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The solutions of one type q-difference functional system

Abstract

In this paper, we study the functional system on q-difference equations, our results can give estimates on the proximity functions and the counting functions of the solutions of q-difference equations system. This implies that solutions have a relatively large number of poles. The main results in this paper concern q-difference equations to the system of q-difference equations.

MSC:30D35, 39B32, 39A13, 39B12.

1 Introduction and main results

A function f(z) is called meromorphic if it is analytic in the complex plane except at isolate poles. In what follows, we assume that the reader is familiar with the basic notion of Nevanlinna’s value distribution theory, see [1] and [2].

Let us consider the q-difference polynomial case. Let d j C for j=1,,n, and let I q be a finite set of multi-indexes γ=( γ 0 ,, γ n ). A q-difference polynomial of a meromorphic function w(z) is defined as follows:

P ( z , w ) = P ( z , w ( q z ) , w ( q 2 z ) , , w ( q n z ) ) = γ I q a γ ( z ) w ( z ) γ 0 w ( q z ) γ 1 w ( q n z ) γ n ,
(1.1)

where qC{0}, and the coefficients a γ (z) are small meromorphic functions with respect to w(z) such that T(r, a γ )=o(T(r,w)) on a logarithmic density 1, denoted by S q (r,w). The total degree of P(z,w) in w(z) and the q-shifts of w(z) is denoted by deg w q (P), and the order of zero of P(z, x 0 , x 1 ,, x n ), as a function of x 0 at x 0 =0, is denoted as ord 0 q (P), which can be found, e.g., in [3]. Moreover, the weight of difference polynomial (1.1) is defined by

K q (P)= max γ I q { j = 1 n γ j } ,

where γ and I q are the same as in (1.1) above. The q-difference polynomial P(z,w) is said to be homogeneous with respect to w(z) if the degree d γ = γ 0 ++ γ n of each term in the sum (1.1) is non-zero and the same for all γ I q .

We recall the following result of Zhang et al. [[4], Theorem 1].

Theorem A Let w(z) be a zero-order meromorphic solution of

H(z,w)P(z,w)=Q(z,w),

where P(z,w) is a homogeneous q-difference polynomial with polynomial coefficients, and H(z,w) and Q(z,w) are polynomials in w(z) with polynomial coefficients having no common factors. If

max { deg w q ( H ) , deg w q ( Q ) deg w q ( P ) } >min { deg w q ( P ) , ord 0 q ( Q ) } ord 0 q (P),

then N(r,w) S q (r,w), where ord 0 q (P) denotes the order of zero of P(z, x 0 , x 1 ,, x n ), as a function of x 0 at x 0 =0.

Now let us introduce some notation. Let q j C{0,} for j=1,,n, and let I and J be a finite set of multi-indexes I=( i 0 ,, i n ) and J=( j 0 ,, j n ). Two q-difference polynomials of a meromorphic function w(z) are defined as follows:

Ω 1 ( z , w 1 , w 2 ) = Ω 1 ( z , w 1 ( z ) , w 2 ( z ) , w 1 ( q 1 z ) , w 2 ( q 1 z ) , , w 1 ( q n z ) , w 2 ( q n z ) ) = i I a i ( z ) k = 1 2 w k ( z ) k i 0 w k ( q 1 z ) k i 1 w k ( q n z ) k i n

and

Ω 2 ( z , w 1 , w 2 ) = Ω 2 ( z , w 1 ( z ) , w 2 ( z ) , w 1 ( q 1 z ) , w 2 ( q 1 z ) , , w 1 ( q n z ) , w 2 ( q n z ) ) = j J b j ( z ) k = 1 2 w k ( z ) k i 0 w k ( q 1 z ) k i 1 w k ( q n z ) k i n ,

where the coefficients a i (z) and b j (z) are small with respect to w 1 (z) and w 2 (z) in the sense that T(r, a i )=o(T(r, w k )) and T(r, b j )=o(T(r, w k )), k=1,2, on a set of logarithmic density 1, as r tends to infinity outside of an exceptional set E of finite logarithmic measure

lim r E [ 1 , r ) d t t <.

The weights of Ω 1 (z, w 1 , w 2 ) and Ω 2 (z, w 1 , w 2 ) in w 1 (z), w 2 (z) are denoted by

λ 11 = max i { l = 0 n i 1 l } , λ 12 = max i { l = 0 n i 2 l }

and

λ 21 = max j { l = 0 n i 1 l } , λ 22 = max j { l = 0 n i 2 l } .

The purpose of this paper is to study the problem of the properties of Nevanlinna counting functions and proximity functions of meromorphic solutions of a type of systems of q-difference equations of the following form:

{ Ω 1 ( z , w 1 , w 2 ) = R 1 ( z , w 1 ) , Ω 2 ( z , w 1 , w 2 ) = R 2 ( z , w 2 ) ,
(1.2)

where

R 1 (z, w 1 )= P 1 ( z , w 1 ) Q 1 ( z , w 1 ) = i = 0 p 1 a i ( z ) w 1 i j = 0 q 1 b j ( z ) w 1 j

and

R 2 (z, w 2 )= P 2 ( z , w 2 ) Q 2 ( z , w 2 ) = i = 0 p 2 c i ( z ) w 2 i j = 0 q 2 d j ( z ) w 2 j ,

the coefficients { a i (z)}, { b i (z)}, { c i (z)}, { d i (z)} are meromorphic functions and small functions. The order of zero of Ω j (z, x 0 ,, x n ), as a function of x 0 at x 0 =0, is denoted by ord 0 ( Ω j ). The q-difference polynomial Ω k (z, w 1 , w 2 ), k=1,2, is said to be homogeneous with respect to w k (z) if the degree d k = i k 0 ++ i k n of each term in the sum is non-zero and the same for all iI. Finally, the order of growth of a meromorphic solution ( w 1 , w 2 ) is defined by

ρ( w 1 , w 2 )=max { ρ ( w 1 ) , ρ 2 ( w 2 ) } ,

where

ρ( w k )= lim sup r log T ( r , w k ) log r ,k=1,2.

In this paper, the main results are as follows.

Theorem 1 Let ( w 1 , w 2 ) be a zero-order meromorphic solution of system (1.2), where Ω k (z, w 1 , w 2 ) (k=1,2) are homogeneous q-difference polynomials in w 1 and w 2 , respectively, with meromorphic coefficients, and P k (z, w k ) and Q(z, w k ), k=1,2, are polynomials in w k (z) with meromorphic coefficients having no common factors. If

max{ q 1 , p 1 λ 11 }>min { λ 11 , ord w 1 ( P 1 ) } ord w 1 ( Ω 1 )+ λ 12
(1.3)

and

max{ q 2 , p 2 λ 22 }>min { λ 22 , ord w 2 ( P 2 ) } ord w 2 ( Ω 2 )+ λ 21 ,
(1.4)

then N(r, w 1 )= S q (r, w 1 ) and N(r, w 2 )= S q (r, w 2 ) cannot hold both at the same time, possibly outside of an exceptional set of finite logarithmic measure.

Theorem 2 Let ( w 1 , w 2 ) be a zero-order meromorphic solution of system (1.2), where Ω k (z, w 1 , w 2 ) (k=1,2) are homogeneous q-difference polynomials in w 1 and w 2 , respectively, with meromorphic coefficients, and P k (z, w k ) and Q(z, w k ), k=1,2, are polynomials in w k (z) with meromorphic coefficients having no common factors,

A=2 λ 11 ( max { p 1 , q 1 + λ 11 } min { λ 11 , ord w 1 ( Ω 1 ) } )

and

B=2 λ 22 ( max { p 2 , q 2 + λ 22 } min { λ 22 , ord w 2 ( Ω 2 ) } ) .

If A<0, B<0 and AB>9 λ 21 λ 12 , then m(r, w k )= S q (r, w k ) (k=1,2), where r runs to infinity outside of an exceptional set of finite logarithmic measure.

2 Some lemmas

Lemma 1 ([5], Theorem 1.2)

Let f(z) be a non-constant zero-order meromorphic function, and qC{0}. Then

m ( r , f ( q z ) f ( z ) ) = S q (r,f).

Lemma 2 ([6], Lemma 4)

If T: R + R + is a piecewise continuous increasing function such that

lim r log T ( r ) log r =0,

then the set

E:= { r : T ( C 1 r ) C 2 T ( r ) }

has logarithmic density 0 for all C 1 >1 and C 2 >1.

3 Proof of Theorem 1

Since Ω k (z, w 1 , w 2 ) are homogeneous in w 1 and w 2 , respectively, it follows by Lemma 1 that

m ( r , Ω 1 ( z , w 1 , w 2 ) w 1 λ 11 ) λ 12 m(r, w 2 )+ S q (r, w 1 )
(3.1)

and

m ( r , Ω 2 ( z , w 1 , w 2 ) w 2 λ 22 ) λ 21 m(r, w 1 )+ S q (r, w 2 )
(3.2)

for all r outside of an exceptional set of finite logarithmic measure. Moreover, from (1.2), we have

T ( r , Ω 1 ( z , w 1 , w 2 ) w 1 λ 11 ) = T ( r , P 1 ( z , w 1 ) Q 1 ( z , w 1 ) w 1 λ 11 ) = ( max { p 1 , q 1 + λ 11 } min { λ 11 , ord w 1 ( P 1 ) } ) T ( r , w 1 ) + S q ( r , w 1 )
(3.3)

and

T ( r , Ω 2 ( z , w 1 , w 2 ) w 2 λ 22 ) = T ( r , P 2 ( z , w 2 ) Q 2 ( z , w 2 ) w 2 λ 22 ) = ( max { p 2 , q 2 + λ 22 } min { λ 22 , ord w 2 ( P 2 ) } ) T ( r , w 2 ) + S q ( r , w 2 ) ,
(3.4)

where r approaches infinity outside of an exceptional set of finite logarithmic measure. By combining (3.1) and (3.3), (3.2) and (3.4), respectively, it follows that

N ( r , Ω 1 ( z , w 1 , w 2 ) w 1 λ 11 ) ( 1 + λ 12 + λ 11 ord w 1 ( Ω 1 ) ) T ( r , w 1 ) λ 12 m ( r , w 2 ) + S q ( r , w 1 )
(3.5)

and

N ( r , Ω 2 ( z , w 1 , w 2 ) w 2 λ 22 ) ( 1 + λ 21 + λ 22 ord w 2 ( Ω 2 ) ) T ( r , w 1 ) λ 21 m ( r , w 1 ) + S q ( r , w 2 ) .
(3.6)

From Lemma 2, we have

N ( r , Ω 1 ( z , w 1 , w 2 ) w 1 ord w 1 ( Ω 1 ( z , w 1 , w 2 ) ) ) ( λ 11 ord w 1 ( Ω 1 ) ) N ( q r , w 1 ) + λ 12 N ( q r , w 2 ) + S q ( r , w 1 ) = ( λ 11 ord w 1 ( Ω 1 ) ) N ( r , w 1 ) + λ 12 N ( r , w 2 ) + S q ( r , w 1 ) + S q ( r , w 2 )

and

N ( r , Ω 2 ( z , w 1 , w 2 ) w 1 ord w 2 ( Ω 2 ( z , w 1 , w 2 ) ) ) ( λ 22 ord w 2 ( Ω 2 ) ) N ( q r , w 2 ) + λ 21 N ( q r , w 1 ) + S q ( r , w 2 ) = ( λ 22 ord w 2 ( Ω 2 ) ) N ( r , w 2 ) + λ 11 N ( r , w 1 ) + S q ( r , w 1 ) + S q ( r , w 2 ) .

Therefore,

N ( r , Ω 1 ( z , w 1 , w 2 ) w 1 λ 11 ) N ( r , Ω 1 ( z , w 1 , w 2 ) w 1 ord w 1 ( Ω 1 ( z , w 1 , w 2 ) ) ) + N ( r , 1 w 1 λ 11 ord w 1 ( Ω 1 ) ) ( λ 11 ord w 1 ( Ω 1 ) ) N ( r , w 1 ) + λ 12 N ( r , w 2 ) + T ( r , 1 w 1 λ 11 ord w 1 ( Ω 1 ) ) + S q ( r , w 1 ) + S q ( r , w 2 ) ( λ 11 ord w 1 ( Ω 1 ) ) N ( r , w 1 ) + λ 12 N ( r , w 2 ) + ( λ 11 ord w 1 ( Ω 1 ) ) T ( r , w 1 ) + S q ( r , w 2 ) + S q ( r , w 2 )
(3.7)

and

N ( r , Ω 2 ( z , w 1 , w 2 ) w 2 λ 22 ) N ( r , Ω 2 ( z , w 1 , w 2 ) w 2 ord w 2 ( Ω 2 ( z , w 1 , w 2 ) ) ) + N ( r , 1 w 2 λ 22 ord w 2 ( Ω 2 ) ) ( λ 22 ord w 2 ( Ω 2 ) ) N ( r , w 2 ) + λ 21 N ( r , w 1 ) + T ( r , 1 w 2 λ 22 ord w 2 ( Ω 2 ) ) + S q ( r , w 1 ) + S q ( r , w 2 ) ( λ 22 ord w 2 ( Ω 2 ) ) N ( r , w 2 ) + λ 21 N ( r , w 1 ) + ( λ 22 ord w 2 ( Ω 2 ) ) T ( r , w 2 ) + S q ( r , w 2 ) + S q ( r , w 2 ) .
(3.8)

Combining (3.5) and (3.7), (3.6) and (3.8), respectively, we have

( 1 + λ 12 + λ 11 ord w 1 ( Ω 1 ) ) T ( r , w 1 ) < ( λ 11 ord w 1 ( Ω 1 ) ) N ( r , w 1 ) + λ 12 T ( r , w 2 ) + ( λ 11 ord w 1 ( Ω 1 ) ) T ( r , w 1 ) + S q ( r , w 1 ) + S q ( r , w 2 )
(3.9)

and

( 1 + λ 21 + λ 22 ord w 2 ( Ω 2 ) ) T ( r , w 2 ) < ( λ 22 ord w 2 ( Ω 2 ) ) N ( r , w 2 ) + λ 21 T ( r , w 1 ) + ( λ 22 ord w 2 ( Ω 2 ) ) T ( r , w 2 ) + S q ( r , w 1 ) + S q ( r , w 2 ) .
(3.10)

Suppose that N(r, w 1 )= S q (r, w 1 ) and N(r, w 2 )= S q (r, w 2 ), according to (3.9) and (3.10), we have

(1+ λ 12 )T(r, w 1 )< λ 12 T(r, w 2 )+ S q (r, w 1 )+ S q (r, w 2 )

and

(1+ λ 21 )T(r, w 2 )< λ 21 T(r, w 1 )+ S q (r, w 1 )+ S q (r, w 2 ).

That is,

( 1 + λ 12 + o ( 1 ) ) T(r, w 1 )< ( λ 12 + o ( 1 ) ) T(r, w 2 )
(3.11)

and

( 1 + λ 21 + o ( 1 ) ) T(r, w 2 )< ( λ 12 + o ( 1 ) ) T(r, w 1 ).
(3.12)

By (3.11) and (3.12), we conclude that

1+ λ 12 +1+ λ 21 +o(1)< λ 12 + λ 21 ,

which is impossible, we prove the assertion.

4 Proof of Theorem 2

It follows by Lemma 1 that

m ( r , Ω 1 ( z , w 1 , w 2 ) w 1 λ 11 ) λ 12 m(r, w 2 )+ S q (r, w 1 )
(4.1)

and

m ( r , Ω 2 ( z , w 1 , w 2 ) w 2 λ 22 ) λ 21 m(r, w 1 )+ S q (r, w 2 )
(4.2)

for all r outside of an exceptional set of finite logarithmic measure.

Suppose now that ( w 1 (z), w 2 (z)) is a finite-order meromorphic solution of (1.2). Denoting C= max j = 1 , , n {| c j |} in Ω 1 (z, w 1 , w 2 ) and Ω 2 (z, w 1 , w 2 ), by Lemma 2, we obtain

N ( r , Ω 1 ( z , w 1 , w 2 ) w 1 λ 11 ) λ 11 ( N ( | q | r , w 1 ) + N ( r , 1 w 1 ) ) + λ 12 ( N ( | q | r , w 2 ) + N ( r , 1 w 2 ) ) + λ 12 N ( r , w 2 ) + S q ( r , w 1 ) + S q ( r , w 2 ) = λ 11 ( N ( r , w 1 ) + N ( r , 1 w 1 ) ) + λ 12 ( N ( r , w 2 ) + N ( r , 1 w 2 ) ) + λ 12 N ( r , w 2 ) + S q ( r , w 1 ) + S q ( r , w 2 )
(4.3)

for all r outside of a set E of finite logarithmic measure. By (4.1) and (4.3), we have

N ( r , Ω 1 ( z , w 1 , w 2 ) w 1 λ 11 ) λ 11 ( N ( r , w 1 ) + N ( r , 1 w 1 ) ) + λ 12 ( N ( r , w 2 ) + N ( r , 1 w 2 ) ) + S q ( r , w 1 ) + S q ( r , w 2 ) λ 12 ( 2 T ( r , w 1 ) m ( r , w 1 ) ) + λ 12 ( 3 T ( r , w 2 ) 2 m ( r , w 2 ) ) + S q ( r , w 1 ) + S q ( r , w 2 )
(4.4)

for all rE. On the other hand, by (4.1) and (4.3),

N ( r , Ω 1 ( z , w 1 , w 2 ) w 1 λ 11 ) + λ 12 m ( r , w 2 ) T ( r , P 1 ( r , w 1 ) w 1 λ 11 Q 1 r , w 1 ) = ( max { p 1 , q 1 + λ 11 } min { λ 11 , ord w 1 ( Ω 1 ) } ) T ( r , w 1 ) + S q ( r , w 1 ) ,
(4.5)

where r lies outside of a set F of finite logarithmic measure. Combining inequalities (4.4) and (4.5) with the assumption in Theorem 2, we have

( max { p 1 , q 1 + λ 11 } min { λ 11 , ord w 1 ( Ω 1 ) } ) T ( r , w 1 ) λ 12 m ( r , w 2 ) + S q ( r , w 1 ) + S q ( r , w 2 ) λ 11 ( 2 T ( r , w 1 ) m ( r , w 1 ) ) + λ 12 ( 3 T ( r , w 2 ) 2 m ( r , w 2 ) ) + S q ( r , w 1 ) + S q ( r , w 2 ) .
(4.6)

Similarly, we obtain

( max { p 2 , q 2 + λ 22 } min { λ 22 , ord w 2 ( Ω 2 ) } ) T ( r , w 2 ) λ 21 m ( r , w 1 ) + S q ( r , w 1 ) + S q ( r , w 2 ) λ 22 ( 2 T ( r , w 2 ) m ( r , w 2 ) ) + λ 21 ( 3 T ( r , w 1 ) 2 m ( r , w 1 ) ) + S q ( r , w 1 ) + S q ( r , w 2 ) .
(4.7)

By (4.6) and (4.7), we obtain

λ 11 m ( r , w 1 ) ( 2 λ 11 ( max { p 1 , q 1 + λ 11 } min { λ 11 , ord w 1 ( Ω 1 ) } ) + o ( 1 ) ) T ( r , w 1 ) + ( 3 λ 12 + o ( 1 ) ) T ( r , w 2 )
(4.8)

and

( ( max { p 2 , q 2 + λ 22 } min { λ 22 , ord w 2 ( Ω 2 ) } ) 2 λ 22 + o ( 1 ) ) T ( r , w 2 ) ( 3 λ 21 + o ( 1 ) ) T ( r , w 1 ) 2 λ 21 m ( r , w 2 ) .
(4.9)

Combining (4.8) and (4.9), we have

λ 11 m ( r , w 1 ) ( 2 λ 11 ( max { p 1 , q 1 + λ 11 } min { λ 11 , ord w 1 ( Ω 1 ) } ) + o ( 1 ) ) T ( r , w 1 ) + 3 λ 12 ( 3 λ 21 + o ( 1 ) ) T ( r , w 1 ) 6 λ 12 λ 21 m ( r , w 1 ) ( max { p 2 , q 2 + λ 22 } min { λ 22 , ord w 2 ( Ω 2 ) } ) 2 λ 22 ,

that is,

( λ 11 6 λ 12 λ 21 B ) m(r, w 1 ) ( A 9 λ 12 λ 21 + o ( 1 ) B ) T(r, w 1 ),
(4.10)

where A=2 λ 11 (max{ p 1 , q 1 + λ 11 }min{ λ 11 , ord w 1 ( Ω 1 )}) and B=2 λ 22 (max{ p 2 , q 2 + λ 22 }min{ λ 22 , ord w 2 ( Ω 2 )}). Combining the assumption and (4.10), we have

m(r, w 1 )= S q (r, w 1 )

for all r outside of EF, a set of finite logarithmic measure.

Similarly, we obtain

m(r, w 2 )= S q (r, w 2 )

for all r outside of EF, we have proved the assertion.

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Acknowledgements

Research was supported by the National Science Foundation of China (11161041), and the Fundamental Research Funds for the Central Universities (No. 31920130006) and Middle-Younger Scientific Research Fund (No. 12XB39).

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Xu, Y., Han, D., Fan, X. et al. The solutions of one type q-difference functional system. Adv Differ Equ 2014, 3 (2014). https://doi.org/10.1186/1687-1847-2014-3

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Keywords

  • functional system
  • q-difference equations
  • zero order
  • difference Nevanlinna theory