Some properties of solutions of a class of systems of complex q-shift difference equations
© Xu and Xuan; licensee Springer. 2013
Received: 24 May 2013
Accepted: 23 July 2013
Published: 9 September 2013
In view of Nevanlinna theory, we study the properties of systems of two types of complex difference equations with meromorphic solutions. Some results of this paper improve and extend previous theorems given by Gao, and five examples are given to show the extension of solutions of the system of complex difference equations.
Keywordsq-shift difference equation transcendental meromorphic function
1 Introduction and main results
In this note, we will investigate the problem of the existence and growth of solutions of complex difference equations. The fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used (see [1–3]). Besides, for the meromorphic function f, denotes any quantity satisfying that for all r outside a possible exceptional set E of finite logarithmic measure , and a meromorphic function is called a small function with respect to f if .
In recent years, difference equations, difference product and q-difference in the complex plane ℂ have been an active topic of study. Considerable attention has been paid to the growth of solutions of difference equations, value distribution and uniqueness of differences analogues of Nevanlinna’s theory [4–8]. Chiang and Feng  and Halburd and Korhonen  established a difference analogue of the logarithmic derivative lemma independently. After their work, a number of results on meromorphic solutions of complex difference equations were obtained.
The structure of this paper is as follows. In Section 1 , some results on growth of solutions of a complex difference equation are listed, and our theorems are given. In Section 2 , we introduce some lemmas. Section 3 is devoted to proving Theorem 1.5. Section 4 is devoted to proving Theorem 1.6. Finally, Section 5 gives some examples to show the accuracy of conclusions of Theorem 1.5.
In 2003, Silvennoinen considered  the growth and existence of meromorphic solutions of functional equations of the form , and obtained the following result.
Theorem 1.1 
where g is an entire function, , are small meromorphic functions with respect to f. Then, g is a polynomial.
Theorem 1.2 
are irreducible rational functions, , , and are small functions.
Theorem 1.3 
then the components and in have at least one rational function, where , .
In 2005, Laine et al.  investigated several higher order difference equations. In particular, they obtained the following result.
Theorem 1.4 
In this paper, we study the question above and the problem of the existence of meromorphic solutions for a system of complex difference equations (3), where is a polynomial, and obtain the following results.
- (i)if and . We have(4)
- (ii)if and , then(5)
if and , then ,
where is the lower order of f.
2 Some lemmas
Lemma 2.1 (Valiron-Mohon’ko )
where and .
Lemma 2.2 
where is a collection of all non-empty subsets of .
Lemma 2.3 
Lemma 2.4 
Let , be monotone increasing functions such that outside of an exceptional set E with the finite linear measure, or , , where is a set of the finite logarithmic measure. Then, for any , there exists such that for all .
Lemma 2.6 
Suppose that (), where μ (), m (), A (), B are constants. Then with , unless and ; and if and , then for any , .
Suppose that (with the notation of (i)) (). Then for all sufficiently large values of r, with , for some positive constant K.
3 The proof of Theorem 1.5
From the assumptions of Theorem 1.5, we know that and are transcendental meromorphic functions.
Hence, (4) holds.
where , are the sets of the finite logarithmic measure.
which implies that (5) is true.
Case 3.3 and . By using the same argument as in Case 3.1, we can get .
From Cases 3.1-3.3, the proof of Theorem 1.5 is completed.
4 The proof of Theorem 1.6
Next, we will prove that . Suppose that , then we can get . For sufficiently small , we have . This contradicts the condition on the transcendency of , .
Thus, the proof of Theorem 1.6 is completed.
5 Some examples for Theorem 1.5
The following examples show that the conclusions (4) and (5) in Theorem 1.5 are sharp.
where , and . This example shows that the equality in (4) can be achieved.
where , and . This example shows that the inequality (4) is true.
where , and . This example shows that the equality in (5) can be achieved.
where , and . This example shows that the inequality in (5) is true.
We have . Thus, it shows that (iii) in Theorem 1.5 is true when , , , and .
The authors thank the referee for his/her valuable suggestions to improve the present article. This project is supported by the NNSF of China (11301233, 61202313) and the Natural Science foundation of Jiangxi Province in China (No. 2010GQS0119, No. 20122BAB201016 and No. 20132BAB211001). The second author is supported in part by the NNSFC (Nos. 11226089, 11201395, 61271370), Beijing Natural Science Foundation (No. 1132013) and The Project of Construction of Innovative Teams and Teacher Career Development for Universities and Colleges Under Beijing Municipality (CIT and TCD20130513).
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