This article is concerned with the existence of entire solutions for the following complex second order partial differential-difference equation
where are constants in and are positive integers. In addition, we also investigate the forms of finite order transcendental entire solutions for several complex second order partial differential-difference equations of Fermat type, and obtain some theorems about the existence and the forms of solutions for the above equations. Meantime, we give some examples to explain the existence of solutions for some theorems in some cases. Our results are some generalizations of the previous theorems given by Qi [
Citation: Wenju Tang, Keyu Zhang, Hongyan Xu. Results on the solutions of several second order mixed type partial differential difference equations[J]. AIMS Mathematics, 2022, 7(2): 1907-1924. doi: 10.3934/math.2022110
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[7] | Hua Wang, Hong Yan Xu, Jin Tu . The existence and forms of solutions for some Fermat-type differential-difference equations. AIMS Mathematics, 2020, 5(1): 685-700. doi: 10.3934/math.2020046 |
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This article is concerned with the existence of entire solutions for the following complex second order partial differential-difference equation
where are constants in and are positive integers. In addition, we also investigate the forms of finite order transcendental entire solutions for several complex second order partial differential-difference equations of Fermat type, and obtain some theorems about the existence and the forms of solutions for the above equations. Meantime, we give some examples to explain the existence of solutions for some theorems in some cases. Our results are some generalizations of the previous theorems given by Qi [
For positive integers , the equation is called as the Fermat equation. In 1995, A. Wiles and R. Taylor [28,29] pointed out that this equation does not admit nontrivial solutions in rational numbers for , and this equation does exist nontrivial rational solution for . The functional equation can be regarded as the Fermat type equation. It has attracted the attention of many mathematics workers in the studying of Fermat type equation. We know that the Fermat type equation has no transcendental meromorphic solutions when (see [5]); and this equation has no transcendental entire solutions when (see [21]). In [10], Iyer pointed out that the entire solutions are of the form , where and is an entire function, no other forms exist. In [36], Yang discussed the Fermat type functional equation
(1.1) |
where are small functions with respect to and obtained
Theorem A. (see [36]). Let be positive integers satisfying . Then there are no nonconstant entire solutions and that satisfy (1.1).
When is replaced by or a differential polynomial of and for Eq (1.1), Yang and Li [37] in 2004 studied the Malmquist type nonlinear differential equational by using Nevanlinna theory of meromorphic functions, and obtained
Theorem B. (see [37]). Let and be nonzero meromorphic functions. Then a necessary condition for the differential equation
to have a transcendental meromorphic solution satisfying is constant.
Theorem C. (see [37]). Let be a positive integer, be constants, be a non-zero constant and let . Then the transcendental meromorphic solution of the following equation
(1.2) |
must have the form , where , is a non-zero constant and satisfies the following equations:
Remark 1.1. Let . From Theorem C, we can see that if is an odd, then Eq (1.2) has transcendental entire solutions. If is an even, then Eq (1.2) has no transcendental entire solutions.
Over the past two decades, with the help of difference Nevanlinna theory for meromorphic functions (see [4,6,7]), the study of the properties of solutions for complex difference equations and complex differential-difference equations has become more and more active, and a series of literatures concerning the existence and forms of solutions for some equations have sprung up (including [15,16,18,19,23,30,31,32,33]).
When is replaced by in Eq (1.2), Liu [15] in 2009 investigated the entire solutions of the equation by using the difference Nevanlinna theory for meromorphic functions and pointed out that the finite order transcendental entire solutions of the equation must satisfy , where , and . Later, Liu, Cao and Cao [17] in 2012 studied the existence of solutions for some complex difference equations and obtained
Theorem D. (see [17,Theorem 1.1]). The transcendental entire solutions with finite order of the equation must satisfy , where is a constant and , is an integer.
Theorem E. (see [17,Theorem 1.3]). The transcendental entire solutions with finite order of
must satisfy , where is a constant and or , is an integer.
In 2019, Liu and Gao [20] further studied the entire solutions of second order differential and difference equation when is replaced by in Theorem E and obtained
Theorem F. (see [20,Theorem 2.1]). Suppose that is a transcendental entire solution with finite order of the complex differential-difference equation
then is a constant, and satisfies
where , and , .
Let us recall some conclusions on the Fermat type equations in several complex variables. Hereinafter, let for any . For the equation in , Li [13] showed that meromorphic solutions must be constant if and only if and have the same zeros. When and in , then any entire solutions of the equation in are necessarily linear ([11]), which was originally investigated by [14,26].
Recently, Xu and Cao [34] investigated the existence of the entire and meromorphic solutions for some Fermat-type partial differential-difference equations and obtained the following theorems.
Theorem G. (see [34,Theorem 1.1]). Let be a constant in . Then the Fermat-type partial differential-difference equation
doesn't have any transcendental entire solution with finite order, where and are two distinct positive integers.
Remark 1.2. In fact, the equation
(1.3) |
admits a finite order transcendental entire solution. For example, let
then is a finite order transcendental entire solution of Eq (1.3).
Theorem H. (see [34,Theorem 1.2]). Let be a constant in . Then any transcendental entire solutions 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 .
In view of Theorem G and Remark 1.2, one question can be raised as follows.
Question 1.1. How to deform the equation can guarantee that the conclusion of Theorem G holds under the condition ?
The forms of equations in Theorem F and Theorem G prompts us to consider the following problems.
Question 1.2. What can be said about the solution of equation if is replaced by
in Theorem G?
Question 1.3. What can be said about the existence and the forms of the entire solution of the equation when is replaced by
in Theorem H?
Motivated by Questions 1.1–1.3, we investigate the existence and the forms of solutions for some second order partial differential-difference equations, by utilizing the Nevanlinna theory and difference Nevanlinna theory of several complex variables (see [3,12]). We give some existence theorems and the forms of entire solutions for some partial differential-difference equations, and also list some examples. Our results are some generalizations of the previous theorems given by Xu and Cao, Liu, Cao and Cao [17,34].
The first theorem is as follows.
Theorem 1.1. If , and be two distinct positive integers, then the Fermat-type partial differential-difference equation
(1.4) |
does not have any transcendental entire solution with finite order.
Remark 1.3. In fact, on the basis of the proof of Theorem 1.1, it is easily to get that the conclusions of Theorem 1.1 still hold if or is replaced by .
Next, we proceed to study the existence and forms of entire solutions of Eq (1.4) for .
Theorem 1.2. Let and . If the second order Fermat-type partial differential-difference equation
(1.5) |
admits a transcendental entire solution with finite order , then has the following form
where are constants in , and satisfy one of the following cases
, and , where , here and below is a integer;
, and .
Two examples are given to explain the existence of solutions for Eq (1.5).
Example 1.1. Let , , and . Thus, the function
satisfies the following equation
Example 1.2. Let , , and . Thus, the function
satisfies the following equation
Corollary 1.1. Let and . If the second order Fermat-type partial differential-difference equation
(1.6) |
admits a transcendental entire solution with finite order , then has the following form
where are constants in , and satisfy one of the following cases
, and , where ;
, and .
Theorem 1.3. Let and . If the second order Fermat-type partial differential-difference equation
(1.7) |
admits a transcendental entire solution with finite order , then has the following form
where are constants in , and satisfy one of the following cases
, and ;
, and .
Theorem 1.4. Let and , . If the second order Fermat-type partial differential-difference equation
(1.8) |
admits a transcendental entire solution with finite order , then has the following form
where are constants in , and satisfy one of the following cases
, and ;
, and .
Theorem 1.5. Let and , . If the second order Fermat-type partial differential-difference equation
(1.9) |
admits a transcendental entire solution with finite order , then has the following form
where are constants in , and satisfy one of the following cases
, and ;
, and .
Remark 1.4. From Theorems 1.1–1.5, the Eq has no nonconstant entire solution for the case , and has no nonconstant meromorphic solutions for the case . Hence based on Theorems 1.1–1.5, an open question is: What will happen for the meromorphic solutions of the Fermat type partial difference-differential equation
in ?
Similar to Examples 1.1 and 1.2, it is easy to give some solutions for Eqs (1.7), (1.8) and (1.9).
Example 1.3. Let , , and . Thus, the function
satisfies the following equation
Example 1.4. Let , , and . Thus, the function
satisfies the following equation
Example 1.5. Let , , and . Thus, the function
satisfies the following equation
Example 1.6. Let , , and . Thus, the function
satisfies the following equation
Example 1.7. Let , , and . Thus, the function
satisfies the following equation
Example 1.8. Let , , and . Thus, the function
satisfies the following equation
To prove Theorem 1.1, the following lemmas should be required.
Lemma 2.1. ([2]). 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 2.2. ([3,12]). Let be a nonconstant meromorphic function on such that , and let . If , then
holds for all outside a set of finite logarithmic measure , where
Remark 2.1. In view of Lemma 2.2, one can get that if is a nonconstant meromorphic function with finite order on such that , for , then
where denotes any quantity satisfying as r sufficiently large outside possibly a set of with finite Lebesgue measure.
The proof of Theorem 1.1: The proof of Theorem 1.1 is very similar to the argument as in Ref. [35]. Assume that is a finite order transcendental entire solution of Eq (1.4), then is transcendental. Thus, in view of (1.4), is also transcendental. Here, two cases will be considered below.
Case 1. . Thus, it follows from Lemma 2.2 that
(2.1) |
holds for all outside of a possible exceptional set with finite logarithmic measure . Thus, by (2.1) and combining with the properties of , we can deduce that
(2.2) |
for all . In view of (2.2), by applying Lemma 2.1 and the Mokhon'ko theorem in several complex variables [8,Theorem 3.4], we have
(2.3) |
for all . This means
(2.4) |
Since is transcendental, so this is a contradiction.
Case 2. . Then . Thus, it follows that . In view of the Nevanlinna second fundamental theorem, Lemma 2.2, and Eq (1.4), we have
(2.5) |
where is a root of .
On the other hand, in view of Eq (1.4), and by applying the Mokhon'ko theorem in several complex variables [8,Theorem 3.4], it yields that
(2.6) |
In view of (2.5)–(2.6) and , it follows
This is impossible since is transcendental.
Case 3. . Then it follows
(2.7) |
Differentiating this equation for , respectively, we have
where is a differential polynomial in and of the form
Divide both sides of the above equation by , it follows that
(2.8) |
By Lemmas 3.1 and 2.2, we have
(2.9) |
and
(2.10) |
Thus, in view of (2.8)–(2.10), it follows
which is a contradiction with the assumption of being transcendental and .
Therefore, this completes the proof of Theorem 1.1.
The following lemmas play the key roles in proving Theorems 1.2–1.5.
Lemma 3.1. ([9,Lemma 3.1]). 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 .
Remark 3.1. Here, is the counting function of the zeros of in , where the simple zero is counted once, and the multiple zero is counted twice.
Lemma 3.2. ([25,27]). 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 3.2. Here, denote to be the order of the counting function of zeros of .
Lemma 3.3. ([22]). 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.
Proof. Suppose that is a finite order transcendental entire solutions of Eq (1.5), then it follows that is transcendental. Otherwise, is not transcendental, this is a contradiction with the condition. Firstly, Eq (1.5) can be represented as the following form
(3.1) |
Since and are transcendental, then by Lemma 3.2 and Lemma 3.3, from (3.1), there exists a nonconstant polynomial in such that
(3.2) |
Thus, in view of (3.2), it yields
(3.3) |
In view of (3.3), we have
(3.4) |
Now, we claim that . If , then Eq (3.4) becomes , 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 . Thus, (3.4) becomes
(3.5) |
Since is a nonconstant polynomial, we have that is not a constant, and
and
Thus, by Lemma 3.1, it yields
(3.6) |
In view of (3.5) and (3.6), it follows
(3.7) |
Here, we claim that , where is a linear function as the form , is a constant in . In fact, since is a nonconstant polynomial, and in view of (3.6) and (3.7), we conclude that , where is a polynomial in , . Thus, it follows from (3.6) that must be a constant in . By combining with , then we have . Thus, is still a linear form of . Hence, we have and . Thus, it follows
(3.8) |
Thus, it follows from (3.8) that
(3.9) |
By observing the second equation in (3.3), we can define the form of as
(3.10) |
By combining with (3.9) and (3.10), it yields
where are constants in satisfying one of the following conditions
(i) , , and ;
(ii) , , and .
Therefore, this completes the proof of Theorem 1.2.
Proof. Suppose that is a finite order transcendental entire solutions of Eq (1.8), then it follows that is transcendental. Otherwise, is not transcendental, this is a contradiction with the condition. Firstly, Eq (1.8) can be rewritten as the following form
(3.11) |
Since and are transcendental, then by Lemma 3.2 and 3.3, it follows from (3.11) that
(3.12) |
where is a nonconstant polynomial in . Thus, in view of (3.12), it yields
(3.13) |
In view of (3.13), we have
(3.14) |
where and .
If , then it follows that
(3.15) |
By making use of the Nevanlinna second fundamental theorem and (3.15), it follows that
(3.16) |
If , in view of (3.15), it yields that , a contradiction. If , then from (3.16), it leads to
outside possibly a set of finite Lebesgue measure. This is a contradiction with the fact
for is a nonconstant polynomial. Thus, it follows that . Similarly, we have . Thus, (3.14) becomes
(3.17) |
Since is a nonconstant polynomial, we have that is not a constant, and
and
Thus, by Lemma 3.1, it yields
(3.18) |
In view of (3.17) and (3.18), it follows
(3.19) |
Since is a nonconstant polynomial, in view of (3.18) and (3.19), similar to the argument as in the proof of Theorem 1.2, we conclude that , where is a linear function as the form , is a constant in . Thus, it follows
(3.20) |
Thus, it follows from (3.20) that
(3.21) |
By observing the second equation in (3.13), we can define the form of as
(3.22) |
By combining with (3.21) and (3.22), it yields
where are constants in satisfying one of the following conditions
(i) , , and ;
(ii) , , and .
Therefore, this completes the proof of Theorem 1.4.
By using the same argument as in the proof of Theorem 1.4, we can easily prove the conclusions of Theorems 1.3 and 1.5.
We can see that Theorem 1.1 is an extension of Theorem G. Meantime, it is also a positive answer to Question 1.1. Moreover, Theorems 1.2–1.5 are the answer to Questions 1.2–1.3. More important, a series of examples show that our results are accurate.
The authors were supported by the National Natural Science Foundation of China (Grant No. 12161074), the Natural Science Foundation of Jiangxi Province in China (Grant No. 20181BAB201001) and the Foundation of Education Department of Jiangxi (GJJ190876, GJJ202303, GJJ201813, GJJ191042) of China.
The authors declare that none of the authors have any competing interests in the manuscript.
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1. | Raju Biswas, Rajib Mandal, Entire solutions for quadratic trinomial partial differential-difference functional equations in , 2024, Accepted, 1450-5444, 10.30755/NSJOM.15512 |