Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 723025, 10 pages
http://dx.doi.org/10.1155/2015/723025

Research Article

The Existence of Meromorphic Solutions of Some Types of Systems of Complex Functional Equations

Hong-Yan Xu,¹ Jin-Lian Wang,² and Hua Wang¹

¹Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China
²Jiangxi Normal University, Nanchang, Jiangxi 330027, China

Received 4 August 2015; Accepted 13 September 2015

Academic Editor: Chris Goodrich

Copyright © 2015 Hong-Yan Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate the existence of transcendental meromorphic solutions of some types of systems of complex functional equations and obtain some results about the existence of meromorphic solutions of such systems. Our results are improvement of the previous theorems given by Gao, Xu, and Zheng, and our examples show that our results are sharp to some extent.

1. Introduction and Main Results

Recently, with the establishment of the differences analogues of Nevanlinna’s theory (see [1–3]), people obtained many interesting theorems about the growth and existence of solutions of difference equations, -difference equations, and so on (see [4–10]). To state some results, we should introduce some basic definition and standard notations. We firstly assume that readers are familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions such as (see Hayman [11], Yang [12], and Yi and Yang [13]). In addition, we also use , , and to denote the order, the exponent of convergence of zeros, and the exponent of convergence of poles of , respectively, and to denote any quantity satisfying for all outside a possible exceptional set of finite logarithmic measure , and a meromorphic function is called small function with respect to , if .

In 2001, Heittokangas et al. considered the existence of solutions of some difference equations and obtained the following results [14].

Theorem 1 (see [14, Propositions and ]). Let . If the following equations

where

with small coefficients , with respect to , admit transcendental meromorphic solutions of finite order, then .

Theorem 2 (see [14]). Let . If (2) with small coefficients , of growth such that . If

then (2) is of the form

where is meromorphic, , and .

In 2002, Gundersen et al. considered the solutions of -difference equation and obtained the following result.

Theorem 3 (see [15]). Suppose that is a transcendental meromorphic solution of a -difference equation of the form

where , and meromorphic coefficients are of growth . If

then (6) is either of the form

In 2012, Zheng and Chen [16] further considered a -difference equation more general than (2) under a condition similar to Theorem 2 on meromorphic solutions and obtained the following result.

Theorem 4 (see [16, Theorem ]). Suppose that is a transcendental meromorphic solution of a -difference equation of the form

where , and is an irreducible rational function in with meromorphic coefficients and of growth such that . If

then (9) is reduced to the form

In 2012, Gao [17] studied the similar problem when (9) is replaced by the following system of functional equations,

where is an entire function, are irreducible rational functions, and the coefficients are small functions, and obtained the following theorems.

Theorem 5 (see [17, Theorem ]). Let be a pair of nonconstant meromorphic solutions of system (12). Then, is a polynomial.

Theorem 6 (see [17, Theorem ]). Let the polynomial be of degree a pair of transcendental meromorphic solutions of system (12), and small functions. If

then system (12) is of the form

where are meromorphic, , and .

Remark 7. We can see that the numerical values of are uncertain in the conclusion of Theorem 6.

After his works, there were some papers focusing on the growth and existence of meromorphic solutions of some types of systems of complex functional equations (see [18–21]).

In this paper, we will further investigate some properties of solutions of the system of complex functional equations. The main purpose of this paper is to find out the relationship between “” and “” and further to confirm the numerical values of , in the conclusion of Theorem 6. Some results are obtained in this paper, and the first main result is about meromorphic solutions with few zeros and poles of a type of system of complex functional equations.

Theorem 8. Let , and let be a pair of nonrational meromorphic solutions of the system

with the meromorphic coefficients of growth , and . If

then system (15) is one of the four following forms:

Corollary 9. Let be a pair of transcendental solutions of (15) with finite order ,    are positive integers and all the other assumptions of Theorem 8 hold.

(i)If    and  , then system (15) is reduced to (18b) only.(ii)If    and  , then system (15) is reduced to (18b) only.

Corollary 10. Let be a pair of transcendental solutions of (15) with finite order , are positive integers and all the other assumptions of Theorem 8 hold. If system (15) is reduced to (18b) only, and

then .

Furthermore, we obtain the following result when system (15) is replaced by the other system.

Theorem 11. Let , and let be a pair of nonrational meromorphic solutions of the system

with the meromorphic coefficients of growth , and . If , satisfy (16) and , where is a polynomial in of degree , then satisfy the following forms:

The researches on the properties of solutions of complex differential equations in the whole complex plane, disc, and angular domain are always interesting in the past several decades (see [22–26]). In 2014, Liu and Dong [27] considered the existence of solutions of some differential-difference equation, where the differential-difference equation is an equation including derivatives, shifts, or differences of , and obtained the following result.

Theorem 12 (see [27, Theorem ]). Let be a positive integer. If the equation

admits a transcendental meromorphic solution of finite order, then . If (22) admits a transcendental entire solution of finite order, then .

Then, we will further consider the system of differential-difference equations with the analogue form of (22) and obtain the following theorem.

Theorem 13. Let be integers. If system

with the meromorphic coefficients of growth , and , admits a pair of transcendental meromorphic solutions of finite order, set , ; then

If (23) admits a pair of transcendental entire solutions of finite order, then

We give the following example to show that Theorem 13 is not valid for a pair of meromorphic solutions with infinite order of (23).

Example 14. Let ; then are a pair of solutions of the system

where is a complex constant and is a positive integer satisfying .

Theorem 15. Under the assumptions of Theorem 13, let be a pair of transcendental meromorphic solutions of system

where of degree . Then

where and

2. Some Lemmas

Lemma 16 (Valiron-Mohon’ko (see [24])). Let be a meromorphic function. Then for all irreducible rational functions in ,

with meromorphic coefficients , the characteristic function of satisfies that

where and .

Lemma 17 (see [14, p. 37]). If a meromorphic function satisfies

then is of regular growth.

Lemma 18 (see [15, p. 127]). The differential field

is algebraically closed in the field of meromorphic functions in the complex plane. That is, any meromorphic function satisfying an algebraic equation over the field actually belongs to .

Remark 19 (see [28, p. 249] and [16, p. 728 Remark]). The following observation

holds for any meromorphic function and any nonzero complex constant . Clearly, we can immediately obtain

Similarly, we also have

Lemma 20 (see [29]). Let be a transcendental meromorphic function and a complex polynomial of degree . For given , let ; then for given and for large enough,

Lemma 21. Suppose that , are meromorphic functions of finite order , are positive integers satisfying (16), and let .
(i) If and

then, satisfies .
(ii) If and

then, satisfies .

Remark 22. The following example shows that the conclusions are not valid if the condition is removed in Lemma 21.

Example 23. Let ; then and . If , are complex constants and satisfy

then we have ; that is, .

Proof. From the assumptions in Lemma 21 and by Hadamard Theorem, we have

where , are polynomials of degree , respectively, and are the canonical products formed with nonzero zeros and poles of ,  , respectively. So, we have

Denoting

then we can rewrite as the following form:

where , are the leading coefficient of and are polynomials of degree .
(i) . By Remark 19, it follows from (16) that

It follows from (44)–(46) that for all . Since , we have .
(ii) . By using the same argument as in (i), we can get easily.
Thus, this completes the proof of Lemma 21.

Lemma 24 (see [2]). Let be a transcendental meromorphic function of finite order and a nonzero complex constant. Then, we have

Lemma 25 (see [13, p. 37] or [12]). Let be a nonconstant meromorphic function in the complex plane and a positive integer. Then

Lemma 26 (see [24, 30]). Let be monotone increasing functions such that outside of an exceptional set with finite linear measure, or , , where is a set of finite logarithmic measure. Then, for any , there exists such that for all .

Lemma 27 (see [31]). Let be a function of , positive, and bounded in every finite interval.
(i) Suppose that , where , , , and are constants. Then with , unless and ; and if and , then, for any , .
(ii) Suppose that (with the notation of (i)) . Then for all sufficiently large values of , with , for some positive constant .

3. Proofs of Theorem 8, Corollaries 9 and 10

3.1. The Proof of Theorem 8

Let . By applying Valiron-Mohon’ko theorem [24] to (15), we have

From (16), we can take constants such that

and then we have

From the definitions of , similar to the above argument, we have

From (16), we know that zeros and poles are Borel exceptions of , and from [32, Satz 13.4], we have that is of regular growth. Hence, there exists that for . So, we can get that

Now, we rewrite system (15) as

without loss of generality, assume that are monic polynomials in with coefficients of growth , and set ; from (54), we have . And because

it follows that

Substituting , to the above equalities and comparing the leading coefficients, we can get

Solving the above system, we get

where From (54)–(58) and , it follows that

Substituting the above system into (15), we have

that is,

where , or

where . By regarding (61) or (62) as an algebraic equation in with coefficients of growth , then it follows by Lemma 18 that . Moreover, if and , we can get a contradiction with the condition that is irreducible in . Thus, we have or . Then it follows from (61) that or .

Similarly, we can get or .

Thus, we complete the proof of Theorem 8.

3.2. The Proof of Corollary 9

Let be a pair of transcendental solutions of (15) of finite order .

(i) If , we can exclude (17a) and (17b) easily by comparing the growth order of two sides of the equation

Next, we will exclude (18a). Suppose that

From the assumptions of Lemma 21 and by Hadamard Theorem, we can write

where , are polynomials of degree , respectively, and are the canonical products formed with nonzero zeros and poles of , respectively. Substituting (65) into (64), we have

Since and , we can get a contradiction easily.

(ii) If , we can exclude (17a) and (18a). By using the same argument as in (i) and combining the condition and , we can exclude (17b).

Therefore, we complete the proof of Corollary 9.

3.3. The Proof of Corollary 10

From the assumptions of Corollary 10 and (18b), we have

We can rewrite (67) as the following form:

where .

Suppose that . Since

by using the similar method as in Lemma 21, from (68), we can get a contradiction with the assumption . Hence .

Thus, we complete the proof of Corollary 10.

4. The Proof of Theorem 11

Let and . By applying Valiron-Mohon’ko theorem [24] to (20), we have

From (70) and the definitions of , similar to the above argument, we have

Since is an increasing function for and , hence, there exists such that for all

where , and . Thus, it follows from the above inequality and Lemma 20 that

From (16), we have as for . Thus, we can get as for ; that is,

Similarly, we have

From (71)–(75), we have

Now, we rewrite system (20) as

and, without loss of generality, assume that are monic polynomials in with coefficients of growth . Thus, by using the same argument as in Theorem 8, we can get the conclusion of Theorem 11 easily.

Therefore, this completes the proof of Theorem 11.

5. Proofs of Theorems 13 and 15

5.1. The Proof of Theorem 13

Let be a pair of transcendental meromorphic solutions of (23); by using Lemmas 16 and 25, it follows from (23) that

By Lemma 24, it follows that

From the above inequalities and the fact that are transcendental functions, then we have and . If are entire functions, we can get and , easily.

Hence, this completes the proof of Theorem 13.

5.2. The Proof of Theorem 15

By using the same argument as in Theorem 13, we can get

Since , by Lemma 20, we can get that, for and sufficiently large ,

where are two sets of finite linear measure. By Lemma 26, we have that, for any given and for sufficiently large ,

that is,

where and . From (83) and (84), we have

And since , we get that . Then it follows from (85) and by Lemma 27 that

where

Set ; then we have

Next, we will prove that . Suppose that ; then we can get that . For sufficiently small , we have . This contradicts the condition on the transcendence of .

Thus, the proof of Theorem 15 is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors were supported by the NSF of China (11171170, 11301233, and 11561033), the Natural Science Foundation of Jiangxi Province in China (20132BAB211001, 20151BAB201008), and the Foundation of Education Department of Jiangxi (GJJ14644) of China.

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