The solutions of one type q-difference functional system
© Xu et al.; licensee Springer. 2014
Received: 25 March 2013
Accepted: 1 December 2013
Published: 6 January 2014
In this paper, we study the functional system on q-difference equations, our results can give estimates on the proximity functions and the counting functions of the solutions of q-difference equations system. This implies that solutions have a relatively large number of poles. The main results in this paper concern q-difference equations to the system of q-difference equations.
MSC:30D35, 39B32, 39A13, 39B12.
1 Introduction and main results
A function is called meromorphic if it is analytic in the complex plane ℂ except at isolate poles. In what follows, we assume that the reader is familiar with the basic notion of Nevanlinna’s value distribution theory, see  and .
where γ and are the same as in (1.1) above. The q-difference polynomial is said to be homogeneous with respect to if the degree of each term in the sum (1.1) is non-zero and the same for all .
We recall the following result of Zhang et al. [, Theorem 1].
then , where denotes the order of zero of , as a function of at .
In this paper, the main results are as follows.
then and cannot hold both at the same time, possibly outside of an exceptional set of finite logarithmic measure.
If , and , then (), where r runs to infinity outside of an exceptional set of finite logarithmic measure.
2 Some lemmas
3 Proof of Theorem 1
which is impossible, we prove the assertion.
4 Proof of Theorem 2
for all r outside of an exceptional set of finite logarithmic measure.
for all r outside of , a set of finite logarithmic measure.
for all r outside of , we have proved the assertion.
Research was supported by the National Science Foundation of China (11161041), and the Fundamental Research Funds for the Central Universities (No. 31920130006) and Middle-Younger Scientific Research Fund (No. 12XB39).
- Hayman W-K: Meromorphic Functions. Clarendon, Oxford; 1964.MATHGoogle Scholar
- Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.View ArticleGoogle Scholar
- Korhonen R: A new Clunie type theorem for difference polynomials. J. Differ. Equ. Appl. 2011, 17(3):387-400. 10.1080/10236190902962244MATHMathSciNetView ArticleGoogle Scholar
- Zhang JC, Wang G, Chen JJ, Zhao RX: Some results on q -difference equations. Adv. Differ. Equ. 2012., 2012: Article ID 191Google Scholar
- Barnett D, Halburd R-G, Korhonen R-J, Morgan W: Nevanlinna theory for the q -difference operator and meromorphic solutions of q -difference equations. Proc. R. Soc. Edinb. A 2007, 137(3):457-474.MATHMathSciNetView ArticleGoogle Scholar
- Hayman W-K: On the characteristic of functions meromorphic in the plane and of their integrals. Proc. Lond. Math. Soc. 1965, 14A: 93-128.MathSciNetView ArticleGoogle 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/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.