The Fixed Points of Solutions of Some <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.714991pt" id="M1" height="12.9124pt" version="1.1" viewBox="-0.0657574 -8.19741 8.94528 12.9124" width="8.94528pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"/></g></svg>-Difference Equations

Zheng, Xiu-Min; Xu, Hong-Yan; Xu, Jun-Feng

doi:https://doi.org/10.1155/2014/262570

Abstract and Applied Analysis

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Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time Scales

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Research Article | Open Access

Volume 2014 | Article ID 262570 | https://doi.org/10.1155/2014/262570

The Fixed Points of Solutions of Some -Difference Equations

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

Academic Editor: Josef Diblík

Received22 Mar 2014

Accepted12 Aug 2014

Published14 Oct 2014

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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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.

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