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The solutions of one type q-difference functional system
Advances in Difference Equations volume 2014, Article number: 3 (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 .
Let us consider the q-difference polynomial case. Let for , and let be a finite set of multi-indexes . A q-difference polynomial of a meromorphic function is defined as follows:
where , and the coefficients are small meromorphic functions with respect to such that on a logarithmic density 1, denoted by . The total degree of in and the q-shifts of is denoted by , and the order of zero of , as a function of at , is denoted as , which can be found, e.g., in . Moreover, the weight of difference polynomial (1.1) is defined by
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].
Theorem A Let be a zero-order meromorphic solution of
where is a homogeneous q-difference polynomial with polynomial coefficients, and and are polynomials in with polynomial coefficients having no common factors. If
then , where denotes the order of zero of , as a function of at .
Now let us introduce some notation. Let for , and let I and J be a finite set of multi-indexes and . Two q-difference polynomials of a meromorphic function are defined as follows:
where the coefficients and are small with respect to and in the sense that and , , on a set of logarithmic density 1, as r tends to infinity outside of an exceptional set E of finite logarithmic measure
The weights of and in , are denoted by
The purpose of this paper is to study the problem of the properties of Nevanlinna counting functions and proximity functions of meromorphic solutions of a type of systems of q-difference equations of the following form:
the coefficients , , , are meromorphic functions and small functions. The order of zero of , as a function of at , is denoted by . The q-difference polynomial , , is said to be homogeneous with respect to if the degree of each term in the sum is non-zero and the same for all . Finally, the order of growth of a meromorphic solution is defined by
In this paper, the main results are as follows.
Theorem 1 Let be a zero-order meromorphic solution of system (1.2), where () are homogeneous q-difference polynomials in and , respectively, with meromorphic coefficients, and and , , are polynomials in with meromorphic coefficients having no common factors. If
then and cannot hold both at the same time, possibly outside of an exceptional set of finite logarithmic measure.
Theorem 2 Let be a zero-order meromorphic solution of system (1.2), where () are homogeneous q-difference polynomials in and , respectively, with meromorphic coefficients, and and , , are polynomials in with meromorphic coefficients having no common factors,
If , and , then (), where r runs to infinity outside of an exceptional set of finite logarithmic measure.
3 Proof of Theorem 1
Since are homogeneous in and , respectively, it follows by Lemma 1 that
for all r outside of an exceptional set of finite logarithmic measure. Moreover, from (1.2), we have
where r approaches infinity outside of an exceptional set of finite logarithmic measure. By combining (3.1) and (3.3), (3.2) and (3.4), respectively, it follows that
From Lemma 2, we have
Combining (3.5) and (3.7), (3.6) and (3.8), respectively, we have
Suppose that and , according to (3.9) and (3.10), we have
By (3.11) and (3.12), we conclude that
which is impossible, we prove the assertion.
4 Proof of Theorem 2
It follows by Lemma 1 that
for all r outside of an exceptional set of finite logarithmic measure.
Suppose now that is a finite-order meromorphic solution of (1.2). Denoting in and , by Lemma 2, we obtain
for all r outside of a set E of finite logarithmic measure. By (4.1) and (4.3), we have
for all . On the other hand, by (4.1) and (4.3),
where r lies outside of a set F of finite logarithmic measure. Combining inequalities (4.4) and (4.5) with the assumption in Theorem 2, we have
Similarly, we obtain
By (4.6) and (4.7), we obtain
Combining (4.8) and (4.9), we have
where and . Combining the assumption and (4.10), we have
for all r outside of , a set of finite logarithmic measure.
Similarly, we obtain
for all r outside of , we have proved the assertion.
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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).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Xu, Y., Han, D., Fan, X. et al. The solutions of one type q-difference functional system. Adv Differ Equ 2014, 3 (2014). https://doi.org/10.1186/1687-1847-2014-3
- functional system
- q-difference equations
- zero order
- difference Nevanlinna theory