The Growth Order of Solutions of Systems Complex Di ff erence Equations

(School of Mathematics and Statistics,Guangdong finance and Economics University,510320)

§1. Introduction

We use the standard notation of the Nevanlinna theory of meromorphic functions(See,for example,[1,2]).Speci fically,we denote the convergence exponent of the zeros of f(z)by λ(f),the convergence exponent of the poles of f(z)by,and the order by ρ(f).

Let c be a fixed non-zero complex number,

for each integer n>2.The paper make clear which quantity is under discussion.Equations written with the above di ff erence operatorsare di ff erence equations.Let E be a subset on the positive real axis.We de fine the logarithmic measure of E to be

A set E∈(1,+∞)is said to have finite logarithmic measure if log(E)<∞

Some authors investigated the problem of the solutions of the growth of an entire or meromorphic solution of a nonlinear algebraic di ff erential equation of the form

where I is a finite set of multi-indices(i0,i1,···in)=(i),and they obtain some results(see[3-10]).

In 2010,Gao Ling-yun in the article[11]investigated the problem of the growth of solution of system of complex di ff erential equations of the form

which

He obtained

Then

§2.The Main Result of This Paper

The purpose of this article is to study the problem of growth order of solutions of a type of systems of di ff erence equations of the from

Then,

{ai(z)},{bj(z)},a(z),and b(z)are polynomials,

Theorem 2.1Let(w1(z),w2(z))be a transcendental entire solution of(2.1),Ωi(z,w1,w2)(i=1,2)be homogeneous on.If one of the following conditions is satis fied:

(1)

(2)then among ϕ(w1)≥ 1,ϕ(w2)≥ 1,at least one of them will be true.

§3.Some Lemmas

Lemma 3.1[1]Ifis an entire function of order ρ(f),then,

Lemma 3.2[10]Let f(z)be a transcendental entire function of order ρ = ρ(f)<1,and η be a fixed,non-zero number;letand z be such that|z|=r,where

if 0<ρ<1,then

where

Lemma 3.3[10]Let f(z)be a meromorphic function of order ρ = ρ(f)<1,Then,for each given ε>0,and integers 0≤ j

Lemma 3.4[9]Let f and h be monotone non-decreasing function on[0,∞)such that g(r) ≤ h(r),for alwhere E ⊂ (1,∞)is a set of finite logarithmic measure.Let α >1 be a given constant.Then,there exists an r0=r0(α)>0 such that g(r)≤ h(αr)for all r ≥ r0.

§4.The Main Result of This Paper

Let(w1(z),w2(z))be a transcendental entire solution of(2.1)with order ρ(w1,w2)<1,By Lemma 3.1,

We prove that for each δ>0,there exists a sequence rn→ ∞,such that rn+1>De fine

which is a union of non-intersecting intervals and clearly has in finite logarithmic measure.We consider all those r that lie in,Then,as v(r,wi)is a non-decreasing function of r,we have

We choose δ and ε,so small that the inequality

holds,where.Hence,

By Lemma 3.2 and taking η=1,we obtain

We rewrite(2.1)as follows:

By Lemma 3.3 and(4.1),we choose r and r=|z|6∈F so big that the inequality

Then we can obtain

By Lemma 3.4 and(4.4),we can get

where ε1,ε2>0 are arbitrarily small numbers.

So

Combining(4.6)and(4.5),we obtain

Asthen,

Similarly,we have

By(4.7)and ϕ(w1)<1,we obtain

So

By(4.8)and ϕ(w2)<1,we obtain

It is in contradiction with the conditions in Theorem 2.1.

§5.Examples 1

Example 1The system of complex di ff erence equations of the form

then(w1,w2)=(z,z2)is a solution of the system of complex di ff erence equations of the form,and ρ(w1)=0,ρ(w2)=0.

So

·[3−3]+2·[3−7]= −8<0.

In this case,It shows that the conditions in Theorem 2.1 is right.

Example 2The system of complex di ff erence equations of the form

then(w1,w2)=(z2,ez)is a solution of the system of complex di ff erence equations of the form(2.1),and ρ(w1)=0,ρ(w2)=1.

·[3−4]+1·[2−2]= −1<0.

In this case,the lower bound in Theorem 2.1 can be reached.

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