- Open Access
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).
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