Some Properties of Meromorphic Solutions of Systems of Complex <i>q</i>-Shift Difference Equations

Xu, Hong-Yan; Liu, Bing-Xiang; Tang, Ke-Zong

doi:https://doi.org/10.1155/2013/680956

Abstract and Applied Analysis

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Volume 2013 | Article ID 680956 | https://doi.org/10.1155/2013/680956

Some Properties of Meromorphic Solutions of Systems of Complex q-Shift Difference Equations

Hong-Yan Xu,¹Bing-Xiang Liu,¹and Ke-Zong Tang¹

Academic Editor: Fasma Diele

Received02 Sept 2012

Accepted29 Mar 2013

Published14 Apr 2013

Abstract

In view of Nevanlinna theory, we study the properties of meromorphic solutions of systems of a class of complex difference equations. Some results obtained improve and extend the previous theorems given by Gao.

1. Introduction and Main Results

The purpose of this paper is to study some properties of meromorphic solutions of complex -shift 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 meromorphic function , a meromorphic function is called small function with respect to if for all outside a possible exceptional set of finite logarithmic measure .

In recent years, it has been a heated topic to study difference equations, difference product, and -difference in the complex plane . There were articles focusing on the growth of solutions of difference equations, value distribution and uniqueness of differences analogues of Nevanlinna’s theory (see [4–9]). Chiang and Feng [10] and Halburd and Korhonen [11] established a difference analogue of the logarithmic derivative lemma independently, and Barnett et al. [5] also established an analogue of the logarithmic derivative lemma on -difference operators. By applying these theorems, a number of results on meromorphic solutions of complex difference and -difference equations were obtained (see [12–19]).

In 2011, Korhonen [20] investigated the properties of finite-order meromorphic solution of the equation where and obtained the following result.

Theorem 1 (see [20]). Let be a finite-order meromorphic solution of (1), where is a homogeneous difference polynomial with meromorphic coefficients and and are polynomials in with meromorphic coefficients having no common factors. If , then , where denotes the order of zero of at with respect to the variable .

Let for , and let be a finite set of multi-indexes . Then a difference polynomial of a meromorphic function is defined as where the coefficients are small with respect to in the sense that as tends to infinity outside of an exceptional set of finite logarithmic measure.

At the same year, Zheng and Chen [21] consider the value distribution of meromorphic solutions of zero order of a kind of -difference equations and obtained the following result which is an extension of Theorem 1.

Theorem 2 (see [21, Theorem 1]). Suppose that is a nonconstant meromorphic solution of zero order of a -difference equation of the form where is a finite index set and for all and . Moreover, suppose that , the and have no common factors, and that all meromorphic coefficients in (3) are of growth of on a set of logarithmic density 1. If then on any set of logarithmic density 1.

Remark 3. The logarithmic density of a set is defined by

Recently, Gao [22–24] and others [25, 26] also investigated the growth and existence of meromorphic solutions of some systems of complex difference equations; one system of complex difference equation is based on (1) and obtained some interesting results.

Inspired by the idea of [21–24, 27], we will investigate the properties of meromorphic solutions of systems of a class of complex -shift difference equations of the form where are two finite sets of multi-indexes , , and are two homogeneous difference polynomials to be defined as The coefficients are small with respect to , in the sense that , , , as tends to infinity outside of an exceptional set of finite logarithmic measure. The weights of , are defined by The coefficients , are meromorphic functions and small functions,

Now, we will show our main results as follows.

Theorem 4. Let be meromorphic solution of systems (7) satisfying . Moreover, suppose that , the and are polynomials in with meromorphic coefficients having no common factors, and that all meromorphic coefficients in (7) are of growth of for all on a set of logarithmic density 1 or outside of an exceptional set of logarithmic density 0. If then and cannot hold both at the same time, for all possibly outside of an exceptional set of logarithmic density 0, where the order of meromorphic solution of systems (7) is defined by

Theorem 5. Let be meromorphic solution of systems (7) satisfying , . Moreover, suppose that , the and are polynomials in with meromorphic coefficients having no common factors, and that all meromorphic coefficients in (7) are of growth of for all on a set of logarithmic density 1 or outside of an exceptional set of logarithmic density 0, and If then , hold for that runs to infinity possibly outside of an exceptional set of logarithmic density 0.

2. Some Lemmas

Lemma 6 (Valiron-Mohon’ko) ([28]). Let be a meromorphic function. Then for all irreducible rational functions in , with meromorphic coefficients , , the characteristic function of satisfies that where and .

Lemma 7 (see [27]). Let be a nonconstant zero-order meromorphic function and . Then on a set of logarithmic density 1 or outside of an exceptional set of logarithmic density 0.

Lemma 8 (see [29]). Let be a transcendental meromorphic function of zero order, and let , be two nonzero complex constants. Then on a set of logarithmic density 1 or outside of a possibly exceptional set of logarithmic density 0.

3. The Proof of Theorem 4

From the definitions of , by Lemma 7, it follows that where , are two sets of logarithmic density 0. By Lemma 6, we have where , are two sets of logarithmic density 0. Thus, from the assumptions of Theorem 4, combining (19) and (21), (20) and (22), respectively, we have

Since , from Lemma 8, we have where , are the sets of logarithmic density 0.

From (23) and (24), it follows that

Suppose now on the contrary to the assertion of Theorem 4 that and , from (25); it follows that that is, From (27), we can get that From the previous inequality, we can get a contradiction.

Therefore, this completes the proof of Theorem 4.

4. The Proof of Theorem 5

Since , from the assumptions concerning the coefficients of systems (7), by Lemma 7, and from the definitions of logarithmic measure and logarithmic density, we have where is a set of logarithmic density 0.

From (29), we have

From (19) and (29), we have From the previous inequality and (30), we have for

By using the same argument as in the previously mentioned, there exists a set of logarithmic density 0, for , and we have From (32) and (33), we have From (34), we have that is, where and . From (14) and (36), we have for all outside of , a set of logarithmic density 0.

Similarly, we can obtain for all possibly outside of , a set of logarithmic density 0.

Thus, this completes the proof of Theorem 5.

Acknowledgments

This work was supported by the NSFC (Grant no. 61202313) and the Natural Science Foundation of Jiangxi Province in China (2010GQS0119, 20122BAB211036, 20122BAB201016, and 20122BAB201044).

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Copyright © 2013 Hong-Yan Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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