Citation: Gavin W. ten Tusscher, Marike M. Leijs, Kees Olie, Adri Ilsen, Tom Vulsma, Janna G. Koppe. Findings on prenatal, lactational and later childhood exposure to dioxins and dioxin-like compounds: a review of the Amsterdam-Zaandam cohort 1987-2005[J]. AIMS Environmental Science, 2015, 2(1): 1-20. doi: 10.3934/environsci.2015.1.1
| [1] | Yeyang Jiang, Zhihua Liao, Di Qiu . The existence of entire solutions of some systems of the Fermat type differential-difference equations. AIMS Mathematics, 2022, 7(10): 17685-17698. doi: 10.3934/math.2022974 |
| [2] | Wenju Tang, Keyu Zhang, Hongyan Xu . Results on the solutions of several second order mixed type partial differential difference equations. AIMS Mathematics, 2022, 7(2): 1907-1924. doi: 10.3934/math.2022110 |
| [3] | Hong Yan Xu, Zu Xing Xuan, Jun Luo, Si Min Liu . On the entire solutions for several partial differential difference equations (systems) of Fermat type in $\mathbb{C}^2$. AIMS Mathematics, 2021, 6(2): 2003-2017. doi: 10.3934/math.2021122 |
| [4] | 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 |
| [5] | Minghui Zhang, Jianbin Xiao, Mingliang Fang . Entire solutions for several Fermat type differential difference equations. AIMS Mathematics, 2022, 7(7): 11597-11613. doi: 10.3934/math.2022646 |
| [6] | Andrey Muravnik . Nonclassical dynamical behavior of solutions of partial differential-difference equations. AIMS Mathematics, 2025, 10(1): 1842-1858. doi: 10.3934/math.2025085 |
| [7] | Yong Liu, Chaofeng Gao, Shuai Jiang . On meromorphic solutions of certain differential-difference equations. AIMS Mathematics, 2021, 6(9): 10343-10354. doi: 10.3934/math.2021599 |
| [8] | Zhenguang Gao, Lingyun Gao, Manli Liu . Entire solutions of two certain types of quadratic trinomial q-difference differential equations. AIMS Mathematics, 2023, 8(11): 27659-27669. doi: 10.3934/math.20231415 |
| [9] | Muhammad Sarwar, Aiman Mukheimer, Syed Khayyam Shah, Arshad Khan . Existence of solutions of fractal fractional partial differential equations through different contractions. AIMS Mathematics, 2024, 9(5): 12399-12411. doi: 10.3934/math.2024606 |
| [10] | Yanyan Cui, Chaojun Wang . Dirichlet and Neumann boundary value problems for bi-polyanalytic functions on the bicylinder. AIMS Mathematics, 2025, 10(3): 4792-4818. doi: 10.3934/math.2025220 |
As is known to all, Nevanlinna theory is an important tool in studying the value distribution of meromorphic solutions on complex differential equations [18]. In recent, with the development of difference analogues of Nevanlinna theory in $ \mathbb{C} $, many scholars paid consideration attention to considering the properties on complex difference equations, by using the difference analogue of the logarithmic derivative lemma given by Chiang and Feng [3], Halburd and Korhonen [8], respectively. In particular, Liu et al. [22,23,24] investigated the existence of entire solutions with finite order of the Fermat type differential-difference equations
|
$ f′(z)2+f(z+c)2=1, $
|
(1.1) |
|
$ f′(z)2+[f(z+c)−f(z)]2=1. $
|
(1.2) |
They proved that the transcendental entire solutions with finite order of Eq (1.1) must satisfy $ f(z) = \sin(z\pm Bi) $, where $ B $ is a constant and $ c = 2k\pi $ or $ c = (2k+1)\pi $, $ k $ is an integer, and the transcendental entire solutions with finite order of Eq (1.2) must satisfy $ f(z) = 1 2\sin(2z+Bi) $, where $ c = (2k+1)\pi $, $ k $ is an integer, and $ B $ is a constant.
The study of complex differential-difference equations in $ \mathbb{C} $ can be traced back to Naftalevich's research [28,29]. He used operator theory and iteration method to consider the meromorphic solutions on complex differential-difference equations. But recently, by using Nevanlinna theory, a number of results on complex differential-difference equations in $ \mathbb{C} $ are rapidly obtained until now, readers can refer to [25,31,32].
Corresponding to Eq (1.1), Gao [5] in 2016 discussed the form of solutions for a class of system of differential-difference equation
|
$ {[f′1(z)]2+f2(z+c)2=1,[f′2(z)]2+f1(z+c)2=1, $
|
(1.3) |
and obtained
Theorem A (see [5,Theorem 1.1]). Suppose that $ (f_1, f_2) $ is a pair of finite order transcendental entire solutions for the system of differential-difference Eq (1.3). Then $ (f_1, f_2) $ satisfies
| $ (f_1, f_2) = (\sin(z-bi), \sin(z-b_1i)) \; or\; (f_1(z), f_2(z)) = (\sin(z+bi), \sin(z+b_1i)), $ |
where $ b, b_1 $ are constants, and $ c = k\pi $, $ k $ is a integer.
Here these conclusions are stated in several complex variables as follows. In many previous articles [13,16,19,25,26,35] about Fermat-type partial differential equations with several complex variables, G. Khavinson [16] pointed out that any entire solutions of the partial differential equations $ \left(\frac{\partial f}{\partial z_1}\right)^2+\left(\frac{\partial f}{\partial z_2}\right)^2 = 1 $ in $ \mathbb{C}^2 $ are necessarily linear. This partial differential equations in real variable case occur in the study of characteristic surfaces and wave propagation theory, and it is the two dimensional eiconal equation, one of the main equations of geometric optics (see [4,6]). Later, Li [20,21] further discussed a series of partial differential equations with more general forms including $ \left(\frac{\partial f}{\partial z_1}\right)^2+\left(\frac{\partial f}{\partial z_2}\right)^2 = e^{g} $, $ \left(\frac{\partial f}{\partial z_1}\right)^2+\left(\frac{\partial f}{\partial z_2}\right)^2 = p $, etc., where $ g, p $ are polynomials in $ \mathbb{C}^2 $, and gave a number of important and interesting results about the existence and the forms of solutions for these partial differential equations.
In 2012, Korhonen [17,Theorem 3.1] gave a logarithmic difference lemma for meromorphic functions in several variables of hyper order strictly less that 2/3. In 2016, Cao and Korhonen [2] improved it to the case for meromorphic functions with hyper order $ < 1 $ in several variables. In 2018, Xu and Cao [39] investigated the existence of the entire and meromorphic solutions for some Fermat-type partial differential-difference equations by utilizing the Nevanlinna theory and difference Nevanlinna theory of several complex variables [2,17], and obtained:
Theorem B (see [39,Theorem 1.1]). Let $ c = (c_1, c_2)\in\mathbb{C}^2 $. Then the Fermat-type partial differential-difference equation
| $ \left(\frac{\partial f(z_1, z_2)}{\partial z_1}\right)^n+f(z_1+c_1, z_2+c_2)^m = 1 $ |
doesn't have any transcendental entire solution with finite order, where $ m $ and $ n $ are two distinct positive integers.
Theorem C (see [39,Theorem 1.2]). Let $ c = (c_1, c_2)\in\mathbb{C}^2 $. Then any transcendental entire solutions with finite order of the partial differential-difference equation
| $ \left(\frac{\partial f(z_1, z_2)}{\partial z_1}\right)^2+f(z_1+c_1, z_2+c_2)^2 = 1 $ |
has the form of $ f(z_1, z_2) = \sin(Az_1+B) $, where $ A $ is a constant on $ \mathbb{C} $ satisfying $ Ae^{iAc_1} = 1 $, and $ B $ is a constant on $ \mathbb{C} $; in the special case whenever $ c_1 = 0 $, we have $ f(z_1, z_2) = \sin(z_1+B) $.
Inspired by the above theorems, the authors [37] in 2020 extended the results of Theorems A, B from the complex Fermat types partial differential difference equations to the Fermat types system of partial differential-difference equations and obtained:
Theorem D (see [37,Theorem 1.1]). Let $ c = (c_1, c_2)\in\mathbb{C}^2 $, and $ m_j, n_j $ $ (j = 1, 2) $ be positive integers. If the following system of Fermat-type partial differential-difference equations
|
$ {(∂f1(z1,z2)∂z1)n1+f2(z1+c1,z2+c2)m1=1,(∂f2(z1,z2)∂z1)n2+f1(z1+c1,z2+c2)m2=1, $
|
(1.4) |
satisfies one of the conditions
$ (i) $ $ m_1m_2 > n_1n_2 $;
$ (ii) $ $ m_j > \frac{n_j}{n_j-1} $ for $ n_j\geq 2 $, $ j = 1, 2 $.
Then system (1.4) does not have any pair of transcendental entire solution with finite order.
Theorem E (see [37,Theorem 1.3]). Let $ c = (c_1, c_2)\in\mathbb{C}^2 $. Then any pair of transcendental entire solutions with finite order for the system of Fermat-type partial differential-difference equations
|
$ \left\{ (∂f1(z1,z2)∂z1)2+f2(z1+c1,z2+c2)2=1,(∂f2(z1,z2)∂z1)2+f1(z1+c1,z2+c2)2=1 \right. $
|
have the following forms
| $ (f_1(z), f_2(z)) = \left(\frac{e^{L(z)+B_1}+e^{-(L(z)+B_1)}}{2}, \frac{A_{21}e^{L(z)+B_1}+A_{22}e^{-(L(z)+B_1)}}{2}\right), $ |
where $ L(z) = a_1z_1+a_2z_2 $, $ B_1 $ is a constant in $ \mathbb{C} $, and $ a_1, c, A_{21}, A_{22} $ satisfy one of the following cases
$ (i) $ $ A_{21} = -i $, $ A_{22} = i $, and $ a_1 = i $, $ L(c) = (2k+\frac{1}{2})\pi i $, or $ a_1 = -i $, $ L(c) = (2k-\frac{1}{2})\pi i $;
$ (ii) $ $ A_{21} = i $, $ A_{22} = -i $, and $ a_1 = i $, $ L(c) = (2k-\frac{1}{2})\pi i $, or $ a_1 = -i $, $ L(c) = (2k+\frac{1}{2})\pi i $;
$ (iii) $ $ A_{21} = 1 $, $ A_{22} = 1 $, and $ a_1 = i $, $ L(c) = 2k\pi i $, or $ a_1 = -i $, $ L(c) = (2k+1)\pi i $;
$ (iv) $ $ A_{21} = -1 $, $ A_{22} = -1 $, and $ a_1 = i $, $ L(c) = (2k+1)\pi i $, or $ a_1 = -i $, $ L(c) = 2k\pi i $.
From Theorems D and E, we can see that there only contains the partial differentiation of the first variable $ z_1 $ of the unknown functions $ f_1, f_2 $ in those systems of partial differential difference equations. Naturally, a question arises: What will happen when the system of the partial differential-difference equations include both the difference $ f_j(z+c) $ and $ \frac{\partial f_j(z_1, z_2)}{\partial z_1}, \frac{\partial f_j(z_1, z_2)}{\partial z_2}, (j = 1, 2) $? In the past two decades, in spite of a number of important and meaningful results about the complex difference equation of single variable and the complex Fermat difference equation were obtained (can be found in [9,10,11,24,31,33]), but as far as we know, there are few results concerning the complex differential and complex difference equation in several complex variables. Further more, it appears that the study of systems of this Fermat type equations in several complex variables has been less addressed in the literature before.
The main purpose of this paper is concerned with the description of the transcendental entire solutions for some Fermat-type equations systems which include both difference operator and two kinds of partial differentials by utilizing the Nevanlinna theory and difference Nevanlinna theory of several complex variables [2,17]. We obtained some results about the existence and the forms of the transcendental entire solutions of some Fermat type systems of partial differential difference equations in $ \mathbb{C}^2 $, which improve the previous results given by Xu and Cao, Xu, Liu and Li, Gao [5,37,38,39,40].
Here and below, let $ z+w = (z_1+w_1, z_2+w_2) $ for any $ z = (z_1, z_2) $ and $ w = (w_1, w_2) $. Now, our main results of this paper are stated below.
Theorem 2.1. Let $ c = (c_1, c_2)\in\mathbb{C}^2 $, and $ m_j, n_j\; (j = 1, 2) $ be positive integers. If the following system of Fermat-type partial differential-difference equations
|
$ {(∂f1(z1,z2)∂z1+∂f1(z1,z2)∂z2)m1+f2(z1+c1,z2+c2)n1=1,(∂f2(z1,z2)∂z1+∂f2(z1,z2)∂z2)m2+f1(z1+c1,z2+c2)n2=1, $
|
(2.1) |
satisfies one of the conditions
$ (i) $ $ n_1n_2 > m_1m_2 $;
$ (ii) $ $ m_j > \frac{n_j}{n_j-1} $ for $ n_j\geq 2 $, $ j = 1, 2 $.
Then system (2.1) does not admit any pair of transcendental entire solution with finite order.
Remark 2.1. Here, $ (f, g) $ is called as a pair of finite order transcendental entire solutions for system
|
$ \left\{ fm1+gn1=1,fm2+gn2=1, \right. $
|
if $ f, g $ are transcendental entire functions and $ \rho(f, g) = \max\{\rho(f), \rho(g)\} < \infty $.
The following examples show system (2.1) admits a transcendental entire solution of finite order when $ m_1 = m_2 = 2 $ and $ n_1 = n_2 = 1 $.
Example 2.1. Let
|
$ \left\{ f1(z1,z2)=5−z214+12(z2−z1)(z1−1)−14[(z2−z1)−1]2+eπi(z2−z1)(2z1−z2)−e2πi(z2−z1),f2(z1,z2)=5−z214+12(z2−z1)(z1−1)−14[(z2−z1)−1]2−eπi(z2−z1)(2z1−z2)−e2πi(z2−z1). \right. $
|
Then $ \rho(f_1, f_2) = 1 $ and $ (f_1, f_2) $ satisfies system (2.1) with $ (c_1, c_2) = (1, 2) $, $ m_1 = m_2 = 2 $ and $ n_1 = n_2 = 1 $.
Theorem 2.2. Let $ c = (c_1, c_2)\in\mathbb{C}^2 $. If $ (f_1, f_2) $ is a pair of transcendental entire solutions with finite order for the system of Fermat-type difference equations
|
$ {(∂f1(z1,z2)∂z1+∂f1(z1,z2)∂z2)2+f2(z1+c1,z2+c2)2=1,(∂f2(z1,z2)∂z1+∂f2(z1,z2)∂z2)2+f1(z1+c1,z2+c2)2=1. $
|
(2.2) |
Then $ (f_1, f_2) $ is of the following forms
|
$ (f1,f2)=(eϕ(z)+B0−e−ϕ(z)−B02(a1+a2),A21eϕ(z)+B0+A22e−ϕ(z)−B02), $
|
where $ \phi(z) = L(z)+H(c_2z_1-c_1z_2) $, $ L(z) = a_1z_1+a_2z_2 $, $ a_1, a_2, B_0 $ is a constant in $ \mathbb{C} $, $ H $ is a polynomial in $ \mathbb{C} $, and
| $ (c_1-c_2)H'\equiv0, \; \; \; (a_1+a_2)^2 = -1, \; \; \; e^{2L(c)} = \pm1, $ |
and $ c, A_{21}, A_{22} $ satisfy one of the following cases
$ (i) $ $ L(c) = 2k\pi i $, here and below, $ k\in \mathbb{Z} $, $ A_{21} = -i $ and $ A_{22} = i $;
$ (ii) $ $ L(c) = (2k+1)\pi i $, $ A_{21} = i $ and $ A_{22} = -i $;
$ (iii) $ $ L(c) = (2k+\frac{1}{2})\pi i $, $ A_{21} = -1 $ and $ A_{22} = -1 $;
$ (iv) $ $ L(c) = (2k-\frac{1}{2})\pi i $, $ A_{21} = 1 $ and $ A_{22} = 1 $.
Here, we only list the following examples to explain the existence of transcendental entire solutions with finite order for system (2.2).
Example 2.2. Let $ a = (a_1, a_2) = (2i, -i) $, $ H = 4\pi^2(z_1-z_2)^2 $, $ A_{21} = -i, A_{22} = i $ and $ B_0 = 0 $. That is
| $ (f_1, f_2) = \left(f(z_1, z_2), f(z_1, z_2)\right), $ |
where
| $ f(z_1, z_2) = \frac{e^{i(2z_1-z_2)+4\pi^2(z_1-z_2)^2}-e^{-i(2z_1-z_2)-4\pi^2(z_1-z_2)^2}}{2i}. $ |
Thus, $ \rho(f_1, f_2) = 2 $ and $ (f_1, f_2) $ satisfies system (2.2) with $ (c_1, c_2) = (2\pi, 2\pi) $.
Example 2.3. Let $ a = (a_1, a_2) = (2i, -i) $, $ H = \pi^n(z_1-z_2)^n, n\in \mathbb{Z}_+ $, $ A_{21} = i, A_{22} = -i $ and $ B_0 = 0 $. That is
| $ (f_1, f_2) = \left(f(z_1, z_2), -f(z_1, z_2)\right), $ |
where
| $ f(z_1, z_2) = \frac{e^{i(2z_1-z_2)+\pi^n(z_1-z_2)^n}-e^{-i(2z_1-z_2)-\pi^n(z_1-z_2)^n}}{2i}. $ |
Thus, $ \rho(f_1, f_2) = n $ and $ (f_1, f_2) $ satisfies system (2.2) with $ (c_1, c_2) = (\pi, \pi) $.
Remark 2.2. From the conclusions of Theorems C and E, there only exists finite order transcendental entire solutions with growth order $ \rho(f_1, f_2) = 1(\rho(f) = 1) $. However, in view of Theorem 2.2, we can see that there exists transcendental entire solution of system (2.2) with growth order $ \rho(f_1, f_2) > 1 $, for example, $ \rho(f_1, f_2) = 2 $ in Example 2.2 and $ \rho(f_1, f_2) = n, n\in \mathbb{Z}_+ $ in Example 2.3, these properties are quite different from the previous results. Hence, our results are some improvements of the previous theorems given by Xu, Liu and Li [37], Xu and Cao[39], Liu Cao and Cao [23].
To state our last result, throughout this paper, let $ s_1 = z_2-z_1 $, $ G_1(s_1) $, $ G_2(s_1) $ be the finite order entire period functions in $ s_1 $ with period $ 2s_0 = 2(c_2-c_1) $, where $ c_1\neq c_2 $, $ G_1(s_1) $, $ G_2(s_1) $ can not be the same at every time occurrence.
Theorem 2.3. Let $ c = (c_1, c_2)\in \mathbb{C}^2 $ and $ c_1\neq c_2 $. Then any pair of transcendental entire solutions with finite order for the system of Fermat-type partial differential-difference equations
|
$ {(∂f1(z1,z2)∂z1+∂f1(z1,z2)∂z2)2+[f2(z1+c1,z2+c2)−f1(z1,z2)]2=1,(∂f2(z1,z2)∂z1+∂f2(z1,z2)∂z2)2+[f1(z1+c1,z2+c2)−f2(z1,z2)]2=1, $
|
(2.3) |
are one of the following forms
(i)
| $ (f_1, f_2) = \left(G_1(s_1)+A_0s_1, G_2(s_1)+A_0s_1\right), $ |
where $ A_0 = \frac{\xi_2+\xi_1}{2(c_2-c_1)} $, $ G_2\left(s_1+s_0\right) = G_1(s_1)+\frac{\xi_2-\xi_1}{2}, $ and $ \xi_1^2 = \xi_2^2 = 1 $;
(ii)
| $ (f_1, f_2) = \left(\xi_3z_1+A_0s_1+G_1(s_1), \xi_3z_1+A_0s_1+G_2(s_1)\right), $ |
where $ A_0 = \frac{\xi_1+\xi_2-2c_1\xi_3}{2(c_2-c_1)} $, $ G_2\left(s_1+s_0\right) = G_1(s_1)+\frac{\xi_1-\xi_2}{2} $, $ \xi_1^2+\xi_3^2 = 1 $, and $ \xi_1 = \pm\xi_2; $
(iii)
| $ (f_1, f_2) = \left(\frac{e^{L(z)+B_1}-e^{-(L(z)+B_1)}}{-4i}+G_1(s_1), \frac{e^{L(z)+B_2}-e^{-(L(z)+B_2)}}{-4i}+G_2(s_1)\right) $ |
where
| $ \; a_1+a_2 = -2i, \; e^{2L(c)} = 1, \; e^{B_1-B_2} = -e^{L(c)}, \; G_2\left(s_1+s_0\right) = G_1(s_1); $ |
$ (iv) $
|
$ (f1,f2)=(z1eL(z)+B1+e−L(z)−B12+G3(s1),D0z1eL(z)+B1+e−L(z)−B12+G4(s1)), $
|
where
| $ a_1+a_2 = 0, \; e^{2L(c)} = 1, $ |
and $ G_3(s_1), G_4(s_1), D_0 $ satisfying one of the following cases:
$ (iv_1) $ $ L(c) = 2k\pi i $, $ D_0 = 1 $,
|
$ G3(s1)=G1(s1)−c1+i2s0s1ea2s1+B1−c1−i2s0s1e−a2s1−B1, $
|
and
|
$ G4(s1)=G2(s1)−c1+i2s0s1ea2s1+B1−c1−i2s0s1e−a2s1−B1; $
|
$ (iv_2) $ $ L(c) = (2k+1)\pi i $, $ D_0 = -1 $,
|
$ G3(s1)=G1(s1)−c1+i2s0s1ea2s1+B1−c1−i2s0s1e−a2s1−B1, $
|
and
|
$ G4(s1)=G2(s1)+c1+i2s0s1ea2s1+B1+c1−i2s0s1e−a2s1−B1; $
|
where $ G_2(s+s_0) = G_1(s) $.
Some examples are listed to exhibit the existence of solutions for system (2.3).
Example 2.4. Let $ G_1(s_1) = -G_2(s_1) = e^{-\pi is_1} $, and $ \xi_1 = -1 $, $ \xi_2 = 1 $. Then it follows that $ A_0 = 0 $ and
| $ (f_1(z_1, z_2), f_2(z_1, z_2)) = \left(e^{-\pi i(z_2-z_1)}, -e^{-\pi i(z_2-z_1)}-1\right). $ |
Thus, $ \rho(f_1, f_2) = 1 $ and $ (f_1, f_2) $ satisfies the system (2.3) with $ (c_1, c_2) = (1, 2) $.
Example 2.5. Let $ \xi_1 = 0 $, $ \xi_2 = 0 $, $ \xi_3 = 1 $ and $ G_1(s_1) = G_2(s_1) = e^{2\pi i s_1} $. Then it follows that $ A_0 = -1 $ and
| $ (f_1, f_2) = \left(2z_1-z_2+e^{2\pi i(z_2-z_1)}, 2z_1-z_2+e^{2\pi i(z_2-z_1)}\right). $ |
Thus, $ \rho(f_1, f_2) = 1 $ and $ (f_1, f_2) $ satisfies system (2.3) with $ (c_1, c_2) = (1, 2) $.
Example 2.6. Let $ L(z) = -i(z_1+z_2) $ and $ G_1(s_1) = -G_2(s_1) = e^{\frac{i}{4}s_1} $. That is
| $ (f_1, f_2) = \left(\frac{e^{-L(z)}-e^{L(z)}}{4i}+e^{\frac{i}{4}(z_2-z_1)}, -\frac{e^{-L(z)}-e^{L(z)}}{4i}-e^{\frac{i}{4}(z_2-z_1)}\right). $ |
Thus, $ \rho(f) = 1 $ and $ (f_1, f_2) $ satisfies system (2.3) with $ (c_1, c_2) = (-\pi, 3\pi) $.
Example 2.7. Let $ a_1 = i $, $ a_2 = -i $, $ L(z) = i(z_1-z_2) $, $ G_1(s_1) = G_2(s_1) = e^{\frac{i}{4}s_1} $ and $ B_1 = 0 $. Then, it follows that
| $ f_1(z_1, z_2) = \frac{z_1}{2}(e^{L(z)}+e^{-L(z)})-\frac{-2\pi+i}{8\pi}(z_2-z_1)e^{L(z)}-\frac{-2\pi-i}{8\pi}(z_2-z_1)e^{-L(z)}+e^{\frac{i}{4}(z_2-z_1)}, $ |
| $ f_2(z_1, z_2) = \frac{z_1}{2}(e^{L(z)}+e^{-L(z)})-\frac{-2\pi+i}{8\pi}(z_2-z_1)e^{L(z)}-\frac{-2\pi-i}{8\pi}(z_2-z_1)e^{-L(z)}-e^{\frac{i}{4}(z_2-z_1)}, $ |
Thus, $ \rho(f_1, f_2) = 1 $ and $ (f_1, f_2) $ satisfies system (2.3) with $ (c_1, c_2) = (-2\pi, 2\pi) $.
Example 2.8. Let $ a_1 = 1 $, $ a_2 = -1 $, $ L(z) = z_1-z_2 $, $ G_1(s_1) = -G_2(s_1) = e^{s_1} $ and $ B_1 = 0 $. Then, it follows that
| $ f_1(z_1, z_2) = \frac{z_1}{2}(e^{L(z)}+e^{-L(z)})+\frac{\pi -2}{4\pi }(z_2-z_1)e^{L(z)}+\frac{\pi +2}{4\pi}(z_2-z_1)e^{-L(z)}+e^{z_2-z_1}, $ |
| $ f_2(z_1, z_2) = -\frac{z_1}{2}(e^{L(z)}+e^{-L(z)})-\frac{\pi-2}{4\pi}(z_2-z_1)e^{L(z)}-\frac{\pi+2}{4\pi}(z_2-z_1)e^{-L(z)}-e^{z_2-z_1}. $ |
Thus, $ (f_1, f_2) $ satisfies system (2.3) with $ (c_1, c_2) = (-\frac{1}{2}\pi i, \frac{1}{2}\pi i) $.
From Theorems 2.1–2.3, we can see that our results are some extension of the previous results given by Xu and Cao [39] from the equations to the systems, and some supplements of the results given by Xu, Liu and Li [37]. More importantly, Examples 2.2 and 2.3 show that system (2.2) can admit the transcendental entire solutions of any positive integer order. However, the conclusions of Theorem C and Theorem E showed that the order of the transcendental entire solutions of the equations must be equal to 1. In fact, this is a very significant difference. Finally, one can find that we only focus on the finite-order transcendental entire solutions of systems (1.4)–(2.2) in this article; thus, the following question can be raised naturally:
Question 3.1. How should the meromorphic solutions of systems (2.2) and (2.3) be characterized?
Similar to the argument as in the proof of Theorems 1.1 and 1.3 in Ref. [37], one can obtain the conclusions of Theorems 2.1 and 2.2 easily. Thus, we only give the proof of Theorem 2.3 as follow. However, the following lemmas play the key roles in proving Theorem 2.3.
Lemma 4.1. ([34,36]) For an entire function $ F $ on $ \mathbb{C}^n $, $ F(0)\neq0 $ and put $ \rho(n_F) = \rho < \infty $. Then there exist a canonical function $ f_F $ and a function $ g_F\in \mathbb{C}^n $ such that $ F(z) = f_F (z)e^{g_F (z)} $. For the special case $ n = 1 $, $ f_F $ is the canonical product of Weierstrass.
Remark 4.1. Here, denote $ \rho(n_F) $ to be the order of the counting function of zeros of $ F $.
Lemma 4.2. ([30]) If $ g $ and $ h $ are entire functions on the complex plane $ \mathbb{C} $ 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.
Lemma 4.3. ([15] or [14,Lemma 3.1]) Let $ f_j(\not\equiv0), j = 1, 2, 3 $, be meromorphic functions on $ \mathbb{C}^m $ such that $ f_1 $ is not constant. If $ f_1+f_2+f_3 = 1 $, and if
| $ \sum\limits_{j = 1}^3\left\{N_2(r, \frac{1}{f_j})+2\overline{N}(r, f_j)\right\} < \lambda T(r, f_1)+O(\log^+T(r, f_1)), $ |
for all $ r $ outside possibly a set with finite logarithmic measure, where $ \lambda < 1 $ is a positive number, then either $ f_2 = 1 $ or $ f_3 = 1 $.
Remark 4.2. Here, $ N_2(r, \frac{1}{f}) $ is the counting function of the zeros of $ f $ in $ |z|\leq r $, where the simple zero is counted once, and the multiple zero is counted twice.
Lemma 4.4. Let $ c = (c_1, c_2) $ be a constant in $ \mathbb{C}^2 $, $ c_1\neq0, c_2\neq0 $ and $ c_2\neq c_1 $. Let $ p(z), \; q(z) $ be two polynomial solutions of the equation
|
$ ∂h∂z1+∂h∂z2=γ0, $
|
(4.1) |
and $ q(z+c)-p(z) = \zeta_1, p(z+c)-q(z) = \zeta_2 $, where $ \zeta_1, \zeta_2, \gamma_0 \in \mathbb{C} $, then $ p(z) = L(z)+B_1 $, $ q(z) = L(z)+B_2 $, where $ L(z) = a_1z_2+a_2z_2 $, $ a_1, a_2, B_1, B_2 \in \mathbb{C} $.
Proof. The characteristic equations of the Eq (4.1) are
| $ \frac{dz_1}{dt} = 1, \; \; \; \frac{dz_2}{dt} = 1, \; \; \; \frac{dh}{dt} = \gamma_0, $ |
Using the initial conditions: $ z_1 = 0, z_2 = s_1 $, and $ h = h(0, s_1): = h_0(s_1) $ with a parameter. Then $ z_1 = t $, $ z_2 = t+s_1 $, and $ h = \int_0^t \gamma_0dt+h_0(s_1) = \gamma_0 t+h_0(s) $, where $ s_1 = z_2-z_1 $, $ h_0(s_1) $ is a function in $ s_1 $. Since $ p(z), q(z) $ are the solutions of (4.1), then it yields that
|
$ p(z1,z2)=p(t,s1)=γ0t+h1(s1),q(z1,z2)=p(t,s1)=γ0t+h2(s1), $
|
(4.2) |
where $ h_1(s_1), h_2(s_1) $ are two polynomials in $ s_1 $. Substituting (4.2) into $ q(z+c)-p(z) = \zeta_1, q(z)-p(z+c) = \zeta_2 $, it leads to
|
$ h2(s1+s0)−h1(s1)=ζ1−γ0c1,h1(s1+s0)−h2(s1)=ζ2−γ0c1. $
|
(4.3) |
Hence, we have
|
$ h1(s1+2s0)=h1(s1)+ε0,h2(s1+2s0)=h2(s1)+ε0, $
|
(4.4) |
where $ \varepsilon_0 = \zeta_1+\zeta_2-2\gamma_0 c_1 $. The fact that $ h_1(s_1), h_2(s_1) $ are polynomials leads to
|
$ h1(s1)=γ1s1+B1,h2(s1)=γ1s1+B2, $
|
(4.5) |
where $ \gamma_1 = \frac{\varepsilon_0}{2s_0} $, $ B_1, B_2\in \mathbb{C} $. By combining with (4.2) and (4.5), it follows that
|
$ p(z1,z2)=γ0t+γ1s1+B1=γ0z1+γ1(z1−z2)+B1=a1z1+a2z2+B1,q(z1,z2)=γ0t+γ1s1+B2=γ0z1+γ1(z1−z2)+B2=a1z1+a2z2+B2, $
|
where $ a_1 = \gamma_0+\gamma_1, \; a_2 = -\gamma_1 $.
Therefore, Lemma 4.4 is proved.
The Proof of Theorem 2.3: Let $ (f_1, f_2) $ be a pair of transcendental entire functions with finite order satisfying system (2.3). In view of [7,27], we know that the entire solutions of the Fermat type functional equation $ f^2+g^2 = 1 $ are $ f = \cos a(z), g = \sin a(z) $, where $ a(z) $ is an entire function. Hence, we only consider the following cases.
(i) Suppose that $ \frac{\partial f_1(z_1, z_2)}{\partial z_1}+\frac{\partial f_1(z_1, z_2)}{\partial z_2} = 0 $. Then it follows from (2.3) that
|
$ f2(z1+c1,z2+c2)−f1(z1,z2)≡ξ1,ξ21=1, $
|
(4.6) |
and
|
$ ∂f2(z+c)∂z1+∂f2(z+c)∂z2=∂f1(z1,z2)∂z1+∂f1(z1,z2)∂z2=0. $
|
(4.7) |
In view of (4.6), (4.7) and (2.3), it yields that
|
$ ∂f2∂z1+∂f2∂z2=0,f1(z1+c1,z2+c2)−f2(z1,z2)≡ξ2,ξ22=1. $
|
(4.8) |
By solving the equations $ \frac{\partial f_j}{\partial z_1}+\frac{\partial f_j}{\partial z_2} = 0 $ $ (j = 1, 2) $, we have
|
$ f1(z1,z2)=g1(s1):=g1(z2−z1),f2(z1,z2)=g2(s1):=g2(z2−z1), $
|
(4.9) |
where $ g_1, g_2 $ are transcendental entire functions of finite order. Substituting (4.9) into (4.6), (4.8), it yields that
|
$ f2(z1+c1,z2+c2)−f1(z1,z2)=g2(s1+s0)−g1(s1)=ξ1,f1(z1+c1,z2+c2)−f2(z1,z2)=g1(s1+s0)−g2(s1)=ξ2. $
|
which implies
|
$ g1(s1+2s0)=g1(s1)+ξ1+ξ2,g2(s1+2s0)=g2(s1)+ξ1+ξ2, $
|
(4.10) |
|
$ g2(s1+s0)=g1(s1)+ξ1,g1(s1+s0)=g2(s1)+ξ2. $
|
(4.11) |
Thus, in view of (4.10) and (4.11), we conclude that
| $ g_1(s_1) = G_1(s_1)+A_0s_1, \; \; \; g_2(s_1) = G_2(s_1)+B_0s_1, $ |
where $ A_0 = B_0 = \frac{\xi_1+\xi_2}{2(c_2-c_1)} $, $ G_1(s_1), G_2(s_1) $ are the finite order entire period functions with period $ 2s_0 $, and
| $ G_2(s_1+s_0) = G_1(s_1)+\frac{\xi_1-\xi_2}{2}, \; \; G_1(s_1+s_0) = G_2(s_1)+\frac{\xi_2-\xi_1}{2}. $ |
Thus, this proves the conclusions of Theorem 2.3 (i).
(ii) Suppose that
|
$ ∂f1(z1,z2)∂z1+∂f1(z1,z2)∂z2=ξ3,ξ3≠0. $
|
(4.12) |
In view of (2.3), we conclude that
|
$ ∂f2(z1,z2)∂z1+∂f2(z1,z2)∂z2=ξ3, $
|
(4.13) |
|
$ f2(z1+c1,z2+c2)−f1(z1,z2)≡ξ1,ξ21+ξ23=1, $
|
(4.14) |
|
$ f1(z1+c1,z2+c2)−f2(z1,z2)≡ξ2,ξ22+ξ23=1. $
|
(4.15) |
The characteristic equations of Eq (4.12) are
| $ \frac{dz_1}{dt} = 1, \; \; \; \frac{dz_2}{dt} = 1, \; \; \; \frac{df_1}{dt} = \xi_3, $ |
Using the initial conditions: $ z_1 = 0, z_2 = s_1 $, and $ f_1 = f_1(0, s_1): = g_1(s_1) $ with a parameter. Thus, we have $ z_1 = t $, $ z_2 = t+s_1 $ and $ f_1(t, s_1) = \int_0^t \xi_3dt+g_1(s_1) = \xi_3t+g_1(s_1) $, where $ s_1 = z_2-z_1 $, $ g_1(s_1) $ is a transcendental entire function of finite order. Hence, it yields that
|
$ f1(z1,z2)=ξ3z1+g1(z2−z1). $
|
(4.16) |
Similar to the same argument for Eq (4.13), we have
|
$ f2(z1,z2)=ξ3z1+g2(z2−z1), $
|
(4.17) |
where $ g_2(s_1) $ is a transcendental entire function of finite order.
Substituting (4.16), (4.17) into (4.14), (4.15), we conclude that
|
$ g2(s1+s0)−g1(s1)=−c1ξ3+ξ1, $
|
(4.18) |
|
$ g1(s1+s0)−g2(s)=−c1ξ3+ξ2. $
|
(4.19) |
which lead to
|
$ g2(s1+s0)=g1(s1)+ξ1−c1ξ3,g1(s1+s0)=g2(s1)+ξ2−c1ξ3,g1(s1+2s0)=g1(s1)+ξ1+ξ2−2c1ξ3,g2(s1+2s0)=g2(s1)+ξ1+ξ2−2c1ξ3. $
|
Since $ g_1(s_1), g_2(s_1) $ are the transcendental entire functions of finite order. then we have
| $ g_1(s_1) = G_1(s_1)+A_0s_1, \; \; \; g_2(s_1) = G_2(s_1)+B_0s_1, $ |
where $ A_0 = B_0 = \frac{\xi_1+\xi_2-2c_1\xi_3}{2(c_2-c_1)} $, $ G_1(s_1), G_2(s_1) $ are the finite order entire period functions with period $ 2s_0 $, and
| $ G_2(s_1+s_0) = G_1(s_1)+\frac{\xi_1-\xi_2}{2}. $ |
Thus, this proves the conclusions of Theorem 2.3 (ii).
(iii) If $ \frac{\partial f_1}{\partial z_1}+\frac{\partial f_1}{\partial z_2} $ is transcendental, then $ f_2(z_1+c_1, z_2+c_2)-f_1(z_1, z_2) $ is transcendental. Here, we can deduce that $ f_1(z_1+c_1, z_2+c_2)-f_2(z_1, z_2) $ and $ \frac{\partial f_2}{\partial z_1}+\frac{\partial f_2}{\partial z_2} $ are transcendental.
Suppose that $ f_1(z_1+c_1, z_2+c_2)-f_2(z_1, z_2) $ is not transcendental. Since $ \frac{\partial f_1}{\partial z_1}+\frac{\partial f_1}{\partial z_2} $ is transcendental, then $ \frac{\partial f_2}{\partial z_1}+\frac{\partial f_2}{\partial z_2} $ is transcendental. In view of (2.3), it yields that $ f_1(z_1+c_1, z_2+c_2)-f_2(z_1, z_2) $ is transcendental, a contradiction.
Suppose that $ \frac{\partial f_2}{\partial z_1}+\frac{\partial f_2}{\partial z_2} $ is not transcendental. In view of (2.3), it follows that $ f_1(z_1+c_1, z_2+c_2)-f_2(z_1, z_2) $ is not transcendental. Thus, it yields that $ \frac{\partial f_1(z+c)}{\partial z_1}+\frac{\partial f_1(z+c)}{\partial z_2} $ is not transcendental. This is a contradiction with $ \frac{\partial f_1}{\partial z_1}+\frac{\partial f_1}{\partial z_2} $ is transcendental.
Hence, if $ \frac{\partial f_1}{\partial z_1}+\frac{\partial f_1}{\partial z_2} $ is transcendental, then $ f_2(z_1+c_1, z_2+c_2)-f_1(z_1, z_2) $, $ f_1(z_1+c_1, z_2+c_2)-f_2(z_1, z_2) $ and $ \frac{\partial f_2}{\partial z_1}+\frac{\partial f_2}{\partial z_2} $ are transcendental. Thus, system (2.3) can be rewritten as
|
$ {(∂f1∂z1+∂f1∂z2+i[f2(z+c)−f1(z1,z2)])(∂f1∂z1+∂f1∂z2−i[f2(z+c)−f1(z1,z2)])=1,(∂f2∂z1+∂f2∂z2+i[f1(z+c)−f2(z1,z2)])(∂f2∂z1+∂f2∂z2−i[f1(z+c)−f2(z1,z2)])=1. $
|
(4.20) |
Since $ f_1, f_2 $ are transcendental entire functions with finite order, then by Lemmas 4.1 and 4.2, there exist two nonconstant polynomials $ p(z), q(z) $ such that
|
$ {∂f1∂z1+∂f1∂z2+i[f2(z1+c1,z2+c2)−f1(z1,z2)]=ep,∂f1∂z1+∂f1∂z2−i[f2(z1+c1,z2+c2)−f1(z1,z2)]=e−p,∂f2∂z1+∂f2∂z2+i[f1(z1+c1,z2+c2)−f2(z1,z2)]=eq,∂f2∂z1+∂f2∂z2−i[f1(z1+c1,z2+c2)−f2(z1,z2)]=e−q. $
|
(4.21) |
Thus, it follows from (4.21) that
|
$ {∂f1(z1,z2)∂z1+∂f1(z1,z2)∂z2=ep(z1,z2)+e−p(z1,z2)2,f2(z1+c1,z2+c2)−f1(z1,z2)=ep(z1,z2)−e−p(z1,z2)2i,∂f2(z1,z2)∂z1+∂f2(z1,z2)∂z2=eq(z1,z2)+e−q(z1,z2)2,f1(z1+c1,z2+c2)−f2(z1,z2)=eq(z1,z2)−e−q(z1,z2)2i, $
|
(4.22) |
which implies
|
$ −i(∂p∂z1+∂p∂z2+i)ep(z)+q(z+c)−i(∂p∂z1+∂p∂z2+i)eq(z+c)−p(z)−e2q(z+c)≡1, $
|
(4.23) |
|
$ −i(∂q∂z1+∂q∂z2+i)eq(z)+p(z+c)−i(∂q∂z1+∂q∂z2+i)ep(z+c)−q(z)−e2p(z+c)≡1. $
|
(4.24) |
Obviously, $ \frac{\partial p}{\partial z_1}+\frac{\partial p}{\partial z_2}\neq -i $. Otherwise, it follows that $ -e^{2q(z+c)}\equiv1 $, this is impossible because $ q(z) $ is a nonconstant polynomial. Similarly, $ \frac{\partial q}{\partial z_1}+\frac{\partial q}{\partial z_2}\neq -i $. Thus, by Lemma 4.3, and in view of (4.23), (4.24), we conclude that
| $ -i\left(\frac{\partial p}{\partial z_1}+\frac{\partial p}{\partial z_2}+i\right)e^{q(z+c)-p(z)}\equiv1, \; or\; -i\left(\frac{\partial p}{\partial z_1}+\frac{\partial p}{\partial z_2}+i\right)e^{p(z)+q(z+c)}\equiv1, $ |
and
| $ -i\left(\frac{\partial q}{\partial z_1}+\frac{\partial q}{\partial z_2}+i\right)e^{p(z+c)-q(z)}\equiv1, \; or\; -i\left(\frac{\partial q}{\partial z_1}+\frac{\partial q}{\partial z_2}+i\right)e^{q(z)+p(z+c)}\equiv1. $ |
Next, we consider the following four cases.
Case 1.
|
$ {−i(∂p∂z1+∂p∂z2+i)eq(z+c)−p(z)≡1,−i(∂q∂z1+∂q∂z2+i)ep(z+c)−q(z)≡1. $
|
(4.25) |
Since $ p(z), q(z) $ are polynomials, then from (4.25), we know that $ \frac{\partial p}{\partial z_1}+\frac{\partial p}{\partial z_2} $ and $ \frac{\partial q}{\partial z_1}+\frac{\partial q}{\partial z_2} $ are constants in $ \mathbb{C} $. Otherwise, we obtain a contradiction from the fact that the right of the above equations is not transcendental, but the left is transcendental. In addition, we get that $ q(z+c)-p(z)\equiv C_1 $ and $ p(z+c)-q(z)\equiv C_2 $, that is, $ p(z+2c)-p(z)\equiv C_1+C_2 $ and $ q(z+2c)-q(z)\equiv C_1+C_2 $. In view of $ c_1\neq c_2 $, then by Lemma 4.4, it means that $ p(z) = L(z)+B_1, q(z) = L(z)+B_2 $, where $ L $ is a linear function as the form $ L(z) = a_1z_1+a_2z_2 $, $ a_1, a_2, B_1, B_2 $ are constants.
By combining with (4.23)–(4.25), we have
|
$ {−i(a1+a2+i)eL(c)+B2−B1≡1,−i(a1+a2+i)eL(c)+B1−B2≡1,−i(a1+a2+i)e−L(c)+B1−B2≡1,−i(a1+a2+i)e−L(c)−B1+B2≡1. $
|
(4.26) |
This means
|
$ (a1+a2+i)2=−1,e2L(c)=1,eB1−B2=−i(a1+a2+i)eL(c). $
|
(4.27) |
Thus, it follows that $ a_1+a_2 = -2i $ or $ a_1+a_2 = 0 $.
If $ a_1+a_2 = -2i $. Thus, $ e^{2L(c)} = 1, e^{B_1-B_2} = -e^{L(c)} $. The characteristic equations of the first equation in (4.22) are
| $ \frac{dz_1}{dt} = 1, \; \; \; \frac{dz_2}{dt} = 1, \; \; \; \frac{df_1}{dt} = \frac{e^{L(z)+B_1}+e^{-L(z)-B_1}}{2}, $ |
Using the initial conditions: $ z_1 = 0, z_2 = s_1 $, and $ f_1 = f_1(0, s_1): = g_0(s_1) $ with a parameter. Thus, it follows that $ z_1 = t $, $ z_2 = t+s_1 $ and
|
$ f1(t,s1)=∫t0e(a1+a2)t+a2s1+B1+e−[(a1+a2)t+a2s1+B1]2dt+g0(s1)=ea2s1+B12∫t0e(a1+a2)tdt+e−(a2s1+B1)2∫t0e−(a1+a2)tdt+g0(s1)=ea2s1+B12(a1+a2)e(a1+a2)t−e−(a2s1+B1)2(a1+a2)e−(a1+a2)t+g1(s1), $
|
where $ g_1(s_1) $ is a finite order entire function, and
| $ g_1(s_1) = g_0(s_1)+\frac{e^{a_2s_1+B_1}}{2(a_1+a_2)} -\frac{e^{-(a_2s_1+B_1)}}{2(a_1+a_2)}. $ |
In view of $ z_1 = t $, $ z_2 = t+s_1 $, we have
|
$ f1(z1,z2)=eL(z)+B1−e−L(z)−B1−4i+g1(s1). $
|
(4.28) |
Similar to the same argument for the third equation in (4.22), we have
|
$ f2(z1,z2)=eL(z)+B2−e−L(z)−B2−4i+g2(s1), $
|
(4.29) |
where $ g_2(s_1) $ is a finite order entire function.
Substituting (4.28), (4.29) into (4.22), and applying (4.27), it yields that
|
$ g2(s1+s0)−g1(s1)=0,g1(s1+s0)−g2(s1)=0. $
|
(4.30) |
Thus, from (4.30), we can deduce that
| $ g_1(s_1) = G_1(s_1), \; \; \; \; g_2(s_1) = G_2(s_1), $ |
where $ G_1(s_1), G_2(s_1) $ are the finite order entire period functions with period $ 2s_0 $, and
| $ G_2(s_1+s_0) = G_1(s_1). $ |
If $ a_1+a_2 = 0 $, then it follows that $ e^{2L(c)} = 1, e^{B_1-B_2} = e^{L(c)} $. In view of $ z_1 = t $, $ z_2 = t+s_1 $, it leads to $ L(z) = a_1z_1+a_2z_2 = a_2s_1 $. The characteristic equations of the first equation in (4.22) are
| $ \frac{dz_1}{dt} = 1, \; \; \frac{dz_2}{dt} = 1, \; \; \frac{df_1}{dt} = \frac{e^{L(z)+B_1}+e^{-L(z)-B_1}}{2}, $ |
Using the initial conditions: $ z_1 = 0, z_2 = s_1 $, and $ f_1 = f_1(0, s): = G_3(s) $ with a parameter. This leads to
|
$ f1(t,s1)=∫t0ea2s1+B1+e−(a2s1+B1)2dt+G3(s1)=t(ea2s1+B12+e−(a2s1+B1)2)+G3(s1)=tea2s1+B1+e−(a2s1+B1)2+G3(s1), $
|
that is
|
$ f1(z1,z2)=z1eL(z)+B1+e−(L(z)+B1)2+G3(s1), $
|
(4.31) |
where $ G_3(s_1) $ is an entire function of finite order. Similarly, we have
| $ f_2(z_1, z_2) = z_1\frac{e^{L(z)+B_2}+e^{-L(z)-B_2}}{2}+G_4(s_1), $ |
where $ G_4(s_1) $ is an entire function of finite order.
If $ e^{L(c)} = 1 $, that is, $ L(c) = 2k \pi i $, which leads to $ e^{B_1-B_2} = 1 $. Thus, it follows that
|
$ f2(z1,z2)=z1eL(z)+B1+e−L(z)−B12+G4(s1). $
|
(4.32) |
Substituting (4.31), (4.32) into the second and fourth equations in (4.22), we have
|
$ G4(s1+s0)−G3(s1)=ea2s1+B1−e−a2s1−B12i−c1ea2s1+B1+e−a2s1−B12,G3(s1+s0)−G4(s1)=ea2s1+B1−e−a2s1−B12i−c1ea2s1+B1+e−a2s1−B12. $
|
that is,
|
$ G4(s1+2s0)−G4(s1)=−(c1+i)ea2s1+B1−(c1−i)e−a2s1−B1,G3(s1+2s0)−G3(s1)=−(c1+i)ea2s1+B1−(c1−i)e−a2s1−B1. $
|
Thus, we can deduce that
|
$ G3(s1)=G1(s1)−c1+i2s0s1ea2s1+B1−c1−i2s0s1e−a2s1−B1,G4(s1)=G2(s1)−c1+i2s0s1ea2s1+B1−c1−i2s0s1e−a2s1−B1, $
|
where $ G_1(s_1), G_2(s_1) $ are the finite order entire period functions with period $ 2s_0 $, and $ G_2(s_1+s_0) = G_1(s_1). $
If $ e^{L(c)} = -1 $, that is $ L(c) = (2k+1)\pi i $, then it follows that $ e^{B_1-B_2} = -1 $, which means
|
$ f2(z1,z2)=−z1eL(z)+B1+e−L(z)−B12+G4(s1). $
|
(4.33) |
Substituting (4.31), (4.33) into the second and fourth equations in (4.22), we have
|
$ G4(s1+s0)−G3(s1)=ea2s1+B1−e−a2s1−B12i−c1ea2s1+B1+e−a2s1−B12,G3(s1+s0)−G4(s1)=−ea2s1+B1−e−a2s1−B12i+c1ea2s1+B1+e−a2s1−B12. $
|
Thus, it yields
|
$ G4(s1+2s0)−G4(s1)=(c1+i)ea2s1+B1+(c1−i)e−a2s1−B1,G3(s1+2s0)−G3(s1)=−(c1+i)ea2s1+B1−(c1−i)e−a2s1−B1. $
|
This leads to
|
$ G3(s1)=G1(s1)−c1+i2s0s1ea2s1+B1−c1−i2s0s1e−a2s1−B1,G4(s1)=G2(s1)+c1+i2s0s1ea2s1+B1+c1−i2s0s1e−a2s1−B1, $
|
where $ G_1(s_1), G_2(s_1) $ are the finite order entire period functions with period $ 2s_0 $, and $ G_2(s+s_0) = G_1(s). $
Case 2.
|
$ {−i(∂p∂z1+∂p∂z2+i)eq(z+c)−p(z)≡1,−i(∂q∂z1+∂q∂z2+i)eq(z)+p(z+c)≡1. $
|
(4.34) |
In view of (4.34), the fact that $ p(z), q(z) $ are polynomials leads to $ q(z+c)-p(z)\equiv C_1 $ and $ q(z)+p(z+c)\equiv C_2 $. This means $ q(z+2c)+q(z)\equiv C_1+C_2 $, this is a contradiction with the assumption of $ q(z) $ being a nonconstant polynomial.
Case 3.
|
$ {−i(∂p∂z1+∂p∂z2+i)ep(z)+q(z+c)≡1,−i(∂q∂z1+∂q∂z2+i)ep(z+c)−q(z)≡1. $
|
(4.35) |
In view of (4.35), the fact that $ p(z), q(z) $ are polynomials leads to $ p(z)+q(z+c)\equiv C_1 $ and $ p(z+c)-q(z)\equiv C_2 $. This means $ p(z+2c)+p(z)\equiv C_1+C_2 $, this is a contradiction with the assumption of $ p_2(z) $ being a nonconstant polynomial.
Case 4.
|
$ {−i(∂p∂z1+∂p∂z2+i)ep(z)+q(z+c)≡1,−i(∂q∂z1+∂q∂z2+i)eq(z)+p(z+c)≡1. $
|
(4.36) |
In view of (4.36), the fact that $ p_1(z), p_2(z) $ are polynomials leads to $ p(z)+q(z+c)\equiv C_1 $ and $ q(z)+p(z+c)\equiv C_2 $, that is, $ p(z+2c)-p(z)\equiv C_1+C_2 $ and $ q(z+2c)-q(z)\equiv C_2+C_1 $. Similar to the same argument in Case 1 of Theorem 2.3, we obtain that $ p(z) = L(z)+B_1, q(z) = -L(z)+B_2 $, where $ L $ is a linear function as the form $ L(z) = a_1z_1+a_2z_2 $, $ a_1, a_2, B_1, B_2 $ are constants. In view of (4.23), (4.24) and (4.36), we have
|
$ {−i(a1+a2+i)e−L(c)+B1+B2≡1,−i(a1+a2+i)eL(c)−B1−B2≡1,i(a1+a2−i)eL(c)+B1+B2≡1,i(a1+a2−i)e−L(c)−B1−B2≡1, $
|
(4.37) |
which implies $ \left(a_1+a_2+i\right)^2 = \left(a_1+a_2-i\right)^2 $, that is, $ a_1+a_2 = 0 $. Thus, we conclude from (4.37) that
|
$ a1+a2=0,e2L(c)=1,eB1+B2=eL(c). $
|
(4.38) |
Similar to the argument as in Case 1 of Theorem 2.3, we get the conclusions of Theorem 2.3 (iv).
Therefore, from Cases 1–4, we complete the proof of Theorem 2.3.
The first author is supported by the Key Project of Jiangxi Province Education Science Planning Project in China (20ZD062), the Key Project of Jiangxi Province Culture Planning Project in China (YG2018149I), the Science and Technology Research Project of Jiangxi Provincial Department of Education (GJJ181548, GJJ180767), the 2020 Annual Ganzhou Science and Technology Planning Project in China. The third author was supported by the National Natural Science Foundation of China (11561033), the Natural Science Foundation of Jiangxi Province in China (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.
| [1] | Koppe JG, Pluim HJ, Olie K. (1989) Breastmilk, PCB's, Dioxins and Vitamin K Deficiency. J Roy Soc Med 82: 416-420. |
| [2] | Touwen BC, Huisjes HJ, Jurgens-van der Zee AD, et al. (1980) Obstetrical condition and neonatal neurological morbidity. An analysis with the help of the optimality concept. Early Hum Dev 4: 207-228. |
| [3] |
Pluim HJ, Slot PC, Olie K, et al. (1992) Diurnal variations in concentrations of PCDDs and PCDFs in human milk. Chemosphere 25: 307-311. doi: 10.1016/0045-6535(92)90547-5
|
| [4] |
Pluim HJ, Wever J, Koppe JG, et al. (1993) Intake and faecal excretion of chlorinated dioxins and dibenzofurans in breastfed infants at different ages. Chemosphere 26: 1947-1952. doi: 10.1016/0045-6535(93)90021-V
|
| [5] |
Pluim HJ, Kramer I, van der Slikke JW, et al. (1993) Levels of PCDDs and PCDFs in human milk: dependence on several parameters and dietary habits. Chemosphere 26: 1889-1895. doi: 10.1016/0045-6535(93)90082-G
|
| [6] |
Pluim HJ, Boersma ER, Kramer I, et al. (1994) Influence of short-term dietary measures on dioxin concentrations in human milk. Environ Health Perspect 102: 968-971. doi: 10.1289/ehp.94102968
|
| [7] | Pluim H, Koppe JG, Olie K, et al. (1992) Effects of dioxins on thyroid function in newborn babies. Lancet 339: 1303. |
| [8] |
Pluim HJ, de Vijlder JJ, Olie K, et al. (1993) Effects of pre- and postnatal exposure to chlorinated dioxins and furans on human neonatal thyroid hormone concentrations. Environ Health Perspect 101: 504-508. doi: 10.1289/ehp.93101504
|
| [9] |
Koopman-Esseboom C, Morse DC, Weisglas-Kuperus N, et al. (1994) Effects of dioxins and polychlorinated biphenyls on thyroid hormone status of pregnant women and their infants. Pediatr Res 36: 468-473. doi: 10.1203/00006450-199410000-00009
|
| [10] |
Pluim HJ, van der Goot M, Olie K, et al. (1996) Missing effects of background dioxin exposure on development of breastfed infants in the first half year of life. Chemosphere 33:1307-1315. doi: 10.1016/0045-6535(96)00268-8
|
| [11] | Pluim HJ, Koppe JG, Olie K, et al. (1994) Clinical laboratory manifestations of exposure to background levels of dioxins in the perinatal period. Acta Paediatr 83: 583-587. |
| [12] | Pluim HJ, van der Slikke JW, Olie K, et al. (1994) Dioxins and vit K status of the newborn. J Environ Sci Heal A 29: 793-802. |
| [13] |
Weisglas-Kuperus N, Sas TCJ, Koopman-Esseboom C, et al. (1995) Ímmunologic effects of background prenatal and postnatal exposure to dioxins and polychlorinated biphenyls in Dutch Infants. Pediatr Res 38: 404-410. doi: 10.1203/00006450-199509000-00022
|
| [14] |
ten Tusscher GW, Steerenberg PA, van Loveren H, et al. (2003) Persistent Hematologic and Immunologic disturbances in 8-year-Old Dutch children Associated with Perinatal Dioxin Exposure. Environ Health Perspect 111: 1519-1523. doi: 10.1289/ehp.5715
|
| [15] |
Leijs MM, Koppe JG, Olie K, et al. (2009) Effects of dioxins, PCBs and PBDEs on immunology and haematology in adolescents. Environ Sci Technol 43: 7946-7951. doi: 10.1021/es901480f
|
| [16] |
Ackerman MF, Gasiewicz TA, Lamm KR, et al. (1989) Selective inhibition of polymorphonuclear neutrophil activity by 2,3,7,8-tetrachlorodibenzo-p-dioxin. Toxicol Appl Pharmacol 101: 470-480. doi: 10.1016/0041-008X(89)90195-6
|
| [17] | Watanabe S, Kitamura K, Otaki M, et al. (2001) Health effects of chronic exposure to polychlorinated dibenzo-p-dioxins, dibenzofurans and co-planar PCBs in Japan. Organohalogen Compounds 53: 136-140. |
| [18] |
Geusau A, Abraham K, Geissler K, et al. (2001) Severe 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD) intoxication: clinical and laboratory effects. Environ Health Perspect 109: 865-869. doi: 10.1289/ehp.01109865
|
| [19] |
McHale CM, Zhang L, Hubbard AE, et al. (2007) Microarray analysis of gene expression in peripheral blood mononuclear cells from dioxin-exposed human subjects. Toxicology 229: 101-113. doi: 10.1016/j.tox.2006.10.004
|
| [20] |
McKinney JD, Fawkes J, Jordan S, et al. (1985) 2,3,7,8 tetrachlorodibenzo-p-dioxin (TCDD) as a potent and persistent thyroxine agonist; A mechanistic model for toxicity based on molecular reactivity. Environ Health Perspect 61: 41-53. doi: 10.1289/ehp.856141
|
| [21] |
Rogan WJ, Gladen BC, Hung KL, et al. (1988) Congenital poisoning by polychlorinated biphenyls and their contaminants in Taiwan. Science 241: 334-336. doi: 10.1126/science.3133768
|
| [22] | Ilsen A, Briet JM, Koppe JG, et al. (1996) Signs of enhanced neuromotor maturation in children due to perinatal load with background levels of dioxins. Follow-up until age 2 years and 7 months. Chemosphere 33: 1317-1326. |
| [23] |
Schellart NAM, Reits D (2008) Influences of perinatal dioxin load to visual motion and oddball stimuli examined with an EEG and MEG analysis. Clin Neurophysiol 119: 1486-1495. doi: 10.1016/j.clinph.2008.03.002
|
| [24] |
ten Tusscher GW, Leijs MM, de Boer LCC, et al. (2014) Neurodevelopmental retardation, as assessed clinically and with magnetoencephalography and electroencephalography,associated with perinatal dioxin exposure. Sci Total Environ 491-492: 235-239. doi: 10.1016/j.scitotenv.2014.02.100
|
| [25] |
Koopman-Esseboom C, Huisman M, Weisglas-Kuperus N, et al. (1994) Dioxin and PCB Levels in Blood and Human Milk in relation to living areas in the Netherlands. Chemosphere 29: 2327-2338. doi: 10.1016/0045-6535(94)90401-4
|
| [26] |
Waalkes MP, Qu W, Tokar EJ, et al. (2014) Lung tumors induced by "whole life" inorganic arsenic exposure at human-relevant doses. Arch Toxicol 88: 1619-1629. doi: 10.1007/s00204-014-1305-8
|
| [27] | Laporte JR. (1977) Effects of dioxin exposure. Lancet 1: 1049-1050. |
| [28] | Pluim HJ, van der Slikke JW, Olie K, et al. (1994) Dioxins and vit K status of the newborn. J Env Sci Health A29: 793-802. |
| [29] |
ten Tusscher GW, Guchelaar HJ, Koch JP, et al. (2008) Perinatal dioxin exposure,cytochrome P-450 activity, liver functions and thyroid hormones at follow-up after 7-12 years. Chemosphere 70: 1865-1872. doi: 10.1016/j.chemosphere.2007.08.001
|
| [30] |
ten Tusscher GW, de Weerdt J, Roos CM, et al. (2001) Decreased lung function associated with perinatal exposure to Dutch background levels of dioxins. Acta Paediatr 90: 1292-1298. doi: 10.1111/j.1651-2227.2001.tb01578.x
|
| [31] | Forouhandeh-Gever M, ten Tusscher GW, Westra M, et al. (1999) Does perinatal exposure to background levels of dioxins have a lasting effect on dentition?. Organohalogen Compounds 44: 279-281. |
| [32] | Alaluusua S, Lukinmaa PL, Torppa J, et al. (1999) Developing teeth as biomarker of dioxin exposure. Lancet 353: 206. |
| [33] |
ten Tusscher GW, Stam GA, Koppe JG (2000) Open chemical combustions resulting in a local increased incidence of orofacial clefts. Chemosphere 40: 1263-1270. doi: 10.1016/S0045-6535(99)00378-1
|
| [34] |
Leijs MM, Teunenbroek TV, Olie K, et al. (2008) Assessment of current serum levels of PCDD/Fs, dl-PCBs and PBDEs in a Dutch cohort with known perinatal PCDD/F exposure. Chemosphere 73: 176-181. doi: 10.1016/j.chemosphere.2008.05.056
|
| [35] |
Fenton SE, Hamm JT, Birnbaum LS, et al. (2002) Persistent abnormalities in the rat mammary gland following gestational and lactational exposure to 2,3,7,8 tetrachlorodibenzo-p-dioxin (TCDD). Toxicol Sci 67: 63-74. doi: 10.1093/toxsci/67.1.63
|
| [36] |
Leijs MM, Koppe JG, Olie K, et al. (2008) Delayed initiation of breast development in girls with higher prenatal dioxin exposure; a longitudinal cohort study. Chemosphere 73: 999-1004. doi: 10.1016/j.chemosphere.2008.05.053
|
| [37] |
den Hond E, Roels HA, Hoppenbrouwers K, et al. (2002) Sexual maturation in relation to polychlorinated aromatic hydrocarbons: Sharpe and Skakkebaek's hypothesis revisited. Environ Health Perspect 110: 771-776. doi: 10.1289/ehp.02110771
|
| [38] |
Dhooge W, den Hond E, Koppen G, et al. (2010) Internal exposure to pollutants and body size in Flemish adolescents and adults: associations and dose-response relationships. Environ Int 36: 330-337. doi: 10.1016/j.envint.2010.01.005
|
| [39] | Luebke RW, Dewitt JC, Germolec DR, et al. (2012) Immunomodulation by persistent organic pollutants, in: Schecter A, Dioxins and Health, 3 eds. Hoboken, New Jersey: Wiley, 171-192. |
| [40] | Li R, Yang Q, Qui X, et al. (2013) Reactive Oxygen Species Alteration of Immune Cells in Local Residents at an Electronic Waste Recycling Site in Northern China. Environ Sci Technol 47: 3344-3352. |
| [41] |
Weisglas-Kuperus N, Patandin S, Berbers GA, et al. (2000) Immunologic effects of background exposure to polychlorinated biphenyls and dioxins in Dutch preschool children. Environ Health Perspect 108: 1203-1207. doi: 10.1289/ehp.001081203
|
| [42] | Leijs MM, ten Tusscher GW, Olie K, et al. (2012) Thyroid Hormone metabolism and environmental chemical exposure. Environ Health 11: S10. |
| [43] | Leijs MM (2010) Toxic Effects of Dioxins and PBDEs in Adolescents. University of Amsterdam, PhD thesis. |
| [44] |
Novelli M, Piaggi S, De TV (2005) 2,3,7,8-Tetrachlorodibenzo-p-dioxin-induced impairment of glucose-stimulated insulin secretion in isolated rat pancreatic islets. Toxicol Lett 156: 307-314. doi: 10.1016/j.toxlet.2004.12.004
|
| [45] |
Piaggi S, Novelli M, Martino L, et al. (2007) Cell death and Impairment of glucose-stimulated insulin secretion induced by 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD) in the beta-cell line INS-1E. Toxicol Appl Pharm 220: 333-340. doi: 10.1016/j.taap.2007.01.017
|
| [46] |
Lee DH, Lee IK, Song K, et al. (2006) A strong Dose-Response Relation Between Serum Concentrations of Persistent Organic Pollutants and Diabetes. Diabetes Care 29: 1638-1644. doi: 10.2337/dc06-0543
|
| [47] |
Avdalovic M, Putney L, Finkbeiner W, et al. (2009) In utero and postnatal exposure to environmental tobacco smoke (ETS) alters alveolar and respiratory bronchioli growth and development in infant monkeys. Toxicol Pathol 37: 256-263. doi: 10.1177/0192623308330788
|
| [48] |
Beech DJ, Sibbons PD, Howard CV, et al. (2000) Terminal bronchiolar duct ending number does not increase post-natally in normal infants. Early Hum Dev 59: 193-200. doi: 10.1016/S0378-3782(00)00095-5
|
| [49] |
Beech DJ, Howard CV, Reed MG, et al. (2000) Unbiased and efficient estimation of the total number of terminal bronchiolar duct endings in lung:a modified physical disector. J Microsc 197: 36-45. doi: 10.1046/j.1365-2818.2000.00636.x
|
| [50] | Guo YL (1999) Human health effects from PCBs and dioxin-like chemicals in the rice-oil poisonings as compared with other exposure episodes. Organohalogen Compounds 42: 241-242. |
| [51] | ten Tusscher GW, Leijs MM, Koppe JG, et al. (2010) Environmental contaminants and lung function - the missing link? in: Leijs MM. Toxic Effects of Dioxins, PCBs and PBDE's in Adolescents, Amsterdam: University of Amsterdam; 127-134. |
| [52] | Pluim HJ, Koppe JG, Olie K, et al. (1994) Clinical laboratory manifestations of exposure to background levels of dioxins in the perinatal period. Acta Paediatr 83: 583-587. |
| [53] | Ilsen A, Briet JM, Koppe JG, et al. (1996) Signs of enhanced neuromotor maturation in children due to perinatal load with background levels of dioxins. Follow-up until age 2 years and 7 months. Chemosphere 33: 1317-1326. |
| [54] |
Koppe JG, Pluim HJ, Olie K, et al. (1991) Breast milk, dioxins and the possible effects on the health of newborn infants. Sci Total Environ 106: 33-41. doi: 10.1016/0048-9697(91)90018-A
|
| 1. | Yuxian Chen, Libing Xie, Hongyan Xu, Kenan Yildirim, Results on Solutions for Several Systems of Partial Differential-Difference Equations with Two Complex Variables, 2023, 2023, 2314-4785, 1, 10.1155/2023/6158435 | |
| 2. | R. Mandal, R. Biswas, On solutions of certain compatible systems of quadratic trinomial Partial differential-difference equations, 2024, 61, 2411-0620, 195, 10.30970/ms.61.2.195-213 | |
| 3. | Raju Biswas, Rajib Mandal, Entire solutions for quadratic trinomial partial differential-difference functional equations in $\mathbb{C}^n$ , 2024, Accepted, 1450-5444, 10.30755/NSJOM.15512 |
Gavin W. ten Tusscher, Marike M. Leijs, Kees Olie, Adri Ilsen, Tom Vulsma, Janna G. Koppe. Findings on prenatal, lactational and later childhood exposure to dioxins and dioxin-like compounds: a review of the Amsterdam-Zaandam cohort 1987-2005[J]. AIMS Environmental Science, 2015, 2(1): 1-20. doi: 10.3934/environsci.2015.1.1
DownLoad: