Abstract and Applied Analysis

Volume 2014 (2014), Article ID 782024, 8 pages

http://dx.doi.org/10.1155/2014/782024

Research Article

Zeros, Poles, and Fixed Points of Meromorphic Solutions of Difference Painlevé Equations

Shuang-Ting Lan and Zong-Xuan Chen

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received 23 January 2014; Accepted 5 March 2014; Published 6 April 2014

Academic Editor: Zhi-Bo Huang

Copyright © 2014 Shuang-Ting Lan and Zong-Xuan Chen. 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 this paper, we mainly study the properties of transcendental meromorphic solutions of difference Painlevé equations and and obtain precise estimations of the exponents of convergence of zeros, poles of and , and of fixed points of for any .

1. Introduction and Main Results

At the beginning of last century, Painlevé, Gambier, and Fuchs classified a large number of second-order differential equations in terms of a characteristic which is now known as the Painlevé property [1–4]. Ablowitz et al. [5] considered discrete equations as delay equations in the complex plane which enabled them to utilize complex analytic methods. They looked at, for instance, difference equations of the type

where is rational in both of its arguments. It is shown that if (1) has at least one nonrational finite-order meromorphic solution, then .

In this paper, we use the basic notions of Nevanlinna's theory (see [6, 7]). In addition, we use the notations

  to denote the order of growth of the meromorphic function ;  and , respectively, to denote the exponents of convergence of zeros and poles of ;  to denote the exponent of convergence of fixed points of .

The quantity is called the deficiency of the value to . Furthermore, we denote by any quantity satisfying for all outside of a set with finite logarithmic measure and by

the field of small functions with respect to . A meromorphic solution of a difference (or differential) equation is called admissible if all coefficients of the equation are in .

Recently, Halburd and Korhonen [8] considered (1) and got the following theorem.

Theorem A. Let be rational in both of its arguments such that its denominator has at least two distinct roots. If the second-order difference equation (1) admits a nonrational meromorphic solution of finite order such that there is a finite real constant , such that for sufficiently large ,

holds, then (1) is a difference Painlevé II equation

where , , and are constants.

Remark 1. If has a pole at , we say the singularity at is of type if and of type if . We denote by the number of type poles (ignoring multiplicities) in the disc . Similarly, the function counts poles of type .

In 2010, Chen and Shon [9] researched the properties of finite-order meromorphic solutions of difference Painlevé I and II equations. They mainly discussed the existence and the forms of rational solutions and value distribution of transcendental meromorphic solutions.

For difference Painlevé III equations, we recall the following.

Theorem B (see [10]). Assume that equation

has an admissible meromorphic solution of hyperorder less than one, where is rational and irreducible in and meromorphic in , then either satisfies a difference Riccati equation

where , , are algebroid functions or (5) can be transformed to one of the following equations:

In (7a), the coefficients satisfy , , , and one of the following:

(1), , ;(2), .

In (7b), and . In (7c), the coefficients satisfy one of the following:

(1) and either or ;(2), , and ;(3), ;(4), .

In (7d), and , .

Zhang and Yang [11] investigated the difference Painlevé III equations (7a)–(7d) with constant coefficients and obtained the following results.

Theorem C. If is a nonconstant meromorphic solution of difference equation (7d), where and is a nonzero constant. Then

(i) cannot be a rational function;(ii).

Lan and Chen [12] studied some difference Painlevé III equations and proved the following.

Theorem D. Suppose that is a nonconstant rational function. Suppose is a transcendental meromorphic solution with finite order of (7d), where . Set . Then

(i) has no Nevanlinna exceptional value;(ii), .

In general, , where is a nonzero constant. For example, has no fixed points, but has infinitely many fixed points and . Combining Theorems C and D, we continue to study properties (including fixed points) of transcendental meromorphic solutions of difference Painlevé III equations (7b) and (7c), and obtain the following.

Theorem 2. Suppose that and are nonconstant polynomials. Suppose is a transcendental meromorphic solution with finite order of difference Painlevé III equation

Then

(i) has at most one Nevanlinna exceptional value;(ii)for any , has infinitely many fixed points, and ;(iii) and ;(iv)if there exists some nonconstant rational function such that , then .

Theorem 3. Suppose that is a nonconstant polynomial. Suppose is a transcendental meromorphic solution with finite order of difference Painlevé III equation

Then

(i) has no Nevanlinna exceptional value;(ii)for any , has infinitely many fixed points, and ;(iii) and ;(iv)if there exists some rational function such that , then .

2. Lemmas for the Proofs of Theorems

Lemma 4 (see [13]). Let be a meromorphic function of finite order and let be a nonzero complex number. Then for each , one has

Lemma 5 (see [13]). Let be a meromorphic function with order , , and let be a fixed nonzero complex number, then for each , one has

Lemmas 4 and 5 show the following.

Lemma 6. Let be a nonzero constant and be a meromorphic function with finite order . Then for each , one has

Lemma 7 (see [14, 15]). Let be a transcendental meromorphic solution of finite order of difference equation

where is a difference polynomial in . If for a meromorphic function , then

Lemma 8 (see [15]). Let be a transcendental meromorphic solution of finite order of difference equation of the form

where , , and are difference polynomials such that the total degree in and its shifts, and . If contains just one term of maximal total degree in and its shifts, then for each ,

possibly outside of an exceptional set of finite logarithmic measure.

3. Proofs of Theorems

Proof of Theorem 2. (i) By (8), we have

Applying Lemma 8 to (17), we have

which yields , that is, .

By (17), we have

By (18), (19), and Lemma 4, we obtain

Thus,

which means that .

Set

Assume that has two Nevanlinna exceptional values . By and , we see that . By Lemma 7, we have and . That is,

Hence,

Since , then is a constant. This contradicts the fact that is a nonconstant polynomial. So, has at most one Nevanlinna exceptional value.

(ii) For any , substituting for in (8), we obtain

Set . Thus, (25) can be written as

Set

Since and are polynomials,

By and Lemma 7, we have , which follows . By and Lemma 5, we have . Thus, .

(iii) By (8), we have

Applying Valiron-Mohon'ko theorem and Lemma 5 to (29), we obtain

Thus,

By (31) and Lemma 4, we have

Therefore, , that is, .

Substituting , into (8), we see

that is,

Let be a zero of , by (33), is a zero of or . Since , then must be a zero of or . Thus, by (21) and Lemma 6, we obtain

Hence, , that is, .

Applying Valiron-Mohon'ko Theorem and Lemma 5 to (34), we deduce

Thus,

By (18), (37), and Lemma 4, we have

Hence, , that is, .

(iv) Suppose that , where is some nonconstant rational function. Now we prove that . Set

By (34), (39), and , we have

Since is a nonconstant rational function, then . By (22) and , we know

Similarly, we obtain . By and Lemma 7, we have

By (18), (20), (40) and (42), we obtain

By (39), we obtain

It sees from Lemma 4 that

Hence,

From (18), (43), (44), (46) and Lemma 4, we deduce that

which yields

By (18) and (48) we have

and by (31),

Then . So, .

Proof of Theorem 3. (i) By (9), we have

Applying Lemma 8 to (51), we have

Thus, , which yields .

Again by (9), we have

From (52), (53), and Lemma 4, we deduce that

which follows

Thus, .

Set

For any , since is a nonconstant polynomial, then . By and Lemma 7, we know that , which means that . Hence, . Combining , , we see that has no Nevanlinna exceptional value.

(ii) For any , substituting for in (9), we see that

Set . Thus, (57) can be written as

Set

Since is a polynomial, then

By this and Lemma 7, we have , which follows . By and Lemma 5, we see that . Thus, .

(iii) Substituting into (9), we have

If is a zero of , by (61), must be a zero of or . Thus, by (55) and Lemma 6, we have

Hence, , that is, .

By (61), we have

Applying Valiron-Mohon'ko theorem and Lemma 5 to (63), we deduce

Hence,

Combining (65) with (52) and Lemma 4, we have

which yields . So, .

By (9), we have

Applying Valiron-Mohon'ko theorem and Lemma 5 to (67), we obtain

Thus,

By (69) and Lemma 4, we see that

Hence, , that is, .

(iv) Suppose that , where is some rational function. Now we prove that . Set

By (63), (71), and , we have

Since is a nonconstant polynomial, is a nonconstant rational function. Then . By (56) and , we know

Similarly, we obtain . By , , and Lemma 7, we have

'By (52), (54), (74), and (72), we obtain

Using the same method as in the proof of (iv) in Theorem 2, we may obtain . By this and (52), we have

and by (69),

Then . So, .

Conflict of Interests

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

Acknowledgment

The project was supported by the National Natural Science Foundation of China (11171119).

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