Entire Solutions for Complex Systems of the Second-Order Partial Differential Difference Equations of Fermat Type

Liu, Si Min; Xu, Hong Yan

doi:https://doi.org/10.1155/2021/4207579

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Research Article | Open Access

Volume 2021 | Article ID 4207579 | https://doi.org/10.1155/2021/4207579

Entire Solutions for Complex Systems of the Second-Order Partial Differential Difference Equations of Fermat Type

Si Min Liu¹and Hong Yan Xu^1,2

Academic Editor: Stanislaw Migorski

Received07 Aug 2020

Accepted08 Jan 2021

Published22 Jan 2021

Abstract

This article is mainly concerned with the existence and the forms of entire solutions for several systems of the second-order partial differential difference equations of Fermat type and . Our results about the existence and the forms of solutions for these systems generalize the previous theorems given by Xu and Cao, Gao, Liu, and Yang. In addition, we give some examples to explain the existence of solutions of this system in each case.

1. Introduction and Main Results

The issue on the existence and form of solutions for Fermat-type equation has attracted considerable attention from many scholars. Especially, Taylor and Wiles [1, 2] pointed out that this equation does not admit a nontrivial solution in rational numbers for , and this equation does admit a nontrivial rational solution for . In fact, the study of this issue should go back to sixty years ago or even earlier; Montel [3] and Gross [4] had pointed out that the entire solutions of the functional equation for are , where is an entire function; for , there are no nonconstant entire solutions.

In 2004, Yang and Li [5] investigated a certain nonlinear differential equational of Malmquist type, by making use of Nevanlinna theory, and obtained the following.

Theorem 1. (See [5]). Let , and be nonzero meromorphic functions. Then, a necessary condition for the differential equation,to have a transcendental meromorphic solution satisfying is .

In the past ten years, Liu and his collaborators investigated the existence of solutions for a series of complex difference equations and complex differential difference equations of Fermat type, by using the difference Nevanlinna theory for meromorphic functions (see [6–8]), and obtained a lot of interesting original results (see [9–11]). In order to be consistent with the following text, here, we only list one of results are given by Liu.

Theorem 2. (See [10], Theorem 9). The transcendental entire solutions with finite order ofmust satisfy , where is a constant and or , where is an integer.

In 2019, Liu and Gao [12] further studied the entire solutions of the second-order differential and difference equation with single complex variable and obtained the following.

Theorem 3. (see [12], Theorem 2.1). Suppose that is a transcendental entire solution with finite order of the complex differential difference equation

Then, is a constant, and satisfieswhere and , where .

In 2016, Gao [13] further investigated the form of solutions for a class of system of differential difference equations corresponding to Theorem 2 and obtained the following.

Theorem 4. (See [13], Theorem 7). Suppose that is a pair of finite-order transcendental entire solutions for the system of differential difference equations

Then, satisfieswhere are constants and , where is a integer.

For the differential difference equations with several complex variables, Xu and Cao [14, 15] recently investigated the existence of the entire and meromorphic solutions for some Fermat-type partial differential difference equations by using the Nevanlinna theory in several complex variables and obtained the following theorems.

Theorem 5. (See [14], Theorem 7). Let be a constant in . Then, the Fermat-type partial differential difference equation,does not have any transcendental entire solution with finite order, where and are two distinct positive integers.

Theorem 6. (See [14], Theorem 8). Let be a constant in . Then, any transcendental entire solution with finite order of the partial differential difference equation,has the form of , where is a constant on satisfying and is a constant on ; in the special case, whenever , we have .

Inspired by the form of the abovementioned equations in Theorems 3–6, a question naturally arises: What will happen about the existence and the form of the solutions when the equations are put into the system and the first-order partial differential is replaced by the second-order partial differentials? For nearly two decades, although there were a lot of important and meaningful results focusing on the solutions of the complex difference equation of single variable (including [8, 16–21]), as far as we all know, there are few literature about the system of the second-order partial differential difference equations of Fermat type in several complex variables. It seems that this topic has never been treated before.

The purpose of this article is concerned with the properties of the solutions for some Fermat-type systems including both the difference operator and the second-order partial differential by making use of the (difference) Nevanlinna theory of several complex variables [22, 23]. We give the existence theorem and the forms of solutions for the Fermat-type systems of the second-order partial differential difference equations, which are generalization of the previous theorems given by Liu, Liu et al., Gao, Xu and Cao, and Xu et al. [9, 10, 13, 14, 24]. Here and below, let for any , and . Now, our main results of this paper are listed as follows.

Theorem 7. Let , and be positive integers, and be constants in that are not zero at the same time. If the following system of Fermat-type partial differential difference equations,satisfies one of the conditions(i)(ii) for , ,then system (9) does not have any pair of transcendental entire solution with finite order.

Remark 1. Here, is called as a pair of finite-order transcendental entire solutions for the systemif are transcendental entire functions and .

Remark 2. By observing the proof of Theorem 7, it is easy to see that the conclusions of Theorem 7 also hold if or of system (9) is replaced by , .

Theorem 8. Let . Then, any pair of transcendental entire solution with finite order for the system of Fermat-type difference equations,is of the following forms:orwhere , , , and is a constant in .

The following examples show the existence of solutions for system (11).

Example 1. Let , , , andwhere is a constant in . Thus, satisfies the following system:

Example 2. Let , , , andwhere is a constant in . Thus, satisfies the following system:

Theorem 9. Let . Then, any pair of transcendental entire solution with finite order for the system of Fermat-type partial differential difference equations,is of the following forms:orwhere , , , and is a constant in .

We also list two examples to exhibit the existence of solutions for system (18).

Example 3. Let , , , andwhere is a constant in . Thus, satisfies the following system:

Example 4. Let , , , andwhere is a constant in . Thus, satisfies the following system:

Remark 3. In fact, in view of the proofs of Theorems 8 and 9, it is easy to get that the conclusions of Theorem 9 still holds, if the system (18) is replaced by the following systems:orrespectively. The only thing that we need to do is to modify the condition to or , respectively.

2. Proof of Theorem 7

To prove Theorem 7, we need the following lemmas.

Lemma 1. (See [25, 26]). Let be a nonconstant meromorphic function on , and let be a multi-index with length . Assume that for some . Then,holds for all outside a set of finite logarithmic measure , where .

Lemma 2. (See [22, 23]). Let be a nonconstant meromorphic function with finite order on such that , and let . Then, for ,holds for all outside a set of finite logarithmic measure .

Proof. The proof of Theorem 7: suppose that is a pair of transcendental entire functions with finite order, satisfying system (9); then, it follows that and are transcendental. Here, the following two cases will be considered:(i)Case 1: : in view of Lemma 2, it yields that hold for all outside of a possible exceptional set of finite logarithmic measure . Thus, it follows from (29) that for all . In view of (30), Lemma 1, and the Mokhon’ko theorem in several complex variables ([27], Theorem 3.4), it yields that for all . Similarly, we have In view of (31) and (32), it yields that Since are transcendental and , this is a contradiction.(ii)Case 2: for , : in view of the Nevanlinna second fundamental theorem, Lemma 2, and system (9), it follows thatwhere is a root of . Similarly, we haveOn the other hand, by the Mokhon’ko theorem in several complex variables ([27], Theorem 3.4), it follows from system (9) thatSimilarly, we haveIn view of (34)–(37) and , it follows thatIt leads to a contradiction with the assumption that are transcendental entire functions.
Therefore, this completes the proof of Theorem 7.

3. Proofs of Theorems 8 and 9

The following lemma plays the key role in proving Theorems 8 and 9.

Lemma 3. (See [28, 29]). For an entire function on , and put . Then, there exist a canonical function and a function such that . For the special case , is the canonical product of Weierstrass.

Remark 4. Here, we denote to be the order of the counting function of zeros of .

Lemma 4. (See [30]). If and are entire functions on the complex plane and is an entire function of finite order, then there are only two possible cases: either(a)the internal function is a polynomial, and the external function is of finite order or else(b)the internal function is not a polynomial but a function of finite order, and the external function is of zero order

Lemma 5. (See [31], Lemma 3). Let be meromorphic functions on such that is not constant and and such thatfor all outside possibly a set with finite logarithmic measure, where is a positive number. Then, either or .

3.1. The Proof of Theorem 8

Suppose that is a pair of transcendental entire solutions with finite order of system (11). System (11) can be represented as follows:

Since are transcendental entire functions with finite order, then by (40), we can see that the functionshave no any zeros and poles. Moreover, by Lemmas 3 and 4, we have that there exist two polynomials such that

In view of (42), it yields thatwhich implies

Now, we claim that . If , then equation (44) becomes , and this is impossible since is a nonconstant polynomial. If and , then , where . Solving this equation, we have , that is, , where is a polynomial in . Thus, it follows that , where is a polynomial in . This is a contradiction with the assumption of being a nonconstant polynomial. Hence, . Similarly, we have and .

In view of Lemma 5 and (44) and (45), it follows that

Now, the four cases will be taken into account below.(i)Case 1. Since are polynomials, from (47), it follows that and , and here and below, are constants. Thus, it yields that and . Hence, we have that , where is a linear function of the form , are constants, and , is a polynomial in in , . Here, we will prove that . If , equation (47) implies that is, where . By comparing the degree of in both sides of the abovementioned 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 (47), we have In addition, in view of (44)–(47), it follows that which means that Thus, we can deduce from (50) and (52) that In view of (43), are of the forms If and , then and . Thus, it follows from (54) and (55) that where , and If and , then and . Thus, it follows from (54) and (55) that If and , then and . Thus, it follows from (54) and (55) that If and , then and . Thus, it follows from (54) and (55) that(ii)Case 2. Since are polynomials, from (61), it follows that and , which imply that , and this is a contradiction with the condition of being a nonconstant polynomial.(iii)Case 3. Since are polynomials, from (62), it follows that and , which imply that , and this is also a contradiction.(iv)Case 4.

Since are polynomials, then from (63), it follows that and . This means that and . Similar to the argument as in case 1 in Theorem 8, we can deduce that , where is a linear function of the form , are constants. Hence, it follows that . Substituting these into (63), we have

In addition, in view of (44)–(47), it follows thatwhich means that

Thus, we can deduce from (63) and (64) that

In view of (43), are of the forms

If and , then and . Thus, it follows from (68) and (69) thatwhere , and

If and , then and . Thus, it follows from (68) and (69) that

Thus, in view of Cases 1–4, this completes the proof of Theorem 8.

3.2. The Proof of Theorem 9

Suppose that is a pair of transcendental entire solutions with finite order of system (18). System (18) can be represented as follows:

Since are transcendental entire functions with finite order, then by (75), we can see that the functionshave no any zeros and poles. Moreover, by Lemmas 3 and 4, we have that there exist two polynomials such that

In view of (77), it yields thatwhich implieswhere

Now, we claim that . If ; then, equation (79) becomes . If , then it yields , and this is a contradiction with the condition of being a nonconstant polynomial. If , we have

By making use of the Mokhon’ko theorem in several complex variables ([27], Theorem 3.4), in view of (82), it follows that

In view of the Nevanlinna second fundamental theorem, (82) and (83), it follows thatoutside possibly a set of finite Lebesgue measure. This is a contradiction with the fact thatfor being a nonconstant polynomial. Hence, . Similarly, we have and .

In view of Lemma 5 and (79) and (80), it follows that

Now, we will consider the four cases below. Case 1. Since are polynomials, from (87), it follows that and . Thus, it yields that and . Hence, similar to the argument as in case 1 of Theorem 8, we have that , where is a linear function of the form , are constants, which means that . Substituting these into (87), we have In addition, in view of (79)–(87), it follows that which means that Thus, we can deduce from (88) and (90) that Similar to the argument as in the proof of Theorem 8 and by combining with (91), we have that is of the form Case 2. Since are polynomials, from (93), it follows that and , which imply that , and this is a contradiction with the condition of being a nonconstant polynomial. Case 3. Since are polynomials, then from (94), it follows that and , which imply that , and this is also a contradiction. Case 4.

Since are polynomials, from (95), it follows that and . This means that and . Thus, similar to the argument as in case 1 of Theorem 2, we can deduce that , where is a linear function of the form , are constants. Hence, it follows that . Substituting these into (95), we have

In addition, in view of (80)–(87), it follows thatwhich means that

Thus, we can deduce from (96) and (98) that

Similar to the argument as in the proof of Theorem 8, and by combining with (99), we can deduce that is of the form

Thus, in view of cases 1–4, this completes the proof of Theorem 8.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest in the manuscript.

Authors’ Contributions

H. Y. Xu. conceptualized the study and wrote the original draft; S. M. Liu and H. Y. Xu. reviewed and edited the manuscript and acquired funding.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11561033), the Natural Science Foundation of Jiangxi Province in China (20181BAB201001), the Foundation of Education Department of Jiangxi (GJJ190876 and GJJ202303) of China, and Shangrao Science and Technology Talent Plan (2020K006).

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Copyright © 2021 Si Min Liu 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.

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