The solutions of one type q difference functional system

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R E S E A R C H

Open Access

The solutions of one type

q

-difference

functional system

Yonglin Xu

1*

_{, Dengli Han}

2

_{, Xiaohong Fan}

3

_{, Gang Wang}

3

_{and Hua Zhong}

3

*_{Correspondence:}

[email protected]

1_{Mathematics and Computer}

College, Northwest Minorities University, Lanzhou, Gansu 730030, China

Full list of author information is available at the end of the article

Abstract

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

MSC: Primary 30D35; secondary 39B32; 39A13; 39B12

Keywords: functional system;q-diﬀerence equations; zero order; diﬀerence Nevanlinna theory

1 Introduction and main results

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

Let us consider theq-diﬀerence polynomial case. Letdj∈Cforj= , . . . ,n, and letIqbe

a ﬁnite set of multi-indexesγ = (γ, . . . ,γn). Aq-diﬀerence polynomial of a meromorphic

functionw(z) is deﬁned as follows:

P(z,w) =Pz,w(qz),wqz, . . . ,wqnz

=

γ∈Iq

aγ(z)w(z)γw(qz)γ· · ·w

qnzγn, (.)

whereq∈C{}, and the coeﬃcientsaγ(z) are small meromorphic functions with respect

tow(z) such thatT(r,aγ) =o(T(r,w)) on a logarithmic density , denoted bySq(r,w). The

total degree ofP(z,w) inw(z) and theq-shifts ofw(z) is denoted bydegq_w(P), and the order of zero ofP(z,x,x, . . . ,xn), as a function ofxatx= , is denoted asordq(P), which can

be found,e.g., in []. Moreover, the weight of diﬀerence polynomial (.) is deﬁned by

Kq(P) =max γ∈Iq

_n

j=

γj

,

whereγ andIqare the same as in (.) above. Theq-diﬀerence polynomialP(z,w) is said

to be homogeneous with respect tow(z) if the degreedγ =γ+· · ·+γnof each term in the

sum (.) is non-zero and the same for allγ ∈Iq.

We recall the following result of Zhanget al.[, Theorem ].

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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

maxdegq_w(H),degq_w(Q) –degq_w(P)>mindegq_w(P),ordq_(Q)–ordq_(P),

then N(r,w)=Sq(r,w),whereordq(P)denotes the order of zero of P(z,x,x, . . . ,xn),as a function of xat x= .

Now let us introduce some notation. Letqj∈C\ {,}forj= , . . . ,n, and letIandJbe a

ﬁnite set of multi-indexesI= (i, . . . ,in) andJ= (j, . . . ,jn). Twoq-diﬀerence polynomials

of a meromorphic functionw(z) are deﬁned as follows:

(z,w,w) =

z,w(z),w(z),w(qz),w(qz), . . . ,w(qnz),w(qnz)

=

i∈I ai(z)



k=

wk(z)kiw_k(q_z)ki· · ·w_k(q_nz)kin

and

(z,w,w) =

z,w(z),w(z),w(qz),w(qz), . . . ,w(qnz),w(qnz)

=

j∈J bj(z)



k=

wk(z)kiwk(qz)ki· · ·wk(qnz)kin,

where the coeﬃcientsai(z) andbj(z) are small with respect tow(z) andw(z) in the sense

thatT(r,ai) =o(T(r,wk)) andT(r,bj) =o(T(r,wk)),k= , , on a set of logarithmic density

, asrtends to inﬁnity outside of an exceptional setEof ﬁnite logarithmic measure

lim

r→∞

E∩[,r) dt

t <∞.

The weights of(z,w,w) and(z,w,w) inw(z),w(z) are denoted by

λ=max i

_n

l= il

, λ=max i

_n

l= il

and

λ=max j

_n

l= il

, λ=max j

_n

l= il

.

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q-diﬀerence equations of the following form:

⎧ ⎨ ⎩

(z,w,w) =R(z,w),

(z,w,w) =R(z,w),

(.)

where

R(z,w) =

P(z,w) Q(z,w)

=

p

i=ai(z)wi q

j=bj(z)wj

and

R(z,w) =

P(z,w) Q(z,w)

=

p i=ci(z)wi q

j=dj(z)wj

,

the coeﬃcients{ai(z)},{bi(z)},{ci(z)},{di(z)}are meromorphic functions and small

func-tions. The order of zero ofj(z,x, . . . ,xn), as a function ofx atx= , is denoted by

ord_(j). Theq-diﬀerence polynomialk(z,w,w),k= , , is said to be homogeneous

with respect towk(z) if the degreedk=ik+· · ·+ikn of each term in the sum is non-zero

and the same for alli∈I. Finally, the order of growth of a meromorphic solution (w,w)

is deﬁned by

ρ(w,w) =max

ρ(w),ρ(w)

,

where

ρ(wk) =lim sup r→∞

logT(r,wk)

logr , k= , .

In this paper, the main results are as follows.

Theorem  Let (w,w) be a zero-order meromorphic solution of system (.), where

k(z,w,w) (k= , )are homogeneous q-difference polynomials in wand w,respectively, with meromorphic coefficients,and Pk(z,wk)and Q(z,wk),k= , ,are polynomials in wk(z) with meromorphic coefficients having no common factors.If

max{q,p–λ}>min

λ,ordw(P)

–ordw() +λ (.)

and

max{q,p–λ}>min

λ,ordw(P)

–ordw() +λ, (.)

then N(r,w) =Sq(r,w)and N(r,w) =Sq(r,w)cannot hold both at the same time,possibly outside of an exceptional set of ﬁnite logarithmic measure.

Theorem  Let (w,w) be a zero-order meromorphic solution of system (.), where

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with meromorphic coeﬃcients,and Pk(z,wk)and Q(z,wk),k= , ,are polynomials in wk(z) with meromorphic coeﬃcients having no common factors,

A= λ–

max{p,q+λ}–min

λ,ordw()

and

B= λ–

max{p,q+λ}–min

λ,ordw()

.

If A< ,B< and AB> λλ,then m(r,wk) =Sq(r,wk) (k= , ),where r runs to inﬁnity outside of an exceptional set of ﬁnite logarithmic measure.

2 Some lemmas

Lemma ([], Theorem .) Let f(z)be a non-constant zero-order meromorphic function,

and q∈C\ {}.Then

m

r,f(qz)

f(z)

=Sq(r,f).

Lemma ([], Lemma ) If T:R+_→_R+_{is a piecewise continuous increasing function} such that

lim

r→∞

logT(r) logr = ,

then the set

E:=r:T(Cr)≥CT(r)

has logarithmic densityfor all C> and C> .

3 Proof of Theorem 1

Sincek(z,w,w) are homogeneous inwandw, respectively, it follows by Lemma  that

m

r,(z,w,w)

wλ_

≤λm(r,w) +Sq(r,w) (.)

and

m

r,(z,w,w)

wλ_

≤λm(r,w) +Sq(r,w) (.)

for allroutside of an exceptional set of ﬁnite logarithmic measure. Moreover, from (.), we have

T

r,(z,w,w)

wλ 

=T

r, P(z,w)

Q(z,w)wλ

=max{p,q+λ}–min

λ,ordw(P)

T(r,w)

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and

whererapproaches inﬁnity outside of an exceptional set of ﬁnite logarithmic measure. By combining (.) and (.), (.) and (.), respectively, it follows that

N

From Lemma , we have

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and

N

r,(z,w,w)

wλ_

≤N

r, (z,w,w)

wordw((z,w,w)) 

+N

r, 

wλ–ordw() 

≤λ–ordw()

N(r,w) +λN(r,w)

+T

r, 

wλ–ordw() 

+Sq(r,w) +Sq(r,w)

≤λ–ordw()

N(r,w) +λN(r,w)

+λ–ordw()

T(r,w) +Sq(r,w) +Sq(r,w). (.)

Combining (.) and (.), (.) and (.), respectively, we have

 +λ+λ–ordw()

T(r,w)

<λ–ordw()

N(r,w) +λT(r,w)

+λ–ordw()

T(r,w) +Sq(r,w) +Sq(r,w) (.)

and

 +λ+λ–ordw()

T(r,w)

<λ–ordw()

N(r,w) +λT(r,w)

+λ–ordw()

T(r,w) +Sq(r,w) +Sq(r,w). (.)

Suppose thatN(r,w) =Sq(r,w) andN(r,w) =Sq(r,w), according to (.) and (.), we

have

( +λ)T(r,w) <λT(r,w) +Sq(r,w) +Sq(r,w)

and

( +λ)T(r,w) <λT(r,w) +Sq(r,w) +Sq(r,w).

That is,

 +λ+o()

T(r,w) <

λ+o()

T(r,w) (.)

and

 +λ+o()

T(r,w) <

λ+o()

T(r,w). (.)

By (.) and (.), we conclude that

 +λ+  +λ+o() <λ+λ,

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4 Proof of Theorem 2

It follows by Lemma  that

m

for allroutside of an exceptional set of ﬁnite logarithmic measure.

Suppose now that (w(z),w(z)) is a ﬁnite-order meromorphic solution of (.). Denoting

for allroutside of a setEof ﬁnite logarithmic measure. By (.) and (.), we have

N

whererlies outside of a setFof ﬁnite logarithmic measure. Combining inequalities (.) and (.) with the assumption in Theorem , we have

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≤λ

T(r,w) –m(r,w)

+λ

T(r,w) – m(r,w)

+Sq(r,w) +Sq(r,w). (.)

Similarly, we obtain

T(r,w)

–λm(r,w) +Sq(r,w) +Sq(r,w)

≤λ

T(r,w) –m(r,w)

+λ

T(r,w) – m(r,w)

+Sq(r,w) +Sq(r,w). (.)

By (.) and (.), we obtain

λm(r,w)

≤λ–

+o()T(r,w)

+λ+o()

T(r,w) (.)

and

– λ+o()

T(r,w)

≤λ+o()

T(r,w) – λm(r,w). (.)

Combining (.) and (.), we have

λm(r,w)

≤λ–

+o()T(r,w)

+ λ(λ+o())T(r,w) – λλm(r,w) (max{p,q+λ}–min{λ,ordw()}) – λ

,

that is,

λ–

λλ B

m(r,w)≤

A–λλ+o()

B

T(r,w), (.)

whereA= λ– (max{p,q+λ}–min{λ,ordw()}) andB= λ– (max{p,q+λ}–

min{λ,ordw()}). Combining the assumption and (.), we have m(r,w) =Sq(r,w)

for allroutside ofE∪F, a set of ﬁnite logarithmic measure. Similarly, we obtain

m(r,w) =Sq(r,w)

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{Mathematics and Computer College, Northwest Minorities University, Lanzhou, Gansu 730030, China.}2_{Laiwu Vocational}

and Technical College, Laiwu, Shandong 271100, China.3_{Shandong Transport Vocational College, Weifang, Shandong}

261206, China.

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).

Received: 25 March 2013 Accepted: 1 December 2013 Published:06 Jan 2014

References

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3. Korhonen, R: A new Clunie type theorem for difference polynomials. J. Differ. Equ. Appl.17(3), 387-400 (2011) 4. Zhang, JC, Wang, G, Chen, JJ, Zhao, RX: Some results onq-difference equations. Adv. Differ. Equ.2012, 191 (2012) 5. Barnett, D, Halburd, R-G, Korhonen, R-J, Morgan, W: Nevanlinna theory for theq-difference operator and meromorphic

solutions ofq-diﬀerence equations. Proc. R. Soc. Edinb. A137(3), 457-474 (2007)

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10.1186/1687-1847-2014-3

Cite this article as:Xu et al.:The solutions of one typeq-difference functional system.Advances in Diﬀerence Equations