Research Article | Open Access
The Existence of Meromorphic Solutions of Some Types of Systems of Complex Functional Equations
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 , Yang , and Yi and Yang ). 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 .
Theorem 1 (see [14, Propositions and ]). Let . If the following equationswhere with small coefficients , with respect to , admit transcendental meromorphic solutions of finite order, then .
In 2002, Gundersen et al. considered the solutions of -difference equation and obtained the following result.
Theorem 4 (see [16, Theorem ]). Suppose that is a transcendental meromorphic solution of a -difference equation of the formwhere , 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  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 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.
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 systemwith the meromorphic coefficients of growth , and . Ifthen 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 systemwith 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  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 equationadmits 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 systemwith 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
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 systemwhere of degree . Then where and
2. Some Lemmas
Lemma 16 (Valiron-Mohon’ko (see )). 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 ). 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) thatIt 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 ). Let be a transcendental meromorphic function of finite order and a nonzero complex constant. Then, we have
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 ). 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 .
3.1. The Proof of Theorem 8
Let . By applying Valiron-Mohon’ko theorem  to (15), we haveFrom (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 thatNow, we rewrite system (15) aswithout 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 getwhere From (54)–(58) and , it follows that
Substituting the above system into (15), we have that is,where , orwhere . 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 .
Next, we will exclude (18a). Suppose thatFrom the assumptions of Lemma 21 and by Hadamard Theorem, we can writewhere , 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.
Therefore, we complete the proof of Corollary 9.
3.3. The Proof of Corollary 10
Thus, we complete the proof of Corollary 10.
4. The Proof of Theorem 11
Let and . By applying Valiron-Mohon’ko theorem  to (20), we haveFrom (70) and the definitions of , similar to the above argument, we haveSince 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 haveFrom (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.