The Study of Solutions of Several Systems of Nonlinear Partial Differential Difference Equations

Xu, Hong Yan; Yu, Meiying; Zhang, Keyu

doi:https://doi.org/10.1155/2022/4449502

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Developments in Functions and Operators of Complex Variables 2022

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Volume 2022 | Article ID 4449502 | https://doi.org/10.1155/2022/4449502

The Study of Solutions of Several Systems of Nonlinear Partial Differential Difference Equations

Hong Yan Xu,^1,2Meiying Yu ,³and Keyu Zhang³

Academic Editor: Sibel Yalçın

Received30 May 2022

Accepted16 Jun 2022

Published12 Jul 2022

Abstract

Our main aim is to describe the entire solutions of several systems of and where are nonzero constants in and are positive integers. We obtain several theorems on the existence and the forms of solutions for these systems, which are some improvements and supplements of the previous theorems given by Xu and Cao, Gao, and Liu and Yang. Moreover, we give some examples to explain the existence of solutions for such systems.

1. Introduction

As everyone knows, the study of the existence of solutions for Fermat type equations has always been an important and interesting problem. The famous Fermat’s Last Theorem has attracted the attention of many mathematical scholars [1, 2]. About 60 years ago or even earlier, Montel [3] and Gross [4] had considered the equation and obtained that the entire solutions of are for the case , where is an entire function, and this equation does not admit any nonconstant entire solution for any positive integer .

With the establishment and rapid development of Nevanlinna value distribution theory for meromorphic functions and theirs difference [5–7], Liu [8] in 2009, Liu et al. [9] in 2012, and Liu and Yang [10] in 2013 studied some complex Fermat type difference and Fermat type differential difference equations and obtained some results.

Theorem 1 (see [9], Theorem 1.1). The transcendental entire solutions with finite order of must satisfy , where is a constant and , with an integer.

Theorem 2 (see [9], Theorem 1.3). The transcendental entire solutions with finite order of must satisfy , where is a constant and or , with an integer.

After that, Gao [11] in 2016 extended Theorem 2 from complex differential difference equation to the system of complex differential difference equations.

Theorem 3 (see [11], Theorem 1.1). Suppose that is a pair of finite-order transcendental entire solutions for the system of differential difference equations Then, satisfies or where are constants and , where is an integer.

In recent, Xu and Cao [12, 13] further discussed the solutions for some Fermat type PDDEs and obtained the following:

Theorem 4 (see [13], Theorem 1.4). Let . Then, any nonconstant entire solution with finite order of the equation has the form of , where is a linear function of the form on such that , and is a constant on .

Theorem 5 (see [13], Theorem 1.1). Let . Then, does not have any transcendental entire solution with finite order, where and are two distinct positive integers.

Theorem 6 (see [13], Theorem 1.2). Let . Then, any transcendental entire solution with finite order of the PDDE has the form of , where is a constant on satisfying and is a constant on ; in the special case whenever , we have .

By analyzing Theorems 3–6, a natural question is as follows: What will happen about the transcendental entire solutions for the system of the PDDEs of Fermat type? Although many scholars have paid considerable attention to the complex difference equation with a single variable and the complex Fermat type difference equation in recent years, a series of important and meaningful results (including [7, 14–22]) were obtained, however, to our knowledge, there were not much results about the complex difference equation in several complex variables. Of course, the references involving the results of systems of complex PDDEs are even less.

This manuscript is aimed at studying the solutions of several Fermat type systems involving both difference operator and partial differential. We establish four theorems on the forms of solutions for several systems of Fermat type PDDEs, which are improvement of the previous theorems given by Liu et al., Gao, and Xu and Cao [8, 9, 11, 13]. We mainly employ the Nevanlinna value distribution theory and difference Nevanlinna theory of several complex variables in this article, and the readers can refer to [23, 24]. Now, we start to state our main results below.

Theorem 7. Let , , and . If the Fermat type PDDE system satisfies one of the conditions (i)(ii) and , then system (9) does not exist any pair of finite-order transcendental entire solution.

Remark 8. Here, we say that is a pair of finite-order transcendental entire solution for if are transcendental entire functions satisfying the above system and .

Remark 9. We list an example to demonstrate that the condition in Theorem 7 cannot be removed. Let where and . Thus, satisfies the system (9) with , , and , .

Theorem 10. Let and . If the system of Fermat type difference equations admits a pair of finite-order transcendental entire solution , then , and have the following forms where , , , and satisfy one of the following cases. (i), and or and , is a integer(ii), and , or and .

Remark 11. From Theorem 10, we can conclude that have the following relationships (i)(ii), where and

Now, two examples can verify the existence of solutions for (12).

Example 1. Let and satisfy , and . Then, the function satisfies the system (12) with , , and .

Example 2. Let and satisfy , and . Then, the function satisfies the system (12) with and .

Theorem 12. Let and . If the system of Fermat type PDDEs admits a pair of finite-order transcendental entire solution , then and is the form of where , is a constant in , and satisfy and one of the following cases (i), and , , , or , , (ii), and , , , or , , (iii), and , , , or , , (iv), and , , , or , , .Here, two examples can verify the existence of solutions for (16).

Example 3. Let , , , , and . That is, and Thus, satisfies the system (16) with , , , , and .

Example 4. Let , , , , and . That is, and Thus, satisfies the system (16) with , , , , and .

Example 5. Let , , , , and . That is, and Thus, satisfies the system (16) with , , , , and .

Example 6. Let , , , , and . That is, and Thus, satisfies the system (16) with , , , , and .

Theorem 13. Let and . Let be a pair of transcendental entire solutions of finite order for the system Then, is one of the forms or where , , and are entire period functions of finite order with period , and satisfy and the following conditions
() if , and if
() or where , , , and ; where are stated as in ((23)) and ((24)), and satisfy Here, five examples can verify the existence of solutions for (22).

Example 7. Let , , , and Thus, satisfies system (22) with , , , , , , and .

Example 8. Let , , , and Thus, satisfies the system (22) with , , , , , , and .

Example 9. Let and Thus, satisfies the system (22) with , , , , , and .

Example 10. Let be of the forms Thus, satisfies the system (22) with , , , , , , and .

Example 11. Let Thus, satisfies system (22) with , , , , , , and .

2. Proof of Theorem 7

Proof. Let be a pair of finite-order transcendental entire functions satisfying (9). Here, let us consider two cases below.

Case 1. . Owing to Refs. [23, 24], we have the following facts that hold for all outside of a possible exceptional set of finite logarithmic measure . Due to the above fact, we have for all . By the Mokhon’ko theorem ([25], Theorem 3.4) and the Logarithmic Derivative Lemma [26], it yields from (35) that for all . Similarly, we also get Thus, we conclude from (36) and (37) that By combining with the condition that and being transcendental functions, we obtain a contradiction.

Case 2. and , . Thus, it is easy to get that . In view of the Nevanlinna second fundamental theorem, the difference logarithmic derivative lemma in several complex variables [23, 24], we thus obtain from (9) that where is a roots of . Similarly, we also have In addition, by applying the Mokhon’ko theorem in several complex variables ([25], Theorem 3.4) for (9), we can conclude Similarly, we also get Due to , it follows from (39)–(42) that and this is a contradiction with being transcendental functions.

Therefore, Theorem 7 is proved.

3. The Proof of Theorem 10

Let be a pair of finite-order transcendental entire functions satisfying (12). We firstly rewrite the system (12) as

By applying the Hadamard factorization theorem (can be found in [27, 28]), then there exist two polynomials such that

Thus, we have from (45) that which implies

By applying [29], Lemma 3.1 (can be found in [30]), for (47) and (48), we have that

Here, four cases will be discussed below.

Case 1. Thus, we can conclude from (50) that and ; here and below, are constants. So, this leads to , where and are constants. Thus, by virtue of (47)–(50)), it yields that which implies In view of (46), let If , i.e., , , then . Thus, If , i.e., , , then . Thus, If , i.e., , , then . Thus, If , i.e., , , then . Thus,

Case 2. Thus, it yields from (58) that and . Hence, we obtain that , and this is a contradiction with is not a constant.

Case 3. Thus, it yields from (59) that and . Hence, we obtain that , and this is a contradiction with is not a constant.

Case 4. Thus, it yields from (60) that and . Hence, we obtian that , where , are constants. By virtue of (47),(48), (60), it yields that which implies In view of (46), let If , i.e., , , then . Thus, If , i.e., , , then . Thus, If , i.e., , , then . Thus, If , i.e., , , then . Thus, Therefore, this completes the proof of Theorem 10.

4. The Proof of Theorem 12

Proof. Let be a pair of finite-order transcendental entire functions satisfying (16). Firstly, (16) may be represented as the following form: By the Hadamard factorization theorem (can be found in [27, 28]), there are two nonconstant polynomials satisfying In view of (69), it yields that which implies Obviously, . Otherwise, . This leads to a contradiction with is not a constant. Similarly, . Thus, due to [29], Lemma 3.1 (can be found in [30]), (71), and (72), we obtain that

Hence, four cases will be discussed below.

Case 1. Thus, it follows from (74) that and . These lead to and . Hence, we obtain that , where , are constants. By combining with (71)–(74), we have and this leads to

Subcase 1. If , then and . Due to (70), we have that

Subcase 2. If , then , and . Due to (70), we have that

Subcase 3. If , then , , and . Due to (70), we have that

Subcase 4. If , then , , and . Due to (70), we have that

Case 2. Thus, it yields from (81) that and . We have that , and this leads to a contradiction with being not constant.

Case 3. Since are polynomials, then from (82), it follows that and . This means , and this is a contradiction because is not a constant.

Case 4. Then, from (83), it yields that and , and this leads to and . Thus, it follows that , where , are constants in . In view of (71), (72), and (83), we have which implies

Subcase 4.1. If , then , , and . By virtue of (70), it follows that

Subcase 4.2. If , then , , and . By virtue of (70), it follows that

Subcase 4.3. If , then and . By virtue of (70), we have that

Subcase 4.4. If , then and . By virtue of (70), we have that

Hence, the proof of Theorem 12 is completed.

5. The Proof of Theorem 13

Proof. Assume that is a pair of finite-order transcendental entire functions satisfying (22). Thus, let us discuss two following cases. (i)Suppose that is transcendental, then is transcendental. Noting that are nonzero constants, we next prove that and are transcendentalSuppose that is not transcendental. Since is transcendental, then and are transcendental. By observing the second equation of (22), we can conclude that is transcendental.
Suppose that is transcendental. If is transcendental, similar to the above argument, and are transcendental. If is not transcendental, it thus leads to that is not transcendental. From (22), we thus get that is not transcendental. Thus, it yields that is not transcendental. This is a contradiction with is transcendental and is not transcendental.
Hence, if is transcendental, then , , and are transcendental. Hence, system (22) can be represented as Thus, by the Hadamard factorization theorem (can be found in [27, 28]), there are two nonconstant polynomials such that In view of (91), it yields that which implies Obviously, . Otherwise, we have that , and this leads to a contradiction since is not a constant. Similarly, . Thus, due to [29], Lemma 3.1 (can be found in [30]), and in view of (93) and (94), we can deduce that

Now, let us consider the following four cases.

Case 1. Then, (96) can lead to that and . Thus, we obtain that and . Hence, we can conclude that , where , are constants. By combining with (93)–(96), we have This means that By combining with (92), have the following forms: where are entire functions of finite order in . Substituting the above expressions into (92), we can deduce that This leads to Due to (101), we have where are entire period functions of finite order with period , and in (102), , if , and , if . Further, in view of (100) and (102), we have ; if , we have .

If , it follows from (97) that . Thus, it yields that

If , it follows from (97) that . Thus, similar to the above argument, we obtain that

If , it follows from (97) that . Thus, we obtain that

Case 2. We thus get from (107) that and . This means , and this yields a contradiction with being not a constant.

Case 3. We thus get from (108) that and . So, we conclude that , and this leads to a contradiction with being not a constant.

Case 4. Then, it follows from (109) that and . These yield that and , which leads to , where , are constants. In view of (93), (94), and (109), we have In view of (110), it follows that By combining with (92), are of the following forms where are finite-order entire functions in . By using the same argument as in Case 1, we have (102).

If , it follows from (110) that . Thus, we can deduce that

If , it follows from (110) that . We have that

If , it follows from (110) that . Thus, we obtain that

Therefore, from (102)–(106) and (113)–(116), we can prove the conclusions (23) and (24) of Theorem 13.

(i)Assume that . Thus, from (22), it follows that

This leads to We thus get from (22) that

By combining with (117) and (118), it yields which implies that

If , then from (120), it follows that which implies that

where are entire period functions of finite order with period , and

If , then from (120), it follows that

where and . Substituting (124) into (119), it follows that , and . Thus, we have (ii)Suppose that . Then, it yields in view of (22) that

where is a transcendental entire function of finite order in . Equation (126) leads to . Thus, due to the second equation in (22), we have

where is a transcendental entire function of finite order in . Combining with (126) and (127), we can deduce that and

This means that

where are entire period functions of finite order with period satisfying

Hence, from (126)–(129), it is easy to get the cases ((26)) and ((27)) of Theorem 13.

Therefore, the proof of Theorem 13 is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that none of the authors have any competing interests in the manuscript.

Authors’ Contributions

Conceptualization was contributed by H. Y. Xu; writing-original draft preparation was contributed by H.Y. Xu and K.Y. Zhang; writing-review and editing was contributed by H. Y. Xu and M.Y. Yu; funding acquisition was contributed by H. Y. Xu and K.Y. Zhang.

Acknowledgments

This work was supported by the National Natural Science Foundation of China 12161074 and the Talent Introduction Research Foundation of Suqian University.

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Copyright © 2022 Hong Yan Xu 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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