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Abstract and Applied Analysis
Volume 2014, Article ID 262570, 7 pages
http://dx.doi.org/10.1155/2014/262570
Research Article

The Fixed Points of Solutions of Some -Difference Equations

1Institute of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
2Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China
3Department of Mathematics, Wuyi University, Jiangmen, Guangdong 529020, China

Received 22 March 2014; Accepted 12 August 2014; Published 14 October 2014

Academic Editor: Josef Diblík

Copyright © 2014 Xiu-Min Zheng 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

The purpose of this paper is to investigate the fixed points of solutions of some -difference equations and obtain some results about the exponents of convergence of fixed points of and , -differences , and -divided differences .

1. Introduction and Main Results

Throughout this paper, we will assume that the readers are familiar with basic notations such as , , and of Nevanlinna theory (see Hayman [1], Yang [2], and Yang and Yi [3]). We use , , and to denote the order, the exponent of convergence of zeros, and the exponent of convergence of poles of , respectively, and we also use the notation to denote the exponent of convergence of fixed points of , which is defined as and to denote any quantity satisfying for all on a set of logarithmic density 1, where the logarithmic density of a set is defined by Throughout this paper, the set of logarithmic density 1 will be not necessarily the same at each occurrence.

Recently, a number of papers (including [49]) focused on complex difference equations, system of complex difference equations, and difference analogues of Nevanlinna theory. Correspondingly, there are many papers focusing on the -difference (or -shift difference) equations, such as  [1016].

In 2013, Zhang [17] investigated the growth of meromorphic solutions of some complex -difference equations and the exponents of convergence of fixed points and zeros of transcendental meromorphic solutions of the second order -difference equation and obtained the following theorem.

Theorem 1 (see [17]). Suppose that is a transcendental meromorphic solution of the equation where , coefficients , , , , , , and are constants, and at least one of is nonzero. Then, and (i) has infinitely many fixed points, and (ii) has infinitely many zeros, whenever .

Our first result of this paper is about the exponents of convergence of fixed points and zeros of transcendental meromorphic solutions of the higher order -difference equation as follows.

Theorem 2. Suppose that is a transcendental meromorphic solution of the equation where , , coefficients (), , , (), are constants, and at least one of is nonzero. Then, and (i) has infinitely many fixed points, and (ii) has infinitely many zeros, whenever .

From Theorem 2, it is a natural question to ask, What will happen if the right-hand side of (4) is a rational function in both arguments?

Regarding the above question, we will investigate the exponents of convergence of fixed points of meromorphic solutions of the -difference equation where , , and are nonzero polynomials, , and . Similar to [18, Page 99], we can call (5) a -Pielou logistic equation, which is a special form of nonautonomous Schröder equations.

Theorem 3. Let , , and be nonzero polynomials such that Set , where and . Then every transcendental meromorphic solution of (5) satisfies the following statements:(i) has infinitely many fixed points and , ;(ii)if , then has infinitely many fixed points and .

We also study fixed points of transcendental meromorphic solutions of the following -difference equations: where , (), and are polynomials and , and obtain the following results.

Theorem 4. Let , , let () be polynomials, and let . If satisfy one of the following conditions:(i)there exists an integer () such that (ii)then every transcendental meromorphic solution of (7) satisfies that has infinitely many fixed points and for .

By using the same argument as that in Theorem 4, we can easily obtain the following theorem.

Theorem 5. Let , , (), and be polynomials and let . If satisfy one of the following conditions:(i) and contain just one term of maximal total degree;(ii)then every transcendental meromorphic solution of (8) satisfies that has infinitely many fixed points and for .

2. Some Lemmas

The following result is a difference counterpart to the standard result due to A. A. Mohon’ko and V. D. Mohon’ko [19].

Lemma 6 (see [20], Theorem 2.2). Let be a nonconstant zero-order meromorphic solution of , where is a -difference polynomial in . If for a slowly moving target , then on a set of logarithmic density 1.

Lemma 7 (see [21, 22]). Let , , and be rational functions, and let , , and (). Then(i)all meromorphic solutions of the equation satisfy ;(ii)all transcendental meromorphic solutions of (13) satisfy .

Lemma 8 (see [17], Theorem 2). Suppose that is a nonconstant meromorphic solution of the equation where () is a complex number, (), , (), , , and () are small functions of , and is irreducible in . Then, and .

Lemma 9 (see [21, page 249] or [23, Theorem 1.1]). Let be a transcendental meromorphic function of zero-order and let be a nonzero complex constant. Then on a set of logarithmic density 1.

3. Proof of Theorem 2

Suppose that is a transcendental meromorphic solution of (4). From the assumptions of Theorem 2, it follows from Lemma 8 that . Thus, . Clearly, we have .

(i) Firstly, we prove that has infinitely many fixed points. Set . Then is transcendental, , and . So, is of zero-order. Then substituting into (4), we get that Set and It follows from (17) that

Suppose that . If , then it follows from (18) that . Thus, the right-hand side of (4) is 0, which is in contradiction with the assumption of Theorem 2. If , it follows from (18) that and Thus, we have from (4) and (19) that which is in contradiction with the assumption of Theorem 2. Hence, we have . By Lemma 6, we get that on a set of logarithmic density 1. Thus, it follows from (21) that on a set of logarithmic density 1. Since is a transcendental meromorphic solution of (4), then it follows from (22) that has infinitely many fixed points.

(ii) From (4), we have Since and from (23), we derive that Thus, it follows from Lemma 6 that on a set of logarithmic density 1; that is, on a set of logarithmic density 1. Since is a transcendental solution of (4), then it follows from (26) that has infinitely many zeros.

Thus, this completes the proof of Theorem 2.

4. Proof of Theorem 3

Suppose that is a transcendental meromorphic solution of (5). Since , , and , , and are polynomials, it follows from Lemma 8 and [11] that is of zero-order.

(i) We first prove that has infinitely many fixed points and . Set . Then is transcendental, , and . Then it follows that is of zero-order. Set Then substituting into (27), we have It follows from (28) that Thus, we derive by (6) and (29) that . Thus, by Lemma 6 and , we have on a set of logarithmic density 1; that is, on a set of logarithmic density 1.

Since is a transcendental meromorphic solution of (5), then it follows from (31) that has infinitely many fixed points.

Next, we prove that has infinitely many fixed points and . From (5), we have By (6), we have . Since is transcendental and , , and are polynomials, we have by (32) the fact that and have the same poles, except possibly finitely many poles. Moreover, we can get that and have at most finitely many common zeros. In fact, suppose that is a common zero of and . Then ; that is, . Substituting it into , we have Thus, this shows that must be the zeros of . Since , , and are polynomials, then has only finitely many zeros. So, and have at most finitely many common zeros. Then it follows from (32) that

From (27), we have Since and , then we have . Thus, it follows from (35) that . Since is transcendental function of zero-order and is a rational function, then we have by Lemma 6 the fact that on a set of logarithmic density 1; that is, on a set of logarithmic density 1. Since is transcendental, we can derive from (34) and (37) that has infinitely many fixed points and .

Now, we prove that has infinitely many fixed points and . From (5), we have where . By Lemma 9, we have . Obviously, , , and . Thus, by using the same argument as in the proof of , we can prove that has infinitely many fixed points and .

Thus, by using the same method as above, we can obtain that has infinitely many fixed points and for .

(ii) Now, we prove that has infinitely many fixed points and By (5) and from , we have Since , is transcendental, and , , and are polynomials, we have by (40) the fact that and have the same poles, except possibly finitely many poles. Moreover, by using the same argument as in (i), we can get that and have at most finitely many common zeros. Then it follows from (40) that From (27), we have where Since , , and are polynomials satisfying (6), then it follows from (43) that is a polynomial of degree . Thus, from (42) we have . Since is transcendental function of zero-order and is a rational function, then we have by Lemma 6 the fact that on a set of logarithmic density 1; that is, on a set of logarithmic density 1. Since is transcendental, we can derive from (41) and (45) that has infinitely many fixed points and .

Thus, this completes the proof of Theorem 3.

5. Proof of Theorem 4

Suppose that is a transcendental meromorphic solution of (7). Since , , and , , are polynomials, by Lemma 7, we see that is of zero-order. Set Thus, it follows from (46) that

(i) Suppose that satisfy condition (9). Then it follows that . Since is a transcendental solution of zero-order, then it follows from Lemma 6 that on a set of logarithmic density 1. So, on a set of logarithmic density 1. Thus, it follows that has infinitely many fixed points and .

Now, we prove that has infinitely many fixed points and . By (7), we derive where . Since is a transcendental meromorphic function of zero-order, then we have by Lemma 9 the fact that is a transcendental and . By , , and (9), we have Thus, by the above proof of , we see that has infinitely many fixed points and . Continuing to use the same method as the above, we can prove that has infinitely many fixed points and for .

(ii) Suppose that satisfy the condition (10).

By using the same argument as the one above, we can prove that has infinitely many fixed points and easily.

Now, we prove that has infinitely many fixed points and . Set Thus, it follows from (10) that In fact, if , replacing by into (53), we have which is in contradiction with the condition (10). Since and is transcendental meromorphic of zero-order, then it follows from (53) and Lemma 6 that has infinitely many fixed points and . Continuing to use the same method as the one above, we can prove that has infinitely many fixed points and for .

Thus, this completes the proof of Theorem 4.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for reading the paper very carefully and making a number of valuable and kind comments which improve the presentation. This work was supported by the NSF of China (11301233 and 61202313), the Natural Science Foundation of Jiangxi Province in China (20132BAB211001, GJJ14644, and GJJ14271), Sponsored Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University of China, Foundation for Distinguished Young Talents in Higher Education of Guangdong China (2013LYM0093), and Training Plan for the Outstanding Young Teachers in Higher Education of Guangdong (Yq 2013159).

References

  1. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964. View at MathSciNet
  2. L. Yang, Value Distribution Theory, Springer, Berlin, Germany, 1993. View at MathSciNet
  3. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic, Dordrecht, The Netherlands; Chinese Original: Science Press, Beijing, China, 2003.
  4. W. Bergweiler and J. K. Langley, “Zeros of differences of meromorphic functions,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 142, no. 1, pp. 133–147, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. Z. Chen, Z. Huang, and X. Zheng, “On properties of difference polynomials,” Acta Mathematica Scientia B, vol. 31, no. 2, pp. 627–633, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. Z. Chen and K. H. Shon, “On zeros and fixed points of differences of meromorphic functions,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 373–383, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. Z. X. Chen and K. H. Shon, “Properties of differences of meromorphic functions,” Czechoslovak Mathematical Journal, vol. 61, no. 136, pp. 213–224, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. Y. Chiang and S. Feng, “On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane,” Ramanujan Journal, vol. 16, no. 1, pp. 105–129, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. I. Laine and C. Yang, “Value distribution of difference polynomials,” Proceedings of Japan Academy A, vol. 83, no. 8, pp. 148–151, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  10. A. Fletcher, J. K. Langley, and J. Meyer, “Slowly growing meromorphic functions and the zeros of differences,” Mathematical Proceedings of the Royal Irish Academy, vol. 109, no. 2, pp. 147–154, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. G. G. Gundersen, J. Heittokangas, I. Laine, and D. Yang, “Meromorphic solutions of generalized Schröder equations,” Aequationes Mathematicae, vol. 63, no. 1-2, pp. 110–135, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. 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, 11 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  13. H. Y. Xu, “On the value distribution and uniqueness of difference polynomials of meromorphic functions,” Advances in Difference Equations, vol. 2013, article 90, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  14. H. Xu, B. Liu, and K. Tang, “Some properties of meromorphic solutions of systems of complex q-shift difference equations,” Abstract and Applied Analysis, vol. 2013, Article ID 680956, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  15. H. Y. Xu, J. Tu, and X. M. Zheng, “Some properties of solutions of complex q-shift difference equations,” Annales Polonici Mathematici, vol. 108, no. 3, pp. 289–304, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. J. Xu and X. Zhang, “The zeros of q-shift difference polynomials of meromorphic functions,” Advances in Difference Equations, vol. 2012, article 200, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. G. Zhang, “Growth of meromorphic solutions of some q-difference equations,” Abstract and Applied Analysis, vol. 2013, Article ID 943209, 6 pages, 2013. View at Google Scholar · View at MathSciNet
  18. S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2005. View at MathSciNet
  19. A. A. Mohon'ko and V. D. Mohon'ko, “Estimates of the Nevanlinna characteristics of certain classes of meromorphic functions, and their applications to differential equations,” Sibirskii Matematicheskii Zhurnal, vol. 15, pp. 1305–1322, 1974 (Russian). View at Google Scholar · View at MathSciNet
  20. D. C. Barnett, R. G. Halburd, W. Morgan, and R. J. Korhonen, “Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations,” Proceedings of the Royal Society of Edinburgh A; Mathematics, vol. 137, no. 3, pp. 457–474, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. W. Bergweiler, K. Ishizaki, and N. Yanagihara, “Meromorphic solutions of some functional equations,” Methods and Applications of Analysis, vol. 5, no. 3, pp. 248–258, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. W. Bergweiler, K. Ishizaki, and N. Yanagihara, “Meromorphic solutions of some functional equations (correction),” Methods and Applications of Analysis, vol. 6, pp. 617–618, 1999. View at Google Scholar
  23. J. Zhang and R. Korhonen, “On the Nevanlinna characteristic of f(qz) and its applications,” Journal of Mathematical Analysis and Applications, vol. 369, no. 2, pp. 537–544, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus