# Notes on Solutions for Some Systems of Complex Functional Equations inℂ2

Authors: Hong Li; Hong Yan Xu
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The purpose of this article is to give the details of finding the transcendental entire solutions with finite order for the systems of nonlinear partial differential-difference equations∂f1z1,z2/∂z1+∂f1z1,z2/∂z2n1+P1zf2z1+c1,z2+c2m1=Q1z,∂f1z1,z2/∂z1+∂f1z1,z2/∂z2n2+P2zf1z1+c1,z2+c2m2=Q2z, where P1z,P2z,Q1z,andQ2z are polynomials in ℂ2; n1,n2,m1,andm2 are positive integers, and c=c1,c2∈ℂ2. We obtain that there exist some pairs of the transcendental entire solutions of finite order for the above system, which is a very powerful supplement to the previous theorems given by Xu and Cao and Xu and Yang.


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## Body

## 1. Introduction

In 1970, Yang [1] proved that the functional equations fn+gm=1 have no nonconstant entire solutions, if m,n are positive integers satisfying 1/m+1/n<1. After this result, with the aid of the Nevanlinna theory and the difference analogues of the Nevanlinna theory (see [2–6]), there were rapid developments on complex differential and difference equations in one and several complex variables. Some classical results and topics in different fields are considered in difference versions, for example, difference Riccati equations, difference Painlevé equations, and difference Fermat equations (see [7–14]). Recently, Cao and Xu [15–17] 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 [18, 19] and obtained the following theorems which is an extension of the previous results given by Liu and his collaborators (see [20–24]).Theorem 1 (see ([16], Theorem 1.1)).
Letc=c1,c2∈ℂ2. Then, the Fermat-type partial differential-difference equation(1)∂fz1,z2∂z1n+fz1+c1,z2+c2m=1,does not have any transcendental entire solution with finite order, where m and n are two distinct positive integers.Theorem 2 (see ([15], Theorem 3.2)).
Letc=c1,c2∈ℂ\0. Suppose that f is a nontrivial meromorphic solution of the Fermat type partial difference equations(2)1fz1+c1,z2+c2m+1fz1,z2m=Az1,z2fz1,z2n,or(3)1fz1+c1,z2+c2m+1fz1+c1,z2m+1fz1,z2+c2m=Az1,z2fz1,z2n,where m∈ℕ,n∈ℕ∪0, and Az1,z2 is a nonzero meromorphic function on ℂ2 with respect to the solution f, that is Tr,A=oTr,f. If δf∞>0, then(4)limsupr⟶∞logTr,fr>0.Remark 3.
Letn=0 and Az1,z2=1, then the above equations become(5)1fz1+c1,z2+c2m+1fz1,z2m=1,1fz1+c1,z2+c2m+1fz1+c1,z2m+1fz1,z2+c2m=1,which can be called as the partial difference equations of Fermat type.In 2020, the first author and his coauthors discussed the transcendental entire solutions with finite order for the systems of partial differential difference equations and gave the conditions on the existence of the finite-order transcendental entire solutions for the following systems, which are some extension and improvements of the previous results given by Xu and Cao and Gao [16, 25].Theorem 4 (see ([26], Theorem 1.2)).
Letc=c1,c2∈ℂ2, and mj,njj=1,2 be positive integers. If the following system of Fermat-type partial differential-difference equations(6)∂f1z1,z2∂z1n1+f2z1+c1,z2+c2m1=1,∂f2z1,z2∂z1n2+f1z1+c1,z2+c2m2=1,satisfies one of the conditions(i)
m1m2>n1n2;(ii)
mj>nj/nj−1 for nj≥2, j=1,2.
Then, system (6) does not have any pair of transcendental entire solution with finite order.Remark 5.
Here,f,g is called as a pair of finite-order transcendental entire solutions for system(7)fn1+gm1=1,fn2+gm2=1,if f,g are transcendental entire functions and ρ=maxρf,ρg<∞.Remark 6.
The conditionmj>nj/nj−1 implies mj>1. Thus, a question rises naturally: what will happen on the existence of transcendental entire solutions with finite order when mj=1,j=1,2 in system (6)?In fact, we give the following example to explain that system (6) has a pair of transcendental entire solutions with finite order when m1=m2=1 and n1=n2=2, that is,(8)∂f1z1,z2∂z12+f2z1+c1,z2+c2=1,∂f2z1,z2∂z12+f1z1+c1,z2+c2=1rgb0.00,0.00,1.00.Example 1.
Let(9)f1z=1−14π2−14z12+12z1z2−πi2z2+z1−πiez2−ez2+12z2−πi2,f2z=1−14π2−14z12+12z1z2−πi2z2−z1−πiez2−ez2−12z2−πi2.
Then,f=f1,f2 is a pair of transcendental entire solutions of system (8) with c1,c2=πi,πi and ρf=1.Corresponding to system (6), we further consider the following system of the partial differential difference equation(10)∂f1z1,z2∂z1+∂f1z1,z2∂z2n1+P1zf2z1+c1,z2+c2m1=Q1z,∂f2z1,z2∂z1+∂f2z1,z2∂z2n2+P2zf1z1+c1,z2+c2m2=Q2z,whereP1z,P2z are two nonzero polynomials in ℂ2 and obtained.Theorem 7.
Letc=c1,c2∈ℂ2, mj,njj=1,2 be positive integers satisfies one of the conditions(i)
m1m2>n1n2;(ii)
mj>nj/nj−1 for nj≥2, j=1,2.
Then, system (10) does not have any pair of transcendental entire solutions with finite order.The following example shows that the conditionsmj>nj/nj−1 for nj≥2 and j=1,2 are precise and the existence of finite-order transcendental entire solutions for the system (10) when n1=n2=2, m1=m2=1 and P1z=P2z=Q1z=Q2z=1, that is,(11)∂f1z1,z2∂z1+∂f1z1,z2∂z22+f2z1+c1,z2+c2=1,∂f2z1,z2∂z1+∂f2z1,z2∂z22+f1z1+c1,z2+c2=1.Example 2.
Let(12)f1z=1−14π2−14z12+12z2−z1z1−πi+z1−πiez2−z1−ez2−z1+12z2−z1−πi2,f2z=1−14π2−14z12+12z2−z1z1−πi−z1−πiez2−z1−ez2−z1−12z2−z1−πi2.
Then,f=f1,f2 is a pair of transcendental entire solutions of system (11) with c1,c2=πi,2πi and ρf=1.Remark 8.
In Sections3 and 4, we give the details proceeding for obtaining a class of finite-order transcendental entire solutions for systems (8) and (11).Next, we continue to discuss the existence of the finite-order transcendental entire solutions for several systems including both the difference operator and the partial differential such as(13)∂f1∂z1n1+f2z1+c1,z2+c2−f1z1,z2m1=1,∂f2∂z1n1+f1z1+c1,z2+c2−f2z1,z2m1=1,(14)∂f1∂z1+∂f1∂z2n1+f2z1+c1,z2+c2−f1z1,z2m1=1,∂f2∂z1+∂f2∂z2n1+f1z1+c1,z2+c2−f2z1,z2m1=1,wherec1,c2 are constants in ℂ. It is easy to find the finite-order transcendental entire solutions for systems (13) and (14). For c2≠0, system (5) has a pair of finite-order transcendental entire solutions f1,f2 of the forms(15)f1=az1+b+d−2ac12c2z2+d2+eπi/c2z2,f2=az1+b+d−2ac12c2z2+b2−eπi/c2z2,and for c2≠c1, system (14) has a pair of finite-order transcendental entire solutions f1,f2 of the forms(16)f1=az1+b+d−2ac12c2−c1z2−z1+d2+eπi/c2−c1z2−z1,f2=az1+b+d−2ac12c2−c1z2−z1+b2−eπi/c2−c1z2−z1,where a,b,d∈ℂ satisfy 1−an1=bm1 and 1−an2=bm2. Furthermore, we can give the finite-order transcendental entire solutions for systems (13) and (14) when n1=n2=2 and m1=m2=1 easily.Example 3.
The function(17)f=f1,f2=z1−z2+eπiz2,z1−z2−eπiz2,is a pair of transcendental entire solutions with ρf=1 for system (13) when c1,c2=1,1, n1=n2=2, and m1=m2=1.Example 4.
The function(18)f=f1,f2=2z1−z2+eπiz2−z1,2z1−z2−eπiz2−z1,is a pair of transcendental entire solutions with ρf=1 for system (14) when c1,c2=1,2, n1=n2=2 and m1=m2=1.Corresponding to systems (13) and (14), we can also obtain the solutions of the following systems(19)∂f1∂z1n1+f2z1+c1,z2+c2−f2z1,z2m1=1,∂f2∂z1n1+f1z1+c1,z2+c2−f1z1,z2m1=1,(20)∂f1∂z1+∂f1∂z2n1+f2z1+c1,z2+c2−f2z1,z2m1=1,∂f2∂z1+∂f2∂z2n1+f1z1+c1,z2+c2−f1z1,z2m1=1,wherec1,c2 are constants in ℂ. In fact, for c2≠0, then systems (19) has a pair of solutions with the forms(21)f1,f2=a1z1+b2−a1c1c2z2+G1z2,a2z1+b1−a2c1c2z2+G2z2,where G1z2,G2z2 are two period functions with period c2, and for s≔z2−z1 and s0≔c2−c1≠0, then system (20) has a pair of solutions with the forms(22)f1,f2=a1z1+b2−a1c1c2−c1s+G1s,a2z1+b1−a2c1c2−c1s+G2s,where G1s,G2s are two period functions with period s0, and a1,a2,c1,c2, d1,andd2 are constants satisfying(23)a1n1+b1m1=1,a2n2+b2m2=1.


## 2. Proof of Theorem7

The following lemmas will be used in this paper.Lemma 9 ([27, 28]).
Letf be a nonconstant meromorphic function on ℂn and let I=i1,⋯,in be a multi-index with length ∣I∣=∑j=1nij. Assume that Tr0,f≥e for some r0. Then,(24)mr,∂Iff=Sr,f,holds for all r≥r0 outside a set E⊂0,+∞ of finite logarithmic measure ∫Edt/t<∞, where ∂If=∂∣I∣f/∂z1i1⋯∂znin.Lemma 10 ([18, 19]).
Letf be a nonconstant meromorphic function with finite order on ℂn such that f0≠0,∞, and let ε>0. Then, for c∈ℂn,(25)mr,fzfz+c+mr,fz+cfz=Sr,f,holds for all r≥r0 outside a set E⊂0,+∞ of finite logarithmic measure ∫Edt/t<∞.Lemma 11 (see [29]).
Letf be a nonconstant meromorphic function on ℂn. Take a positive integer m and take polynomials of f and its partial derivatives:(26)Pf=∑p∈Iapfp0∂i1fp1⋯∂ilfpl,p=p0,⋯,pl,Qf=∑q∈Icqfq0∂j1fq1⋯∂jlfql,q=q0,⋯,ql,Bf=∑k=0mbkfk,where I,J are finite sets of distinct elements and ap,cq,andbk are meromorphic functions on ℂn with bm≡0. Assume that f satisfies the equation(27)BfQf=Pf,such that Pf,Qf, and Bf are differential polynomials, that is, their coefficients a satisfy mr,a=Sr,f. If degPf≤m=degBf, then(28)mr,Qf=Sr,f,holds for all r possibly outside of a set E with finite logarithmic measure.Proof.
Letf1,f2 be a pair of transcendental entire functions with finite-order satisfying system (10). Here, we will discuss two following cases.Case 1.
n1n2>m1m2. In view of Lemma 10, the following conclusions that(29)mr,fjz1,z2fjz1+c1,z2+c2=Sr,fj,j=1,2,
holds for allr>0 outside of a possible exceptional set Ej⊂1,+∞ of finite logarithmic measure ∫Ejdt/t<∞. Thus, we can deduce from (29) that(30)Tr,fjz1,z2=mr,fjz1,z2≤mr,fjz1,z2fz1+c1,z2+c2+mr,fjz1+c1,z2+c2+log2=mr,fjz1+c1,z2+c2+Sr,fj=Tr,fjz1+c1,z2+c2+Sr,fj,j=1,2,
for allr∈E≕E1∪E2. By using Lemma 9 and Lemma 11, it follows from (30) that(31)n1Tr,f2z1,z2≤n1Tr,f2z1+c1,z2+c2+Sr,f2≤Tr,P1zf2z1+c1,z2+c2n1+Sr,f2=Tr,∂f1∂z1+∂f1∂z2m1−Q1z+Sr,f2=m1Tr,∂f1∂z1+∂f1∂z2+Ologr+Sr,f2+Sr,f1=m1mr,∂f1∂z1+∂f1∂z2+Ologr+Sr,f2+Sr,f1≤m1mr,∂f1/∂z1+∂f1/∂z2f1+mr,f1+Ologr+Sr,f1+Sr,f2=m1Tr,f1+Ologr+Sr,f1+Sr,f2,
for allr∈E. Similarly, we have(32)n2Tr,f1≤m2Tr,f2+Ologr+Sr,f1+Sr,f2,r∈E.
In view of (31) and (32), it yields(33)n1n2−m1m2Tr,fj≤Ologr+Sr,f1+Sr,f2,r∈E.
In view ofn1n2>m1m2, this is impossible since f1,f2 are transcendental entire functions.Case 2.
mj>nj/nj−1, nj≥2, j=1,2. In view of the Nevanlinna second fundamental theorem concerning small functions, Lemma 10, and system (11), we can deduce that(34)m1Tr,∂f1∂z1+∂f1∂z2=Tr,∂f1∂z1+∂f1∂z2m1+Sr,f1≤N¯r,1∂f1/∂z1+∂f1/∂z2m1+N¯r,1∂f1/∂z1+∂f1/∂z2m1−Q1z+Sr,f1≤N¯r,1∂f1/∂z1+∂f1/∂z2m1+N¯r,1P1zf2z1+c1,z2+c2n1+Sr,f1≤N¯r,1∂f1/∂z1+∂f1/∂z2+N¯r,1f2z1+c1,z2+c2+Ologr+Sr,f1≤Tr,∂f1∂z1+∂f1∂z2+Tr,f2z1+c1,z2+c2+Ologr+Sr,f1+Sr,f2,that is,(35)m1−1Tr,∂f1∂z1+∂f1∂z2≤Tr,f2z+c+Ologr+Sr,f1+Sr,f2.
Similarly, we have(36)m2−1Tr,∂f2∂z1+∂f2∂z2≤Tr,f1z+c+Ologr+Sr,f1+Sr,f2.
On the other hand, in view of system (10) and Lemma 10, it follows that(37)n1Tr,f2z1+c1,z2+c2+Ologr=Tr,P1zf2z1+c1,z2+c2n1+Sr,f2=Tr,∂f1∂z1+∂f1∂z2m1−Q1z+Sr,f2=m1Tr,∂f1∂z1+∂f1∂z2+Ologr+Sr,f1+Sr,f2.
Similarly, we have(38)n2Tr,f1z1+c1,z2+c2=m2Tr,∂f2∂z1+∂f2∂z2+Ologr+Sr,f1+Sr,f2.
In view of (35)–(38), we obtain that(39)n1−m1m1−1Tr,f2z1+c1,z2+c2≤Ologr+Sr,f1+Sr,f2,n2−m2m2−1Tr,f1z1+c1,z2+c2≤Ologr+Sr,f1+Sr,f2.
The fact thatmj>nj/nj−1 can lead to a contradiction since f1,f2 are transcendental entire functions.
Therefore, this completes the proof of Theorem7.


## 3. Entire Solutions for System (8)

Now, the details that we obtain a pair of finite-order transcendental entire solutions for system (8) will be given below.Letf1,f2 be a pair of finite-order transcendental entire solutions for system (8). Differentiating both equations in system (8) for z1, we deduce(40)2∂f1z1,z2∂z1∂2f1z1,z2∂z12+∂f2z1+c1,z2+c2∂z1=0,2∂f2z1,z2∂z1∂2f2z1,z2∂z12+∂f1z1+c1,z2+c2∂z1=0.LetF1z1,z2=∂f1z1,z2/∂z1 and F2z1,z2=∂f2z1,z2/∂z1, then it follows from (18) that(41)2F1z1,z2∂F1z1,z2∂z1=−F2z1+c1,z2+c2,2F2z1,z2∂F2z1,z2∂z1=−F1z1+c1,z2+c2rgb0.00,0.00,1.00.By Lemmas9–11, it yields that ∂Fjz1,z2/∂z1=Sr,fj for j=1,2. Thus, we can assume that(42)∂F1z1,z2∂z1=a1,∂F2z1,z2∂z1=a2,wherea1,a2∈ℂ. Solving Equation (42), we have(43)F1z1,z2=a1z1+φ1z2,F2z1,z2=a2z1+φ2z2,whereφ1z2,φ2z2 are finite-order transcendental entire functions in z2. Due to Equations (41) and (42), we obtain that(44)F1z=−12a1F2z+c,F2z=−12a2F1z+c.Substituting (43) into (44), we can deduce that(45)a1z1+φ1z2=−12a1a2z1+a2c1−12a1φ2z2+c2,a2z1+φ2z2=−12a2a1z1+a1c1−12a2φ1z2+c2,which implies that a13=a23=−1/8. It would be well if a1=a2=−1/2. So, it follows that(46)F1z1,z2=−12z1+φ1z2,F2z1,z2=−12z1+φ2z2,φ1z2+c2=φ2z2+12c1,φ2z2+c2=φ1z2+12c1.This means that(47)φ1z2+2c2−φ1z2=c1,φ1z2+2c2−φ1z2=c1,which imply(48)φ1z2=G1z2+c12c2z2,φ2z2=G2z2+c12c2z2,whereG1z2,G2z2 are finite-order entire period function with period 2c2 satisfying G2z2+c2=G1z2.Solving the following system(49)∂f1z1,z2∂z1=F1z1,z2=−12z1+φ1z2,∂f2z1,z2∂z1=F2z1,z2=−12z1+φ2z2,we obtain that(50)f1z1,z2=−14z12+z1φ1z2+ψ1z2,f2z1,z2=−14z12+z1φ2z2+ψ2z2,whereψ1z2,ψ2z2 are finite-order entire functions in z2. Substituting (50) into (8), and combining with the periodicity of φ1z2 and φ2z2, it yields that(51)−12z1+c12c2z2+G1z22−14z1+c12+z1+c1φ2z2+c2+ψ2z2+c2=1,−12z1+c12c2z2+G2z22−14z1+c12+z1+c1φ1z2+c2+ψ1z2+c2=1.Thus, we have(52)ψ2z2+c2=1−14c12−c122c2z2−c1G1z2−c12c2z2+G1z22,ψ1z2+c2=1−14c12−c122c2z2−c1G2z2−c12c2z2+G2z22,which mean that(53)ψ2z2=1+14c12−c122c2z2−c1G2z2−c12c2z2−c2+G2z22,(54)ψ1z2=1+14c12−c122c2z2−c1G1z2−c12c2z2−c2+G1z22.In view of (48)–(54), it follows that(55)f1z1,z2=1+14c12−14z12+c12c2z1z2−c122c2z2+z1−c1G1z2,−c12c2z2−c2+G1z22,f2z1,z2=1+14c12−14z12+c12c2z1z2−c122c2z2+z1−c1G2z2,−c12c2z2−c2+G2z22,whereG1z2,G2z2 are finite-order transcendental entire period functions with period 2c2 satisfying G2z2+c2=G1z2. Substituting f1,f2 into system (2), it is easy to confirm that f1,f2 is a solution of system (8).


## 4. Entire Solutions for System (11)

Letf1,f2 be a pair of finite-order transcendental entire solutions of system (13). Next, the detail that we obtain one form of f1,f2 is listed as follows. Differentiating system (13) for z1,z2, respectively, we have(56)2F1z1,z2∂F1z1,z2∂z1+∂F1z1,z2∂z2+F2z1+c1,z2+c2=0,2F2z1,z2∂F2z1,z2∂z1+∂F2z1,z2∂z2+F1z1+c1,z2+c2=0,where(57)F1z1,z2=∂f1z1,z2∂z1+∂f1z1,z2∂z2,forj=1,2.In view of Lemmas9–11, it follows that ∂Fjz1,z2/∂z1=Sr,fj for j=1,2. For the convenience, assume that(58)∂Fjz1,z2∂z1+∂Fjz1,z2∂z2=bj,j=1,2,wherebj∈ℂ. The characteristic equations for Equation (58) are(59)dz1dt=1,dz2dt=1,dFjdt=bj.In view of the initial conditions:z1=0,z2=s, and Fj=Fj0,s≔Fjs with a parameter s, we thus obtain the following parametric representation for the solutions of the characteristic equations: z1=t, z2=t+s,(60)Fj=∫0t‍bjdt+μjs=bjt+μjs,whereμjs,j=1,2 are entire functions with finite order in s. Thus, it follows that(61)Fjz1,z2=bjz1+μjz2−z1,j=1,2.In view of (56) and (58), it follows that(62)2b1F1z1,z2=−F2z1+c1,z2+c2,2b2F2z1,z2=−F1z1+c1,z2+c2.Substituting (61) into (62), we have that(63)2b12z1+2b1μ1s=−b2z1+c1−μ2s+s0,2b22z1+2b1μ1s=−b1z1+c1−μ1s+s0,wheres=z2−z1 and s0≔c2−c1. This implies that b13=b23=−1/8. Let us assume that b1=b2=−1/8. Thus, it yields that(64)μ2s+s0=μ1s+12c1,μ1s+s0=μ2s+12c1.This means(65)μjs=Gjs+τs,j=1,2,(66)G2s+s0=G1s,whereG1s,G2s are finite-order transcendental entire period functions with period 2s0, and τ=c1/2c2−c1. Then, in view of (61) and (65), we deduce(67)Fjz1,z2=−12z1+Gjz2−z1+τz2−z1,j=1,2,that is,(68)∂fjz1,z2∂z1+∂fjz1,z2∂z2=−12z1+Gjz2−z1+τz2−z1.By making use of the characteristic equations for Equation (68) again, let(69)dz1dt=1,dz2dt=1,dfjdt=−12z1+Gjz2−z1+τz2−z1.In view of the initial conditions:z1=0,z2=s, and fj=fj0,s≔fjs with a parameter s, we can deduce that the parametric representation for the solutions of the characteristic equations: z1=t, z2=t+s, and(70)fj=∫0t−12t+Gjs+τsdt+νjs=−14t2+tGjs+τs+νjs,j=1,2,whereνjs is an entire function with finite order in s. Substituting t=z1 and s=z2−z1 into the above form, we have that(71)fjz1,z2=−14z12+z1Gjz2−z1+a3z2−z1+νjz2−z1,j=1,2.Substituting (71) into (13), and combining with the periodicity of Gjs, it follows that(72)ν1s=1−14c12−c1G2s−s0−τc1s−s0−G2s−s0+τs−s02,(73)ν2s=1−14c12−c1G1s−s0−τc1s−s0−G1s−s0+τs−s02.Thus, in view of (66) and (71)–(73), we obtain that a pair of entire solutions of system (13) are of the forms(74)f1z1,z2=1+14c12−14z12+c12c2−c1z2−z1z1−c1+z1−c1G1z2−z1−G1z2−z1+c12c2−c1z2−z1−c2−c12,f2z1,z2=1+14c12−14z12+c12c2−c1z2−z1z1−c1+z1−c1G2z2−z1−G2z2−z1+c12c2−c1z2−z1−c2−c12,whereG1s,G2s are finite-order transcendental entire period functions with period 2s0 and satisfy (66). Let(75)G1s=eπi/c2−c1s,G2s=−eπi/c2−c1s.Thus,f1,f2 is a pair of finite-order transcendental entire solutions of system (13).



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*Source: 5424284-2021-07-02.xml*