Results on Solutions for Several Systems of Partial Differential-Difference Equations with Two Complex Variables

Chen, Yuxian; Xie, Libing; Xu, Hongyan

doi:https://doi.org/10.1155/2023/6158435

Journal of Mathematics

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Review Article | Open Access

Volume 2023 | Article ID 6158435 | https://doi.org/10.1155/2023/6158435

Results on Solutions for Several Systems of Partial Differential-Difference Equations with Two Complex Variables

Yuxian Chen,¹Libing Xie,¹and Hongyan Xu^2,3

Academic Editor: Kenan Yildirim

Received06 Mar 2023

Revised10 May 2023

Accepted30 Aug 2023

Published12 Sept 2023

Abstract

This article is to describe the entire solutions of some partial differential-difference equations and systems. Some theorems about the forms of transcendental entire solutions with finite order for several high-order partial differential-difference equations (or systems) of the Fermat type with two complex variables are obtained. Moreover, some examples are provided to explain that our results are precise to some extent.

1. Introduction

Fermat’s last theorem, as everyone knows, states that the Fermat equation does not have nontrivial rational solutions for [1]. In 1960s, Gross [2, 3] considered the Fermat-type functional equationand they obtained that equation (1) has entire solutions for the form and meromorphic solutions for the form , , where is entire and is meromorphic. From then on, many authors further explored these problems when is the differential or difference operators of . In 2004, Yang and Li [4] discussed the existence of transcendental meromorphic solution for the Fermat-type differential equation , where is the nonzero meromorphic function. In the same year, Li [5] considered the entire solutions for Fermat-type partial differential equations. In recent years, there exist many results for Fermat-type differential-difference equations (see [6–12]) with the aid of the difference Nevanlinna theory with one complex variable. Around 2012, Liu et al. [8, 9, 13] investigated the solutions for differential-difference equations and and proved that the finite-order transcendental entire solutions for these equations are of the forms and , respectively, under some conditions on . Gao [14] in 2016 discussed the solutions for a system of differential-difference equations , corresponding to the equations mentioned in [8, 13] and obtained that the pair of finite-order transcendental entire solutions of this system is of the forms or , where are constants, , and is a integer.

Besides, after Li’s results [5], Fermat-type partial differential equations, including , , and , have been studied (see [15–17]), and a number of results concerning the existence and forms of solutions for partial differential equations in several complex variables have been studied. In very recent years, with the rapid development of the difference Nevanlinna theory, Xu and Cao [18] discussed the existence of the entire solutions for the Fermat-type partial differential-difference equation:in . They pointed out that (i) when and are two distinct positive integers, equation (2) cannot have any transcendental entire solution with finite order; (ii) when , equation (2) has the transcendental entire solution with finite order. Moreover, Xu and Cao [18] also obtained the following results.

Theorem 1 (see [18]). Let . If the partial differential-difference equation isthere exist transcendental entire solutions with finite order, which have the form:where satisfies and and is a polynomial in one variable such that . In the special case whenever , there is .

In 2021, Xu et al. [19–21] further investigated the entire solutions for partial differential-difference equations with more general form and obtained the following.

Theorem 2 (see [21]). Let and . If the partial differential-difference equation isthere exist the transcendental entire solution with finite order, and the solution has the following two forms:(i) where is a transcendental entire function with finite order in satisfying(ii)where and satisfies

In 2020, Xu et al. [17] extended these results from equations to Fermat-type systems of partial differential-difference equations and obtained the following.

Theorem 3 (see ([20], Theorem 1.3)). Let . Then, any pair of transcendental entire solutions with finite order for the system of Fermat-type partial differential-difference equations exists:which have the following forms:where , is a constant in , and satisfy one of the following cases(i),, and ,, or ,(ii),, and ,, or ,(iii), , and ,, or , (iv), , and ,, or ,

However, according to above Theorems 1–3, we find that the authors mainly considered first-order partial differential equations and rarely considered second-order or more higher order partial differential equations. In this paper, based on their results, we further consider some more generalized questions, considering mixing higher order partial differential-difference equations (system). Thus, we consider the following two questions.

Question 4. How to describe the solution for PDDEs when first-order partial differential-difference equations in Theorems 1 and 2 are replaced by some high-order partial differential-difference equations?

Question 5. What happens about the solutions of the system of PDDEs when the system in Theorem 3 includes the second-order partial derivative and second-order mixed partial derivative?
Based on the above questions, we will investigate the entire solutions for some second-order and high-order partial differential-difference equations (systems) in this paper and obtain some results which will be listed in Section 2 by using the methods of previous articles [17, 19–21]. Now, we first introduce some lemmas to prove our main results.

Lemma 6 (see [22, 23]). For an entire function on , we put . Then, there exist a canonical function and a function such that . For the special case, is the canonical product of Weierstrass.

Lemma 7. (see [24]). If and are entire functions on the complex plane and is an entire function of finite order, then there are only two possible cases:(a)The internal function is a polynomial, and the external function is of finite order.(b)Or else the internal function is not a polynomial but a function of finite order, and the external function is of zero order.

Lemma 8 (see [25]). Let be meromorphic functions on such that is not constant, and , and such that,for all outside possibly a set with finite logarithmic measure, where is a positive number. Then, either or .

2. Main Results and Some Examples

First, we assume that readers are very familiar with some basic notations and theorems of Nevanlinna value distribution theory (see [7, 26, 27]). Besides, let denote the order of growth of , whereand let for any . In this paper, based on the results of Xu and Cao [18] and Liu and Xu [15, 17], we further obtain some more generalized results, considering mixing higher order partial differential-difference equations (system). We obtain the following.

Theorem 9. Let and . If the partial differential equation,has the transcendental entire solution with , then has the following two forms:(i) where is a transcendental entire function with in satisfying(ii)where is a transcendental entire function with in satisfying

Furthermore, when the second-order partial derivative is replaced with the -th order partial derivative in equation (12), we obtain the following.

Theorem 10. Let and . If the partial differential equation,has the transcendental entire solution with , then has the following two forms:(i) where is a transcendental entire function of finite order in satisfying(ii)where , , and is a transcendental entire function with , in satisfying

Remark 11. Based on the above two results, we easily see that Theorem 9 is especially an example of Theorem 10 for the case .
The following two examples are sufficient to show that our results are correct.

Example 1. Let . Then, is a transcendental entire solution with , of (i) of equations (12) and (15) with .

Example 2. Let andThen, is a transcendental entire solution with , of (ii) of equations (12) and (15) with .

Theorem 12. Let . If the partial differential equation,has the transcendental entire solution with , then has the following form:where and are constants satisfying and .

The following example shows the existence of the finite-order transcendental entire solutions for equation (19).

Example 3. Let andThen, is a transcendental entire solution of equation (19) with and .
Finally, we continue to investigate the transcendental entire solutions of a certain system of Fermat-type second-order partial differential-difference equations and obtain the following.

Theorem 13. Let . If the system of the partial differential equation,has the transcendental entire solution with , then must be of the following forms:where and is constant in .

Example 4. Let ; thus, we have following four cases: Case 1: let , , and Thus, is a pair of the transcendental entire solution of equation (22) with . Case 2: let , , and Thus, is a pair of the transcendental entire solution of equation (22) with . Case 3: let , , and Thus, is a pair of the transcendental entire solution of equation (22) with . Case 4: let , , and Thus, is a pair of the transcendental entire solution of equation (22) with .

3. Proof of Main Results

3.1. The Proof of Theorem 9

Let be a transcendental entire solution with of equation (12). Two cases will be considered below.

Case 14. If is a constant, we setOn the basis of equation (12), we can see that is also a constant, and thus, we setThis leads to . In view of (28) and (29), and taking partial derivations on both two sides of equations (28) and (29) for the variables , respectively, and noting that the fact , it yields thatThese mean thatThe characteristic equations of (31) areLet the initial conditions be and with a parameter , and we then can obtain the following parametric representation for solutions of the characteristic equations: ,Noting that , we haveSubstituting (34) into (28) or (29), we obtain thatTherefore, this completes the proof of of Theorem 9.

Case 15. If is not a constant, we first write equation (12) as the following form:which indicates that both,have no poles and zeros. Thus, by Lemmas 6 and 8, there exists a polynomial such thatwhich lead toIn view of (39) and (40), we getTaking partial derivations on both sides of equations (39) and (40) for the variables , respectively, and combining with the fact , it yields thatThus, it follows from (41) and (42) thatwhereNext, we will prove that and cannot be equal to 0. Obviously, and cannot hold at the same time. Otherwise, we have , and this is a contradiction. If and , then it follows from (43) thatDue to the fact that is a nonconstant polynomial, we have that , , and . In view of the Nevanlinna second main theorem in several complex variables and (46), we haveThis is impossible. If and , similarly, we can get a contradiction. Consequently, these cases can conclude that and . Due to the fact that and cannot be constant and by Lemma 8, it follows from (43) and (44) thatandTherefore, (48) and (49) indicate that is a constant (let ). Thus, it means that , where is a linear function as the form are constants. Thus, we havewhere . This leads toThus, we have from (42) thatthat is,where . The characteristic equations of (53) areBy using the initial conditions and with a parameters, we can obtain the following parametric representation for solutions of the characteristic equations: ,where is an entire function with finite order in satisfyingThus, it yields that.Substituting (57) into (39) and combining with (51), we can deduce that satisfiesTherefore, from Cases 14 and 15, the proof of the theorem is completed.

3.2. The Proof of Theorem 10

Similar to the proof of Theorem 9, we will consider two cases below.

Case 16. If and are constants, by the same discussion of Theorem 9, we haveThus, this leads to . In view of (59) and (60), and taking partial derivations on both two sides of equations (61) and (62) for the variables , respectively, and combining with the fact , it yields thatwhich implies thatThe characteristic equations of (63) areBy using the initial conditions , and with a parameter , we can obtain the following parametric representation for solutions of the characteristic equations: ,In view of the above conditions, we haveSubstituting (66) into (59) or (60), we obtainwhich completes the proof of of Theorem 10.

Case 17. If and are not constants, similar to the proof of Theorem 9, we can getwhich lead toTaking the partial derivations on both two sides of equations (68) and (69) for the variables , respectively, by combining with the fact , it yields thatThus, it follows from (70) and (71) thatwhereNext, we will prove that and cannot be equal to 0. Obviously, and cannot hold at the same time. Otherwise, we have , and this is impossible. Similar to the cases about and in the proof of Theorem 9, we can conclude that and . Since and cannot be constant, by Lemma 8, it follows from (72) and (73) thatandThen, (75) and (76) imply that , where is a constant in . Thus, we get that , where is a linear function as the form are constants. Thus, we havewhere , and this leads toIn view of (71), we haveThis implieswhere . The characteristic equations of (80) areBy using the initial conditions and with parameters, we can obtain the following parametric representation for solutions of the characteristic equations: ,where is an entire function with finite order in such thatThus, it yields thatSubstituting (84) into (68) and combining with (76), we can deduce that satisfiesTherefore, from Cases 16 and 17, the proof of this theorem is completed.

3.3. The Proof of Theorem 12

We suppose that is a finite-order transcendental entire solution of equation (19). Apparently, and are not constants; otherwise, there is a contradiction with the assumption. In view of the proof of Theorem 9, we get

Differentiating both sides of equation (86) for the variables , respectively, and combining (86) and (87), this leads to

Thus, it follows from (88) thatwhere

Next, we will prove that and cannot be equal to 0. Obviously, and cannot hold at the same time. Otherwise, we have , and this is a contradiction. If and , then it follows from (89) that

Similar to the cases about and in the proof of Theorem 9, we can get that and . Since and cannot be constant, and by Lemma 8, it follows from (89) and (90) thatand

Hence, (93) and (94) imply that , where is a constant in . Thus, it means that , where is a linear function as the form are constants. Thus, we havewhere , and this leads to

Therefore, the proof of this theorem is completed.

4. The Proof of Theorem 13

We suppose that is a pair of finite-order transcendental entire functions satisfying system (22). First, system (22) can be rewritten as

Since are finite-order transcendental entire functions, then there exist two nonconstant polynomials such that

In view of (98), it yields that.which impliesor

By using the Nevanlinna second main theorem in several complex variables, we can deduce that and cannot be equal to 0. Similarly, and cannot also be equal to 0. Thus, by Lemma 8, (100) and (101), we have.

Now, we will discuss four cases below.

Case 18. Due to the fact that are polynomials, it follows from (103) that and , and here and below, are constants. Thus, it yields that and . This leads to and , where () and is a polynomial in in .
Following, we will illustrate that . Supposing that , equation (103) impliesthat is,where . By comparing the degree of on both sides of the above equation, we have , that is, . Thus, the form of is still the linear form of , which means that . Thus, this means that . Substituting these into (103), we haveMoreover, based on (100)-(103), we can see thatwhich means thatThus, we can deduce from (106) and (108) thatNoting that and , then we have and . In view of (106) and (108), we get . Thus, it follows from (99) thatandIf , then and . Thus, it follows from (110) and (111) thatIf , then and . Thus, it follows from (110) and (111) that

Case 19. From (114), we get that and , which deduce that . Noting that are nonconstant polynomials, we obtain a contradiction.

Case 20. From (115), it follows that and , which leads to . Noting that are nonconstant polynomials, we obtain a contradiction.

Case 21. Similar to the argument as in Case 18, we can deduce that , where (). Hence, it follows that . Substituting these into (116), we haveMoreover, based on (100)–(103), we can see thatwhich means thatThus, we can deduce from (117) and (119) thatIn view of and , then we have and . By combining with (117) and (119), we also get . In view of (99), we haveIf , then and . Thus, it follows from (121) and (122) thatIf , then and . Thus, it follows from (121) and (122) that.Therefore, the proof of Theorem 12 is completed.

5. Conclusion

In view of Theorems 9, 10, 12, and 13, we give the exact form of entire solutions for some second-order and higher order (mixed) partial differential equations (systems). These results are some improvements of the previous results, which are mainly concerning with first-order partial differential equations. Meantime, some examples (including Examples 2, 3, and 4) show that our results are precise to some extent.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Y. X. Chen and H. Y. Xu were responsible for conceptualization. Y. X. Chen and L. B. Xie were responsible for writing the original draft. Y. X. Chen, L. B. Xie, and H. Y. Xu were responsible for writing, reviewing, and editing. Y. X. Chen and L. B. Xie were responsible for funding acquisition.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (12161074) and the Foundation of Education Department of Jiangxi (\#GJJ191042, \#GJJ202303, \#GJJ212305, and \#GJJ2202228) in China.

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Copyright © 2023 Yuxian Chen 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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