equation}

If $p(z)\equiv0$, then $P_q(z,0)=-(az+b)\not\equiv0$.

If $k=0$ and $a_0\equiv\alpha\in \mathbb{C}\setminus \{0\}$,
then $P_q(z,0)=2\alpha^2-(az+b)-c\alpha\not\equiv0$.

If $k\geq 1$ and $a_k\neq 0$ is a constant. Then, we have from \eqref{e9} that
\begin{equation} \label{e10}
  P_q(z,0)=[p(qz)+p(\frac{z}{q})]p(z)-(az+b)-cp(z)
=(q^k+\frac{1}{q^k})a_k^2z^{2k}+\dots.
\end{equation}
Since $|q|\neq 1$, we have $q^k+\frac{1}{q^k}\neq0$, then
$P_q(z,0)\not\equiv0$. Thus, we have by Lemma \ref{lem2.2} that
$$
m(r,\frac{1}{g})=S(r,g).
$$
Then, we obtain
\begin{equation} \label{e11}
  N\Big(r,\frac{1}{f(z)-p(z)}\Big)
=N\big(r,\frac{1}{g(z)}\big)
=T(r,g)+S(r,g)=T(r,f)+S(r,f).
\end{equation}
Hence, it follows  that $\lambda(f-p) =\sigma(f)$.

If $a=0$ and $p(z)=\beta\not\in E$, then we have
$$
P_q(z,0)=2\beta^2-c\beta-b\not\equiv0.$$
Set $g(z)=f(z)-\beta$, by using the same argument as above, we can obtain $\lambda(f-\beta) =\sigma(f)$. Therefore, we can obtain that
the Borel exceptional values of $f(z)$ can only come from the set $E=\{z| 2z^2-cz-b=0\}$.

(ii) From \eqref{e5}, we have
\begin{equation} \label{e12}
  f(z)[f(qz)+f(\frac{z}{q})]=az+b+cf(z).
\end{equation}
It follows from Lemma \ref{lem2.1} that
\begin{equation} \label{e13}
  m\big(r,f(qz)+f(\frac{z}{q})\big)=S(r,f).
\end{equation}
By applying Lemma \ref{lem2.4} for \eqref{e5}, we have
\begin{equation} \label{e14}
  T\big(r,f(qz)+f(\frac{z}{q})\big)=T(r,f)+S(r,f).
\end{equation}
And by Lemma \ref{lem2.3} we obtain
\begin{equation}
  N\big(r,f(qz)+f(\frac{z}{q})\big)
\leq N(r,f(qz))+N\big(r,f(\frac{z}{q})\big)=2(1+o(1))N(r,f)
\end{equation}
on a set of lower logarithmic density 1. Thus, combining  \eqref{e13}
and \eqref{e14}, we have
$$
T(r,f)\leq 2(1+o(1))N(r,f)+S(r,f).
$$
 Hence, we have
\begin{equation} \label{e16}
  \sigma(f(z))\leq \lambda\big(\frac{1}{f(z)}\big).
\end{equation}

Next, we  prove that $ \lambda(\frac{1}{\Delta_qf(z)})\geq
 \lambda(\frac{1}{f(z)})$. Set $z=qw$, then we can rewrite \eqref{e5} as the form
\begin{equation} \label{e17}
  f(q^2w)+f(w)=\frac{aqw+b}{f(qw)}+c.
\end{equation}
Then it follows that
\begin{equation} \label{e18}
  f(qw)[f(q^2w)+f(w)]=aqw+b+cf(qw).
\end{equation}
Since $\Delta_qf(w)=f(qw)-f(w)$, we have $f(qw)=\Delta_qf(w)+f(w)$ and
$f(q^2w)=\Delta_qf(qw)+\Delta_qf(w)+f(w)$.
Substituting them into \eqref{e18}, we obtain
$$
[\Delta_qf(w)+f(w)][\Delta_qf(qw)+\Delta_qf(w)+2f(w)]
=(aqw+b)+c[\Delta_qf(w)+f(w)],
$$
i.e.,
\begin{equation} \label{e19}
\begin{aligned}
  -2f(w)^2=&[\Delta_qf(qw)+3\Delta_qf(w)-c]f(w)-(aqw+b)\\
&+[\Delta_qf(qw)+\Delta_qf(w)-c]\Delta_qf(w).
\end{aligned}
\end{equation}
Since $f(z)$ is a zero order transcendental meromorphic function and $z=qw$,
by Lemma \ref{lem2.3}, we obtain that $f(w)$ is of zero order. Thus,
by Lemma \ref{lem2.3} again, we have that $\Delta_qf(w),\Delta_qf(qw)$ are of zero order.
Set $\Delta_q^2f(w):=\Delta_q(\Delta_qf(w))$, so we have
$\Delta_qf(qw)=\Delta_q^2f(w)+\Delta_qf(w)$.
Since $\Delta_qf(w)$ is of zero order, and by Lemma \ref{lem2.3} we have
\begin{equation} \label{e20}
  N(r,\Delta_q^2f(w))\leq 2N(r,\Delta_qf(w))+S(r,f).
\end{equation}
It follows that
\begin{equation} \label{e21}
N(r,\Delta_qf(qw))\leq 3N(r,\Delta_qf(w))+S(r,f).
\end{equation}
 Thus, from (19) and \eqref{e21} we have
\begin{align*}
 2N(r,f(w))
=&N\Big(r,[\Delta_qf(qw)+3\Delta_qf(w)-c]f(w)-(aqw+b)\\
& +[\Delta_qf(qw)+\Delta_qf(w)-c]\Delta_qf(w)\Big)\\
\leq& N(r,f(w))+9N(r,\Delta_qf(w))+O(\log r)+S(r,f).
\end{align*}
 That is,
\begin{equation} \label{e22}
 N(r,f(w))\leq 9N(r,\Delta_qf(w))+S(r,f).
\end{equation}
Then, it follows that
\begin{equation}
  \lambda\Big(\frac{1}{\Delta_qf(w)}\Big)
\geq \lambda\Big(\frac{1}{f(w)}\Big).
\end{equation}
Since $f(w)$ is of zero order, by Lemma \ref{lem2.3} and $z=qw$ we have
\begin{gather*}
 \lambda\Big(\frac{1}{\Delta_qf(w)}\Big)
= \lambda\Big(\frac{1}{\Delta_qf(\frac{z}{q})}\Big)
= \lambda\Big(\frac{1}{\Delta_qf(z)}\Big),
\\
\lambda\Big(\frac{1}{f(w)}\Big)=\lambda\Big(\frac{1}{f(\frac{z}{q})}\Big)
=\lambda\Big(\frac{1}{f(z)}\Big).
\end{gather*}
Hence,
\begin{equation} \label{e24}
 \lambda\Big(\frac{1}{\Delta_qf(z)}\Big)\geq
 \lambda\Big(\frac{1}{f(z)}\Big).
\end{equation}
From this inequality and  \eqref{e16}  we have
\begin{equation} \label{e25}
   \lambda\Big(\frac{1}{\Delta_qf(z)}\Big)\geq
 \lambda\Big(\frac{1}{f(z)}\Big)\geq\sigma(f(z)).
\end{equation}
So, we have $T(r,\Delta_qf(z))\leq 2T(r,f(z))+S(r,f)$ by Lemma \ref{lem2.3}; that is,
$\sigma(f(z))\geq \sigma(\Delta_qf(z))$.
 Thus, combining this and \eqref{e25}, we have
$$
 \lambda\Big(\frac{1}{f(z)}\Big)= \lambda\Big(\frac{1}{\Delta_qf(z)}\Big)
=\sigma(\Delta_qf(z))=\sigma(f(z)).
$$
The proof of Theorem \ref{thm1.1} is complete.

\section{Proof of Theorem \ref{thm1.2}}

Suppose that $f(z)$ is a zero order transcendental meromorphic
solution of \eqref{e6}. We will consider the following two cases.


(i) $a\neq0$.  If $p(z)$ is a polynomial of degree $k$ and $p(z)=a_kz^k+\dots$. 
Let $g_1(z)=f(z)-p(z)$. Substituting $f(z)=g_1(z)+p(z)$ into equation \eqref{e6},
 we have
$$
  g_1(qz)+p(qz)+g_1(\frac{z}{q})+p(\frac{z}{q})
=\frac{(az+b)[g_1(z)+p(z)]+c}{1-[g_1(z)+p(z)]^2}.
$$
It follows that
\begin{equation} \label{e26}
\begin{aligned}
  P_q(z,g_1):=&[g_1(qz)+p(qz)+g_1(\frac{z}{q})+p(\frac{z}{q})][g_1(z)+p(z)]^2\\
  &+(az+b)[g_1(z)+p(z)]+c\\
  &-[g_1(qz)+p(qz)+g_1(\frac{z}{q})+p(\frac{z}{q})]=0.
\end{aligned}
\end{equation}
From this equality, we have
\begin{equation} \label{e27}
P_q(z,0)=[p(qz)+p(\frac{z}{q})]p(z)^2+(az+b)p(z)+c-p(qz)-p(\frac{z}{q}).
\end{equation}

If $k=0$ and $a_0\equiv\alpha\in \mathbb{C}\setminus \{0\}$, 
then $P_q(z,0)=2\alpha^3+\alpha(az+b)+c-2\alpha\not\equiv0$.

If $k\geq 1$ and $a_k\neq 0$ is a constant. Then, we have from \eqref{e26} that
\begin{align*}
  P_q(z,0)&=[p(qz)+p(\frac{z}{q})]p(z)^2+(az+b)p(z)+c-p(z)-p(\frac{z}{q})\\
  &=\big(q^k+\frac{1}{q^k}\big)a_k^3z^{3k}+\dots.
\end{align*}
Since $|q|\neq 1$, we have $q^k+\frac{1}{q^k}\neq0$, then  
$P_q(z,0)\not\equiv0$. Thus, we have by Lemma \ref{lem2.2} that
$$
m(r,\frac{1}{g_1})=S(r,g_1).
$$
 Then, we obtain
\[
  N\Big(r,\frac{1}{f(z)-p(z)}\Big)
=N\Big(r,\frac{1}{g_1(z)}\Big)=T(r,g_1)+S(r,g_1)=T(r,f)+S(r,f).
\]
It follows hat $\lambda(f-p) =\sigma(f)$.

If $a=0$ and $p(z)=\beta\not\in E$, then we have
$$
P_q(z,0)=2\beta^3+(b-2)\beta+c\not\equiv0.$$
Set $g_1(z)=f(z)-\beta$, by using the same argument as above, we can obtain $\lambda(f-\beta) =\sigma(f)$. Therefore, we can obtain that
the Borel exceptional values of $f(z)$ can only come from the set $E=\{z| 2z^3+(b-2)z+c=0\}$.

If $c\neq 0$, then we have from \eqref{e6} that
$$
P_q(z,f):=f(z)^2[f(qz)+f(\frac{z}{q})]+(az+b)f(z)+c-f(qz)-f(\frac{z}{q}).
$$
Hence, we obtain
$$
P_q(z,0)\equiv c\not\equiv0.
$$
Using a similar method as above, we obtain $\lambda(f)=\sigma(f)$.

(ii) From \eqref{e6}, we have
\begin{equation}
 f(z)^2[f(qz)+f(\frac{z}{q})]=f(qz)+f(\frac{z}{q})-(az+b)f(z)-c.
\end{equation}
It follows from \eqref{e12} that
\begin{equation}
  m\Big(r,f(qz)+f(\frac{z}{q})\Big)=S(r,f).
\end{equation}

By applying Lemma \ref{lem2.4} for \eqref{e6}, we have
\begin{equation} \label{e31}
  T\Big(r,f(qz)+f(\frac{z}{q})\Big)=2T(r,f)+S(r,f).
\end{equation}
By Lemma \ref{lem2.3} we obtain
\begin{equation} \label{e32}
  N\Big(r,f(qz)+f(\frac{z}{q})\Big)
\leq N(r,f(qz))+N\Big(r,f(\frac{z}{q})\Big)=2(1+o(1))N(r,f)
\end{equation}
on a set of lower logarithmic density 1. Thus, combining  
(31) and (32), we have
$$ 
T(r,f)\leq 2(1+o(1))N(r,f)+S(r,f).
$$
 Hence,
\begin{equation} \label{e33}
  \sigma(f(z))\leq \lambda\Big(\frac{1}{f(z)}\Big).
\end{equation}

Next, we  prove that $ \lambda\Big(\frac{1}{\Delta_qf(z)}\Big)\geq
 \lambda\Big(\frac{1}{f(z)}\Big)$. Set $z=qw$, then we can rewrite \eqref{e6}
in the form
\begin{equation} \label{e34}
  f(q^2w)+f(w)=\frac{(aqw+b)f(qw)+c}{1-f(qw)^2}.
\end{equation}
Then it follows that
\begin{equation} \label{e35}
  f(qw)^2[f(q^2w)+f(w)]=f(q^2w)+f(w)-(aqw+b)f(qw)-c.
\end{equation}
Since $\Delta_qf(w)=f(qw)-f(w)$, we have
 $f(qw)=\Delta_qf(w)+f(w)$ and $f(q^2w)=\Delta_qf(qw)+\Delta_qf(w)+f(w)$.
Substituting these two equalities in \eqref{e35}, we obtain
\begin{align*}
& [\Delta_qf(w)+f(w)]^2[\Delta_qf(qw)+\Delta_qf(w)+2f(w)]\\
&=\Delta_qf(qw)+\Delta_qf(w)+2f(w)-(aqw+b)[\Delta_qf(w)+f(w)]-c,
\end{align*}
Thus, we obtain
\begin{equation}
  -2f(w)^3=A(w)f(w)+B(w)\Delta_qf(w)+C(w),
\end{equation}
where
\begin{gather*}
\begin{aligned}
  A(w)=&[\Delta_qf(qw)+5\Delta_qf(w)]f(w)+4(\Delta_qf(w))^2\\
  &+2\Delta_qf(w)\Delta_qf(qw)+aqw+b-2,
\end{aligned}\\
  B(w)=\Delta_qf(qw)\Delta_qf(w)+(\Delta_qf(w))^2+(aqw+b)-1,\\
  C(w)=c-\Delta_qf(qw).
\end{gather*}
 Thus, by Lemma \ref{lem2.3} and from \eqref{e21} we have
\begin{align*}
 3N(r,f(w))=&N(r,A(w)f(w)+B(w)\Delta_qf(w)+C(w))\\
\leq& 2N(r,f(w))+19N(r,\Delta_qf(w))+O(\log r)+S(r,f).
\end{align*}
 That is,
\begin{equation}
 N(r,f(w))\leq 19N(r,\Delta_qf(w))+S(r,f).
\end{equation}
Then, it follows from (37) that
\begin{equation}
  \lambda\Big(\frac{1}{\Delta_qf(w)}\Big)\geq \lambda\Big(\frac{1}{f(w)}\Big).
\end{equation}
Since $f(w)$ is of zero order, by Lemma \ref{lem2.3} and $z=qw$ we have
\begin{gather*}
 \lambda\Big(\frac{1}{\Delta_qf(w)}\Big)
= \lambda\Big(\frac{1}{\Delta_qf(\frac{z}{q})}\Big)
= \lambda\Big(\frac{1}{\Delta_qf(z)}\Big), \\
\lambda\Big(\frac{1}{f(w)}\Big)
=\lambda\Big(\frac{1}{f(\frac{z}{q})}\Big)
=\lambda\Big(\frac{1}{f(z)}\Big).
\end{gather*}
Hence, we obtain
\begin{equation} \label{e39}
 \lambda\Big(\frac{1}{\Delta_qf(z)}\Big)\geq
 \lambda\Big(\frac{1}{f(z)}\Big).
\end{equation}

Thus, from this inequality and \eqref{e33} we have
\begin{equation} \label{e40}
   \lambda\Big(\frac{1}{\Delta_qf(z)}\Big)\geq
 \lambda\Big(\frac{1}{f(z)}\Big)\geq\sigma(f(z)).
\end{equation}
Then we have $T(r,\Delta_qf(z))\leq 2T(r,f(z))+S(r,f)$ by Lemma \ref{lem2.3};
 that is, $\sigma(f(z))\geq \sigma(\Delta_qf(z))$. 
Thus, combining this and \eqref{e40}, we have
$$
 \lambda\Big(\frac{1}{f(z)}\Big)
= \lambda\Big(\frac{1}{\Delta_qf(z)}\Big)=\sigma(\Delta_qf(z))
=\sigma(f(z)).
$$
The proof of Theorem \ref{thm1.2} is complete.

\section{Proof of Theorem \ref{thm1.3}}

For the convenience of the reader, we use the notation form the proof of 
Theorem \ref{thm1.1}(i). Suppose that $f(z)$ is a zero order transcendental
 meromorphic solution of \eqref{e7}.  We will consider the following two cases.

(i) Let $a=0$ and $p(z)=\beta\not\in E$. Using the same methods as in 
the proof of Theorem \ref{thm1.1}(i),
we have
$$
P_q(z,0)=2\beta^3-b\beta-c\not\equiv0.
$$
Thus,  $\lambda(f-\beta)=\sigma(f)$. Hence, the Borel exceptional values 
of $f(z)$ can only come from the set $E =\{z |2z^3-bz-c =0\}$.


(ii) From \eqref{e7}, we have
\begin{equation}
 f(z)^2[f(qz)+f(\frac{z}{q})]=(az+b)f(z)+c.
\end{equation}
It follows from Lemma \ref{lem2.1} that
\begin{equation}  \label{e42}
  m\Big(r,f(qz)+f(\frac{z}{q})\Big)=S(r,f).
\end{equation}
By Lemma \ref{lem2.3} we obtain
\begin{equation}  \label{e43}
  N\Big(r,f(qz)+f(\frac{z}{q})\Big)
\leq N(r,f(qz))+N\Big(r,f(\frac{z}{q})\Big)
=2(1+o(1))N(r,f)
\end{equation}
on a set of lower logarithmic density 1.

If $c\neq0$, by applying Lemma \ref{lem2.4} for \eqref{e7}, we have
\begin{equation} \label{e44}
  T\Big(r,f(qz)+f(\frac{z}{q})\Big)=2T(r,f)+S(r,f).
\end{equation}
Thus, it follows from \eqref{e42}--\eqref{e44} that
\begin{equation} \label{e45}
T(r,f)\leq (1+o(1))N(r,f)+S(r,f).
\end{equation}

If $c=0$, by applying Lemma \ref{lem2.4} for \eqref{e7} and since 
$|a|+|b|+|c|=|a|+|b|\neq 0$,  we have
\begin{equation} \label{e46}
  T\Big(r,f(qz)+f(\frac{z}{q})\Big)=T(r,f)+S(r,f).
\end{equation}
Thus, it follows from \eqref{e42}, \eqref{e43} and \eqref{e46} that
\begin{equation} \label{e47}
 T(r,f)\leq 2(1+o(1))N(r,f)+S(r,f).
\end{equation}
From \eqref{e45} and \eqref{e47} we have
\begin{equation} \label{e48}
  \sigma(f(z))\leq \lambda\Big(\frac{1}{f(z)}\Big).
\end{equation}

Next, we  prove that $ \lambda\Big(\frac{1}{\Delta_qf(z)}\Big)\geq
 \lambda\Big(\frac{1}{f(z)}\Big)$. 
Set $z=qw$, by using the same argument as in Theorem \ref{thm1.2}(ii), we have
\begin{equation} \label{e49}
  -2f(w)^3=A(w)f(w)+B(w)\Delta_qf(w)+C(w),
\end{equation}
where
\begin{gather*}
  A(w)=[\Delta_qf(qw)+5\Delta_qf(w)]f(w)+4(\Delta_qf(w))^2
  +2\Delta_qf(w)\Delta_qf(qw)-aqw-b,\\
  B(w)=[\Delta_qf(qw)+\Delta_qf(w)]\Delta_qf(w)-aqw-b,\\
  C(w)=-c.
\end{gather*}
Thus, by Lemma \ref{lem2.3} and from \eqref{e21} we have
\begin{align*}
 3N(r,f(w))
=&N(r,A(w)f(w)+B(w)\Delta_qf(w)+C(w))\\
\leq& 2N(r,f(w))+15N(r,\Delta_qf(w))+O(\log r)+S(r,f).
\end{align*}
 That is,
\begin{equation} \label{e50}
 N(r,f(w))\leq 15N(r,\Delta_qf(w))+S(r,f).
\end{equation}
Then, it follows  that
\begin{equation}
  \lambda\Big(\frac{1}{\Delta_qf(w)}\Big)
\geq \lambda\Big(\frac{1}{f(w)}\Big).
\end{equation}
Since $f(w)$ is of zero order, by Lemma \ref{lem2.3} and $z=qw$ we have
\begin{align*}
 \lambda\Big(\frac{1}{\Delta_qf(w)}\Big)
= \lambda\Big(\frac{1}{\Delta_qf(\frac{z}{q})}\Big)
= \lambda\Big(\frac{1}{\Delta_qf(z)}\Big), \\
\lambda\Big(\frac{1}{f(w)}\Big)=\lambda\Big(\frac{1}{f(\frac{z}{q})}\Big)
=\lambda\Big(\frac{1}{f(z)}\Big).
\end{align*}
Hence, we obtain
\begin{equation}
 \lambda\Big(\frac{1}{\Delta_qf(z)}\Big)\geq
 \lambda\Big(\frac{1}{f(z)}\Big).
\end{equation}
Thus,  by \eqref{e48}  we have
\begin{equation} \label{e53}
   \lambda\Big(\frac{1}{\Delta_qf(z)}\Big)\geq
 \lambda\Big(\frac{1}{f(z)}\Big)\geq\sigma(f(z)).
\end{equation}
So, we have $T(r,\Delta_qf(z))\leq 2T(r,f(z))+S(r,f)$ by 
Lemma \ref{lem2.3}; that is, $\sigma(f(z))\geq \sigma(\Delta_qf(z))$.
Thus, combining this and \eqref{e53}, we
have
$$
 \lambda\Big(\frac{1}{f(z)}\Big)=
 \lambda\Big(\frac{1}{\Delta_qf(z)}\Big)
 =\sigma(\Delta_qf(z))=
\sigma(f(z)).
$$
The proof of Theorem \ref{thm1.3} is complete.

\subsection*{Acknowledgments}
This work was supported by the National Natural Science Foundation
of China (11561033, 61373174, 11301233, 61662037),
the Natural Science Foundation of Jiangxi Province in China
(20151BAB201008, 20171BAB201002),
the Natural Science Foundation of Shaanxi Province in China (2017JM1001),
and the Foundation of Education Department of Jiangxi (GJJ160914,GJJ150902)
 of China.

The authors want to thank the anonymous referees for reading the manuscript 
very carefully  and making a number of valuable comments which improved 
this article.

\begin{thebibliography}{00}

\bibitem{1} D. C. Barnett, R. G. Halburd, R. J. Korhonen, W. Morgan;
\emph{Nevanlinna theory for the $q$-difference operator and
meromorphic solutions of $q$-difference equations}, Proc. Roy. Soc.
Edin. Sect. A Math., 137 (2007), 457-474.

\bibitem{ch08} Z. X. Chen;
\emph{On properties of meromorphic solutions for some difference equations}, 
Kodai Math. I., 34 (2011), 244-256.

\bibitem{cs11} Z. X. Chen, K. H. Shon;
\emph{Properties of differences of
meromorphic functions}, Czechoslovak Math. J., 61 (136) (2011), 213-224.

\bibitem{ch10} Z. X. Chen;
\emph{Value distribution of meromorphic solutions of certain difference 
Painlev\'e equations}, J. Math. Anal. Appl., 364 (2010), 556-566.

\bibitem{3}Y. M. Chiang, S. J. Feng;
 \emph{On the Nevanlinna characteristic of $f(z + \eta)$ and difference equations 
in the complex plane}, Ramanujan J., 16 (2008), 105-129.

\bibitem{fokas}  A. S. Fokas;
 \emph{From continuous to discrete Painlev\'e equations},
J. Math. Anal. Appl., 180 (1993), 342-360.

 \bibitem{gw} L. Y. Gao, M. L. Wang;
 \emph{The transcendental meromorphic solutions of composite functional equations},
 J. Jiangxi Norm. Univ. Nat. Sci., 40 (6) (2016), 587-590.

\bibitem{grammat} B. Grammaticos, F. W. Nijhoff, A. Ramani;
 \emph{Discrete Painlev\'e equations, The Painlev\'e property},
CRM Ser. Math. Phys., Springer, New York, 1999, pp. 413-516.

\bibitem{gund} G. G. Gundersen, J. Heittokangas, I. Laine, J. Rieppo, D. Q. Yang;
 \emph{Meromorphic solutions of generalized Schr\"{o}der
equations}, Aequationes Math. 63 (2002), 110-135.

\bibitem{5} R. G. Halburd, R. J. Korhonen;
\emph{Difference analogue of the lemma on the logarithmic derivative with 
applications to difference equations}, J. Math. Anal. Appl., 314 (2006), 477-487.

\bibitem{hklondon} R. G. Halburd, R. J. Korhonen;
\emph{Finite order solutions and the discrete Painlev\'e equations},
Proc. London Math. Soc. 94 (2007), 443-474.

\bibitem{7} R. G. Halburd, R. J. Korhonen;
\emph{Nevanlinna theory for the difference operator}, 
Ann. Acad. Sci. Fenn. Math., 31 (2006), 463-478.

\bibitem{8} W. K. Hayman;
 \emph{Meromorphic Functions}, The Clarendon Press,
Oxford, 1964.

\bibitem{jh} J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, K. Tohge;
 Complex difference equations of Malmquist type, Comput. Methods Funct. Theory,
 1 (2001), 27-39.

\bibitem{12} I. Laine;
\emph{Nevanlinna theory and complex differential equations}, 
Walter de Gruyter, Berlin, 1993.

\bibitem{laineyang} I. Laine, C. C. Yang;
 \emph{Clunie theorems for difference and $q$-difference polynomials},
 J. London Math. Soc., 76 (2) (2007), 556-566.

\bibitem{14} I. Laine, C. C. Yang;
\emph{Value distribution of difference polynomials}, 
Proc. Japan Acad. Ser. A, 83 (2007), 148-151.

\bibitem{li}  G. Li, M. Chen;
 \emph{Infinite horizon linear quadratic optimal control for stochastic 
difference time-delay systems}, Adv. Difference Equations, 2015 (2015), Art. 14.

\bibitem{liao1} L. W. Liao;
 \emph{The new developments in the research of nonlinear complex differential 
equations}, J Jiangxi Norm. Univ. Nat. Sci., 39 (4) (2015), 331-339.

\bibitem{ly} L. W. Liao, C. C. Yang;
\emph{Some new and old (unsolved) problems and
conjectures on factorization theory, dynamics and
functional equations of meromorphic functions}, J Jiangxi Norm. Univ.
Nat. Sci. 41 (3) (2017), 242-247.

\bibitem{pain} P. Painlev\'e;
 \emph{M\'emoire sur les \'equations diff\'erentielles dont 
l'int\'egrale g\'en\'erale est uniforme}, Bull. Soc. Math. France,
 28 (1900), 201-261.

\bibitem{pc} C. W. Peng, Z. X. Chen;
\emph{On properties of meromorphic solutions
for difference Painlev\'e equations}, Advances in Difference Equation, 
2015 (2015), no. 123, pp. 1-15.

\bibitem{qiyang} X. G. Qi, L. Z. Yang;
\emph{Properties of meromorphic solutions of $q$-difference equations}, 
Electronic Journal of Differential Equations,  2015 (2015), No. 59, pp. 1-9.

\bibitem{wangj} J. Wang, K. Xia, F. Long;
\emph{The poles of meromorphic solutions of Fermat type
differential-difference equations}, J Jiangxi Norm. Univ.
Nat. Sci., 40 (5) (2016), 497-499.

\bibitem{19} L. Yang;
 \emph{Value distribution theory}, Springer-Verlag. Berlin, 1993.

\bibitem{20} H. X. Yi, C. C. Yang;
\emph{Uniqueness theory of meromorphic functions}, Kluwer Academic Publishers,
 Dordrecht, 2003; Chinese original: Science Press, Beijing, 1995.

\bibitem{zhangkorhonen} J. L. Zhang, R. Korhonen;
\emph{On the Nevanlinna characteristic of $f(qz)$ and its applications}, 
J. Math. Anal. Appl., 369 (2010), 537-544.
\bibitem{zhch}
X. M. Zheng and Z. X. Chen;
\emph{Some properties of meromorphic solutions of $q$-difference equations}, 
J. Math. Anal. Appl., 361 (2010), 472-480.

\bibitem{zhch2} X. M. Zheng, Z. X. Chen;
\emph{On properties of $q$-difference equations}, 
Acta Mathematica Scientia, 32B (2) (2012), 724-734.

\end{thebibliography}

\end{document}