On Meromorphic Solutions of Nonlinear Complex Differential Equations
1 Introduction and Main Results
Let w(z)be meromorphic in the complex plane,and we use the standard notations of Nevanlinna theory of meromorphic function(see[1]–[2]),which are also introduced as follows for convenient to read:
![pagenumber_ebook=55,pagenumber_book=149](https://qkk.lw71.com/uploads/image/2018/2/52d8c2504f17ca70a8d85328b61e7c51.png)
![pagenumber_ebook=56,pagenumber_book=150](https://qkk.lw71.com/uploads/image/2018/2/3fbcceceb43e4afee8ae7973e01be5ea.png)
where n(r,w)counts the number of the pole of w(z)in|z|≤r,each pole according to its multiplicity,and(r,w)counts the number of distinct poles of w(z)in|z|≤r.We call an error term and denote by S(r,w)any quantity satisfying
![pagenumber_ebook=56,pagenumber_book=150](https://qkk.lw71.com/uploads/image/2018/2/0d260789b937a53cc84d1435173e6a4a.png)
as r→∞,possibly outside a set of r of finite linear measure.
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Some authors investigated the existence of meromorphic solutions of algebraic differential equations and obtained many meaningful results(see[3]–[11]).Gackstatter and Laine[3]considered the following differential equation
![pagenumber_ebook=56,pagenumber_book=150](https://qkk.lw71.com/uploads/image/2018/2/b02a1ad04dea2a1f2f0bacadb836bace.png)
They gave the conjecture that:the differential equation(1.1)does not possess any admissible solutions if 1≤m≤n−1.For simplicity,we denotew is called an admissible solution of(1.1)if S(r)=o(T(r,w)).
He and Laine[6]proved the conjecture in[3].While they obtained the following theorem:
Theorem A[6]The differential equation(1.1)does not possess any admissible solutions for 1≤m≤n−1.
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Gao[7]investigated the following high-order differential equation:
![pagenumber_ebook=56,pagenumber_book=150](https://qkk.lw71.com/uploads/image/2018/2/5dce35dd56927ee29283a7bddc4515a9.png)
where
![pagenumber_ebook=56,pagenumber_book=150](https://qkk.lw71.com/uploads/image/2018/2/9735b955670657a785e860f562c49154.png)
and
![pagenumber_ebook=56,pagenumber_book=150](https://qkk.lw71.com/uploads/image/2018/2/430155ec189fb86e34f18d7c8606bde5.png)
He obtained the result as below:
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Theorem B[7] Let w(z)be a meromorphic solution of(1.2).If
![pagenumber_ebook=56,pagenumber_book=150](https://qkk.lw71.com/uploads/image/2018/2/628042bc7553075af31d71be663e40ce.png)
then w(z)is a non-admissible solution.
Remark 1.1 From the proof of Theorem B,we know that p−s−>0,if p<m.Therefore,the condition p≥m can be removed(see[7]).
Motivated by[7],we reinvestigate meromorphic solution of(1.2)when
![pagenumber_ebook=56,pagenumber_book=150](https://qkk.lw71.com/uploads/image/2018/2/87aa4ece305c7605c86180435fd9aa62.png)
More precisely,one of our results can be stated as follows:
where
![pagenumber_ebook=56,pagenumber_book=150](https://qkk.lw71.com/uploads/image/2018/2/b53e8dc97a35e6e44e92cab95f455dca.png)
then w(z)is a non-admissible solution.
Clearly,Theorem 1.1 reduces to Theorem B when λ=0.
Example 1.1 Obviously,w(z)=ezis an admissible solution of the following differential equation:
![pagenumber_ebook=57,pagenumber_book=151](https://qkk.lw71.com/uploads/image/2018/2/3878e2fb727fbbaaa28b337acb7d47d1.png)
It is easy to see that
![pagenumber_ebook=57,pagenumber_book=151](https://qkk.lw71.com/uploads/image/2018/2/66961c5ac3af2e0af0fb6abe79d5dc20.png)
and
![pagenumber_ebook=57,pagenumber_book=151](https://qkk.lw71.com/uploads/image/2018/2/34514785a05995da9bbb673d04784dcf.png)
but
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![pagenumber_ebook=57,pagenumber_book=151](https://qkk.lw71.com/uploads/image/2018/2/63d88e351c9e342c20f8352813e8938f.png)
Example 1.2 Clearly,w(z)=ezis the non-admissible solution of following differential equation:
![pagenumber_ebook=57,pagenumber_book=151](https://qkk.lw71.com/uploads/image/2018/2/4d8fa8402ef5ce59db9499555cafa404.png)
here
![pagenumber_ebook=57,pagenumber_book=151](https://qkk.lw71.com/uploads/image/2018/2/eaf9969bb90ae16d199d0f077bae4d50.png)
and we have
![pagenumber_ebook=57,pagenumber_book=151](https://qkk.lw71.com/uploads/image/2018/2/73f217b254ad2d3acb73b71fec623a24.png)
Examples 1.1 and 1.2 imply that the conditions in Theorem 1.1 are necessary and sharp.
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We mention for the existence of solutions,growth of solutions or the form of systems of differential equations(see[12]–[17]).Particularly,Su and Li[15]considered the following systems of complex differential equation:
![pagenumber_ebook=57,pagenumber_book=151](https://qkk.lw71.com/uploads/image/2018/2/2a0e168022609ae77a733a1a6388d2a4.png)
Theorem 1.1 Let w(z)be a meromorphic solution of(1.2).If
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![pagenumber_ebook=57,pagenumber_book=151](https://qkk.lw71.com/uploads/image/2018/2/a72092ab8fd55a95220f00e6c355316e.png)
and
![pagenumber_ebook=57,pagenumber_book=151](https://qkk.lw71.com/uploads/image/2018/2/c84ec35318163f1700694567e6c4191f.png)
They proved the existence of non-admissible solutions of the above equations.
In this paper,we mainly concern the following systems of nonlinear complex algebraic differential equations:
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![pagenumber_ebook=57,pagenumber_book=151](https://qkk.lw71.com/uploads/image/2018/2/097af5ae707250ef1b35d27b8484c5db.png)
Lemma 2.3[10]Let w,a0,a1,···,akbe meromorphic functions in|z|< ∞ and T(r,aj)=S(r,w)(j=1,2,···,k).Then
![pagenumber_ebook=58,pagenumber_book=152](https://qkk.lw71.com/uploads/image/2018/2/7a6eeae220fa3380e602e732340134c6.png)
and
![pagenumber_ebook=58,pagenumber_book=152](https://qkk.lw71.com/uploads/image/2018/2/fb3c8197b2f8f1d5e53fcfd13737a60f.png)
are differential polynomials with meromorphic coefficients{ck(z)},{dl(z)},
![pagenumber_ebook=58,pagenumber_book=152](https://qkk.lw71.com/uploads/image/2018/2/d48ecb154aabcd5ce6ff2f683f691a58.png)
and
![pagenumber_ebook=58,pagenumber_book=152](https://qkk.lw71.com/uploads/image/2018/2/b8fd93d06c1e87c3d438d757793eae09.png)
are differential monomials.
Now we introduce two examples to show that some conditions in Theorem 1.2 are necessary.
One of main results can be stated as below:
Theorem 1.2 Let(w1,w2)be a meromorphic solution of(1.3).If p<m2,q<m1,s≤p−2,t≤q−2,and
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![pagenumber_ebook=58,pagenumber_book=152](https://qkk.lw71.com/uploads/image/2018/2/8eee0a4ef1f24157123c0c7b2c7eb055.png)
then(w1,w2)is a non-admissible meromorphic solution.
Definition 1.1 Let(w1(z),w2(z))be a meromorphic solution of(1.3),S(r)be the sum of all characteristic functions of coefficients in(1.3).If wk(z)satis fies S(r)=o(T(r,wk(z))(k=1,2)outside a possible exception set E with finite linear measure,we call that(w1,w2)is an admissible solution,or else,non-admissible solution.
Example 1.3 Consider the following systems of differential equations:
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![pagenumber_ebook=58,pagenumber_book=152](https://qkk.lw71.com/uploads/image/2018/2/c9c4ea8874baa2df379791e35c369e12.png)
where(w1,w2)=(e2z,ez)is a solution of the differential equations containing two nonadmissible components,and
![pagenumber_ebook=58,pagenumber_book=152](https://qkk.lw71.com/uploads/image/2018/2/28b92dcac6dd6ba3344bbecc43804b5d.png)
Obviously,we have
![pagenumber_ebook=58,pagenumber_book=152](https://qkk.lw71.com/uploads/image/2018/2/279c23a495e1108961fe69b1203c47cb.png)
and
![pagenumber_ebook=58,pagenumber_book=152](https://qkk.lw71.com/uploads/image/2018/2/6c5a260ee3dd06bc300d3e43836d1955.png)
Example 1.4 is an admissible solution of the following equations:
![pagenumber_ebook=59,pagenumber_book=153](https://qkk.lw71.com/uploads/image/2018/2/6883c759c30945d7fe924d4063c4b755.png)
which satis fies
![pagenumber_ebook=59,pagenumber_book=153](https://qkk.lw71.com/uploads/image/2018/2/0b62ed9a97ca2c55f46b0a6720a353e3.png)
One can easily see that
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![pagenumber_ebook=59,pagenumber_book=153](https://qkk.lw71.com/uploads/image/2018/2/554f13e070d0388d87de99217975ccfc.png)
but
![pagenumber_ebook=59,pagenumber_book=153](https://qkk.lw71.com/uploads/image/2018/2/94d9f7bb2cec88f71d2ae6b79852035f.png)
Generally,we have
Theorem 1.3 Let(w1,w2)be a meromorphic solution of(1.3).If
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![pagenumber_ebook=59,pagenumber_book=153](https://qkk.lw71.com/uploads/image/2018/2/d5db475bb7e5a71fd75d9662a74f82cb.png)
and
![pagenumber_ebook=59,pagenumber_book=153](https://qkk.lw71.com/uploads/image/2018/2/a47d9e40515c3c0967a23b7f39c6827a.png)
where λ is a constant with λ < 1,T(r)=min{T(r,w1),T(r,w2)},and I is a subset of(0,+∞)with in finite linear measure,then(w1,w2)is a non-admissible meromorphic solution.
In fact,Theorem 1.2 is a special case of Theorem 1.3,that is λ=0.
In this paper,We also discuss the following systems of differential equations:
![pagenumber_ebook=59,pagenumber_book=153](https://qkk.lw71.com/uploads/image/2018/2/8b9d52233be7931796aad13f439e726c.png)
where
![pagenumber_ebook=59,pagenumber_book=153](https://qkk.lw71.com/uploads/image/2018/2/36ba1ab967cffc0d433392a5b57811ae.png)
and
![pagenumber_ebook=59,pagenumber_book=153](https://qkk.lw71.com/uploads/image/2018/2/9328d466844d1f349017cee77c853d3e.png)
are differential monomials,ap(z) 0,bq(z)
0,as(z)
0,bt(z)
0,{ai(z)},{bj(z)}are meromorphic coefficients.Concerning the existence of meromorphic solution,we have
Theorem 1.4 Let(w1,w2)be a meromorphic solution of(1.4).If p<m2,q<m1,s≤p−2,t≤q−2 and m2m1,then(w1,w2)is a non-admissible meromorphic solution.
Corollary 1.1 Let(w1,w2)be a meromorphic solution of(1.4).If p<m2,q<m1,s<p−2,t<q−2 and m2=m1,then(w1,w2)is a non-admissible solution.
Based on Gao’s results(see[11]and[17]),we apply Theorem 1.2 to discuss(1.4).Precisely,we get
Theorem 1.5 Let(w1,w2)be an admissible meromorphic solution of(1.3).If p<m2,q<m1,and
![pagenumber_ebook=60,pagenumber_book=154](https://qkk.lw71.com/uploads/image/2018/2/2413a61bd48661ae7703ecbf3bca694d.png)
then(1.3)takes one of the following forms:
![pagenumber_ebook=60,pagenumber_book=154](https://qkk.lw71.com/uploads/image/2018/2/51b284235ffeaabdb2f791709bde8b74.png)
where
2 Preliminaries
In this section,we introduce some lemmas.They would bring a great convenience during our analysis.Namely,
Lemma 2.1[4]Let w,b0,b1,···,bkbe meromorphic functions in|z| < ∞,bk 0,then there hold:
![pagenumber_ebook=60,pagenumber_book=154](https://qkk.lw71.com/uploads/image/2018/2/385cd5f5a3ac7ea657cf6ed3e40ae8b9.png)
Lemma 2.2[5]Let g0(z)and g1(z)be meromorphic functions in|z|<∞and linearly independent over C.Set g0(z)+g1(z)=Φ.Then we have
![pagenumber_ebook=60,pagenumber_book=154](https://qkk.lw71.com/uploads/image/2018/2/897065be47c2ee601f23a4b9b8021298.png)
where
![pagenumber_ebook=60,pagenumber_book=154](https://qkk.lw71.com/uploads/image/2018/2/865cd46012db6f19e09a5d0522933070.png)
where p > s,q > t,{(k)},{(l)}are two finite index sets,ap(z) 0,bq(z)
0,as(z)
0,bt(z)
0,ai(z)and bj(z)are meromorphic functions,
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![pagenumber_ebook=61,pagenumber_book=155](https://qkk.lw71.com/uploads/image/2018/2/cd4710cc2dfba03c213aa72039ec4f4e.png)
If(w1,w2)is an admissible solution,then A1,B1and A2,B2are linearly independent respectively.
![pagenumber_ebook=61,pagenumber_book=155](https://qkk.lw71.com/uploads/image/2018/2/a5379b8b8cf31e943b0597e6b1798d0d.png)
Then
![pagenumber_ebook=61,pagenumber_book=155](https://qkk.lw71.com/uploads/image/2018/2/d935ee4c642b28c9ff961529081f0334.png)
Lemma 2.4 Let(w1,w2)be a meromorphic solution of(1.3).Putting
The proof is based on the definition of linear independence,we omit the detail here.
Lemma 2.5 Let(w1,w2)be a meromorphic admissible solution of(1.4).If p<m2,q<m1,then
![pagenumber_ebook=61,pagenumber_book=155](https://qkk.lw71.com/uploads/image/2018/2/c327ee4a5a747ed29d3ea3e47bf7b6cb.png)
Proof. Let(w1,w2)be a meromorphic admissible solution of(1.4).By Lemma 2.1,we get
![pagenumber_ebook=61,pagenumber_book=155](https://qkk.lw71.com/uploads/image/2018/2/d91933cdcc353d65e39360cca43d40ea.png)
That is,
![pagenumber_ebook=61,pagenumber_book=155](https://qkk.lw71.com/uploads/image/2018/2/cca2506c70ef8485c442b1f31d238226.png)
Similarly,we have
![pagenumber_ebook=61,pagenumber_book=155](https://qkk.lw71.com/uploads/image/2018/2/5465aaaefbfe079424a04e8ee38a02ca.png)
By(2.1)and(2.2),we gain
![pagenumber_ebook=61,pagenumber_book=155](https://qkk.lw71.com/uploads/image/2018/2/84705bc8b7b7a746add420ee7a473a38.png)
Since m1−q>0,m2−p>0,we conclude
![pagenumber_ebook=61,pagenumber_book=155](https://qkk.lw71.com/uploads/image/2018/2/b31b0a329b7b1a73e213839b87a0cd20.png)
We finish the proof of the lemma.
3 Proof of Theorem 1.1
Firstly,let w(z)be an admissible meromorphic solution of the differential equation(1.2).Denote
![pagenumber_ebook=61,pagenumber_book=155](https://qkk.lw71.com/uploads/image/2018/2/9389a3995cd82279318ff48440f2e6d9.png)
Analogous to Lemma 2.4,we maybe easily prove that A and B are linearly independent.Meanwhile,we can see from Lemma 2.2 that
![pagenumber_ebook=62,pagenumber_book=156](https://qkk.lw71.com/uploads/image/2018/2/290ff07547dc3835381606b21c6150e2.png)
Now,we estimate very term on the right-hand side of(3.1).By using Lemmas 2.1 and 2.3,we get
![pagenumber_ebook=62,pagenumber_book=156](https://qkk.lw71.com/uploads/image/2018/2/27f7ab9bd6aa74528efa819f6dcba5d2.png)
Since
![pagenumber_ebook=62,pagenumber_book=156](https://qkk.lw71.com/uploads/image/2018/2/d01466297c9915afc24ad778f9e97a39.png)
that is,
![pagenumber_ebook=62,pagenumber_book=156](https://qkk.lw71.com/uploads/image/2018/2/128ff10ea8292313ac941dff7110a474.png)
and since
![pagenumber_ebook=62,pagenumber_book=156](https://qkk.lw71.com/uploads/image/2018/2/bf30b847b1eba3bf382d91c7581e6d3f.png)
by(3.3),we have
![pagenumber_ebook=62,pagenumber_book=156](https://qkk.lw71.com/uploads/image/2018/2/c63c390937251b84e5a63ab0e4fa55f8.png)
Thus
![pagenumber_ebook=62,pagenumber_book=156](https://qkk.lw71.com/uploads/image/2018/2/87a009f64e44681e277fa9443970e1a9.png)
Inserting(3.2),(3.4)–(3.9)into(3.1),we obtain
![pagenumber_ebook=62,pagenumber_book=156](https://qkk.lw71.com/uploads/image/2018/2/f1b2a2d4b0f7773e5d45146bf5ce80a7.png)
Since
![pagenumber_ebook=62,pagenumber_book=156](https://qkk.lw71.com/uploads/image/2018/2/fe1e7abd47edf404b496dbf4b660b805.png)
we conclude
![pagenumber_ebook=62,pagenumber_book=156](https://qkk.lw71.com/uploads/image/2018/2/dca7e7ea23adf0f7e5a1d8b09bfd3387.png)
Noting that
![pagenumber_ebook=63,pagenumber_book=157](https://qkk.lw71.com/uploads/image/2018/2/f6035900afce06b6f294fc946ef4f7f4.png)
we have
![pagenumber_ebook=63,pagenumber_book=157](https://qkk.lw71.com/uploads/image/2018/2/537a2c37ce482bb08c140cae85e70995.png)
which is a contradiction.Therefore,w(z)is not an admissible solutions of(1.2)and the proof of the theorem is completed.
4 Proof of Theorem 1.3
Suppose that(w1,w2)is a meromorphic admissible solution of the systems of differential equations(1.3).In view of Lemmas 2.2 and 2.4,we have
![pagenumber_ebook=63,pagenumber_book=157](https://qkk.lw71.com/uploads/image/2018/2/b0d466e6745e0a3bc61c46316d6a3c84.png)
Applying Lemmas 2.1 and 2.3,we estimate every term in(4.1).Since
![pagenumber_ebook=63,pagenumber_book=157](https://qkk.lw71.com/uploads/image/2018/2/534aff88799965f511899d4b33c80888.png)
then
![pagenumber_ebook=63,pagenumber_book=157](https://qkk.lw71.com/uploads/image/2018/2/0638054f7e3997532ff9fd0676adb8d5.png)
Meanwhile,since
![pagenumber_ebook=63,pagenumber_book=157](https://qkk.lw71.com/uploads/image/2018/2/89183eb54d76ff8f7f77b81de3012ce2.png)
then we have
![pagenumber_ebook=63,pagenumber_book=157](https://qkk.lw71.com/uploads/image/2018/2/e7aecf992e5ea8a55c69a44e891810c6.png)
and
![pagenumber_ebook=63,pagenumber_book=157](https://qkk.lw71.com/uploads/image/2018/2/c033e50e4ed5ae18599d8069620e242d.png)
Therefore
![pagenumber_ebook=64,pagenumber_book=158](https://qkk.lw71.com/uploads/image/2018/2/d7c05d44f340e61aa410af170e9fd0ec.png)
Inserting(4.2)–(4.9)into(4.1),we obtain
![pagenumber_ebook=64,pagenumber_book=158](https://qkk.lw71.com/uploads/image/2018/2/14e28ee15e3ac6e821af6522d72888ea.png)
If T(r,w1)≤T(r,w2),then
![pagenumber_ebook=64,pagenumber_book=158](https://qkk.lw71.com/uploads/image/2018/2/6675e629bb6de67955681b4b0760973d.png)
Otherwise,if T(r,w1)≥T(r,w2),then
![pagenumber_ebook=64,pagenumber_book=158](https://qkk.lw71.com/uploads/image/2018/2/fcd775039924258291c2339e4cce6dce.png)
Meanwhile,noting that
![pagenumber_ebook=64,pagenumber_book=158](https://qkk.lw71.com/uploads/image/2018/2/70ff39b78a99df4f4e0aa34bdd425b7f.png)
we obtain
![pagenumber_ebook=64,pagenumber_book=158](https://qkk.lw71.com/uploads/image/2018/2/b8f5d51323be66cc1e9b398a5c7f5db4.png)
Proceeding similarly as the above proof,we can conclude
![pagenumber_ebook=64,pagenumber_book=158](https://qkk.lw71.com/uploads/image/2018/2/58dcafa9085964d520137fa366f05501.png)
In view of p<m2,q<m1,we get
![pagenumber_ebook=64,pagenumber_book=158](https://qkk.lw71.com/uploads/image/2018/2/5ef166e234043fcbb5687fb568738cb4.png)
and
![pagenumber_ebook=64,pagenumber_book=158](https://qkk.lw71.com/uploads/image/2018/2/1b8abcecdfb1c58e5d262f3f6348eec6.png)
Meanwhile,since s≤p−2,t≤q−2 and 0≤λ<1,we have one of the inequalities(4.10)and(4.11)does not always hold.This shows that our supposition is impossible.The proof of Theorem 1.3 is completed.
5 Proof of Theorem 1.4
We suppose that(w1,w2)is a meromorphic admissible solution of(1.4).Let
![pagenumber_ebook=64,pagenumber_book=158](https://qkk.lw71.com/uploads/image/2018/2/93c008264f731f8c1551fba1c48c7d0a.png)
Then
![pagenumber_ebook=64,pagenumber_book=158](https://qkk.lw71.com/uploads/image/2018/2/0943fd5bcec8b91f65c9e690100a53fd.png)
Moreover,A1,C1and A2,C2are linearly independent respectively by Lemma 2.4.
Similarly to the proceeding of the analysis of Theorem 1.3 and together with the relations
![pagenumber_ebook=64,pagenumber_book=158](https://qkk.lw71.com/uploads/image/2018/2/7b59c89e71e1bdf52e6532654313e3f7.png)
we can derive that
![pagenumber_ebook=65,pagenumber_book=159](https://qkk.lw71.com/uploads/image/2018/2/057e07028c40b2f003d54a60a539e99d.png)
That is,
![pagenumber_ebook=65,pagenumber_book=159](https://qkk.lw71.com/uploads/image/2018/2/03ec21c357675910bff1f05417378295.png)
Similarly,we can get
![pagenumber_ebook=65,pagenumber_book=159](https://qkk.lw71.com/uploads/image/2018/2/0436d32066991033e954957aaae3e800.png)
Using Lemma 2.5,we can immediately deduce that
![pagenumber_ebook=65,pagenumber_book=159](https://qkk.lw71.com/uploads/image/2018/2/1a74bbec9deb69fad9250fdb948a4fb9.png)
Noting that p<m2,q<m1,one can see from(5.1)and(5.2)that
![pagenumber_ebook=65,pagenumber_book=159](https://qkk.lw71.com/uploads/image/2018/2/fba7deba096f88ee4f196d4d9bdd7385.png)
and
![pagenumber_ebook=65,pagenumber_book=159](https://qkk.lw71.com/uploads/image/2018/2/2b1f97cd4ff00c990db54224a14e8dd8.png)
It is impossible.Because m2m1,at least one is a positive constant between
and
This ends the proof of Theorem 1.4.
6 Proof of Theorem 1.5
We assume that the systems of differential equations(1.3)does not have the forms(1.5),(1.6)or(1.7),and admits an admissible solution(w1,w2).We can rewrite equation(1.3)as
![pagenumber_ebook=65,pagenumber_book=159](https://qkk.lw71.com/uploads/image/2018/2/679d46860b6b06c3002e354265249b05.png)
where
![pagenumber_ebook=65,pagenumber_book=159](https://qkk.lw71.com/uploads/image/2018/2/224121007dbc249deea521ac9997c8ca.png)
By Theorem 1.2,we know that the systems of differential equations(6.1)has no admissible solution,which is a contradiction with the condition of Theorem 1.5.Hence,we complete the proof of Theorem 1.5.
References
[1]Hayman W K.Meromorphic Functions.London:Oxford University Press,1964.
[2]Yang C C,Yi H X.Uniqueness Theorey of Meromorphic Functions.Beijing:Science Press,2004.
[3]Gackstatter F,Laine I.Zur Theorie der gewohnlichen differentialalgleichungen im komplexen(in German).Ann.Polon.Math.,1980,38:259–287.
[4]Toda N.On the conjecture of Gackstatter and Laine concerning the differential eqautionKodai Math.J.,1983,6:238–249.
[5]Toda N.On the growth of meromorphic solutions of some higher order differential equations.J.Math.Soc.Japan,1986,38(3):439–451.
[6]He Y Z.Laine I,The Hayman-Miles theorem and the differential equation(y′)n=R(z,y).Analysis,1990,10:439–451.
[7]Gao L Y.The existence of algebraic solutions of higher-order algebraic differential equations.J.Math.(Wuhan),2003,23(3):381–384.
[8]Gao L Y.Admissible meromorphic solutions of a type of higher-order algebraic differential equation.J.Math.Res.Exposition,2003,23(3):387–396.
[9]Gao L Y.On the behavior of meromorphic solutions of some higher-order algebraic differential equations.J.of Math.,2002,22(2):241–244.
[10]Xiao X Z,He Y Z.On Malmquist theorem of algebraic differential equations(in Chinese).Chinese Sci.Bull.,1982,10:583–586.
[11]Chen Z X,Shon K H.On the growth of solutions of a class of higher order differential equations.Acta Math.Sci.Ser.B Engl.Ed.,2004,24:52–60.
[12]Tu Z H,Xiao X Z.On the meromorphic solutions higher-order system of algebraic differential equations.Complex Variables Theory Appl.,1990,15:197–209.
[13]Su X F,Gao L Y.On the order of the solutions of systems of complex algebraic differential equations.Chin.Quart.J.of Math.,2011,26(2):196–199.
[14]Song S G,Shu H W.The adimissible solutions of algebraic differential equation system(in Chinese).J.of Math.,2008,28(6):685–688.
[15]Su X F,Li X M.On non-admissible meromorphic solutions of syestems of non-linear complex algebraic differential equations(in Chinese).Shandong Univ.Nat.Sci.,2012,47(8):39–41.
[16]Gao L Y.A result of systems of nonlinear complex algebraic differential equations.Acta Math.Sci.Ser.B Engl.Ed.,2010,30:1507–1513.
[17]Gao L Y.Expression of meromorphic solution of systems of algebraic differential equations with exponential coeffients.Acta Math.Sci.,2011,31B(2):541–548.