The study of solutions for several systems of PDDEs with two complex variables

: The purpose of this article is to describe the properties of the pair of solutions of several systems of Fermat-type partial di ﬀ erential di ﬀ erence equations. Our theorems exhibit the forms of ﬁ nite order trans-cendental entire solutions for these systems, which are some extensions and improvement of the previous theorems given by Xu, Cao, Liu, etc. Furthermore, we give a series of examples to show that the existence conditions and the forms of transcendental entire solutions with ﬁ nite order of such systems are precise.


Introduction
Let us first recall the classical results that the entire solutions of the functional equation (1.1) are = = f a z g a z cos , sin ( ) ( ) was proved by Gross [1], where a z ( ) is an entire function.This simple-looking nonlinear functional equation (1.1) can be called as the Fermat-type functional equation, analogous with the equation + = x y z 2 2 2 in Fermat's last theorem in number theory.As a matter of fact, we can find that the study of the Fermat-type functional equations can be tracked back to more than 60 years ago or even earlier [2,3].
In the past 30 years, there were lots of research focusing on the solutions of functional equation (1.1), readers can refer to [4][5][6][7][8][9][10][11][12][13][14][15][16][17].For example, Khavinson [11] in 1995 proved that any entire solutions of the partial differential equations (1.2) in 2 are necessarily linear.It should be noted that equation (1.2) is called as eiconal equation.Later, Saleeby [18,19] further proved that the entire solution of equation (1.2) is of the form ( ) .After theirs works, Li and co-authors [20][21][22] further discussed a series of deformation forms of Fermat-type partial differential equations and gave a number of important and interesting results about the existence and the forms of solutions for these partial differential equations.
Theorem A. [20, Corollary 2.3] ( ) be arbitrary polynomials in C 2 .Then, f is an entire solution of the equation if and only if = + + f cz cz c 3 is a linear function, where c j 's are constants, and exactly one of the following holds: and P is a constant satisfying that = c P 1 . Then, any transcendental entire solution with finite order of the PDEE ) ), where A is a constant on satisfying , and B is a constant on ; as a special case, whenever = c 0 1 , we have In 2020, the author of this article and his colleagues [33] studied the finite order transcendental entire solutions when equation (1.4) turns to the system of Fermat-type PDDEs and obtained Theorem C.
Theorem C. [33,Theorem 1.3] ) .Then, any pair of transcendental entire solutions with finite order for the system of Fermat-type PDEEs have the following forms: , where = + L z az a z ( ) , B 1 is a constant in , and a c A A , , , 2 2 satisfy one of the following cases: , and = a i , and = a i , and = a i , and = a i To the best of our knowledge, there are few results about the study of systems of this Fermat-type PDDE with several complex variables.Moreover, it appears that the study of such fields has not been addressed in the literature before.Based on these, we are mainly concerned with the solutions of complex Fermat-type PDDEs, and describe the existence and form of the pair of the finite order transcendental solutions of the systems of PDDEs with constant coefficients and where α β μ λ λ λ c c , , , , , , , 1 2 1 2 are constants in .Obviously, equation (1.4) and system (1.5) are the special cases of systems (1.6) and (1.7).The article is organized as follows.We will introduce our main results about the existence and the forms of entire solutions for (1.6) and (1.7) in Section 2, which generalize the previous theorems given by .Meantime, we give a series of examples to explain that our results about the forms of solutions of such systems are precise.The proofs of Theorems 1.6 and 1.7 are given in Sections 4 and 5, respectively.

Results and examples
The first main theorem is about the existence and the forms of the solutions for system (1.6).
, and α β μ λ , , , be nonzero constants in .Let f z z f z z , , , )) be a pair of transcendental entire solution with finite order of system (1.6).Then, f z z f z z , , , )) must satisfy one of the following cases: and one of the following cases:

, then
The study of solutions for several systems of PDDEs with two complex variables  3 where ϑ z 2 ( ) is a finite order period entire function with period c 2 2 , and ∈ a a b b γ D D , , , , , , satisfy (2.1), (2.2) and ; .
The following examples show the existence of transcendental entire solutions of system (1.6).
) is a pair of finite order transcendental entire solution of system (1.6 Example 2.5.Let ) is a pair of finite order transcendental entire solution of system (1.
) is a transcendental entire solution of (1.6

(
) are a pair of finite order transcendental entire solution of the following system: ) must be of the form where ∈ a a b b , , , or The study of solutions for several systems of PDDEs with two complex variables  5 ( ) be a pair of the finite order transcendental entire solutions of system (1.7).Then, f f , ( ) must satisfy one of the following cases: (i) where and one of the following cases: , and where ϑ s 1 ( ) is a finite order period entire functions with period s 2 0 , and a a b b γ D D , , , , , , 2), (2.8), and (2.9) satisfy (2.2), (2.8), and ; satisfy (2.2), (2.8), and .
The following examples show the existence of transcendental entire solutions of system (1.7).
) is a pair of finite order transcendental entire solution of system (1.7 .
) is a pair of finite order transcendental entire solution of system (1.7 ) ) is a pair of finite order transcendental entire solution of system (1.7 Example 2.10.( ) ) is a pair of finite order transcendental entire solution of system (1.7 2 , and ( ) ) , μ λ λ , , 1 2 be nonzero constants, and ) are a pair of finite order transcendental entire solution of the following system: The study of solutions for several systems of PDDEs with two complex variables  7 ) must be of the form where ∈ a a b b , , , ) and 3 Some lemmas The following lemmas play the key role in proving our results.[3] If g and h are entire functions on the complex plane and g h ( ) is an entire function of finite order, then there are only two possible cases: either (a) the internal function h is a polynomial and the external function g is of finite order; or else (b) the internal function h is not a polynomial but a function of finite order, and the external function g is of zero order.
, be meromorphic functions on m such that f 1 is not constant, and , and such that for all r outside possibly a set with finite logarithmic measure, where < λ 1 is a positive number.Then, either , where the simple zero is counted once, and the multiple zero is counted twice.
where ξ is a constant, = θ z j , 1,2 j 2 ( ) are finite order entire functions, and ( ) is a pair of solutions of system ( ) can be expressed as the form of where θ z 2 ( ) are finite order entire period function with the period c 2 2 and γ is a constant such that 1, e e .
D D γc c ξ Proof.Substituting g g , 1 2 into system (3.1),we have which implies Noting that ψ z j 2 ( ) is a polynomial, we will consider two cases as follows.
Case 1. Suppose that ( ) ( ) is a constant.In view of (3.3), it follows that ) is a nonzero constant for = j 1, 2. Set In view of (3.3), we can deduce that where ) are constants and ) are finite order entire period functions with period c ( ) ( ) ( ) The study of solutions for several systems of PDDEs with two complex variables  9 ( ) ( ) such that This leads to where which implies that (3.13) In view of (3.10) and (3.12), we have ( ) be a pair of transcendental entire solutions of finite order for system (1.6).Thus, we will consider the following two cases.
are all nonconstant.Otherwise, if one of these terms is a constant, we can deduce that constant.This is a contradiction.Thus, we can rewrite (1.6) as the form Since f f , 1 2 are entire functions, it follows that not exist zeros and poles.By Lemmas 3.1 and 3.2, there exist two nonconstant polynomials p z q z , ( ) ( ) in e , e .
The above equations lead to In view of (4.17) and (4.18), we can deduce that e .
q z c q z c p z p z 2 In view of (4.16) and (4.17), we have  q z p z c q z q z p z c q z p z c 2 q z p z c q z q z p z c q z Now we will consider four cases as follows.
In view of (4.24), it follows that , where d d , 1 2 are constants in .Thus, it yields that , , , 1 2 1 2 are constants.Substituting p z q z , ( ) ( ) into (4.24),we have The study of solutions for several systems of PDDEs with two complex variables  13 By combining with ≠ λ 0 and ≠ c 0 2 , it follows from (4.25) that ≤ H deg 1 s .Thus, we still write p z q z , ( ) ( ) as the forms of . In view of (4.22)-(4.24),we have L c b b Thus, we can deduce from (4.26) that , solving equations (4.16) and (4.18), we have where ϑ z ϑ z , ( ) ( ) are finite order entire functions and ϕ z ϕ z , ( ) ( ) are polynomials in z 2 .Substituting (4.28) and (4.29) in (4.17 where ϑ z 2 ( ) is a finite order period entire function with period c 2 2 , and γ D D , , , we have where ϑ z 2 ( ) is a finite order period entire function with period c 2 2 , and γ D D , , , solving equations (4.16) and (4.18), we have where ϑ z 2 ( ) is a finite order period entire function with period c 2 2 , and γ D D , , 1 2 satisfy (4.11) and (4.27).Similar to the above argument, we have Thus, it follows that where d d , 1 2 are constants.Hence, we have , which is a contradiction with the assumption of q z ( ) being nonconstant polynomial in 2 .
Thus, it follows that where d d , 1 2 are constants.Hence, we have , which is a contradiction with the assumption of p z ( ) being nonconstant polynomial in 2 .
p z q z c p z q z p z c q z In view of (  ( ) .In view of (4.37), it follows ( ) ( ) are finite order entire functions and ϕ z ϕ z , ( ) be a pair of transcendental entire solutions of finite order for system (1.7).Thus, we will consider the following two cases.
In view of (1.7), it follows that where η 2 is a constant satisfying (4.3).From (5.1) and (5.2), we have This shows that (5.4) then, we have (4.7).The characteristic equations of (5.1) are Using the initial conditions: 1 , and ) with a parameter s, we obtain the following parametric representation for the solutions of the characteristic equations: = z λ t where φ s 1 2 .Similarly, solving equation (5.4), we have where φ s 1 2 .Substituting (5.6) and (5.7) in (5.2) and (5.5), we have φ s s φ s c φ s s φ s c Thus, it yields that where D D , Moreover, from (4.7) and (5.8), we have (4.12).Thus, from (5.6), (5.7), and (5.9), using the same argument as in the proof of Theorem 2.1 (i), we can obtain the conclusions of Theorem 2.2 (i). (ii ( ) is a constant.This is a contradiction.Thus, similar to the argument as in the proof of Theorem 2.1 (ii), there exists two nonconstant polynomials p z q z , ( ) ( ) in 2 such that In view of (5.11) and (5.14), we have The study of solutions for several systems of PDDEs with two complex variables  17 q z q z p z c q z q z q z p z c q z p z c 1 2 1 2 2 q z q z p z c q z q z q z p z c q z 1 2 1 2  where ϑ s ϑ s , ( ) ( ) are finite order entire functions and ϕ s ϕ s , , solving equations (5.11) and (5.13), similar to the argument as in case + ≠ λ a λ a μ p z p z q z c p z q z q z p z c q z 1 2 1 2 , which is a contradiction with the assumption of q z ( ) being nonconstant polynomial in 2 .
Case 3. p z p z q z c p z q z q z p z c q z

Lemma 3 . 1 .
[27,28] For an entire function F on n , ) .For the special case = n 1, f F is the canonical product of Weierstrass.Remark 3.1.Here denote ρ n F ( ) to be the order of the counting function of zeros of F .Lemma 3.2.

2
are constants.Similar to the argument as in Case 1, we can obtain that = of (4.22), (4.23), and (4.36), it follows ) are polynomials in z 2 .Similar to the above argument in Case 1, it follows from (4.39) and (4.40) that completes the proof of Theorem 2.1.□ 5 The proof of Theorem 2.2 Proof.Let f f , 1 2