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Mathematical Computation June 2014, Volume 3, Issue 2,
PP.49-54
Meromorphic Solutions of Nonlinear Difference
Equations Xiongying Li #, Binhui Wang
College of Economics Jinan University, Guangzhou, Guangdong
510632, P.R.China
#Email: [email protected]
Abstract
In this paper, using the Nevanlinna value distribution theory of
meromorphic functions and some skills of difference equations,
we investigate the growth order of meromorphic solutions of
nonlinear complex difference equations, and obtain some results
which are more precise and more general.
Keywords: Malmquist Type, Meromorphic Solution, Value
Distribution, Complex Difference Equations
1 INTRODUCTION
In what follows, we assume the reader is familiar with the
standard notions of Nevanlinnas value distribution theory
as the proximity function ( , )m r , the integrated counting
function ( , )N r , the characteristic function ( , )T r , see
e.g. [1, 2]. Recently, there have been renewed interests in
difference equations in the complex plane C [4, 5, 7-8, 10-
11, 13-19]. In particularly, Ablowitz, Halburd and Herbst [4]
used the notion of order of growth of meromorphic
functions in the sense of classical Nevanlinna theory [19]
investigated the second order non-linear difference equations
in C. They obtained next result.
Theorem A ([4]) If a complex difference equation
1
0
( )
( 1) ( 1)
( )
pi
ii
qj
jj
a z
z z
b z
With rational coefficients { },{ }i ja b admits a transcendental
meromorphic solution of finite order, then
max{ , } 2p q .
In 2001, Heittokangas [8] had considered a type of difference
equation, they obtained next result.
Theorem B ([8]) Let \ {0}jc C ; j = 1, 2, , n. If a complex
difference equation
1
1
0
( )
( )
( )
pi
ini
j qj jj
j
a z
z c
b z
With rational coefficients {ai} (i = 0, 1,, p); {bj}(j = 0,1,,q)
admits a transcendental
meromorphic solution of finite order, then max{ , }p q n .
By the extend of the Malmquist theorem, we realize that the
rational function R(z; w) of z can be reduced as the
polynomials of z for single difference equation, but for the
difference equations, it is different, the following example
1 can explain it.
Example 1 For a system of difference equations
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41 2
2 21 1
41 2
2 22 2
( 1) ( 1)
( 1)
( 1) ( 1)
( 1)
z z e
z
z z e
z
1 2( , ) ( , )z ze e is an admissible solution, but the
right-hand sides of the system is not the polynomials of w1 or
w2.
Remark 1 The example 1 shows that systems have not analogous
results, the existence of admissible solutions of
complex difference equation is different from the existence of
admissible solutions of complex difference equations.
Let jc C for j = 1, , n, difference polynomials 1 1 2( , , )z ;
2 1 2( , , )z ; 3 1 2( , , )z ; 4 1 2( , , )z are
defined as
0 1
2
1 1 2 11( )
( , , ) ( ) ( ) ( ( )) ...( ( ))k k kni i i
i k k k nki
z a z z c z c
0 1
2
2 1 2 11( )
( , , ) ( ) ( ) ( ( )) ...( ( ))k k knj j j
j k k k nkj
z b z z c z c
0 1
2
3 1 2 11( )
( , , ) ( ) ( ) ( ( )) ...( ( ))k k kns s s
s k k k nks
z c z z c z c
0 1
2
4 1 2 11( )
( , , ) ( ) ( ) ( ( )) ...( ( ))k k knt t t
t k k k nkt
z d z z c z c
where the coefficients { } { } { } { }i j s ta b c d are small
with respect to w1;w2,i.e.
( ) ( ) ( ) ( )( , ) ( ( , )) ( , )) ( ( , )) ( , ) ( ( , )) ( ,
)) ( ( , ))i l j l s l t lT r a o T r T r b o T r T r c o T r T r d
o T r ; l = 1, 2,as r
tends to infinity, outside of an exceptional set E of finite
logarithmic measure E
dt
t to denote finite logarithmic
measure.
The weight of 1 1 2( , , )z ; 2 1 2( , , )z ; 3 1 2( , , )z ; 4
1 2( , , )z are defined by
1 1 10 0
max{max{ },max{ }}n n
l ll l
i j
2 2 20 0
max{max{ },max{ }}n n
l ll l
s t
12 2 20 0
max{max{ },max{ }}n n
l ll l
i j
21 1 10 0
max{max{ },max{ }}n n
l ll l
s t
In this paper, we will study the existence of meromorphic
solutions of difference equations of the following form
1 1 21 1
2 1 2
3 1 22 2
4 1 2
( , , )( , )
( , , )( , , )
( , )( , , )
zR z
zz
R zz
Where
21
1 20 01 1 2 2
1 1 2 21 21 1 2 2
1 20 0
( ) ( )( , ) ( , )
( , ) , ( , )( , ) ( , )
( ) ( )
pPi i
i ii i
q qj j
j jj j
a z c zP z P z
R z R zQ z Q z
b z d z
The coefficients { ( )},{ ( )},{ ( )},{ ( )}i j s ta z b z c z d
z are meromorphic functions and small, where
1 1 2 20 0p q p qa b c d ( ) ( ) ( ) ( )( ) ( , ) ( , ) ( , ) (
, )
( , ) ( , ) ( , ) ( , )i j s t
i j i j
S r T r a T r b T r c T r d
T r a T r b T r c T r d
The growth order of meromorphic solutions (w1, w2) is defined
by
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1 2 1 2( , ) max{ ( ), ( )} , log ( , )
( ) lim sup , 1,2log
kk
r
T rk
r
Definition 1 Let 1 2( ( ), ( ))z z be a set of meromorphic
solutions of (1).If the component k of (w1; w2) satisfies
the following condition:
( ) ( ( , ))( , )iS r o T r r r I ,
We say that wk is an admissible component of a set of solutions
of (1), where I is the set of finite logarithmic
measure.
Our results are:
Theorem 1 Let (w1; w2) be an admissible meromorphic solution of
finite order p of (1),
1 1 1 2 2 2max( , ) ,max( , )p q p q .
Then
1 1 1 2 2 2 12 21[max( , ) ][max( , ) ]p q p q :
Remark 2: The following example 2 shows that the upper bound in
Theorem 1 can be reached.
Example 2 For the system of difference equations:
22
2 41 121
2 42 2
(z 1) 1
(z 1)
(z 1) 1
(z 1)
1 2( , ) ( , )z ze e is an admissible meromorphic solution, we
have
1 2 12 21 1 1 2 2=2 2 = =2 max{ , } max{ , } 4;p q p q ,
1 1 1 2 2 2 12 21[max , ][max , ] 4p q p q
Remark 3: The following example 3 shows that 1 2,w w of
admissible solution in Theorem 2 cannot be omitted.
Example 3 For the system of difference equations:
31 2 5 2
1 1 142
32 1 4
221 2
1 1 14 5 5 4 1,
( 1)
1 1 11
1 ( 1)
w z w zw z w w z
w z z
w z w zw
w z w z z
21 2, ,w w z z is a non-admissible solution. In this case
1 2 12 21 1 1 2 23, 3, 1,max , 5,max , 4;p q p q
1 1 1 2 2 2 12 21max , max , 2 1 .p q p q
Theorem 2 Let 1 2,w w be a meromorphic solution of finite order
_ of (1).If one of the following inequalities holds
1 1 1 2 2 2max , ,max , ,p q p q
Then the components 1 2,w w in the meromorphic solution 1 2,w w
are either admissible or non-admissible.
2 SOME LEMMAS
Lemma 1([3]) Let
0
0
,
pi
ii
qj
jj
a z w
R z w
b z w
be an irreducible rational function in w z with the
meromorphic
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coefficients ia z and jb z .If w z is a meromorphic function,
then
, , max , , , , .i jT r R z w p q T r w O T r a T r b Lemma
2([10]) Let w z be a nonconstant meromorphic function of finite
order, 1 2 1 2, ,c c C c c , and 0 1 . Then
1
,, , ,w z c T r w
m r o S r ww z r
Where the exceptional set E associated to 1 ,S r w is of finite
logarithmic measure.
Lemma 3 Let 1 2,w w be of finite order, , , , , , , 1,2,k ki jT
r a o T r w T r b o T r w k
1 1 21
2 1 2
, ,,
, ,
z w w
z w w
Where
10
10
2
1 1 2 112
2 1 2 11
, , ,
, , ,
j jnj
j jnj
i ii
j j j nii j
u uu
j j j nuu j
z w w a z w w z c w z c
z w w b z w w z c w z c
Then
1 1 1 12 2 1 1 1 2 1 2( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( ,
).T r T r w T r w S r w S r w S r w S r w
Proof: Proceeding similarly as in the proof the Lemma 2:4 in
[2], we can verify the assertion.
Lemma 4([3]) If the following conditions are satisfied:
1 1( ) ( ( , )),S r o T r w r I
2 2( , ) ( ( )),( ),T r w O S r r I
Then
2 1 1 2( , ) ( ( , )),( ),T r w o T r w r I I
Where I1; I2 are exceptional set of finite logarithmic measure,
S(r) is as in Definition 1.
3 PROOF OF THEOREMS
3.1 Proof of Theorem 1
We apply Lemma 1 and Lemma 3 to difference equation (1), and
obtain
1 1 1 1 1 12 2 1 1 1 2 1 2max{ , } ( , ) ( , ) ( , ) ( , ) ( , )
( , ) ( , ), (2)p q T r w T r w T r w S r w S r w S r w S r w
2 2 2 2 2 21 1 1 1 1 2 1 2max{ , } ( , ) ( , ) ( , ) ( , ) ( , )
( , ) ( , ), (3)p q T r w T r w T r w S r w S r w S r w S r w
For all r outside of a possible exceptional set 1 2I I with
finite logarithmic measure. It follows from (2) and (3) that
1 1 1 1 12 2[max{ , } (1)] ( , ) ( (1)) ( , ), (4)p q o T r w o
T r w
2 2 2 2 21 1[max{ , } (1)] ( , ) ( (1)) ( , ), (5)p q o T r w o
T r w
Combining (4) and (5), we have
1 1 1 2 2 2 12 21[max{ , } ][max{ , } ] .p q p q
This proves Theorem 1.
3.2 Proof of Theorem 2
By Lemma 1 and Lemma 3, we get
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1 1 12 2 1 1 1 2 1 2 1 1 1( , ) ( , ) ( , ) ( , ) ( , ) ( , )
max{ , } ( , ), (6)T r w T r w S r w S r w S r w S r w p q T r
w
2 2 21 1 1 1 1 2 1 2 2 2 2( , ) ( , ) ( , ) ( , ) ( , ) ( , )
max{ , } ( , ), (7)T r w T r w S r w S r w S r w S r w p q T r
w
If the component w1 is admissible and the component w2 is not
admissible, then the inequality (6) becomes
21 1 1 12
1 1
( , ) ( )max{ , } ( (1)) ,
( , ) ( , )
T r w S rp q o
T r w T r w
According to Lemma 4, as r tends to , possibly outside a set of
a finite logarithmic measure, we get
1 1 1max{ , }p q
It is in contradiction with the first inequality of the
conditions in Theorem 2.
If the component w2 is admissible, the component w1 is not
admissible, then the inequality (7) becomes
12 2 2 21
2 2
( , ) ( )max{ , } ( (1)) ,
( , ) ( , )
T r w S rp q o
T r w T r w
as r tends to ,possibly outside a set of a finite logarithmic
measure, we get 2 2 2max{ , }p q
It is in contradiction with the second inequality of the
conditions in Theorem 2.
Therefore, the components 1 2,w w in the meromorphic solution 1
2,w w are either admissible or non-admissible.
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AUTHORS
Xiongying Li (1987- ), female, born at Meizhou, Guangdong,
PH.D., major in statistical.
Email: [email protected]
Binhui Wang (1965- ), male, born at Shanxi, professor, major
in
statistical.
