Meromorphic solutions of nonlinear difference equations

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: lixiongying2818@163.com

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 Nevanlinna’s 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, 1011, 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 p

 ( z  1)   ( z  1) 

 ai ( z ) i

i 1 q

 b j ( z ) j

j 0

With rational coefficients {ai },{b j } admits a transcendental meromorphic solution of finite order, then

max{ p, q}  2 .

In 2001, Heittokangas [8] had considered a type of difference equation, they obtained next result.

Theorem B ([8]) Let c j  C \ {0} ; j = 1, 2, … , n. If a complex difference equation p

n

 (z  c j )  j 1

 ai ( z ) i

i 1 q

 b j ( z ) j

j 0

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 - 49 www.ivypub.org/mc

 1 ( z  1)  2 ( z  1) e4  2   12 ( z  1) 1   ( z  1)   ( z  1) e4 1 2    22 ( z  1) 22 

(1 , 2 )  (e z , e z ) 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 c j  C for j = 1, …, n, difference polynomials 1 ( z, 1 , 2 ) ; 2 ( z, 1 , 2 ) ; 3 ( z, 1 , 2 ) ; 4 ( z, 1 , 2 ) are defined as 2

1 ( z, 1 , 2 )   ai ( z )  (k )i (k ( z  c1 ))i ...(k ( z  cn ))i k0

k1

kn

k 1

(i )

2

2 ( z, 1 , 2 )   b j ( z )  (k ) j (k ( z  c1 )) j ...(k ( z  cn )) j k0

k1

kn

k 1

( j)

2

3 ( z, 1 , 2 )   cs ( z )  (k ) s (k ( z  c1 )) s ...(k ( z  cn )) s k0

k1

kn

k 1

(s)

2

4 ( z, 1 , 2 )   dt ( z )  (k )t (k ( z  c1 ))t ...(k ( z  cn ))t k0

where

the

coefficients

k1

kn

k 1

(t )

{ai }、 {b j }、 {cs }、 {dt }

are

small

with

respect

to

w1;w2,i.e.

T (r, a(i ) )  o(T (r, l ))，T (r, b( j ) ))  o(T (r, l ))，T (r, c( s) )  o(T (r, l ))，T (r, d(t ) ))  o(T (r, l )) ; l = 1, 2,as tends to infinity, outside of an exceptional set E of finite logarithmic measure E

r

dt   to denote finite logarithmic t

measure.

The weight of 1 ( z, 1 , 2 ) ; 2 ( z, 1 , 2 ) ; 3 ( z, 1 , 2 ) ; 4 ( z, 1 , 2 ) are defined by n

n

l 0 n

l 0 n

1  max{max{ i1l }, max{ j1l }}

n

n

l 0 n

l 0 n

2  max{max{ s2l }, max{ t2l }}

12  max{max{ i2l }, max{ j2l }} 21  max{max{ s1l }, max{ t1l }} l 0

l 0

l 0

l 0

In this paper, we will study the existence of meromorphic solutions of difference equations of the following form  1 ( z, 1 , 2 )  R1 ( z, 1 )   2 ( z, 1 , 2 )   ( z,  ,  ) 1 2  3  R2 ( z, 2 )   4 ( z, 1 , 2 )

Where p2

P1

R1 ( z, 1 ) 

P1 ( z, 1 )  Q1 ( z, 1 )

 ai ( z )1i

i 0 q1

 b j ( z )1j

, R2 ( z, 2 ) 

P2 ( z, 2 )  Q2 ( z, 2 )

j 0

The

coefficients

{ai ( z)},{b j ( z)},{cs ( z)},{dt ( z)}

a p1bq1  0，c p 2 dq 2  0

 ci ( z )2i

i 0 q2

 d j ( z )2j

j 0

are

meromorphic

functions

S (r )   T (r , a(i ) )  T (r , b( j ) )  T (r , c( s ) )  T (r , d (t ) )   T (r, ai )  T (r, b j )  T (r, ci )  T (r, d j )

The growth order of meromorphic solutions (w1, w2) is defined by - 50 www.ivypub.org/mc

and

small,

where

log T (r , k ) , k  1, 2 log r Definition 1 Let (1 ( z), 2 ( z)) be a set of meromorphic solutions of (1).If the component k of (w1; w2) satisfies the following condition:

   (1 , 2 )  max{ (1 ),  (2 )} ,  (k )  lim sup r 

S (r )  o(T (r, i ))(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), max( p1 , q1 )  1 , max( p2 , q2 )  2 .

Then [max( p1 , q1 )  1 ][max( p2 , q2 )  2 ]  1221 :

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 (z  1) 1   2  1 (z  1) 14  2  1 (z  1)  1  22 (z  1) 24

(1 , 2 )  (e z , e z ) is an admissible meromorphic solution, we have

1 =2，,2  2，12 =21 =2， max{ p1, q1}  max{ p2 , q2 }  4; [max  p1 , q1  1 ][max  p2 , q2   2 ]  4  1221

Remark 3: The following example 3 shows that  w1 , w2  of admissible solution in Theorem 2 cannot be omitted.

Example 3 For the system of difference equations:

 w1, w2    z 2 , z 

 w13  z  1 w2  z  1 1 5  4 w1   4 z  5 w12  5w1  4 z  1,   w2 ( z  1) z  3 w z  1 w z  1     1 2 1   w4  1  w1  z  1 w2 ( z  1) z 2 2 is a non-admissible solution. In this case

1  3, 2  3, 12  21  1,max  p1 , q1  5,max  p2 , q2   4; Theorem 2 Let  w1 , w2 

max  p1 , q1  1  max  p2 , q2   2   2  1  1221. be a meromorphic solution of finite order _ of (1).If one of the following inequalities holds

max  p1, q1  1 ,max  p2 , q2   2 , Then the components w1 , w2 in the meromorphic solution

 w1, w2  are either admissible or non-admissible.

2 SOME LEMMAS p

Lemma 1([3]) Let R  z , w  

 ai  z  wi

i 0 q

 bj  z  w

be an irreducible rational function in w  z  with the meromorphic

j

j 0

- 51 www.ivypub.org/mc





coefficients ai  z  and b j  z  .If w  z  is a meromorphic function, then







T  r , R  z, w   max  p, qT  r , w  O  T  r , ai    T r , b j .

Lemma 2([10]) Let w  z  be a nonconstant meromorphic function of finite order, c1 , c2  C, c1  c2 , and 0    1 .

Then  w z  c   T  r , w  m  r ,   o    S1  r , w ,  w z    r   Where the exceptional set E associated to S1  r , w is of finite logarithmic measure.









Lemma 3 Let w1 , w2 be of finite order, T r , ai   o T  r , wk   , T r , b j   o T  r , wk   , k  1, 2,

1 

1  z, w1 , w2 

2  z, w1 , w2 

,

Where 2

1  z, w1 , w2    a i   z  w j i 

ij0

j 1 2

2  z, w1 , w2    b u   z  w j u 

j 1

 w j  z  c1  

uj0

i j1

 w j  z  c1  

u j1

 w j  z  cn 

i jn

,

 w j  z  cn 

u jn

,

Then T (r, 1 )  1T (r, w1 )  12T (r, w2 )  S1 (r, w1 )  S1 (r, w2 )  S (r, w1 )  S (r, w2 ).

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: S (r )  o(T (r, w1 )), r  I1 T (r, w2 )  O(S (r )),(r  I 2 ),

Then T (r, w2 )  o(T (r, w1 )),(r  I1  I 2 ),

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 max{ p1 , q1}T (r, w1 )  1T (r, w1 )  12T (r, w2 )  S1 (r, w1 )  S1 (r, w2 )  S (r, w1 )  S (r, w2 ), (2) max{ p2 , q2 }T (r, w2 )  2T (r , w2 )  21T (r , w1 )  S1 (r, w1 )  S1 (r, w2 )  S (r , w1 )  S (r , w2 ), (3) For all r outside of a possible exceptional set I1  I 2 with finite logarithmic measure. It follows from (2) and (3) that [max{ p1 , q1}  1  o(1)]T (r, w1 )  (12  o(1))T (r, w2 ), (4) [max{ p2 , q2 }  2  o(1)]T (r, w2 )  (21  o(1))T (r, w1 ), (5)

Combining (4) and (5), we have [max{ p1 , q1}  1 ][max{ p2 , q2 }  2 ]  1221.

This proves Theorem 1.

3.2 Proof of Theorem 2

By Lemma 1 and Lemma 3, we get - 52 www.ivypub.org/mc

1T (r, w1 )  12T (r, w2 )  S1 (r, w1 )  S1 (r, w2 )  S (r, w1 )  S (r, w2 )  max{ p1, q1}T (r, w1 ), (6) 2T (r, w2 )  21T (r, w1 )  S1 (r, w1 )  S1 (r, w2 )  S (r, w1 )  S (r, w2 )  max{ p2 , q2 }T (r, w2 ), (7) If the component w1 is admissible and the component w2 is not admissible, then the inequality (6) becomes max{ p1 , q1}  (1  o(1))  12

T (r , w2 ) S (r )  , T (r , w1 ) T (r , w1 )

According to Lemma 4, as r tends to ∞, possibly outside a set of a finite logarithmic measure, we get max{ p1 , q1}  1

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

T (r , w1 ) S (r )  , T (r , w2 ) T (r , w2 ) as r tends to ∞,possibly outside a set of a finite logarithmic measure, we get max{ p2 , q2 }  2 max{ p2 , q2 }  (2  o(1))  21

It is in contradiction with the second inequality of the conditions in Theorem 2.

Therefore, the components w1 , w2 in the meromorphic solution  w1 , w2  are either admissible or non-admissible.

REFERENCES [1]

Yi Hongxun,Yang C C.Theory of the uniqueness of meromorphic functions(in Chinese)[M]. Beijing:Science Press,1995.

[2]

Gao Lingyun.Systems of Complex Difference Equations of Malmquist Type [J].Acta Math.Sinica, 2012.

[3]

Gao Lingyun. On admissible solutions of two types of systems of differential equations.

[4]

Acta Math Sinica(in Chinese),2000,43(1):149-156.[4] M. J. Ablowitz, R. Halburd, and B. Herbst, On the extension of the PainlevLe property to difference equations[J],Nonlinearity, 2000,13(3):889-905.

[5]

W. Bergweiler and J. K. Langley,Zeros of differences of meromorphic functions, Mathematical Proceedings of the Cambridge Philosophical Society[J], 2007,142(1):133-147.

[6]

Y.-M. Chiang and S.-J. Feng,On the Nevanlinna characteristic of f(z + _) and difference equations in the complex plane[J],Ramanujan Journal, 2008,16(1):105-129.

[7]

G. G. Gundersen, J. Heittokangas etc., Meromorphic solutions of generalized Schroder[J],Aequationes Mathematicae, 2002,63(12):110-135.

[8]

J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and K. Tohge, Complex difference equations of Malmquist type[J],Computational Methods and Function Theory, 2001,1(1):27-39.

[9]

R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations,Journal of Mathematical Analysis and Applications[J],2006,314(2):477-487.

[10] R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operator, Annales Academia Scientiarium Fennica[J].

Mathematica, 2006,31(2):463-478.

[11] R. G. Halburd and R. J. Korhonen,Existence of finite-order meromorphic solutions as a detector of integrability in difference equations[J],Physica D, 2006,218(2):191-203.

[12] K. Ishizaki and N. Yanagihara,Wiman-Valiron method for difference equations[J],Nagoya Mathematical Journal,2004,175:75-102.

[13] I. Laine, J. Rieppo, and H. Silvennoinen, Remarks on complex difference equations [J], Computational Methods and Function Theory, 2005, 5(1):77-88.

[14] V. Gromak, I. Laine, and S. Shimomura, Painleve Differential Equations in the Complex Plane[J], vol. 28 of Studies in Mathematics, de Gruyter, New York, NY, USA, 2002.

[15] G. Weissenborn,On the theorem of Tumura and Clunie,The Bulletin of the London Mathematical Society[J], 1986,18(4):371-373.

[16] Bergweiler, W., Langley, J.K. Zeros of differences of meromorphic functions [J]. Math. Proc. Camb. Philos. Soc. 2007, 142(1):133-147.

[17] Chiang, Y.M., Ruijsenaars, S.N.M. On the Nevanlinna order of meromorphic solutions to linear analytic difference equations [J].

Stud. Appl. Math. 2006,116:257-287.

[18] Ruijsenaars, S.N.M.: First order analytic difference equations and integrable quantum systems [J].Math.Phys.1997, 40:1069-1146.

- 53 www.ivypub.org/mc

[19] R.Korhonen,A new Clunie type theorem for difference polynomials[J]. Difference Equ. Appl.2011, 3:387-400.

AUTHORS Xiongying Li (1987- ), female, born at Meizhou, Guangdong,

Binhui Wang (1965- ), male, born at Shanxi, professor, major in

PH.D., major in statistical.

statistical.

Email: lixiongying2818@163.com

- 54 www.ivypub.org/mc