# The properties of solutions of certain type of difference equations

- Xiaoguang Qi
^{1}Email author, - Jia Dou
^{2}and - Lianzhong Yang
^{3}

**2014**:256

https://doi.org/10.1186/1687-1847-2014-256

© Qi et al.; licensee Springer. 2014

**Received: **23 April 2014

**Accepted: **17 September 2014

**Published: **3 October 2014

## Abstract

In this paper, we shall utilize Nevanlinna value distribution theory to study the solvability of the difference equations of the form: $f{(z)}^{n}+p(z){({\mathrm{\Delta}}_{c}f)}^{m}=r(z){e}^{q(z)}$ and $f{(z)}^{n}+p(z){e}^{q(z)}{({\mathrm{\Delta}}_{c}f)}^{m}=r(z)$, and we shall study the growth of their entire solutions. Moreover, we will give a number of examples to show that the results in this paper are the best possible in certain senses. This article extends earlier results by Liu *et al.* (Czechoslov. Math. J. 61:565-576, 2011; Ann. Pol. Math. 102:129-142, 2011).

**MSC:**39A05, 30D35.

## Keywords

## 1 Introduction and main results

In this paper, we use the basic notions in Nevanlinna theory of meromorphic functions, as found in [1]. In addition, we use $\delta (f)$, $\lambda (f)$, and $\lambda (\frac{1}{f})$ to denote the order, and the exponents of the convergence of zeros and poles of a meromorphic function $f(z)$, respectively. We define difference operators as ${\mathrm{\Delta}}_{c}f=f(z+c)-f(z)$, where *c* is a non-zero constant.

where $L(f)$ is a linear differential polynomial in *f* with polynomial coefficients, $p(z)$ is a non-vanishing polynomial, $h(z)$ is an entire function, and *n* is an integer such that $n\ge 4$. Subsequently, several papers have appeared in which the solutions of (1) are studied. The reader is invited to see [3–5].

where $L(z,f)$ is a finite sum of product of *f*, derivatives of *f*, and their shifts. They obtained the following result.

**Theorem A** *Let* $n\ge 4$ *be an integer*, $L(z,f)$ *be a linear differential*-*difference polynomial of* *f*, *with small meromorphic coefficients*, *and* $h(z)$ *be a meromorphic function of finite order*. *Then* (2) *possesses at most one admissible transcendental entire solution of finite order*, *unless* $L(z,f)$ *vanishes identically*. *If such a solution* $f(z)$ *exists*, *then* $f(z)$ *is of the same order as* $h(z)$.

has no transcendental entire solution of finite order, where $P(z)$, $Q(z)$ are polynomials.

we get the following result in [11].

**Theorem B** *Let* $P(z)$, $Q(z)$ *be polynomials*, *n* *and* *m* *be integers satisfying* $n>m\ge 0$. *Then* (3) *has no transcendental entire solution of finite order*.

In fact, the special case of (3) with $n=m$, and $p(z)=q(z)=1$, can be viewed as the Fermat type functional equation. It is well known that (3) has no transcendental entire solutions when $n\ge 3$, which can be seen in [12].

**Theorem 1.1**

*Consider the non*-

*linear difference equation of the form*

*where* $p(z)\not\equiv 0$, $q(z)$, $r(z)$ *are polynomials*, *n* *and* *m* *are positive integers*. *Suppose that* $f(z)$ *is a transcendental entire function of finite order*, *not of period* *c*. *If* $n>m$, *then* $f(z)$ *cannot be a solution of* (4).

**Remarks**(1) In case $n\le m$, Theorem 1.1 is not true. In the special case that $m=n=1$, the function $f(z)={e}^{z}$ solves

- (2)
From Theorem 1.1, we get the following result: Let $f(z)$ be a transcendental entire function of finite order, then for $n>m$, $f{(z)}^{n}+p(z){({\mathrm{\Delta}}_{c}f)}^{m}$ assumes zero infinitely often, where $p(z)\not\equiv 0$ is polynomial.

where $p(z)\not\equiv 0$, $q(z)$, $r(z)$ are polynomials. In fact, (5) may have solutions. For example, the function $f(z)=z{e}^{z}$ is a solution of the equation $f{(z)}^{3}-\frac{{z}^{3}}{2\pi i}{e}^{2z}{\mathrm{\Delta}}_{2\pi i}f=0$. One solution of the equation $f(z)-{e}^{z}\frac{1}{2\pi i}{\mathrm{\Delta}}_{2\pi i}f=z$ is the function $f(z)={e}^{z}+z$. In addition, the function $f(z)={e}^{z}+z$ can solve $f(z)+{e}^{z}\frac{1}{4\pi}{({\mathrm{\Delta}}_{2\pi i}f)}^{2}=z$ as well. Hence, here we just study the order of the growth of the solutions. We state our findings as follows.

**Theorem 1.2** *Let* $p(z)\not\equiv 0$, $q(z)$, $r(z)$ *be polynomials*, *n* *and* *m* *be positive integers satisfying* $n>m$. *Let* $f(z)$ *be finite order entire solutions of* (5), *then* $\delta (f)=degq(z)$.

**Remark** We will give the following examples to show that the assumption that $n>m$ in Theorem 1.2 is sharp. Clearly, $f(z)={e}^{z}$ is a solution of the equation $f(z)-\frac{1}{2}{\mathrm{\Delta}}_{ln2}f=0$. The function $f(z)=4{e}^{8\pi iz}-{e}^{4\pi iz}+z$ solves $f(z)-{({\mathrm{\Delta}}_{-\frac{1}{4}}f)}^{2}=z-\frac{1}{16}$. However, $\delta (f)\ne degq(z)$ in the above two examples.

**Corollary 1.3**

*Let*$p(z)\not\equiv 0$, $q(z)$, $r(z)$

*be polynomials*, $n>1$.

*Let*$f(z)$

*be finite order entire solutions of the equation*

*then* $\delta (f)=degq(z)$.

According to (6), we will give a further discussion on the existence of meromorphic solutions. We get:

**Theorem 1.4**

*Let*$p(z)\not\equiv 0$, $q(z)$, $r(z)$

*be polynomials*.

*Assume one of the following assertions holds*:

- (i)
$n>1$, $f(z)$

*is a finite order meromorphic function*(*not entire*), - (ii)
$n>1$, $r(z)\equiv 0$,

*and*$f(z)$*is a finite order entire function with infinitely many zeros*, - (iii)
$n\ge 1$, $r(z)\not\equiv 0$,

*and*$f(z)$*is a finite order entire function satisfying*$\lambda (f)<\delta (f)$.

*Then* $f(z)$ *cannot be a solution of* (6).

**Remarks**(1) From Theorem 1.4, we know that the solution of the equation $f{(z)}^{n}+p(z){e}^{q(z)}{\mathrm{\Delta}}_{c}f=0$ must be expressed as $f(z)=\alpha (z){e}^{\beta (z)}$, where $\alpha (z)$ and $\beta (z)$ are polynomials.

- (2)
The condition that $n>1$ in Theorem 1.4 is sharp. In fact, if $n=1$, then we know the function $f(z)=\frac{{e}^{z}}{z}$ is a solution of the equation $f(z)+(\frac{z}{2\pi i}+1){\mathrm{\Delta}}_{2\pi i}f=0$. Moreover, the function $f(z)=cosz$ can solve $f(z)+\frac{1}{2}{\mathrm{\Delta}}_{\pi}f=0$.

- (3)
Some ideas of this paper are from [13].

## 2 Preliminary lemmas

**Lemma 2.1** [[14], Theorem 2.1]

*Let*$f(z)$

*be a meromorphic function of finite order*,

*and let*$c\in \mathbb{C}$,

*then*

**Remark** From Lemma 2.1, we know $T(r,f)=T(r,f(z+c))+S(r,f)$ when $f(z)$ is an entire function of finite order.

**Lemma 2.2** [[15], Theorem 2.4.2]

*Let*$f(z)$

*be a transcendental meromorphic solution of*

*where* $A(z,f)$, $B(z,f)$ *are differential polynomials in* *f* *and its derivatives with small meromorphic coefficients* ${a}_{\lambda}$, *in the sense of* $m(r,{a}_{\lambda})=S(r,f)$ *for all* $\lambda \in I$. *If* $d(B(z,f))\le n$, *then* $m(r,A(z,f))=S(r,f)$.

**Lemma 2.3** [[1], Theorem 1.56]

*Let*${f}_{j}(z)$ ($j=1,2,3$)

*be meromorphic functions that satisfy*

*If*${f}_{1}(z)$

*is not a constant*,

*and*

*where* $\lambda <1$ *and* $T(r)={max}_{1\le j\le 3}\{T(r,{f}_{j})\}$, *then either* ${f}_{2}(z)\equiv 1$ *or* ${f}_{3}(z)\equiv 1$.

**Lemma 2.4** [[1], Theorem 1.51]

*Suppose that*${f}_{j}(z)$ ($j=1,\dots ,n$) ($n\ge 2$)

*are meromorphic functions and*${g}_{j}(z)$ ($j=1,\dots ,n$)

*are entire functions satisfying the following conditions*.

- (1)
${\sum}_{j=1}^{n}{f}_{j}(z){e}^{{g}_{j}(z)}\equiv 0$.

- (2)
$1\le j<k\le n$, ${g}_{j}(z)-{g}_{k}(z)$

*are not constants for*$1\le j<k\le n$. - (3)
*For*$1\le j\le n$, $1\le h<k\le n$,$T(r,{f}_{j})=o\left\{T(r,{e}^{{g}_{h}-{g}_{k}})\right\},\phantom{\rule{1em}{0ex}}r\to \mathrm{\infty},r\notin E,$

*where* $E\subset (1,\mathrm{\infty})$ *is of finite linear measure*.

*Then* ${f}_{j}(z)\equiv 0$.

## 3 Proof of Theorem 1.1

*c*. Differentiating (4) and eliminating ${e}^{q(z)}$, we have

*A*is a non-zero constant. Substituting $f(z)$ into (4), we get

*c*, which contradicts the assumption. Hence, $A\ne 1$. Let $g={e}^{\frac{q(z)}{n}}$, then ${({\mathrm{\Delta}}_{c}f)}^{m}$ can be expressed as ${\sum}_{i=0}^{m}\left(\genfrac{}{}{0ex}{}{m}{i}\right){(-1)}^{i}h{(z)}^{i}h{(z+c)}^{m-i}g{(z)}^{i}g{(z+c)}^{m-i}$. Furthermore, from Lemma 2.1, we have

which contradicts the condition that $n>m$. Therefore, we conclude that $n{f}^{\prime}(z)-({q}^{\prime}(z)+\frac{{r}^{\prime}(z)}{r(z)})f(z)\not\equiv 0$. We discuss the following two cases.

which is a contradiction.

Suppose ${z}_{0}$ is a zero of $f(z)$ with multiplicity *k*. If ${z}_{0}$ is a zero of $r(z)$ as well, then the contribution to $N(r,\frac{1}{f})$ is $S(r,f)$. Assuming that ${z}_{0}$ is not a zero of $r(z)$, we will discuss the two subcases:

Subcase 1. Suppose ${z}_{0}$ is a zero of $H(z)$ with multiplicity *t*. From (13), by simple calculations, we know that $k=1+t\le 2t$, which means that ${z}_{0}$ is a contribution of $S(r,f)$ to $N(r,\frac{1}{f})$ by (12).

Subcase 2. Suppose ${z}_{0}$ is not a zero of $H(z)$. By (13), we get ${k}^{2}-k=0$, then such a zero of $f(z)$ must be simple and we find that ${q}^{\prime}+\frac{{r}^{\prime}}{r}+(m+1)\frac{{H}^{\prime}}{H}$ must vanish at ${z}_{0}$. That implies that ${z}_{0}$ makes a contribution of $S(r,f)$ to $N(r,\frac{1}{f})$ by (12).

Plugging the above equation into (10), similar to (9), we get $T(r,r{e}^{q}-p{({\mathrm{\Delta}}_{c}f)}^{m})\le mT(r,f)+S(r,f)$. Comparing both sides of (9), we get a contradiction. If ${f}_{3}(z)\equiv 1$, then we get $f{(z)}^{m+1}\equiv r(z){e}^{q(z)}$, which means that ${\mathrm{\Delta}}_{c}f\equiv 0$, and this contradicts the assumption.

Therefore, from the discussions above, we find that a transcendental finite order entire function $f(z)$ which is not of period *c*, cannot be a solution of (4). The proof of Theorem 1.1 is finished completely.

## 4 Proof of Theorem 1.2

which means that $(n-m)m(r,f)\le m(r,{e}^{q})+S(r,f)$. From the assumption that $n>m$ and the above inequality, we conclude that $\delta (f)\le degq(z)$.

Now we show that $\delta (f)=degq(z)$. Suppose to the contrary that $\delta (f)<degq(z)$. Then $\delta ({f}^{n}+p{e}^{q}{({\mathrm{\Delta}}_{c}f)}^{m})=degq(z)>0$ from Remark below Lemma 2.1, and $\delta (r)=0$. This is a contradiction by (5).

## 5 Proof of Theorem 1.4

*c*cannot be a solution of (6). Actually, if $f(z)$ is a period function, then the order of $f(z)$ satisfies $\delta (f)\ge 1$. At the same time, we get $f{(z)}^{n}=r(z)$, which is impossible. It remains to consider the function which is not of period

*c*.

- (i)
Suppose that $f(z)$ is a finite order meromorphic solution of (6), and ${z}_{0}$ is a pole of $f(z)$ with multiplicity

*t*. Then we get ${z}_{0}+c$ is a pole of $f(z)$ with multiplicity $\ge nt$. By calculation, we see that ${z}_{0}+kc$ is a pole of $f(z)$ with multiplicity ${n}^{k}t$, where*k*is a positive integer. Since $n>1$, we conclude that $\lambda (f)=\mathrm{\infty}$,*i.e.*, $\delta (f)=\mathrm{\infty}$. This is a contradiction. - (ii)
Suppose that a finite order entire solution $f(z)$ has infinitely many zeros and $r(z)\equiv 0$. We assume that all zeros of $p(z)$ are in $D=\{z:|Rez|\le M,|Imz|\le M\}$, where $M>0$ is some constant. From the assumption, we know that $f(z)$ has infinitely many zeros which are not in

*D*. If ${z}_{0}\notin D$ satisfies $f({z}_{0})=0$, then from (6) we know that ${z}_{0}+kc$ are zeros of $f(z)$. Moreover, we have ${z}_{0}+kc$ are not in*D*, as $k\to \mathrm{\infty}$. In the same way as (i), we get $\lambda (\frac{1}{f})=\mathrm{\infty}$,*i.e.*, $\delta (f)=\mathrm{\infty}$, which contradicts the assumption that $\delta (f)<\mathrm{\infty}$. - (iii)
Suppose $f(z)$ is an entire solution of finite order satisfying $\lambda (f)<\delta (f)$, then we know $\delta (f)\ge 1$ by the conclusion of Corollary 1.3. Hence, from the Hadamard factorization theorem, $f(z)$ can be written as $f(z)=\alpha (z){e}^{\beta (z)}$, where $\beta (z)$ is a non-constant polynomial.

where $\nu (z)$ is a polynomial such that $deg\nu (z)\le q-1$.

Since $r(z)\not\equiv 0$, we get $\gamma {(z)}^{n}+p(z)\mu (z)(\gamma (z+c){e}^{\nu (z)}-\gamma (z))\not\equiv 0$. Comparing the characteristics function of both sides of the above equation, we get a contradiction.

Applying Lemma 2.4 to the above equation, we get $r(z)\equiv 0$, which contradicts the assumption.

where $\delta (\phi )<k$, and $T(r,\phi )=S(r,f)$. Since $r(z)\not\equiv 0$, we get $\varphi (z)+p(z){e}^{q(z)}(\varphi (z+c)\phi (z)-\varphi (z))\not\equiv 0$. Comparing the characteristics function of both sides of the above equation, we get a contradiction.

## Declarations

### Acknowledgements

The authors thank the referee for his/her valuable suggestions to improve the present paper. This work was supported by the National Natural Science Foundation of China (No. 11301220 and No. 11371225) and the Tianyuan Fund for Mathematics (No. 11226094), the NSF of Shandong Province, China (No. ZR2012AQ020) and the Fund of Doctoral Program Research of University of Jinan (XBS1211).

## Authors’ Affiliations

## References

- Yang CC, Yi HX:
*Uniqueness Theory of Meromorphic Functions*. Kluwer Academic, Dordrecht; 2003.View ArticleMATHGoogle Scholar - Yang CC: On entire solutions of a certain type of nonlinear differential equation.
*Bull. Aust. Math. Soc.*2001, 64: 377-380. 10.1017/S0004972700019845View ArticleMathSciNetMATHGoogle Scholar - Heittokangas J, Korhonen R, Laine I: On meromorphic solutions of certain nonlinear differential equations.
*Bull. Aust. Math. Soc.*2002, 66: 331-343. 10.1017/S000497270004017XMathSciNetView ArticleMATHGoogle Scholar - Li P, Yang CC: On the nonexistence of entire solution of certain type of nonlinear differential equations.
*J. Math. Anal. Appl.*2006, 320: 827-835. 10.1016/j.jmaa.2005.07.066MathSciNetView ArticleMATHGoogle Scholar - Li P: Entire solution of certain type of differential equations.
*J. Math. Anal. Appl.*2008, 344: 253-259. 10.1016/j.jmaa.2008.02.064MathSciNetView ArticleMATHGoogle Scholar - Yang CC, Laine I: On analogies between nonlinear difference and differential equations.
*Proc. Jpn. Acad., Ser. A, Math. Sci.*2010, 86: 10-14. 10.3792/pjaa.86.10MathSciNetView ArticleMATHGoogle Scholar - Liu K, Liu XL, Yang LZ: Existence of entire solutions of nonlinear difference equations.
*Czechoslov. Math. J.*2011, 61: 565-576. 10.1007/s10587-011-0075-1View ArticleMathSciNetMATHGoogle Scholar - Liu K, Cao TB: Entire solutions of Fermat type
*q*-difference differential equations.*Electron. J. Differ. Equ.*2013., 2013: Article ID 59Google Scholar - Peng CW, Chen ZX: On a conjecture concerning some nonlinear difference equations.
*Bull. Malays. Math. Sci. Soc.*2013, 36: 221-227.MathSciNetMATHGoogle Scholar - Wen ZT, Heittokangas J, Laine I: Exponential polynomials as solutions of certain nonlinear difference equations.
*Acta Math. Sin.*2012, 28: 1295-1306. 10.1007/s10114-012-1484-2MathSciNetView ArticleMATHGoogle Scholar - Qi XG: Value distribution and uniqueness of difference polynomials and entire solutions of difference equations.
*Ann. Pol. Math.*2011, 102: 129-142. 10.4064/ap102-2-3View ArticleMathSciNetMATHGoogle Scholar - Gross F: On the equation ${f}^{n}+{g}^{n}=1$.
*Bull. Am. Math. Soc.*1966, 72: 86-88. 10.1090/S0002-9904-1966-11429-5View ArticleMATHGoogle Scholar - Laine I: A note on value distribution of difference polynomials.
*Bull. Aust. Math. Soc.*2010, 81: 353-360. 10.1017/S000497270900118XMathSciNetView ArticleMATHGoogle Scholar - Halburd R, Korhonen R: Nevanlinna theory for the difference operator.
*Ann. Acad. Sci. Fenn., Math.*2006, 31: 463-478.MathSciNetMATHGoogle Scholar - Laine I:
*Nevanlinna Theory and Complex Differential Equations*. de Gruyter, Berlin; 1993.View ArticleMATHGoogle Scholar

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