- Open Access
The properties of solutions of certain type of difference equations
Advances in Difference Equations volume 2014, Article number: 256 (2014)
In this paper, we shall utilize Nevanlinna value distribution theory to study the solvability of the difference equations of the form: and , 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).
1 Introduction and main results
In this paper, we use the basic notions in Nevanlinna theory of meromorphic functions, as found in . In addition, we use , , and to denote the order, and the exponents of the convergence of zeros and poles of a meromorphic function , respectively. We define difference operators as , where c is a non-zero constant.
In 2001, Yang  started to study the existence and uniqueness of finite order entire solutions of the following type of non-linear differential equation:
where is a linear differential polynomial in f with polynomial coefficients, is a non-vanishing polynomial, is an entire function, and n is an integer such that . Subsequently, several papers have appeared in which the solutions of (1) are studied. The reader is invited to see [3–5].
Recently, Yang and Laine  considered the existence of the non-linear differential-difference equation of the form
where is a finite sum of product of f, derivatives of f, and their shifts. They obtained the following result.
Theorem A Let be an integer, be a linear differential-difference polynomial of f, with small meromorphic coefficients, and be a meromorphic function of finite order. Then (2) possesses at most one admissible transcendental entire solution of finite order, unless vanishes identically. If such a solution exists, then is of the same order as .
In particular, it is shown in  that the equation
has no transcendental entire solution of finite order, where , are polynomials.
we get the following result in .
Theorem B Let , be polynomials, n and m be integers satisfying . Then (3) has no transcendental entire solution of finite order.
In fact, the special case of (3) with , and , can be viewed as the Fermat type functional equation. It is well known that (3) has no transcendental entire solutions when , which can be seen in .
It is natural to ask what happens if or is a transcendental entire function in (3). Corresponding to this question, we will investigate the finite order entire (meromorphic) solution of
Theorem 1.1 Consider the non-linear difference equation of the form
where , , are polynomials, n and m are positive integers. Suppose that is a transcendental entire function of finite order, not of period c. If , then cannot be a solution of (4).
Remarks (1) In case , Theorem 1.1 is not true. In the special case that , the function solves
Moreover, the entire function is a solution of the following equation:
From Theorem 1.1, we get the following result: Let be a transcendental entire function of finite order, then for , assumes zero infinitely often, where is polynomial.
In the following, we will consider the properties of the solutions of the equation
where , , are polynomials. In fact, (5) may have solutions. For example, the function is a solution of the equation . One solution of the equation is the function . In addition, the function can solve 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 , , be polynomials, n and m be positive integers satisfying . Let be finite order entire solutions of (5), then .
Remark We will give the following examples to show that the assumption that in Theorem 1.2 is sharp. Clearly, is a solution of the equation . The function solves . However, in the above two examples.
Corollary 1.3 Let , , be polynomials, . Let be finite order entire solutions of the equation
According to (6), we will give a further discussion on the existence of meromorphic solutions. We get:
Theorem 1.4 Let , , be polynomials. Assume one of the following assertions holds:
, is a finite order meromorphic function (not entire),
, , and is a finite order entire function with infinitely many zeros,
, , and is a finite order entire function satisfying .
Then cannot be a solution of (6).
Remarks (1) From Theorem 1.4, we know that the solution of the equation must be expressed as , where and are polynomials.
The condition that in Theorem 1.4 is sharp. In fact, if , then we know the function is a solution of the equation . Moreover, the function can solve .
Some ideas of this paper are from .
2 Preliminary lemmas
Lemma 2.1 [, Theorem 2.1]
Let be a meromorphic function of finite order, and let , then
Remark From Lemma 2.1, we know when is an entire function of finite order.
Lemma 2.2 [, Theorem 2.4.2]
Let be a transcendental meromorphic solution of
where , are differential polynomials in f and its derivatives with small meromorphic coefficients , in the sense of for all . If , then .
Lemma 2.3 [, Theorem 1.56]
Let () be meromorphic functions that satisfy
If is not a constant, and
where and , then either or .
Lemma 2.4 [, Theorem 1.51]
Suppose that () () are meromorphic functions and () are entire functions satisfying the following conditions.
, are not constants for .
For , ,
where is of finite linear measure.
3 Proof of Theorem 1.1
If is a constant or , then the conclusion follows from Theorem B. It remains to consider the case is a non-constant polynomial and . Assume is a transcendental entire solution of (4), which is finite order, not of period c. Differentiating (4) and eliminating , we have
If , then we have . Writing , where satisfies and A is a non-zero constant. Substituting into (4), we get
Clearly, if , then is a period function with period c, which contradicts the assumption. Hence, . Let , then can be expressed as . Furthermore, from Lemma 2.1, we have
From (8) and (9), we get
which contradicts the condition that . Therefore, we conclude that . We discuss the following two cases.
Case 1. . Rewrite (7) in the following forms:
Applying Lemma 2.1, Lemma 2.2, and the lemma on the logarithmic derivative to the above two equations, we obtain
which is a contradiction.
Case 2. . For this case, (4) and (7) now take the following forms:
Set . Dividing (11) by , we have
Since is entire, applying Lemma 2.1 and the lemma on the logarithmic derivative to the above equation, we conclude that
Differentiating , we have
The above equation can be rewritten in the following form:
Suppose is a zero of with multiplicity k. If is a zero of as well, then the contribution to is . Assuming that is not a zero of , we will discuss the two subcases:
Subcase 1. Suppose is a zero of with multiplicity t. From (13), by simple calculations, we know that , which means that is a contribution of to by (12).
Subcase 2. Suppose is not a zero of . By (13), we get , then such a zero of must be simple and we find that must vanish at . That implies that makes a contribution of to by (12).
In a word, . Therefore, we can assume to be of the form , where is a non-constant polynomial, and is an entire function satisfying by the Hadamard factorization theorem. Substituting this expression into (10), we get
which may be written in the form
From Lemma 2.3, we have either
If , then we get
Plugging the above equation into (10), similar to (9), we get . Comparing both sides of (9), we get a contradiction. If , then we get , which means that , and this contradicts the assumption.
Therefore, from the discussions above, we find that a transcendental finite order entire function 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
Applying Lemma 2.1 to (5), we get
which means that . From the assumption that and the above inequality, we conclude that .
Now we show that . Suppose to the contrary that . Then from Remark below Lemma 2.1, and . This is a contradiction by (5).
5 Proof of Theorem 1.4
First, we confirm that one period function with period c cannot be a solution of (6). Actually, if is a period function, then the order of satisfies . At the same time, we get , which is impossible. It remains to consider the function which is not of period c.
Suppose that is a finite order meromorphic solution of (6), and is a pole of with multiplicity t. Then we get is a pole of with multiplicity . By calculation, we see that is a pole of with multiplicity , where k is a positive integer. Since , we conclude that , i.e., . This is a contradiction.
Suppose that a finite order entire solution has infinitely many zeros and . We assume that all zeros of are in , where is some constant. From the assumption, we know that has infinitely many zeros which are not in D. If satisfies , then from (6) we know that are zeros of . Moreover, we have are not in D, as . In the same way as (i), we get , i.e., , which contradicts the assumption that .
Suppose is an entire solution of finite order satisfying , then we know by the conclusion of Corollary 1.3. Hence, from the Hadamard factorization theorem, can be written as , where is a non-constant polynomial.
Case 1. . From the conclusion of Corollary 1.3, we know . So we can rewrite in the following form:
where is a non-zero constant, , and . From the conclusion of Corollary 1.3, we conclude that
where is a non-zero constant, , and . By (6), (15), and (16), we have
where is a polynomial such that .
Subcase 1. If , then (17) can be expressed as
Since , we get . Comparing the characteristics function of both sides of the above equation, we get a contradiction.
Subcase 2. If , then we rewrite (17) in the form
Applying Lemma 2.4 to the above equation, we get , which contradicts the assumption.
Case 2. . Comparing both sides of (6), we obtain by Remark below Lemma 2.1. If , then in the same way as Case 1, the conclusion follows. If , then can be written as , where , and . Substituting this expression into (6), we get
where , and . Since , we get . Comparing the characteristics function of both sides of the above equation, we get a contradiction.
Yang CC, Yi HX: Uniqueness Theory of Meromorphic Functions. Kluwer Academic, Dordrecht; 2003.
Yang CC: On entire solutions of a certain type of nonlinear differential equation. Bull. Aust. Math. Soc. 2001, 64: 377-380. 10.1017/S0004972700019845
Heittokangas J, Korhonen R, Laine I: On meromorphic solutions of certain nonlinear differential equations. Bull. Aust. Math. Soc. 2002, 66: 331-343. 10.1017/S000497270004017X
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.2005.07.066
Li P: Entire solution of certain type of differential equations. J. Math. Anal. Appl. 2008, 344: 253-259. 10.1016/j.jmaa.2008.02.064
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.86.10
Liu K, Liu XL, Yang LZ: Existence of entire solutions of nonlinear difference equations. Czechoslov. Math. J. 2011, 61: 565-576. 10.1007/s10587-011-0075-1
Liu K, Cao TB: Entire solutions of Fermat type q -difference differential equations. Electron. J. Differ. Equ. 2013., 2013: Article ID 59
Peng CW, Chen ZX: On a conjecture concerning some nonlinear difference equations. Bull. Malays. Math. Sci. Soc. 2013, 36: 221-227.
Wen ZT, Heittokangas J, Laine I: Exponential polynomials as solutions of certain nonlinear difference equations. Acta Math. Sin. 2012, 28: 1295-1306. 10.1007/s10114-012-1484-2
Qi XG: Value distribution and uniqueness of difference polynomials and entire solutions of difference equations. Ann. Pol. Math. 2011, 102: 129-142. 10.4064/ap102-2-3
Gross F: On the equation .Bull. Am. Math. Soc. 1966, 72: 86-88. 10.1090/S0002-9904-1966-11429-5
Laine I: A note on value distribution of difference polynomials. Bull. Aust. Math. Soc. 2010, 81: 353-360. 10.1017/S000497270900118X
Halburd R, Korhonen R: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Math. 2006, 31: 463-478.
Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.
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).
The authors declare that they have no competing interests.
XQ completed the main part of this paper, JD and LY corrected the main theorems. All authors read and approved the final manuscript.
About this article
Cite this article
Qi, X., Dou, J. & Yang, L. The properties of solutions of certain type of difference equations. Adv Differ Equ 2014, 256 (2014). https://doi.org/10.1186/1687-1847-2014-256
- meromorphic functions
- difference equation
- finite order