Journal of Mathematics

Journal of Mathematics / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 3781942 | https://doi.org/10.1155/2020/3781942

Bu Sheng Li, Rui Ying, Xiu Min Zheng, Hong Yan Xu, "Results on Solutions for Several -Painlevé Difference Equations concerning Rational Solutions, Zeros, and Poles", Journal of Mathematics, vol. 2020, Article ID 3781942, 10 pages, 2020. https://doi.org/10.1155/2020/3781942

Results on Solutions for Several -Painlevé Difference Equations concerning Rational Solutions, Zeros, and Poles

Academic Editor: Basil K. Papadopoulos
Received30 May 2020
Revised24 Jul 2020
Accepted03 Aug 2020
Published01 Sep 2020

Abstract

In this article, we discuss the problem about the properties on solutions for several types of -difference equations and obtain some results on the exceptional values of transcendental meromorphic solutions with zero order, their -differences , and divided differences . In addition, we also investigated the condition on the existence of rational solution for a class of -difference equations. Our theorems are some extensions and supplement to those results given by Liu and Zhang and Qi and Yang.

1. Introduction and Main Results

All the time, Painlevé equations have attracted much interest due to the reduction of solution equations, which are solvable by inverse scattering transformations, and they often occur in many physical situations: plasma physics, statistical mechanics, and nonlinear waves. The study of Painlevé equations has spanned more than one hundred years (see [13]).

Around 2006, Halburd and Korhonen [4, 5] and Ronkainen [6] used Nevanlinna theory to discuss the following equations:where is rational in and meromorphic in , respectively, and they singled out the following difference equations:where satisfy some conditions. In these equations, equation (2) is called as the difference Painlevé equation, equation (3) is called as the difference Painlevé equation, and the last four equations are called as the difference Painlevé equations.

In the last decade or so, there were a lot of papers focusing on the properties of solutions for difference Painlevé equations (see [711]). For example, Chen and Shon [12] in 2010 considered the difference Painlevé I equation (3) and obtained the following theorem.

Theorem 1 (see [12], Theorem 4). Let be constants, where are not both equal to zero. Then, the following holds:(i)If , then (3) has no rational solution.(ii)If and , then (3) has a nonzero constant solution , where satisfies .

The other rational solution satisfies , where and are relatively prime polynomials and satisfy .

In 2013 and 2018, Zhang and Yi [11] and Du et al. [13] studied the difference Painlevé equations with the constant coefficients and obtained the result as follows.

Theorem 2 (see [11, 13]). If is a transcendental finite-order meromorphic solution ofwhere and are constants, then the following holds:(i).(ii)If , then .(iii)For any , .(iv).

Ramani et al. [14] in 2003 investigated the existence of transcendental solution of equationwhich is called as difference Painlevé equations and obtained the result as follows.

Theorem 3 (see [14]). If the second-order difference equation (9) admits a nonrational meromorphic solution of finite order, then and .

Of late, many mathematicians paid considerable attention to the value distribution of solutions for complex -difference equations, which are formed by replacing the -difference , with of meromorphic function in some expressions concerning complex difference equations, by utilizing the logarithmic derivative lemma on -difference operators given by Barnett et al. [15] in 2007 (see [1626]). For example, Qi and Yang [27] considered the following equation:which can be seen as -difference analogues of (2) and obtained the result as follows.

Theorem 4 (see [27], Theorem 1). Let be a transcendental meromorphic solution with zero order of equation (10) and be three constants such that cannot vanish simultaneously. Then, the following holds:(i) has infinitely many poles.(ii)If and any , then has infinitely many zeros.(iii)If and takes a finite value finitely often, then is a solution of .

In 2018, Liu and Zhang [28] further investigated the following equation:and obtained the result as follows.

Theorem 5 (see [28], Theorem 1). Let be a transcendental meromorphic solution with zero order of (11) and be three constants such that cannot vanish simultaneously. Then, the following holds:(i) has infinitely many poles.(ii)For any finite value , if , then has infinitely many zeros.(iii)If and has finite zeros, then is a solution of .

Motivated by the idea [27, 28], a natural question is what is the result if we give -difference analogues of (9). For this question, our main aim of this article is further to investigate some properties of meromorphic solutions for some -Painlevé difference equations. It seems that this topic has never been treated before.

In what follows, it should be assumed that the readers are familiar with the fundamental results and the standard notations in the theory of Nevanlinna value distribution (see Hayman [29], Yang [30], and Yi and Yang [31]). Let be a meromorphic function, and we denote , and to be the order, the exponent of convergence of zeros, and the exponent of convergence of poles of , respectively, and denote to be the exponent of convergence of fixed points of , which is defined by

In addition, we use denotes any quantity satisfying for all on a set of logarithmic density 1, and the logarithmic density of a set is defined by

Now, our main results are listed as follows.

Theorem 6. Let be an irreducible rational function, and let and , and where are polynomials with and .

(i)Suppose that and are even numbers or zero. If equation (14) has an irreducible rational solution , where are polynomials with and , then the following holds:(ii)Suppose that and are even numbers or zero. If equation (14) has an irreducible rational solution , where are polynomials with and , then(iii)If is an odd number, then equation (14) has no rational solution.

Theorem 7. For and , let be a transcendental meromorphic solution with zero order of equationwhere is a constant. Let . Then, the following holds:(i)Both and have no nonzero finite Nevanlinna exceptional value.(ii)If , then and have infinitely many fixed points and for any nonzero constant .

Theorem 8. For and , and let be a transcendental meromorphic solution with zero order of equationwhere is a nonconstant rational function satisfying that is not a constant. Then, the following holds:(i)Both and have no Nevanlinna exceptional value.(ii) has infinitely many poles and zeros, and .(iii) has infinitely many fixed points and .

2. Proof of Theorem 6

Proof:. assume that (14) has a rational solution and has poles . Then, can be represented in the following form:where and are constants; are poles of with multiplicity , respectively.(i)Suppose that and are even numbers. Then, in view of (14) and (19), it yieldsIf , then for , it followsHowever, as ; thus, from (20), we can get a contradiction easily.If , then let , and it leads towhere is a nonzero constant. Thus, let ; in view of (20), we also get a contradiction. So, it follows . Thus, assume that , where . As , it yieldswhere is a constant, and it follows now in view of (20) thatas . Since , then . Hence, it follows from (24) thatNext, assume that . As , it followswhere is a constant. If , then by using the same argument as above, we get a contradiction. If , then we assume that . By using the same argument as above, we concludeas . Thus, if , then in view of (27), we can get a contradiction; if , then we have(ii)Suppose that and are even numbers. Then, in view of (14) and (19), we get (20).If , then for , it leads toHowever, as ; thus, from (20), we can get a contradiction easily.If , then let , it followswhere is a nonzero constant. Thus, let ; in view of (20), we also get a contradiction. Thus, . We rewrite (14) as the following form:Denotewhere , and are all nonnegative integers. Thus, in view of (31) and (32), we can deduceSince , then . Thus, by combining with this and (33), we haveand .(iii)If , then is an odd number. Assume that is a rational solution of (14). In view of the conclusion of Theorem 6 (i), it follows . This means a contradiction with the assumption that is an odd number. Thus, (14) has no rational solution.If , then is an odd number. Similar to the above argument, we also conclude that (14) has no rational solution.
Therefore, this completes the proof of Theorem 6.

3. Proof of Theorem 7

We first introduce some notations and some basic results about Nevanlinna theory, which can be used in Section 3 and Section 4. Let be a meromorphic function in , the Nevanlinna characteristic , which encodes information about the distribution of values of on the disk , is defined by

The proximity function is defined bywhere andwhere is the number of poles of in the circle , counted according to multiplicities.

Let , and the deficiency of with respect to is defined by

If , then the complex number is called the Nevanlinna exceptional value. And the order , the exponent of convergence of zeros , and the exponent of convergence of poles of are defined by

Besides, we also use some properties of such aswhere are meromorphic functions and and require some lemmas as follows.

Lemma 1 (see [15], Theorem 2). Let be a nonconstant zero-order meromorphic solution of , where is a -difference polynomial in . If for slowly moving target , thenwhere denotes any quantity satisfying for all on a set of logarithmic density 1.

Remark 1. For , a polynomial in and finitely many of its -shifts with meromorphic coefficients in the sense that their Nevanlinna characteristic functions are on a set of logarithmic density 1 and can be called as a -difference polynomial of .

Lemma 2 (see [24], Theorems 1 and 3). Let be a nonconstant zero-order meromorphic function and . Then,on a set of lower logarithmic density 1.

The proof of Theorem 7:. (i) suppose that is a transcendental meromorphic solution of equation (17), then in view of (17), letFor any given constant and with a view of , it followsIn view of and by Lemma 1, we conclude that . This leads towhich implies . Thus, has no nonzero finite Nevanlinna exceptional value.
Since is of zero order and , then by Lemma 2, it follows , which means that if of zero order. In view of (17), it followsWith (17) subtraction, it leads toFrom (17), we see that . Otherwise, it leads to , a contradiction. Thus, the above equality meansthat is,DenoteFor any given constant , then from (50), we have . Hence, with a view of , it follows . Thus, by Lemma 1, we have , and this leads towhich implies that . Thus, has no nonzero finite Nevanlinna exceptional value:(ii)Replacing by in (17), we haveLet , it yieldsSetThus, it follows , and with a view of , we have . By applying Lemma 1, it yields . Thus, in view of Lemma 2, this leads towhich implies that has infinitely many fixed points and .
In view of (48), set andthen . Since , then . By applying Lemma 1, we have . Thus, in view of Lemma 2, this leads towhich implies that has infinitely many fixed points and .
Therefore, the proof of Theorem 7 is completed.

4. Proof of Theorem 8

Lemma 3 (see [15], Theorem 1). Let be a nonconstant zero-order meromorphic function and . Then,

Lemma 4 (see [17], Theorem 2.5). Let be a transcendental meromorphic solution of order zero of a q-difference equation of the formwhere , 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,

Proof of Theorem 8:. (i) suppose that is a transcendental meromorphic solution of equation (18). We firstly prove that has no Nevanlinna exceptional value. Equation (18) can be rewritten asSet . In view of (61) and Lemma 2, it followsAnd with a view of , we thus conclude that is transcendental and of order zero, and is small with respect to .
Replacing by in (18), it followsBy combining with (18), we haveSince , then it yieldsSubstituting (65) into (64), we obtainthat is,By applying Lemma 4 for (67), it follows . This leads toThus, in view of Lemma 2, it yieldsThis showswhich implies .
SetFor any constant , we haveSince is not a constant, then . By Lemma 1, it followsThis meanswhich implies that for any constant .
In view of (18) and Lemma 3, we haveThus, we can conclude from (18), (75), and Lemma 2 thatOn the contrary, we can see that the zero of is the zero of , and the zero of is also the zero of . Indeed, if is a zero of , that is, , then it follows , and this shows that is a zero of ; if is a zero of , that is, , then it follows , and this shows that is a zero of . Thus, by combining with (76), we concludeThis shows that ; thus, by combining with