Meromorphic solutions of three certain types of non-linear difference equations

1.   Introduction

In 1964, Hayman [6,P69] extended and improved the results of Tumura ^[16] and Clunie ^[1], and obtained the following theorem.

Theorem 1.1. (See ^[6]) Suppose that $f(z), \; g(z)$ are non-constant meromorphic function and satisfy the equation

where $Q_{d}(z, f)$ is a differential polynomial in $f$ of degree $d\leq n-1$. If

then $g(z) = (\gamma_{0}(z)+f(z))^{n}$, where $\gamma_{0}(z)$ is a small function of $f(z)$.

With the establishment of difference analogue of the lemma of the logarithmic derivative, the Clunie and Mohon'ko lemmas, see Halburd and Korhonen ^[8], Chiang and Feng ^[5], more and more researchers ^{[2,3,4,13,15,17,20]} come to study the properties of complex difference equations or complex differential-difference equations. In particular, there has been a renewed interest ^{[11,12,14,17]} in solvability and existence for entire or meromorphic solutions of Eq $(1.1)$. Replacing the differential polynomial $Q_{d}(z, f)$ in Eq $(1.1)$ by a differential-difference polynomial, and satisfying the condition $(1.2)$, Chen et al. ^[4] proved the following result.

Theorem 1.2. (See ^[4]) Let $n\geq 2$ be an integer. Suppose that $f(z)$ is a non-constant finite order meromorphic solution of

where $Q_{d}(z, f)$ is a differential-difference polynomial in $f$ with degree $d\leq n-2$, $u(z)$ is a non-zero rational function, $v(z)$ is a non-constant polynomial. If $N(r, f) = S(r, f)$, then

Remark 1.1. With other conditions fixed, if $Q_{d}(z, f)$ is a difference polynomial in $f$, the Theorem 1.2 is still valid. If the condition $d\leq n-2$ is omitted, then the Theorem 1.2 is impossible. For example, $f(z) = 1+e^{z}$ is an entire solution of equation $f^{2}(z)-f(z+ln2) = e^{2z}$, where $n = 2$, $d = n-1 = 1$. So it's natural to ask what will happen to the solutions of Eq $(1.3)$ when $d = n-1$? In this paper, we study this problem and obtain the following result.

Theorem 1.3. Let $n\geq 2$ be an integer, $P_{d}(z, f)$ be a difference polynomial in $f$ of degree $d\leq n-1$ with polynomial coefficients, $u(z)$ be a non-zero polynomial, $v(z)$ be a non-constant polynomial. If $f(z)$ is a transcendental meromorphic solution of equation

and $N(r, f) = S(r, f)$, then $\sigma(f) = \deg v(z)$, and one of the following holds:

$(1)$ $f^{n}(z) = u(z)e^{v(z)}$ and $P_{d}(z, f) = 0$;

$(2)$ If $\varphi(z) = (u'(z)+u(z)v'(z))f(z)-nu(z)f'(z)\not\equiv 0$, then $T(r, f) = N_{1)}\left(r, \frac{1}{f}\right)+S(r, f)$. Furthermore, if $\varphi(z)$ is a non-zero polynomial, then $f(z) = \gamma_{0}(z)+p(z)e^{q(z)}$, where $\gamma_{0}(z), p(z), q(z)$ are non-zero polynomials satisfying $p^{n}(z) = u(z), \; nq(z) = v(z)$.

Example 1.1. $f(z) = z+e^{z}$ is a solution of the following difference equation

Here, $u(z) = 1$, $v(z) = 3z$, $n = 3$, $\varphi = (u'+uv')f-3uf' = 3(z-1)$. It implies that the solution satisfying Theorem 1.3 does exist.

If $g(z) = p_{1}e^{\lambda z}+p_{2}e^{-\lambda z}$ and the condition $(1.2)$ is not met, what will happen to the solutions of Eq $(1.1)$? Li ^[11] obtained the following result.

Theorem 1.4. (See ^[11]) Let $n\geq 2$ be an integer, $Q_{d}(z, f)$ be a differential polynomial in $f$ of degree $d\leq n-1$, and let $\lambda, p_{1}, p_{2}$ be non-zero constants. If $f(z)$ is a transcendental meromorphic solution of the equation

and $N(r, f) = S(r, f)$, then $f(z) = c_{0}(z)+c_{1}e^{\frac{\lambda}{n}z}+c_{2}e^{-\frac{\lambda}{n}z}$, where $c_{0}(z)$ is small function of $f(z)$, $c_{j}$ are constants satisfying $c_{j}^{n} = p_{j}, j = 1, 2$.

It's natural to ask: what will happen to the solutions of Eq $(1.5)$ when $Q_{d}(z, f)$ is a difference polynomial in $f$ of degree $d\leq n-1$? In this paper, we consider this problem and obtain the following result.

Theorem 1.5. Let $n\geq 2$ be an integer, $P_{d}(z, f)$ be a difference polynomial in $f$ of degree $d\leq n-1$ with polynomial coefficients, $\lambda, p_{1}, p_{2}$ be non-zero constants. If $f(z)$ is a transcendental meromorphic solution of equation

and $N(r, f) = S(r, f)$, then $\sigma(f) = 1$, and one of the following holds:

$(1)$ $f(z) = c_{1}e^{\frac{\lambda}{n}z}+c_{2}e^{-\frac{\lambda}{n}z}$;

$(2)$ If $\varphi(z) = \lambda^{2}f-n^{2}f''\not\equiv 0$, then $T(r, f) = N_{2)}\left(r, \frac{1}{f}\right)+S(r, f)$. Furthermore, if $\varphi(z)$ is a non-zero polynomial, then $f(z) = \gamma_{0}(z)+c_{1}e^{\frac{\lambda}{n}z}+c_{2}e^{-\frac{\lambda}{n}z}$, where $\gamma_{0}(z)$ is a non-zero polynomial, $c_{j}$ are constants satisfying $c_{j}^{n} = p_{j}, j = 1, 2$.

Example 1.2. $f(z) = z+e^{z}+e^{-z}$ is a solution of the following difference equation

Here, $\lambda = 2$, $n = 2$, $\varphi = \lambda^{2}f-n^{2}f'' = 4z$. It implies that the solution satisfying Theorem 1.5 does exist.

If $g(z) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}$ and the condition $(1.2)$ is not met, what will happen to the solutions of Eq $(1.1)$? Liu et al. ^[14] obtained the following result.

Theorem 1.6. (See ^[14]) Let $n\geq 2$ be an integer, $Q_{d}(z, f)$ be a differential polynomial in $f$ of degree $d\leq n-1$ with polynomial coefficients, and let $p_{j}, \alpha_{j}\; (j = 1, 2)$ be non-zero constants satisfying

If equation

admits a meromorphic solution $f(z)$ satisfying $N(r, f) = S(r, f)$, then one of the following holds:

$(1)$ $f(z) = \gamma_{1}(z)+c_{1}e^{\alpha_{1}z}$, $\frac{\alpha_{1}}{\alpha_{2}} = \frac{n}{t}$,

$(2)$ $f(z) = \gamma_{2}(z)+c_{2}e^{\alpha_{2}z}$, $\frac{\alpha_{1}}{\alpha_{2}} = \frac{t}{n}$, where $\gamma_{j}(z)$ are small functions of $f(z)$, $c_{j}$ are constants satisfying $c_{j}^{n} = p_{j}, j = 1, 2$.

What will happen to the solutions of Eq $(1.8)$ when $Q_{d}(z, f)$ is a difference polynomial in $f$ of degree $d\leq n-1$? In this paper, we study this problem and obtain the following result.

Theorem 1.7. Let $n\geq 2$ be an integer, $P_{d}(z, f)$ be a differential polynomial in $f$ of degree $d\leq n-1$ with polynomial coefficients, and let $p_{j}, \alpha_{j}\; (j = 1, 2)$ be non-zero constants satisfying the condition $(1.7)$. If $f(z)$ is a meromorphic solution of equation

and $N(r, f) = S(r, f)$, then $\sigma(f) = 1$ and $T(r, f) = N_{2)}\left(r, \frac{1}{f}\right)+S(r, f)$. If $\varphi(z) = \alpha_{1}\alpha_{2}f(z)-n(\alpha_{1}+\alpha_{2})f'(z)+n^{2}f''(z)$ is a non-zero polynomial and then one of the following holds:

$(1)$ $f(z) = \gamma_{0}(z)+c_{1}e^{\frac{\alpha_{1}}{n}z}$, $\frac{\alpha_{1}}{\alpha_{2}} = \frac{n}{t}$;

$(2)$ $f(z) = \gamma_{0}(z)+c_{2}e^{\frac{\alpha_{2}}{n}z}$, $\frac{\alpha_{1}}{\alpha_{2}} = \frac{t}{n}$, where $\gamma_{0}(z)$ is a non-zero polynomial, $c_{j}$ are constants satisfying $c_{j}^{n} = p_{j}, j = 1, 2$.

Remark 1.2. The condition $(1.7)$ in Theorem 1.7 is necessary, see example 1.3.

Example 1.3. $f(z) = e^{z}+e^{2z}$ is a solution of the following difference equation

Here, $\alpha_{1} = 4$, $\alpha_{2} = 3$, $n = 2$, $\frac{\alpha_{1}}{\alpha_{2}} = \frac{2n}{n+1}$.

Remark 1.3. The following examples 1.4 and 1.5 show that the solution satisfying Theorem $1.7$ does exist.

Example 1.4. $f(z) = 1+e^{z}$ is a solution of the following difference equation

Here, $\alpha_{1} = 3$, $\alpha_{2} = 2$, $n = 3$, $\frac{\alpha_{1}}{\alpha_{2}} = \frac{n}{n-1}$, $\varphi = \alpha_{1}\alpha_{2}f-n(\alpha_{1}+\alpha_{2})f'+n^{2}f'' = 6$.

Example 1.5. $f(z) = 1-e^{z}$ is a solution of the following difference equation

Here, $\alpha_{1} = 1$, $\alpha_{2} = 3$, $n = 3$, $\frac{\alpha_{1}}{\alpha_{2}} = \frac{n-2}{n}$, $\varphi = \alpha_{1}\alpha_{2}f-n(\alpha_{1}+\alpha_{2})f'+n^{2}f'' = 3$.

In this paper, we assume familiarity with the basic results and standard notations of Nevanlinna theory ^[6,9,19]. $f$ is meromorphic function in the whole complex plane $\mathbb{C}$. In addition, we use $\sigma(f)$ to denote the order of growth of $f$. For simplicity, we denote by $S(r, f)$ any quantify satisfying $S(r, f) = o(T(r, f)),$ as $r\to\infty$, outside of a possible exceptional set of finite logarithmic measure. A meromorphic function $\varphi$ defined in $\mathbb{C}$ is called a small function of $f$ if $T(r, \varphi) = S(r, f)$. Let $N_{k)}\left(r, \frac{1}{f}\right)$ denote the counting function of those zeros of $f$ (counting multiplicity) whose multiplicities are not greater than $k$, and let $N_{(k}\left(r, \frac{1}{f}\right)$ denote the counting function of those zeros of $f$ (counting multiplicity) whose multiplicities are not less than $k$.

2.   Some Lemmas

Lemma 2.1. (See [5,Corollary $2.6$]) Let $\eta_{1}, \eta_{2}$ be two complex numbers such that $\eta_{1}\neq\eta_{2}$ and let $f(z)$ be a finite order meromorphic function. Let $\sigma$ be the order of $f(z)$, then for each $\varepsilon > 0$, we have

Lemma 2.2. (See [7,Corollary $3.3$]) Let $f$ be a non-constant finite order meromorphic solution of

where $P(z, f)$ and $Q(z, f)$ are difference polynomials in $f$ with small meromorphic coefficients, and let $c\in \mathbb{C}$, $\delta < 1$. If the total degree of $Q(z, f)$ as a polynomial in $f$ and its shifts is $\leq n$, then

for all $r$ outside of a possible exceptional set $E$ with finite logarithmic measure $\int_{E}\frac{dr}{r} < \infty$.

Using the same methods as in proof of [9,Lemma $2.4.2$] and Lemma 2.1, we have a similar conclusion as follows.

Lemma 2.3. Let $f$ be a non-constant finite order meromorphic solution of

where $P(z, f)$ and $Q(z, f)$ are differential-difference polynomials in $f$ with small meromorphic coefficients. If the total degree of $Q(z, f)$ as a polynomial in $f$ and its derivatives and its shifts is $\leq n$, then

for all $r$ outside of a possible exceptional set with finite logarithmic measure.

Lemma 2.4. Suppose that $\alpha(z), \beta(z), \varphi(z)$ are non-zero polynomials and satisfy the differential equation

where $\deg\varphi(z)\geq\deg\alpha(z)\geq\deg\beta(z)$. Then Eq $(2.1)$ has a special solution $\gamma_{0}(z)$ which is a non-zero polynomial.

Proof. Now we distinguish two cases below. Case $1$: $\deg \alpha(z) = \deg \beta(z)$; Case $2$: $\deg \alpha(z) > \deg \beta(z)$.

Case 1. In this case, we consider three subcases. Subcase $1.1$. If $\alpha, \beta, \varphi$ are non-zero constants, then $\gamma_{0} = \frac{\varphi}{\alpha}$. Subcase $1.2$. If $\alpha, \beta$ are non-zero constants, and $\varphi$ is non-constant polynomial. Let

where $a_{n}(\neq 0), a_{n-1}, \ldots, a_{0}$ are constants. Assuming

and using the method of undetermined coefficients, it follows from Eq $(2.1)$ that

So, we get the expression of the special solution $\gamma_{0}(z)$ and $\deg\gamma_{0}(z) = \deg\varphi(z)$. Subcase $1.3$. If $\alpha, \beta$ are non-constant polynomials, then $\varphi(z)$ must be non-constant polynomial. Using the method of undetermined coefficients and Eq $(2.1)$, similar to the proof of subcase $1.2$, we can also obtain the expression of the special solution $\gamma_{0}(z)$ and $\deg\gamma_{0}(z) = \deg\varphi(z)-\deg\alpha(z)$.

Case 2. If $\deg \alpha(z) > \deg \beta(z)$, then $\varphi(z)$ must be are non-constant polynomials. By using the method of undetermined coefficients and equation $(2.1)$, similar to the proof of subcase $1.2$, we can also obtain the expression of the special solution $\gamma_{0}(z)$ and $\deg\gamma_{0}(z) = \deg\varphi(z)-\deg\alpha(z)$.

Lemma 2.5. (See [3,Lemma $2.6$]) Let $n\geq 1$ be an integer, $\lambda$ be a non-zero constant and $\varphi(z)$ be a non-zero polynomial. Then the differential equation

has a special solution $\gamma_{0}(z)$ which is a non-zero polynomial.

Lemma 2.6. Let $n\geq 2$ be an integer, $P_{d}(z, f)$ be a difference polynomial in $f$ of degree $d\leq n-1$ with small meromorphic coefficients, and let $p_{j}, \alpha_{j}\; (j = 1, 2)$ be non-zero constants satisfying $\alpha_{1}\neq\alpha_{2}$. If $f(z)$ is a meromorphic solution of equation

and $N(r, f) = S(r, f)$, then $\sigma(f) = 1$.

Proof. Set

where $I$ is a finite set of the index $\mu$, $t_{\mu}$, $l_{\mu j}$ $(\mu\in I, j = 1, \ldots, t_{\mu})$ are natural numbers, $\delta_{\mu j}$ $(\mu\in I, j = 1, \ldots, t_{\mu})$ are distinct complex constants. Denote $g_{\mu j}(z): = \frac{f(z+\delta_{\mu j})}{f(z)}$ and substitute this equality into $(2.3)$ yields

where $l_{\mu} = \sum\limits_{j = 1}^{t_{\mu}}l_{\mu j}$, $d = \max\limits_{\mu\in I}\{l_{\mu}\}$, $b_{k}(z) = \sum\limits_{l_{\mu} = k}\left(a_{\mu}(z)\prod\limits_{j = 1}^{t_{\mu}}g_{\mu j}^{l_{\mu j}}(z)\right)$ $(k = 0, \ldots, d)$. By applying Lemma 2.1, we get $m(r, b_{k}(z)) = S(r, f)$ $(k = 0, \ldots, d)$. Differentiating $(2.4)$ yields

By using Lemma 2.1 and the lemma of the logarithmic derivative, we get $m(r, c_{k}(z)) = S(r, f)$ $(k = 0, \ldots, d)$. Noting that $m(r, b_{k}(z)) = S(r, f)$ $(k = 0, \ldots, d)$ and by induction, we obtain

Combining $(2.2)$ and $(2.6)$, and noting that $N(r, f) = S(r, f)$, we get

and

From the above two inequalities, we have

Then $\sigma(f) = 1$.

Lemma 2.7. (See [14,Lemma $2.5$]) Let $n\geq 1$ be an integer, $\alpha_{1}, \alpha_{2}$ be non-zero constants satisfying $\alpha_{1}\neq\alpha_{2}$, and let $\varphi(z)$ be a non-vanishing polynomial. Then the differential equation

has a special solution $\gamma_{0}(z)$ which is a non-vanishing polynomial.

Lemma 2.8. (See [7,p.$247$]) Suppose that $f(z)$ is a transcendental meromorphic function and $K > 1$. Then there exists a set $M(K)$ of upper logarithmic density at most $\delta(K) = \min\{(2e^{K-1}-1)^{-1}, (1+e(K-1))exp(e(1-K))\}$ such that for every positive integer $k$, we have

Remark 2.1. By Lemma 2.8, we see that if $f$ is a transcendental meromorphic function, and if $\varphi$ satisfying $T(r, \varphi^{(k)}) = S_{1}(r, f)$, then $T(r, \varphi) = S_{1}(r, f)$, where $S_{1}(r, f)$ is defined to be any quantity such that for any positive number $\varepsilon$ there exists a set $E(\varepsilon)$ whose upper logarithmic density is less than $\varepsilon$, and $S_{1}(r, f) = o(T(r, f))$ as $r\to\infty$, $r\not\in E(\varepsilon)$.

Lemma 2.9. (See [18,Theorem $1.51$]) Suppose that $f_{1}, f_{2}, \ldots, f_{n}(n\geq 2)$ are meromorphic functions and $g_{1}, g_{2}, \ldots, g_{n}$ are entire functions satisfying the following conditions:

$(1)$ $\sum_{j = 1}^{n}\limits f_{j}e^{g_{j}}\equiv 0.$

$(2)$ $g_{j}-g_{k}$ are not constants for $1\leq j < k\leq n$.

$(3)$ For $1\leq j\leq n, 1\leq h < k\leq n$,

where $E\subset [1, \infty)$ is finite linear measure $\int_{E} dr < \infty$ or finite logarithmic measure $\int_{E}\frac{dr}{r} < \infty$. Then $f_{j}\equiv 0\; (j = 1, \ldots, n)$.

Lemma 2.10. (See [10,Lemma $3$]) Suppose that $h$ is a nonconstant meromorphic function satisfying

Let $f = a_{0}h^{p}+a_{1}h^{p-1}+\cdots+a_{p}$, $g = b_{0}h^{q}+b_{1}h^{q-1}+\cdots+b_{q}$ be polynomials in $h$ with coefficients $a_{0}, a_{1}, \ldots, a_{p}, b_{0}, b_{1}, \ldots, b_{q}$ being small functions of $h$ and $a_{0}b_{0}a_{p}\not\equiv 0$. If $q\leq p$, then $m(r, g/f) = S(r, f)$.

3.   Proof of Theorem 1.3

Proof. Combining $(1.4)$, $(2.6)$ and $N(r, f) = S(r, f)$, we get

and

From the above two inequalities, we get

Then $\sigma(f) = \deg v(z)$.

Denote $P_{d}(z, f) = P$. By differentiating $(1.4)$, we have

Eliminating $e^{v}$ from $(1.4)$ and $(3.1)$, we have

where

By Lemma 2.3, we have

Noting that $N(r, f) = S(r, f)$, then $T(r, \varphi) = S(r, f)$. We consider two cases as follows.

Case 1. If $\varphi\equiv 0$, then $(u'+uv')f = nuf'$, that is $\frac{f'}{f} = \frac{u'+uv'}{nu} = \frac{1}{n}\left(\frac{u'}{u}+v'\right)$. By integration, we get $f^{n} = cue^{v}\; (c\neq 0)$. Substituting this equality into $(1.4)$ yields $\left(1-\frac{1}{c}\right)f^{n} = -P$. If $c\neq 1$, by Lemma 2.2, we have $m(r, f) = S(r, f)$. Noting that $N(r, f) = S(r, f)$, then $T(r, f) = S(r, f)$, a contradiction. Therefore $c = 1$, and $f(z) = p(z)e^{q(z)}$, where $p(z), q(z)$ are non-zero polynomials satisfying $p^{n}(z) = u(z), \; nq(z) = v(z)$.

Case 2. If $\varphi\not\equiv 0$, and $z_{0}$ is a multiple zero of $f$, it follows from $(3.3)$ that $z_{0}$ is a zero of $\varphi$. Then

We claim that $u'+uv'\not\equiv 0$, otherwise $f' = -\frac{\varphi}{nu}$, then $T(r, f') = S_{1}(r, f)$, by Lemma 2.8, we have $T(r, f) = S_{1}(r, f)$, which is impossible. Rewrite $(3.3)$ as $\frac{1}{f} = \frac{1}{\varphi}\left[(u'+uv')-nu\frac{f'}{f}\right]$, by the lemma of the logarithmic derivative and $T(r, \varphi) = S(r, f)$, we get

Combining $(3.4)$ and $(3.5)$, we get

If $\varphi$ is a non-zero polynomial and $\deg(u'+uv')\geq\deg(nu)$, by Lemma 2.4, we have

where $\gamma_{0}(z), p(z), q(z)$ are non-zero polynomials. Substituting $(3.6)$ into $(1.4)$, and using Lemma 2.9, we get $p^{n}(z) = u(z)$, $nq(z) = v(z)$.

This completes the proof of Theorem $1.3$.

4.   Proof of Theorem 1.5

Proof. Suppose that $f$ is a transcendental meromorphic solution of finite order of Eq $(1.6)$ and $N(r, f) = S(r, f)$, by Lemma 2.6, we get $\sigma(f) = 1$. Denote $P_{d}(z, f) = P$. Differentiating $(1.6)$ gives

Differentiating $(4.1)$ yields

Combining $(1.6)$ and $(4.1)$, we get

Combining $(1.6)$ and $(4.2)$, we get

Eliminating $(f')^{2}$ from $(4.3)$ and $(4.4)$, we have

where

and

$Q(z, f)$ is a differential-difference polynomial in $f$ of of degree $2n-1$. By Lemma 2.3, we have $m(r, \varphi) = S(r, f)$. Noting that $N(r, f) = S(r, f)$, then

We distinguish two cases below:

Case 1. If $\varphi\equiv 0$, that is

which has two fundamental solutions $f_{1}(z) = e^{\frac{\lambda}{n}z}$, $f_{2}(z) = e^{-\frac{\lambda}{n}z}$. Then the general solution of Eq $(4.9)$ can be expressed as

Substituting the above formula into $(1.6)$, and applying Lemma 2.9, we obtain $c_{j}^{n} = p_{j}, j = 1, 2$.

Case 2. If $\varphi\not\equiv 0$, and $z_{0}$ is a multiple zero of $f$ whose multiplicities are not less than $3$, it follows from $(4.6)$ that $z_{0}$ is a zero of $\varphi$. Then

Rewriting $(4.6)$ as $\frac{1}{f} = \frac{1}{\varphi}\left(\lambda^{2}-n^{2}\frac{f''}{f}\right)$, by the lemma of the logarithmic derivative and $(4.8)$, we get

It follows from $(4.10)$ and $(4.11)$ that

If $\varphi$ is a non-zero polynomial, by Lemma 2.5, we have

where $\gamma_{0}(z)$ is a non-zero polynomial, $c_{j}\; (j = 1, 2)$ are constants. Substituting $(4.12)$ into $(1.6)$, and using Lemma 2.9, we get $c_{j}^{n} = p_{j}, j = 1, 2$.

This completes the proof of Theorem $1.5$.

5.   Proof of Theorem 1.7

Proof. Suppose that $f$ is a transcendental meromorphic solution of finite order of Eq $(1.9)$ and $N(r, f) = S(r, f)$, by Lemma 2.6, we get $\sigma(f) = 1$. Denote $P_{d}(z, f) = P$. Differentiating $(1.9)$ gives

Eliminating $e^{\alpha_{1}z}$ and $e^{\alpha_{2}z}$ from $(1.4)$ and $(4.1)$, respectively, we have

Differentiating $(5.3)$ yields

Eliminating $e^{\alpha_{1}z}$ from $(5.3)$ and $(5.4)$, we have

where

is a differential-difference polynomial in $f$ of degree $n-1$. It follows from $(5.2)$ and $(5.3)$ that

where

is a differential-difference polynomial in $f$ of degree $2n-1$. Eliminating $(f')^{2}$ from $(5.5)$ and $(5.7)$, we have

where

and

is a differential-difference polynomial in $f$ of degree $2n-1$. It follows from $(1.9)$, $(2.6)$ and $N(r, f) = S(r, f)$ that

It follows from $(2.4)$, $(2.5)$ and $(5.2)$ that

Combining $(5.12)$ and $(5.13)$, we have

Substituting $(2.4)$ into $(1.9)$ yields

Without loss of generality, we assume that $b_{0} = P_{d}(z, 0)\not\equiv 0$. Otherwise, we make the transformation $\hat{f} = f-c$ for a suitable constant c satisfying $c^{n}+P_{d}(z, c)\not\equiv 0$. Then $(1.9)$ is changed to the form $(\hat{f})^{n}(z)+P^{*}_{d}(z, \hat{f}) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}$, where $P^{*}_{d}(z, \hat{f})$ is a difference polynomial in $\hat{f}$ of degree at most $n-1$ with polynomial coefficients, and $P^{*}_{d}(z, 0) = c^{n}+P_{d}(z, c)\not\equiv 0$. Noting that $b_{0} = P_{d}(z, 0)\not\equiv 0$, by applying Lemma 2.10, it follows from $(1.7)$ that

It follows from $(5.14)-(5.16)$ that

then

Without loss of generality, if $\frac{\alpha_{1}}{\alpha_{2}} = \frac{t}{n}$, for some $t\in\{1, 2, \ldots, n-1\}$, then

From $(5.17)$ and the above equality, we have

By applying Lemma 2.1 and the lemma of the logarithmic derivative, from $(5.9)$, $(5.17)$ and $(5.18)$, we get $m(r, \varphi) = S(r, f)$. Noting that $N(r, f) = S(r, f)$, then $T(r, \varphi) = S(r, f)$.

We consider two cases as follows.

Case 1. If $\varphi\equiv 0$, that is $\alpha_{1}\alpha_{2}f-n(\alpha_{1}+\alpha_{2})f'+n^{2}f''\equiv 0$. The corresponding characteristic equation $\alpha_{1}\alpha_{2}-n(\alpha_{1}+\alpha_{2})\lambda+n^{2}\lambda^{2} = 0$ has two roots $\frac{\alpha_{1}}{n}$, $\frac{\alpha_{2}}{n}$, we get

where $c_{j}\; (j = 1, 2)$ are constants. Noting that $f(z)$ is a transcendental meromorphic solution of equation $(1.9)$ and satisfies $m(r, 1/f) = S(r, f)$, then $c_{1}c_{2}\neq 0$. Substituting $(5.19)$ into $(1.9)$ yields

From $(1.7)$ and Lemma 2.9, at least one of $\binom{j}{n}c_{1}^{j}c_{2}^{n-j}e^{\left(\frac{j}{n}\alpha_{1}+\frac{n-j}{n}\alpha_{2}\right)z}$ $(j = 1, \ldots, n-1)$ is zero, then $c_{1}c_{2} = 0$, which is impossible.

Case 2. If $\varphi\not\equiv 0$, and $z_{0}$ is a multiple zero of $f$ whose multiplicities are not less than $3$, it follows from $(5.10)$ that $z_{0}$ is a zero of $\varphi$. Then

It follows from $(5.17)$ and $(5.20)$ that

If $\varphi$ is a non-zero polynomial, by applying Lemma 2.6, we have

where $\gamma_{0}(z)$ is a non-zero polynomial, $c_{j}\; (j = 1, 2)$ are constants. Noting that $f(z)$ is a transcendental meromorphic solution of Eq $(1.9)$ and satisfies $m(r, 1/f) = S(r, f)$, then $c_{1}, c_{2}$ are not complete zeroes. If $c_{1}c_{2}\neq 0$, substituting $(5.21)$ into $(1.9)$ yields

From $(1.7)$ and Lemma 2.9, at least one of $\binom{j}{n}c_{1}^{j}c_{2}^{n-j}e^{\left(\frac{j}{n}\alpha_{1}+\frac{n-j}{n}\alpha_{2}\right)z}$ $(j = 1, \ldots, n-1)$ is zero, then $c_{1}c_{2} = 0$, which is impossible. Then we get $f(z) = \gamma_{0}(z)+c_{1}e^{\frac{\alpha_{1}}{n}z}$ or $f(z) = \gamma_{0}(z)+c_{2}e^{\frac{\alpha_{2}}{n}z}$.

This completes the proof of Theorem $1.7$.

6.   Conclusions

Using the Nevalinna theory of meromorphic functions, this paper study the meromorphic solutions of three Clunie-Tumura types of non-linear difference equations and get the exact forms of the meromorphic solutions of these difference equations with some added conditions. Improvements and extensions of some results in the literature are presented.

Acknowledgments

The authors would like to thank the referee for his/her thorough reviewing with constructive suggestions and comments to the paper. This research was supported by the National Natural Science Foundation of China (Nos: 12001117, 12001503, 11801093) and by the National Natural Science Foundation of Guangdong Province (No. 2018A030313267).

Conflict of interest

The authors declare no conflict of interest.