Abstract and Applied Analysis

Volume 2013 (2013), Article ID 680956, 6 pages

http://dx.doi.org/10.1155/2013/680956

Research Article

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

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

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China

Received 2 September 2012; Accepted 29 March 2013

Academic Editor: Fasma Diele

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.

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).

References

W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964.

L. Yang, Value Distribution Theory, Springer, Berlin, Germany, 1993.

C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, vol. 557 of Mathematics and its Applications, Kluwer Academic, Dordrecht, The Netherlands, 2003.

M. J. Ablowitz, R. Halburd, and B. Herbst, “On the extension of the Painlevé property to difference equations,” Nonlinearity, vol. 13, no. 3, pp. 889–905, 2000.

D. C. Barnett, R. G. Halburd, W. Morgan, and R. J. Korhonen, “Nevanlinna theory for the -difference operator and meromorphic solutions of -difference equations,” Proceedings of the Royal Society of Edinburgh A, vol. 137, no. 3, pp. 457–474, 2007.

G. G. Gundersen, J. Heittokangas, I. Laine, J. Rieppo, and D. Yang, “Meromorphic solutions of generalized Schröder equations,” Aequationes Mathematicae, vol. 63, no. 1-2, pp. 110–135, 2002.

J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and K. Tohge, “Complex difference equations of Malmquist type,” Computational Methods and Function Theory, vol. 1, no. 1, pp. 27–39, 2001.

J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and J. Zhang, “Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity,” Journal of Mathematical Analysis and Applications, vol. 355, no. 1, pp. 352–363, 2009.

J. Zhang and R. Korhonen, “On the Nevanlinna characteristic of and its applications,” Journal of Mathematical Analysis and Applications, vol. 369, no. 2, pp. 537–544, 2010.

Y.-M. Chiang and S.-J. Feng, “On the Nevanlinna characteristic of and difference equations in the complex plane,” Ramanujan Journal, vol. 16, no. 1, pp. 105–129, 2008.

R. G. Halburd and R. J. Korhonen, “Difference analogue of the lemma on the logarithmic derivative with applications to difference equations,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 477–487, 2006.

W. Bergweiler, K. Ishizaki, and N. Yanagihara, “Growth of meromorphic solutions of some functional equations. I,” Aequationes Mathematicae, vol. 63, no. 1-2, pp. 140–151, 2002.

Z.-X. Chen, “Growth and zeros of meromorphic solution of some linear difference equations,” Journal of Mathematical Analysis and Applications, vol. 373, no. 1, pp. 235–241, 2011.

R. G. Halburd and R. J. Korhonen, “Finite-order meromorphic solutions and the discrete Painlevé equations,” Proceedings of the London Mathematical Society, vol. 94, no. 2, pp. 443–474, 2007.

R. G. Halburd and R. J. Korhonen, “Nevanlinna theory for the difference operator,” Annales Academiæ Scientiarum Fennicæ, vol. 31, no. 2, pp. 463–478, 2006.

I. Laine, J. Rieppo, and H. Silvennoinen, “Remarks on complex difference equations,” Computational Methods and Function Theory, vol. 5, no. 1, pp. 77–88, 2005.

H. Silvennoinen, Meromorphic Solutions of Some Composite Functional Equations, vol. 132 of Annales Academiae Scientiarum Fennicae, Mathematica, Dissertationes, Helsinki, 2003.

J. Wang, “Growth and poles of meromorphic solutions of some difference equations,” Journal of Mathematical Analysis and Applications, vol. 379, no. 1, pp. 367–377, 2011.

X.-M. Zheng and Z.-X. Chen, “Some properties of meromorphic solutions of -difference equations,” Journal of Mathematical Analysis and Applications, vol. 361, no. 2, pp. 472–480, 2010.

R. Korhonen, “A new Clunie type theorem for difference polynomials,” Journal of Difference Equations and Applications, vol. 17, no. 3, pp. 387–400, 2011.

X. M. Zheng and Z. X. Chen, “Value distribution of meromorphic solutions of some -difference equations,” Journal of Mathematical Research and Exposition, vol. 31, no. 4, pp. 698–704, 2011.

L. Gao, “On meromorphic solutions of a type of system of composite functional equations,” Acta Mathematica Scientia B, vol. 32, no. 2, pp. 800–806, 2012.

L. Y. Gao, “Systems of complex difference equations of Malmquist type,” Acta Mathematica Sinica, vol. 55, no. 2, pp. 293–300, 2012.

L. Gao, “Estimates of -function and -function of meromorphic solutions of systems of complex difference equations,” Acta Mathematica Scientia B, vol. 32, no. 4, pp. 1495–1502, 2012.

H. Wang, H. Y. Xu, and B. X. Liu, “The poles and growth of solutions of systems of complex difference equations,” Advances in Difference Equations, vol. 2013, article 75, 2013.

H.-Y. Xu, T.-B. Cao, and B.-X. Liu, “The growth of solutions of systems of complex q-shift difference equations,” Advances in Difference Equations, vol. 2012, article 216, 2012.

K. Liu and X.-G. Qi, “Meromorphic solutions of -shift difference equations,” Annales Polonici Mathematici, vol. 101, no. 3, pp. 215–225, 2011.

I. Laine, Nevanlinna Theory and Complex Differential Equations, vol. 15 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1993.

H. Y. Xu and J. Tu, “Some properties of meromorphic solutions of -shift difference equations,” Journal of Mathematical Study, vol. 45, no. 2, pp. 124–132, 2012.