Journal of Function Spaces
Volume 2018 (2018), Article ID 8318570, 10 pages
https://doi.org/10.1155/2018/8318570

Research Article

Some Properties of Solutions for Some -Difference Equations Containing Painlevé Equation

Peng Jun Zhao¹ and Hong Yan Xu²

¹School of Mathematics and Computer Applications, Shangluo University, Shangluo, Shaanxi 726000, China
²Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China

Correspondence should be addressed to Hong Yan Xu; xhyhhh@126.com

Received 4 January 2018; Revised 7 March 2018; Accepted 28 March 2018; Published 7 May 2018

Academic Editor: Richard I. Avery

Copyright © 2018 Peng Jun Zhao and Hong Yan Xu. 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 existence and growth of meromorphic solutions for some -difference equations are studied, and some estimates for the exponent of convergence of poles of , , , and are also obtained. Our theorems are improvements and extensions of the previous results.

1. Introduction and Main Results

In 1900, Painlevé [1] first studied the differential equations, which were called differential Painlevé equations later. Moreover, at the beginning of last century, differential Painlevé equations had been an important research subject in the field of the mathematics and physics. They occur in many physical situations, such as plasma physics, statistical mechanics, and nonlinear waves.

In the 1990s, the discrete Painlevé equations had become important and interest research problems (see [2, 3]). For example, let , , and be constants and , it is usual that

are called the special discretization of discrete , and

is called the special discretization of the discrete .

Of late, with the development of Nevanlinna theory, Chiang and Feng [4] and Halburd and Korhonen [5] established independently those results about the difference analog of the lemma on the logarithmic derivative, and there has been an increasing interest in studying complex difference equations. And there were a number of papers (see [4, 6, 7]) concerning complex difference equations and difference analogs of Nevanlinna theory, by applying the results of Chiang and Feng [4] and Halburd and Korhonen [5]. For example, Halburd and Korhonen [5, 8, 9] used Nevanlinna theory to analyze the following equation:

where is rational in and meromorphic in , and we single out the difference of Painlevé and equations such as

In recent, Laine and Yang, Zhang and Korhonen, and Zheng and Chen further investigated the value distribution of -difference operator of meromorphic functions, by utilizing the analog of Logarithmic Derivative Lemma on -difference operators given by Barnett et al. [10]. Moreover, during the last decades, considerable attention has been paid to -difference operators, -difference equations, by replacing the -difference , with of a meromorphic function in some complex difference equations and complex difference operators (see [11–26]).

Throughout this paper, a term “meromorphic” will always mean meromorphic in the complex plane . Hereinafter, we will use some basic results and the standard notations of Nevanlinna theory (see [27–29]). For a meromorphic function , we use , , and to denote the order, the exponent of convergence of zeros, and the exponent of convergence of poles of , respectively, and let be the exponent of convergence of fixed points of , which is defined by

Besides, we use to denote any quantity satisfying for all outside a possible exceptional set of finite logarithmic measure

and a meromorphic function is called a small function with respect to if , and we use to denote the field of small functions relative to .

In 2010, Chen and Shon [30] considered the difference Painlevé I equation (4) and obtained the following theorem.

Theorem 1 (see [30, Theorem 4]). Let , , and be constants, where , are not both equal to zero. Then
(i) if , then (4) has no rational solution;
(ii) if , and , then (4) has a nonzero constant solution , where satisfies .
The other rational solution satisfies , where and are relatively prime polynomials and satisfy .

In 2015, the properties of solutions of a certain type of difference equation were further investigated by Li and Huang [31], and some results were obtained as follows.

Theorem 2 (see [31, Theorem 3.1]). Suppose that equation

where , admits a finite-order transcendental meromorphic solution . Then
(i) ;
(ii) has no Borel exceptional value;
(iii) if , then the exponent of convergence of fixed points of satisfies .

In the same year, Qi and Yang [32] discussed the following equation:

which can be seen as -difference analogs of (4), and obtained some properties of the zeros of , where is a solution of (10) and .

Inspired by the idea of Li and Huang [31] and Qi and Yang [32], our main purpose is further to investigate some properties of meromorphic solutions for some -difference equations which are different from (10) to a certain extent, and the following theorems are obtained.

Theorem 3. Let , and . If

admits a zero-order transcendental meromorphic solution , then
(i) has infinitely many poles and zeros, also has infinitely many poles, and

and further, if , then each of , , has infinitely many poles, and

(ii) if , then has infinitely many fixed points and the exponent of convergence of fixed points of satisfies .

Theorem 4. Let , and , and assume that is a zero-order transcendental meromorphic solution of equation

Then
(i) has infinitely many poles and zeros, also has infinitely many poles, and

and further, if , then each of has infinitely many poles, and

(ii) if , then has infinitely many fixed points and the exponent of convergence of fixed points of satisfies .

Theorem 5. Let , and be identically vanishing simultaneously. And is a zero-order transcendental meromorphic solution of equation

Then
(i) has infinitely many poles and zeros, also has infinitely many poles, and

and further, if , then each of has infinitely many poles, and

(ii) if , then has infinitely many fixed points and the exponent of convergence of fixed points of satisfies .

2. Some Lemmas

Let the logarithmic density of a set be defined by

Definition 6 (see [10]). For , if a polynomial in includes finitely many of its -shifts with meromorphic coefficients in the sense that their Nevanlinna characteristic functions are on a set of logarithmic density 1, then it can be called a -difference polynomial of .

Lemma 7 (see [13, Theorem 2.5]). Let be a transcendental meromorphic solution of order zero of a -difference equation of the form

where , , and ) are -difference polynomials such that the total degree   in and its -shifts, whereas . Moreover, we assume that contains just one term of maximal total degree in and its -shifts. Then

on a set of logarithmic density 1.

Lemma 8 (see [10, Theorem 2.2]). Let be a nonconstant zero-order meromorphic solution of , where is a -difference polynomial in . If for slowly moving target , then

on a set of logarithmic density 1.

Remark 9 (see [10]). Let and be meromorphic functions of zero-order such that on a set of logarithmic density 1. Then is called a slowly moving target or a small function with respect to .

Lemma 10 (see [22, Theorems 1.1 and 1.3]). Let be a nonconstant zero-order meromorphic function and . Then

on a set of lower logarithmic density 1.

Lemma 11 (Valiron-Mohon’ko, see [33]). Let be a meromorphic function. Then for all irreducible rational functions in ,

with meromorphic coefficients , , the characteristic function of satisfies

where and .

Lemma 12 (see [10, Theorem 1.1]). Let be a nonconstant zero-order meromorphic function and . Then

on a set of logarithmic density 1.

3. The Proof of Theorem 3

We first assume that is of zero-order and a transcendental meromorphic solution of (11).

(i) In view of (11), it follows that

Hence, we conclude from (28) and Lemma 7 that

on a set of logarithmic density 1. Since is of zero-order, then we can deduce by Lemma 10 that

on a set of lower logarithmic density 1.

Since and , by applying Lemma 11 for (11), we obtain

Thus, it follows from (29) to (31) that

on a set of logarithmic density 1. Hence, it means that has infinitely many poles and

On the other hand, it yields from (11) that

Since , then . Thus, by Lemma 8, it follows that

on a set of logarithmic density 1. Hence,

on a set of logarithmic density 1. Therefore, has infinitely many zeros and .

Next, we will prove that . Let , then (11) can be rewritten as the form

Then it follows from (37) that

Moreover, it yields and , where . Substituting them into (38), we conclude

that is,

From the conditions of Theorem 3, and by Lemma 10, we conclude that is of zero order. Thus, we conclude that , are of zero order and , and , where . Since is of zero order, then

on a set of logarithmic density 1. Thus, we can deduce from (41) that

on a set of logarithmic density 1. Moreover, it follows from (40) and (42) that

on a set of logarithmic density 1; that is,

on a set of logarithmic density 1. Then, it means that has infinitely many poles and

Since is of zero order, then we conclude from Lemma 10 that

Hence, it yields

Thus, by (33) and (47) we conclude that

on a set of logarithmic density 1: that is, . Thus, combining this and (48), we conclude that has infinitely many poles and

Next, we prove that

At first, it can be seen that and are mutually prime polynomials in , where . In fact, taking and , it yields . Thus, in view of (11) and Lemma 11, we deduce

that is,

Hence, by Lemma 12, it yields

Besides, in view of (11), it follows that

and by Lemmas 10 and 11, it yields

Thus, it follows that

Finally, similar to the above argument, we have

Thus, we can conclude from (53), (56), and (57) that each of , , has infinitely many poles and

(ii) Set . Then is a zero-order transcendental meromorphic function with and . Substituting into (11), we have

Since , then by Lemma 8, it yields

on a set of logarithmic density 1. By using the same argument as in the proof of Theorem 3(i), we conclude

on a set of logarithmic density 1, which implies has infinitely many fixed points and .

Therefore, this completes the proof of Theorem 3.

4. The Proof of Theorem 4

We assume that is a zero-order transcendental meromorphic solution of (14).

(i) In view of (14), it follows that

Since , then it follows that

Thus, we can conclude by Lemma 8 that

on a set of logarithmic density 1. Hence,

on a set of logarithmic density 1. Thus, it means that has infinitely many zeros and .

On the other hand, it follows from (14) that

Then in view of (66) and Lemma 7, we have

And by applying Lemma 11 for (14), it yields

Then, we conclude by Lemma 10 that

on a set of lower logarithmic density 1. Thus, combining (67) and (68), it follows that

Hence, has infinitely many poles and

Next, we will prove that . Let ; then (14) can be rewritten as the following form:

Then from (72) it follows that

and and . Substituting them into (73), we have

Thus, it follows that

where

Hence, it follows by Lemma 10 and from (75) that

that is,

Therefore, it follows from (78) that has infinitely many poles and

Since is of zero-order, by using the same argument as in the proof of Theorem 3, we conclude that has infinitely many poles and

Here, we will prove that

Similarly, let , , and ; then . Hence, it can be seen that and are mutually prime polynomials in . Thus, from (14) and by Lemma 11, it follows that

that is,

Hence, in view of Lemmas 8 and 12, it yields

Besides, in view of (14), we have

and then by Lemma 11, it follows that

Thus, by Lemma 10 and the above equalities, it yields

Finally, similar to the above argument, it follows that

Thus, in view of (84), (87), and (88), we can deduce that each of , , has infinitely many poles and

(ii) Set . Thus, is a zero-order transcendental meromorphic function with and . Substituting into (14), it follows that

Since , then in view of Lemma 8, it follows that

on a set of logarithmic density 1. By using the same argument as in the proof of Theorem 3(i), it follows that

on a set of logarithmic density 1, which implies that has infinitely many fixed points and .

Therefore, this completes the proof of Theorem 4.

5. The Proof of Theorem 5

We first assume that is a zero-order transcendental meromorphic solution of (17).

(i) In view of (17), it follows that

Thus, we conclude from (93) and Lemma 7 that

And by Lemma 10 it follows that

on a set of lower logarithmic density 1.

If , then by applying Lemma 11 for (17), it follows that

Hence, from (94) to (96), it yields

If and , by using the same argument as in the proof of Theorem 3(i), it follows that

Therefore, from (97) and (98), it means that has infinitely many poles and

Now, we will prove that as follows.

Set , by using the same argument as in Theorem 3(i), it yields

where

Hence, we conclude by Lemma 10 and from (100) that

that is,

Thus, it follows from (103) that

By using the same argument as in the proof of Theorem 3(i), we can conclude that has infinitely many poles, and

(ii) Let . Thus, is a zero-order transcendental meromorphic function with and . Substituting into (17), it follows that

Since , then it yields by Lemma 8 that

on a set of logarithmic density 1. By using the same argument as in the proof of Theorem 3(i), we have

on a set of logarithmic density 1, which implies that has infinitely many fixed points and .

Therefore, this completes the proof of Theorem 5.

Conflicts of Interest

The authors declare that they have no conflicts of interest in the manuscript.

Authors’ Contributions

Hong Yan Xu and Peng Jun Zhao completed the main part of this article, and Hong Yan Xu and Peng Jun Zhao corrected the main theorems. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11561033), the Natural Science Foundation of Jiangxi Province in China (20151BAB201008), and the Foundation of Education Department of Jiangxi (GJJ160914, GJJ170759, GJJ161603, and GJJ170788) of China.

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