The properties of solutions of certain type of difference equations
© Qi et al.; licensee Springer. 2014
Received: 23 April 2014
Accepted: 17 September 2014
Published: 3 October 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.
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].
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 .
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 .
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).
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.
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.
According to (6), we will give a further discussion on the existence of meromorphic solutions. We get:
, 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).
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]
Remark From Lemma 2.1, we know when is an entire function of finite order.
Lemma 2.2 [, Theorem 2.4.2]
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]
where and , then either or .
Lemma 2.4 [, Theorem 1.51]
, are not constants for .
- (3)For , ,
where is of finite linear measure.
3 Proof of Theorem 1.1
which contradicts the condition that . Therefore, we conclude that . We discuss the following two cases.
which is a contradiction.
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).
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
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
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.
where is a polynomial such that .
Since , we get . 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 , which contradicts the assumption.
where , and . Since , we get . Comparing the characteristics function of both sides of the above equation, we get a contradiction.
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).
- 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 .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
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.