Abstract and Applied Analysis
Volume 2014 (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

Xiu-Min Zheng,¹ Hong-Yan Xu,² and Jun-Feng Xu³

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 [4–9]) 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  [10–16].

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).

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