In this paper, we investigate some systems of the Fermat type differential-difference equations with polynomial coefficients and obtain the condition for the existence of finite order transcendental entire solutions and the expression for the entire solutions. We also give some corresponding examples.
Citation: Yeyang Jiang, Zhihua Liao, Di Qiu. The existence of entire solutions of some systems of the Fermat type differential-difference equations[J]. AIMS Mathematics, 2022, 7(10): 17685-17698. doi: 10.3934/math.2022974
[1] | Hong Li, Keyu Zhang, Hongyan Xu . Solutions for systems of complex Fermat type partial differential-difference equations with two complex variables. AIMS Mathematics, 2021, 6(11): 11796-11814. doi: 10.3934/math.2021685 |
[2] | 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 |
[3] | 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 |
[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] | 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 |
[6] | Guowei Zhang . The exact transcendental entire solutions of complex equations with three quadratic terms. AIMS Mathematics, 2023, 8(11): 27414-27438. doi: 10.3934/math.20231403 |
[7] | 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 |
[8] | Zhiyong Xu, Junfeng Xu . Solutions to some generalized Fermat-type differential-difference equations. AIMS Mathematics, 2024, 9(12): 34488-34503. doi: 10.3934/math.20241643 |
[9] | Fengrong Zhang, Linlin Wu, Jing Yang, Weiran Lü . On entire solutions of certain type of nonlinear differential equations. AIMS Mathematics, 2020, 5(6): 6124-6134. doi: 10.3934/math.2020393 |
[10] | Nan Li, Jiachuan Geng, Lianzhong Yang . Some results on transcendental entire solutions to certain nonlinear differential-difference equations. AIMS Mathematics, 2021, 6(8): 8107-8126. doi: 10.3934/math.2021470 |
In this paper, we investigate some systems of the Fermat type differential-difference equations with polynomial coefficients and obtain the condition for the existence of finite order transcendental entire solutions and the expression for the entire solutions. We also give some corresponding examples.
It is assumed that the reader is familiar with the standard notations and basic results of Nevanlinna's value distribution theory (see [2,10]). Especially, we use the notions σ(f) to denote the order of growth of the meromorphic function f(z) and S(r,f) to denote any quantity that satisfies S(r,f)=o(T(r,f)), where r→∞ outside of a possible exceptional set of finite logarithmic measure.
The celebrated Fermat's last theorem [17] elaborates that it do not exist nonzero rational numbers x,y and an integer n≥3 such that xn+yn=1. The equation x2+y2=1 does admit nontrivial rational solutions. Replacing x,y in it by entire or meromorphic functions f,g, Fermat type functional equations were studied by Gross [8] and many others thereafter. In 2004, Yang and Li[15] did some related research and they obtained the following result.
Theorem A. [15] Let n be a positive integer, a(z),b0(z),b1(z),⋯,bn−1(z) are polynomials, and bn(z)≡bn be a nonzero constant. Let L(f)=∑nk=0bk(z)f(k). If a(z)≢0, then a transcendental meromorphic solution of the equation
f2+(L(f))2=a(z), |
must have the form f(z)=12(P(z)eR(z)+Q(z)e−R(z)), where P(z),Q(z),R(z) are polynomials, and P(z)Q(z)=a(z).
Recently, as the difference analogues of Nevanlinna's theory are being investigated [2], there are many interests in the complex analytic properties of meromorphic solutions of complex difference equations, and many results on the complex linear or nonlinear difference equations are got rapidly, such as [2,9,11]. In particular, some results on the solutions of the Fermat type functional equation are obtained[11,12,13].
In 2012, Liu et al.[11] investigated entire solutions with finite order of the Fermat type differential-difference equation and obtained the following result.
Theorem B. [11] The transcendental entire solutions with finite order of the differential-difference equation
f′(z)2+f(z+c)2=1, |
must satisfy f(z)=sin(z+iB), where B is a constant and c=2kπ or c=2kπ+π,k is an integer.
In 2012, Gao had discussed the existence or growth of some types of systems of complex difference equations, and obtained some results [3,4,5,6,7]. Especially, Gao [7] investigated the existence of transcendental entire functions for the system of the nonlinear differential-difference equations
{w′1(z)2+P1(z)2w2(z+c)2=Q1(z),w′2(z)2+P2(z)2w1(z+c)2=Q2(z), |
and obtained the following interesting result.
Theorem C. [7] Suppose that (w1(z),w2(z)) are the transcendental entire solutions for the above differential-difference equations, σ(w1,w2)<∞, then P1(z)=A,P2(z)=B,AB≠0, and
w1(z)=c11eaz+b1−c12e−az−b12a,w2(z)=c21eaz+b2−c22e−az−b22a, |
where a4=A2B2,A,B,bj,cjk,k,j=1,2 are all constants.
The order of growth of meromorphic solutions (f,g) of the system of the nonlinear differential-difference equations is defined by
σ(f,g)=max{σ(f),σ(g)},σ(f)=lim supr→∞log+T(r,f)logr. |
In 2019, Liu and Gao [13] investigated the existence of entire functions for the nonlinear differential-difference equation
ω″2(z)+ω(z+c)2=Q(z), | (1.1) |
where Q(z) is a nonzero polynomials and they obtained the following result.
Theorem D. [13] There is a transcendental entire solution w(z) with finite order for the differential-difference equation (1.1), then Q(z)=c1c2 is a constant, and w(z) must satisfy
w(z)=c1eaz+b+c2e−az−b2a2, |
where a is a constant such that a4=1,b and b′ are arbitrary constants, c=ln(−ia2)+2kπia,k is an integer.
Inspired by the above theorems, we [14] consider the existence of entire functions for the nonlinear differential-difference
ω″(z)2+[P(z)(d1ω(z+c)+d0ω(z))]2=Q(z), | (1.2) |
where P(z) and Q(z) be nonzero polynomials, d0,d1 are nonzero constants and we obtained the following result.
Theorem E. [14] Let P(z) and Q(z) be nonzero polynomials, d0,d1 are nonzero constants such that d0=±d1, then there is no finite order entire solution ω(z) satisfying the nonlinear differential-difference equation (1.2).
More interestingly, we find that for d1=1,d0=0, the Eq (1.2) has a transcendental entire solution with the following form
ω(z)=q1eaz+b−q2e−(az+b)2a2, |
where P(z)≡A(≠0) and Q(z)=q1q2(≠0), a4=A2,ac=ikπ2,b∈C,q1,q2,c are nonzero complex constants, k∈Z.
A natural and interesting question is, what we can say about the system of the differential-difference equation
{f″(z)2+[P1(z)(d1g(z+c)+d0g(z))]2=Q1(z),g″(z)2+[P2(z)(d1f(z+c)+d0f(z))]2=Q2(z), | (1.3) |
where Pj(z)(j=1,2), Qi(z)(i=1,2) are nonzero polynomials and d0,d1 are nonzero constants?
In this paper, we investigate the the existence of finite order transcendental entire solutions for the Fermart type of the system of differential-difference equations (1.3), and we obtain the following result.
Theorem 1.1. Let Pj(z)(j=1,2), Qi(z)(i=1,2) be nonzero polynomials, d0,d1 be constants such that d0≠±d1. Ifthe Fermart type of systems of differential-difference equation (1.3) has transcendental entire solutions (f(z),g(z)) such that σ(f,g)<∞, then Pj(z)≡Pj(j=1,2) are nonzero constants and (f(z),g(z)) can be expressed as
{f(z)=S∗1(z)eaz+b1+S∗2(z)e−az−b12i,g(z)=T∗1(z)eaz+b2+T∗2(z)e−az−b22, |
where a4=P1P2(d0+d1eac)2,d0+d1eac≠0, either ac=kπi,k∈Z or −2d0d1=e−ac+eac, and S∗i(z),T∗i(z)(i=1,2) are polynomials related to Qi(z)(i=1,2),b1 and b2 are arbitrary constants.
Example 1.1. Let a=2i,Pj=12,S1=S2=−4,T1=4i,T2=−4i,d0=−d1=4, then the solutions of the systems of differential-difference equation
{f″(z)2+[12(4g(z−π2)−4g(z))]2=16g″(z)2+[12(4f(z−π2)−4f(z))]2=16 |
has the transcendental entire solutions
{f(z)=−4ei(2z−π2)−4e−i(2z−π2)−8=ei(2z−π2)+e−i(2z−π2)2=sin2z,g(z)=4iei(2z+π2)−4ie−i(2z+π2)8i2=ei(2z+π2)−e−i(2z+π2)2i=cos2z. |
Corollary 1.1. Let Pj(z)(j=1,2) be nonconstant polynomials, then the Fermart type of systems of differential-difference equation (1.3) has no finite order transcendental entire solutions (f(z),g(z)).
Remark 1.1. We want to prove a stronger conclusion that Si(z),Ti(z)(i=1,2) are constants too, unfortunately we fail. In some special cases, we obtain the ideal result, such as d1=1,d0=0.
In fact, we investigate the differential-difference equation
{f″(z)2+[P1(z)g(z+c)]2=Q1(z),g″(z)2+[P2(z)f(z+c)]2=Q2(z), | (1.4) |
and obtain the following result.
Theorem 1.2. Let Pj(z)(j=1,2) and Qi(z)(i=1,2) be nonzero polynomials. Ifthe Fermart type of systems of differential-difference equation (1.5) has transcendental entire solutions (f(z),g(z)) such that σ(f,g)<∞, then Pj(z)≡Pj(j=1,2),Q1(z)=β1β2 and Q2(z)=α1α2, where Pi,αi,βi(i=1,2) are nonzero constants. Furthermore, (f(z),g(z)) can be expressed as
{f(z)=β1eaz+b1+β2e−az−b12a2,g(z)=α1eaz+b2+α2e−az−b22a2, | (1.5) |
where a8=P21P22 and ac=12kπi,k∈Z,b1 and b2 are arbitrary constants.
Example 1.2. Let a=2i,Pj=±4,α11=α12=β11=β12=1, then the solutions of the systems of differential-difference equation
{f″(z)2+16g(z+c)2=1g″(z)2+16f(z+c)2=1 | (1.6) |
must have the form
{f(z)=e2iz+b1+e−2iz−b1−8=ei(2z−ib1)+e−i(2z−ib1)−8=−14cos(2z−ib1),g(z)=e2iz+b2+e−2iz−b2−8=ei(2z−ib2)+e−i(2z−ib2)−8=−14cos(2z−ib2). |
Furthermore, we take c=14π, then (−14cos(2z−π2),−14cos(2z+π2)) is a pair of entire solutions of the Eq (1.6).
Remark 1.2. Let a=i,Pj=±i,α11=α12=β11=β12=1,c=12π then
{f(z)=eiz−π2i+e−iz+π2i−2=ei(z−π2)+e−i(z−π2)−2=−cos(z−π2)g(z)=eiz+π2i+e−iz−π2i−2=ei(z+π2)+e−i(z+π2)−2=−cos(z+π2) |
are the solutions of the systems of the differential-difference equation
{f″(z)2−g(z+c)2=0,g″(z)2−f(z+c)2=0. | (1.7) |
This example indicates that the Eq (1.4) maybe also have the solutions of the form as (1.5) when Qk(z)≡0(k=1,2).
Corollary 1.2. Let Pj(z)(j=1,2) and Qi(z)(i=1,2) be nonconstant polynomials, then the Fermart type of systems of differential-difference equation (1.4) has no finite order transcendental entire solutions (f(z),g(z)).
Next, the last question is what happens when d0=±d1 in the Eq (1.3)? When d0=±d1, we can rewrite the Eq (1.3) as the following equations
{f″(z)2+P1(z)2[g(z+c)+qg(z)]2=Q1(z),g″(z)2+P2(z)2[f(z+c)+qf(z)]2=Q2(z), | (1.8) |
where q=±1, and obtain the following theorem.
Theorem 1.3. Let Pj(z)(j=1,2) and Qi(z)(i=1,2) be nonzero polynomials. Ifthe Fermart type of systems of differential-difference equation (1.8) has transcendental entire solutions (f(z),g(z)) such that σ(f,g)<∞, then Pj(z)≡Pj(j=1,2),Q1(z)=β1β2 and Q2(z)=α1α2, where Pi,αi,βi(i=1,2) are nonzero constants. Furthermore, (f(z),g(z)) can be expressed as
{f(z)=β1eaz+b1+β2e−az−b12a2,g(z)=α1eaz+b2+α2e−az−b22a2, | (1.9) |
where a4=4P1P2,b1 and b2 are arbitrary constants. Especially, for ∀k∈Z,
(i) ac=2kπi, when q=1;
(ii) ac=(2k+1)πi, when q=−1.
To prove our theorems, the following lemmas are used play the key roles in proving our main theorems.
Lemma 2.1. [16] Suppose that n≥2, and let fj(z)(j=1,⋯,n) be meromorphic functions and gj(z)(j=1,⋯,n) be entire functions such that
(i) ∑nj=1fj(z)egj(z)≡0;
(ii) when 1≤j<k≤n,gj(z)−gk(z) is not a constant;
(iii) when 1≤j≤n,1≤h<k≤n,
T(r,fj)=o{T(r,egh−gk)}(r→∞,r∉E), |
where E⊂(1,∞) is of finite logarithmic measure.
Then fj(z)≡0(j=1,⋯,n).
Lemma 2.2. [1] Suppose Q(z) is nonzero entire function, P(z) is nonzero polynomial, h(z) is not constant polynomial and satisfy
(Q′(z)±Q(z)h′(z))P(z)−Q(z)P′(z)≡0 | (2.1) |
then Q(z) is a transcendent entire function.
Lemma 2.3. [16] Let n≥3 and fj(z)(j=1,⋯,n) be meromorphic functions satisfying ∑nj=1fj(z)=1 such that fk(z)(k=1,⋯,n−1) being nonconstant. If fn(z)≢0 and
n∑j=1N(r,1fj)+(n−1)n∑j=1¯N(r,fj)<(λ+o(1))T(r,fk), |
where λ<1 and k=1,⋯,n−1, then fn(z)≡1.
Proof. Suppose that (f(z),g(z)) is a transcendental entire solution with σ(f,g)<∞ satisfying (1.3).
Equation (1.3) can be rewritten as follows
{P1(z)[d1g(z+c)+d0g(z)]+if″(z)=S1(z)eh1(z),P1(z)[d1g(z+c)+d0g(z)]−if″(z)=S2(z)e−h1(z),g″(z)+iP2(z)[d1f(z+c)+d0f(z)]=T1(z)eh2(z),g″(z)−iP2(z)[d1f(z+c)+d0f(z)]=T2(z)e−h2(z), | (3.1) |
where h1(z) and h2(z) are nonconstant polynomials, Q1(z)=S1(z)S2(z),Q2(z)=T1(z)T2(z),Si(z)(i=1,2) and Tj(z)(j=1,2) are nonzero polynomials. Then we obtain that
{f″(z)=S1(z)eh1(z)−S2(z)e−h1(z)2i,d1f(z+c)+d0f(z)=T1(z)eh2(z)−T2(z)e−h2(z)2iP2(z), | (3.2) |
and
{g″(z)=T1(z)eh2(z)+T2(z)e−h2(z)2,d1g(z+c)+d0g(z)=S1(z)eh1(z)+S2(z)e−h1(z)2P1(z). | (3.3) |
By the second equation of (3.2), we see that
d1f″(z+c)+d0f″(z)=A1(z)eh2(z)+A2(z)e−h2(z)2iP2(z)3, | (3.4) |
where
A1(z)=(H′1+H1h′2)P2−2P′2H1,A2(z)=−(H′2−H2h′2)P2+2P′2H2, |
and H1=T′1P2+T1h′2P2−T1P′2,H2=T′2P2−T2h′2P2−T2P′2.
Combining the above several equations, we have that
a1(z)eh2(z)−h1(z+c)+a2(z)e−h1(z)−h2(z)−d1S1(z+c)d1S1(z)eh1(z+c)−h1(z)+d1S2(z+c)d0S1(z)e−h1(z+c)−h1(z)+S2(z)S1(z)e−2h1(z)≡1, | (3.5) |
where
a1(z)=A1(z)d0S1(z)P2(z)3,a2(z)=A2(z)d0S1(z)P2(z)3. |
To facilitate discussion, we rewrite the Eq (3.5) as
A1(z)eh2(z)−h1(z+c)+A2(z)e−h1(z)−h2(z)+A3(z)eh1(z+c)−h1(z)+A4(z)e−h1(z+c)−h1(z)+A5(z)e−2h1(z)+A6(z)eh0(z)≡0, | (3.6) |
where h0(z)≡0 and A3(z)=d1S1(z+c)P2(z)3,A4(z)=d1S2(z+c)P2(z)3, A5(z)=d0S2(z)P2(z)3,A6(z)=d0S1(z)P2(z)3.
Similar discussion to the Eq (3.3), we obtain that
b1(z)eh1(z)+h2(z)+b2(z)eh2(z)−h1(z)−d1T1(z+c)d0T2(z)eh2(z+c)+h2(z)−d1T2(z+c)d0T2(z)eh2(z)−h2(z+c)−T1(z)T2(z)e2h2(z)≡1, | (3.7) |
where
b1(z)=B1(z)d0T2(z)P1(z)3,b2(z)=B2(z)d0T2(z)P1(z)3. |
and
B1(z)eh1(z)+h2(z)+B2(z)eh2(z)−h1(z)+B3(z)eh2(z+c)+h2(z)+B4(z)eh2(z)−h2(z+c)+B5(z)e2h2(z)+B6(z)eh0(z)≡0, | (3.8) |
where h0(z)≡0 and
B1(z)=(U′1+U1h′1)P1−2P′1U1,B2(z)=(U′2−U2h′1)P1−2P′1U2.B3(z)=−d1T1(z+c)P1(z)3,B4(z)=−d1T2(z+c)P1(z)3,B5(z)=−d0T1(z)P1(z)3,B6(z)=−d0T2(z)P1(z)3, |
denoting U1=S′1P1+S1h′1P1−S1P′1 and U2=S′2P1−S2h′1P1−S2P′1.
From the formula (3.2), using the fact that f″(z) and d1f(z+c)+d0f(z) having the same finite order of growth and f(z) transcendental, we can deduce easily that degh1=degh2≥1. We claim that degh1=degh2=1. Conversely, we assume that degh1≥2 and degh2≥2. Since Pi(z),Si(z),Ti(z),hi(z)(i=1,2) are polynomials, Aj(z)(j=1,2,3,4,5,6) and Bk(z)(k=1,2,3,4,5,6) are polynomials, too. Thus,
T(r,Ai)=o(eh2(z)+h1(z+c)−h0(z)),T(r,Ai)=o(eh2(z)+h1(z+c)−2h1(z+c)),⋯T(r,Ai)=o(e2h1(z+c)−h0(z)),T(r,Bi)=o(eh1(z)+h2(z+c)−h0(z)),⋯T(r,Bi)=o(e2h2(z)−h0(z)),i=1,2,3,4,5,6. |
By Lemma 2.1, we see that
A1(z)≡0,A2(z)≡0,B1(z)≡0,B2(z)≡0. |
i.e.,
(H′1+H1h′2)P2−2P′2H1≡0,(H′2−H2h′2)P2+2P′2H2≡0,(U′1+U1h′1)P1−2P′1U1≡0,(U′2−U2h′1)P1+2P′1U2≡0. | (3.9) |
Noting that Si(z)(i=1,2) and Tj(z)(j=1,2) are polynomials, by Lemma 2.2, we see that T′iP2±Tih′2P2−TiP′2≢0(i=1,2) and S′iP1±Sih′1P1−SiP′1≢0(i=1,2). Hence Hi≢0(i=1,2) and Ui≢0(i=1,2). Thus there is only a item that it has the largest degree in Ai(z),Bi(z)(i=1,2). For example, T1(z)h′2(z)2P2(z)2 is the only item that it has the largest degree in A1(z). Then Ai(z)≢0,Bi(z)≢0(i=1,2). It contradicts the Eq (3.9).
Now we consider that degh1=1 and degh2=1. Let h1(z)=a1z+b1 and h2(z)=a2z+b2, by (3.2) or (3.3), we easily obtain that a1=a2. Hence, we assume that h1(z)=az+b1 and h2(z)=az+b2.
We rewrite the Eq (3.5) as
[a2(z)e−b1−b2−ac−d1S2(z+c)d0S1(z)e−2b1−ac−S2(z)S1(z)e−2b1]e−2az+[a1(z)eb2−b1−d1S1(z+c)d0S1(z)eac]≡1, |
hence
{a1(z)eb2−b1−d1S1(z+c)d0S1(z)eac≡1,a2(z)e−b1−b2−ac−d1S2(z+c)d0S1(z)e−2b1−ac−S2(z)S1(z)e−2b1≡0. | (3.10) |
Noticing the expressions of a1(z) and a2(z), we see that
A1(z)=eb1−b2P2(z)3[d0S1(z)+d1S1(z+c)eac], | (3.11) |
A2(z)=eb2−b1P2(z)3[d0S2(z)+d1S2(z+c)e−ac]. | (3.12) |
Similar discussion to the Eq (3.7), we obtain that
B1(z)=eb2−b1P1(z)3[d0T1(z)+d1T1(z+c)eac], | (3.13) |
B2(z)=eb1−b2P1(z)3[d0T2(z)+d1T2(z+c)e−ac]. | (3.14) |
Firstly we claim that d0+d1eac≠0 and d0+d1e−ac≠0. Conversely, either d0+d1eac=0 and d0+d1e−ac=0, or one of d0+d1eac and d0+d1e−ac is not equal 0. On the one hand, if d0+d1eac=0 and d0+d1e−ac=0, it contradicts the condition that d0≠±d1. On the other hand, without losing generality, we assume that d0+d1eac≠0 and d0+d1e−ac=0. Let degPj(z)=pj,degSj(z)=sj,degTj(z)=tj(j=1,2), then
deg[d0S1(z)+d1S1(z+c)eac]=s1−1,deg[d0T1(z)+d1T1(z+c)eac]=t1−1,deg[d0S2(z)+d1S2(z+c)e−ac]=s2,deg[d0T2(z)+d1T2(z+c)e−ac]=t2. |
Comparing the degree of the polynomials on two sides of the Eqs (3.11) and (3.13), we have that t1+2p2=s1−1+3p2, and s1+2p1=t1−1+3p1, i.e., p1+p2=2. Comparing the degree of the polynomials on two sides of the Eq (3.12) and (3.14), we have that t2+2p2=s2+3p2, and s2+2p1=t2+3p1, i.e., p1+p2=0. It is a contradiction.
Secondly we claim that P1(z) and P2(z) are constant functions. We set degPj(z)=pj,degSj(z)=sj,degTj(z)=tj(j=1,2). Comparing the degree of the polynomials on two sides of the Eqs (3.11) and (3.13), we obtain that t1+2p2=s1+3p2, and s1+2p1=t1+3p1, i.e., p1+p2=0. Since degPj(z)=pj≥0, then p1=p2=0. The claim is proved, i.e., P1(z)≡P1 and P2(z)≡P2 are constants. Furthermore, we see that degS1(z)=degT1(z) and degS2(z)=degT2(z). Rewriting (3.11)–(3.14) as follows:
T″1(z)+2aT′1(z)+a2T1(z)=P2eb1−b2[d0S1(z)+d1S1(z+c)eac], | (3.15) |
2aT′2(z)−T″2(z)−a2T2(z)=P2eb2−b1[d0S2(z)+d1S2(z+c)e−ac], | (3.16) |
S″1(z)+2aS′1(z)+a2S1(z)=P1eb2−b1[d0T1(z)+d1T1(z+c)eac], | (3.17) |
2aS′2(z)−S″2(z)−a2S2(z)=P1eb1−b2[d0T2(z)+d1T2(z+c)e−ac]. | (3.18) |
Let
T1(z)=αszs+αs−1zs−1+⋯+α1z1+α0,αs≠0,S1(z)=βszs+βs−1zs−1+⋯+β1z1+β0,βs≠0. | (3.19) |
Comparing the leading coefficients of polynomials on both sides of the Eqs (3.15) and (3.17), we obtain that a2αs=P2eb2−b1(d0+d1eac)βs,a2βs=P1eb1−b2(d0+d1eac)αs. Thus
a4=P1P2(d0+d1eac)2. | (3.20) |
Similarly, from (3.16) and (3.18), we have that
a4=P1P2(d0+d1e−ac)2. | (3.21) |
By (3.20) and (3.21), we obtain that (d0+d1eac)2=(d0+d1e−ac)2. Thus, either ac=kπi,k∈Z or −2d0d1=e−ac+eac.
Combining h1(z)=az+b1 and h2(z)=az+b2 and integrating the first equation of (3.2) and (3.3) twice, we have that
{f(z)=∫∫(S1(z)eaz+b1+S2(z)e−az−b1)dzdz2i=S∗1(z)eaz+b1+S∗2(z)e−az−b12i,g(z)=∫∫(T1(z)eaz+b2+T2(z)e−az−b2)dzdz2=T∗1(z)eaz+b2+T∗2(z)e−az−b22, |
where S∗i(z),T∗i(z)(i=1,2) are polynomials related to Qi(z)(i=1,2).
Thus, Theorem 1.1 is proved.
Proof. We do the same proof as in the proof in Theorem 1.1, and easily obtain that P1(z)≡P1 and P2(z)≡P2 are constants and degS1(z)=degT1(z) and degS2(z)=degT2(z). Noticing that d1=1,d0=0 and rewriting (3.15)–(3.18) as follows.
2aT′2(z)−T″2(z)−a2T2(z)=P2eb2−ac−b1S2(z+c), | (3.22) |
T″1(z)+2aT′1(z)+a2T1(z)=P2eb1+ac−b2S1(z+c), | (3.23) |
2aS′2(z)−S″2(z)−a2S2(z)=P1eb1−ac−b2T2(z+c), | (3.24) |
S″1(z)+2aS′1(z)+a2S1(z)=P1eb2+ac−b1T1(z+c). | (3.25) |
Let
T2(z)=αszs+αs−1zs−1+⋯+α1z1+α0,αs≠0,S2(z)=βszs+βs−1zs−1+⋯+β1z1+β0,βs≠0. |
Comparing the leading coefficients of polynomials on both sides of the Eqs (3.22) and (3.24), we obtain that −a2αs=P2eb2−ac−b1βs,−a2βs=P1eb1−ac−b2αs. Hence
a4=P1P2e−2ac. | (3.26) |
Similarly, from (3.23) and (3.25), we have that
a4=P1P2e2ac. | (3.27) |
By (3.26) and (3.27), we see that
a8=P21P22,ac=12kπi,k∈Z. | (3.28) |
Multiplying (3.22) and (3.24), we get
[2aT′2(z)−T″2(z)−a2T2(z)][2aS′2(z)−S″2(z)−a2S2(z)]=−P1P2e−2acT2(z+c)S2(z+c). |
Combining with (3.26), we rewrite the above equation as follow,
a4T2(z)S2(z)−2a3[T′2(z)S2(z)+T2(z)S′2(z)]+H2s−2(z)=a4T2(z+c)S2(z+c), | (3.29) |
where H2s−2(z)=4a2S′2(z)T′2(z)−S″2(z)(2aT′2(z)−a2T2(z)) −T″2(z)(2aS′2(z)−a2S2(z))+T″2(z)S″2(z) is a polynomial with degree 2s−2.
Next we prove that T2(z) and S2(z) are constants. Conversely, we assume that degS2(z)=degT2(z)=s≥1, then 2s−1≥1.
By the definition of T2(z) and S2(z), we easily have that
T′2(z)=sαszs−1+(s−1)αs−1zs−2+⋯+α1,S′2(z)=sβszs−1+(s−1)βs−1zs−2+⋯+β1,T2(z+c)=αszs+(scαs+αs−1)zs−1+⋯+cα1+α0,S2(z+c)=βszs+(scβs+βs−1)zs−1+⋯+cβ1+β0. |
Comparing the coefficients of the terms of the degree equivalent to 2s−1 of the two sides of the Eq (3.29), we see that
a4(αsβs−1+αs−1βs)−2a3(sαsβs+sαsβs)=a4[αs(scβs+βs−1)+βs(scαs+αs−1)]. |
This gives ac=−2, which contradicts with (3.28). Similarly, we obtain that T1(z) and S1(z) are constants. Setting
T1(z)≡α1,T2(z)≡α2,S1(z)≡β1,S2(z)≡β2. |
According h1(z)=az+b1 and h2(z)=az+b2, we have that
f(z)=β1eaz+b1+β2e−az−b12a2,g(z)=α1eaz+b2+α2e−az−b22a2. |
Thus, Theorem 1.2 is proved.
Proof. Without losing generality, we only consider that q=1.
We do the same proof as in the proof in Theorem 1.1, and easily obtain that 1+eac≠0,P1(z)≡P1 and P2(z)≡P2 are constants and degS1(z)=degT1(z),degS2(z)=degT2(z). Noticing that d1=1,d0=0 and rewriting (3.15)–(3.18) as follows:
2aT′2(z)−T″2(z)−a2T2(z)=P2eb2−b1[S2(z)+S2(z+c)e−ac], | (3.30) |
T″1(z)+2aT′1(z)+a2T1(z)=P2eb1−b2[S1(z)+S1(z+c)eac], | (3.31) |
2aS′2(z)−S″2(z)−a2S2(z)=P1eb1−b2[T2(z)+T2(z+c)e−ac], | (3.32) |
S″1(z)+2aS′1(z)+a2S1(z)=P1eb2−b1[T1(z)+T1(z+c)eac]. | (3.33) |
Let
T2(z)=αszs+αs−1zs−1+⋯+α1z1+α0,αs≠0,S2(z)=βszs+βs−1zs−1+⋯+β1z1+β0,βs≠0. |
Comparing the leading coefficients of polynomials on both sides of the Eqs (3.30) and (3.32), we obtain that −a2αs=P2eb2−b1βs(1+e−ac),−a2βs=P1eb1−b2αs(1+e−ac). Thus
a4=P1P2(1+e−ac)2. | (3.34) |
Similarly, from (3.31) and (3.33), we have that
a4=P1P2(1+eac)2. | (3.35) |
By (3.34) and (3.35), we see that (1+eac)2=(1+e−ac)2, i.e., 1+eac=±(1+e−ac), hence e2ac=1 or eac+e−ac=2. By eac+e−ac=2, we obtain that 1+eac=0, that contradicts 1+eac≠0. Thus e2ac=1, i.e., ac=lπi,l∈Z. Noticing that 1+eac≠0, we see that l is a even number. Hence ac=2kπi,k∈Z.
Multiplying (3.30) and (3.32), noticing that ac=2kπi, i.e., e−ac=1, we get
[2aT′2(z)−T″2(z)−a2T2(z)][2aS′2(z)−S″2(z)−a2S2(z)]=P1P2[T2(z)+T2(z+c)][S2(z)+S2(z+c)]. |
Combining with (3.34), we rewrite the above equation as follow,
a4T2(z)S2(z)−2a3[T′2(z)S2(z)+T2(z)S′2(z)]+H2s−2(z)=P1P2[T2(z)+T2(z+c)][S2(z)+S2(z+c)], | (3.36) |
where H2s−2(z)=4a2S′2(z)T′2(z)−S″2(z)(2aT′2(z)−a2T2(z)) −T″2(z)(2aS′2(z)−a2S2(z))+T″2(z)S″2(z) is a polynomial with degree not higher than 2s−2.
Next we prove that T2(z) and S2(z) are constants. Conversely, we assume that degS2(z)=degT2(z)=s≥1, then 2s−1≥1. We easily have that
T′2(z)=sαszs−1+(s−1)αs−1zs−2+⋯+α1,S′2(z)=sβszs−1+(s−1)βs−1zs−2+⋯+β1,T2(z+c)=αszs+(scαs+αs−1)zs−1+⋯+cα1+α0,S2(z+c)=βszs+(scβs+βs−1)zs−1+⋯+cβ1+β0. |
Comparing the coefficients of the terms of the degree equivalent to 2s−1 of the two sides of the Eq (3.36), we see that
a4(αsβs−1+αs−1βs)−2a3(sαsβs+sαsβs)=4P1P2[αs(scβs+βs−1)+βs(scαs+αs−1)]. |
This gives ac=−2, which contradicts with ac=2kπi. Similarly, we obtain that T1(z) and S1(z) are constants. Setting
T1(z)≡α1,T2(z)≡α2,S1(z)≡β1,S2(z)≡β2. |
According with h1(z)=az+b1 and h2(z)=az+b2, we have that
f(z)=β1eaz+b1+β2e−az−b12a2,g(z)=α1eaz+b2+α2e−az−b22a2. |
Thus, Theorem 1.3 is proved.
The authors would like to thank the referees for many valuable comments and suggestions. This work was supported by the Natural Science Foundation of Jiangxi Province (No. 20202BABL211002 and No.20212BAB201012).
The authors declare no conflict of interest.
[1] | M. F. Chen, Z. S. Gao, Entire function solutions of a certain type of nonlinear differential equation, (Chinese), Acta Mathematica Scientia, 36 (2016), 297–306. http://doi.org/10.3969/j.issn.1003-3998.2016.02.008 doi: 10.3969/j.issn.1003-3998.2016.02.008 |
[2] | Z. X. Chen, Complex differences and difference equations, Beijing: Science Press, 2014. |
[3] | L. Y. Gao, On meromorphic solutions of a type of system of composite functional equations, Acta Math. Sci., 32 (2012), 800–806. https://doi.org/10.1016/s0252-9602(12)60060-5 doi: 10.1016/s0252-9602(12)60060-5 |
[4] | L. Y. Gao, Estimates of N-function and m-function of meromorphic solutions of systems of complex difference equations, Acta Math. Sci., 32 (2012), 1495–1502. https://doi.org/10.1016/S0252-9602(12)60118-0 doi: 10.1016/S0252-9602(12)60118-0 |
[5] | L. Y. Gao, Systems of complex difference equations of Malmquist type, (Chinese), Acta Mathematica Scientia, Chinese Series, 55 (2012), 293–300. |
[6] | L. Y. Gao, On entire solutions of two types of systems of complex differential-difference equations, Acta Math. Sci., 37 (2017), 187–194. https://doi.org/10.1016/S0252-9602(16)30124-2 doi: 10.1016/S0252-9602(16)30124-2 |
[7] | L. Y. Gao, On solutions of a type of systems of complex differential-difference equations, (Chinese), Chinese Annals of Mathematics, 38 (2017), 23–30. https://doi.org/10.16205/j.cnki.cama.2017.0003 doi: 10.16205/j.cnki.cama.2017.0003 |
[8] | F. Gross, On the equation fn+gn=1, Bull. Amer. Math. Soc., 72 (1966), 86–88. https://doi.org/10.1090/S0002-9904-1966-11429-5 doi: 10.1090/S0002-9904-1966-11429-5 |
[9] | Y. Y. Jiang, L. Liao, Z. X. Chen, The value distribution of meromorphic solutions of some second order nonlinear difference equation, J. Appl. Anal. Comput., 8 (2018), 32–41. https://doi.org/10.11948/2018.32 doi: 10.11948/2018.32 |
[10] | I. Laine, Nevanlinna theory and complex differential equations, Berlin: Walter de Gruyter, 1993. https://doi.org/10.1515/9783110863147 |
[11] | K. Liu, T. B. Cao, H. Z. Cao, Entire solutions of Fermat type differential-difference equations, Arch. Math., 99 (2012), 147–155. https://doi.org/10.1007/s00013-012-0408-9 doi: 10.1007/s00013-012-0408-9 |
[12] | K. Liu, I. Laine, L. Yang, Complex delay-differential equations, De Gruyter, 2021. https://doi.org/10.1515/9783110560565 |
[13] | M. Liu, L. Gao, Transcendental solutions of systems of complex differential-difference equations, (Chinese), Sci. Sin. Math., 49 (2019), 1633–1654. |
[14] | D. Qiu, Y. Y. Jiang, The existence of entire solutions of some nonlinear differential-difference equations, submitted for publication. |
[15] | C. C. Yang, P. Li, On the transcendental solutions of a certain type of nonlinear differential equations, Arch. Math., 82 (2004), 442–448. https://doi.org/10.1007/s00013-003-4796-8 doi: 10.1007/s00013-003-4796-8 |
[16] | C. C. Yang, H. X. Yi, Theory of the uniqueness of meromorphic function, (Chinese), Beijing: Science Press, 1995. |
[17] | A. Wiles, Modular elliptic curves and Fermats last theorem, Ann. Math., 141 (1995), 443–551. https://doi.org/10.2307/2118559 doi: 10.2307/2118559 |
1. | Xue-Ying Zhang, Ze-Kun Xu, Wei-Ran Lü, On Entire Function Solutions to Fermat Delay-Differential Equations, 2022, 11, 2075-1680, 554, 10.3390/axioms11100554 | |
2. | Guowei Zhang, The exact transcendental entire solutions of complex equations with three quadratic terms, 2023, 8, 2473-6988, 27414, 10.3934/math.20231403 |