In view of Nevanlinna theory, we study the properties of meromorphic solutions of systems of a class of complex difference equations. Some results obtained improve and extend the previous theorems given by Gao.

1. Introduction and Main Results

The purpose of this paper is to study some properties of meromorphic solutions of complex q-shift difference equations. The fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used (see [1–3]). Besides, for meromorphic function f, a meromorphic function a(z) is called small function with respect to f if T(r,a(z))=o(T(r,f))=S(r,f) for all r outside a possible exceptional set E of finite logarithmic measure limr→∞∫[1,r)∩E(dt/t)<∞.

In recent years, it has been a heated topic to study difference equations, difference product, and q-difference in the complex plane ℂ. There were articles focusing on the growth of solutions of difference equations, value distribution and uniqueness of differences analogues of Nevanlinna’s theory (see [4–9]). Chiang and Feng [10] and Halburd and Korhonen [11] established a difference analogue of the logarithmic derivative lemma independently, and Barnett et al. [5] also established an analogue of the logarithmic derivative lemma on q-difference operators. By applying these theorems, a number of results on meromorphic solutions of complex difference and q-difference equations were obtained (see [12–19]).

In 2011, Korhonen [20] investigated the properties of finite-order meromorphic solution of the equation
(1)H(z,ω)P(z,ω)=Q(z,ω),
where P(z,ω)=P(z,ω(z),ω(z+c1),…,ω(z+cn)),c1,…,cn∈ℂ and obtained the following result.

Theorem 1 (see [<xref ref-type="bibr" rid="B19">20</xref>]).

Let ω(z) be a finite-order meromorphic solution of (1), where P(z,ω) is a homogeneous difference polynomial with meromorphic coefficients and H(z,ω) and Q(z,ω) are polynomials in ω(z) with meromorphic coefficients having no common factors. If max{degω(H),degω(Q)-degω(P)}>min{degω(P),
ord
0(Q)}-
ord
0(P), then N(r,ω)≠S(r,ω), where
ord
0(P) denotes the order of zero of P(z,x0,x1,…,xn) at x0=0 with respect to the variable x0.

Let cj∈ℂ for j=1,…,n, and let I be a finite set of multi-indexes λ=(λ0,…,λn). Then a difference polynomial of a meromorphic function ω(z) is defined as
(2)P(z,ω)=P(z,ω(z),ω(z+c1),…,ω(z+cn))=∑λ∈Icλ(z)w(z)λ0w(z+c1)λ1⋯ω(z+cn)λn,
where the coefficients cλ(z) are small with respect to ω(z) in the sense that T(r,cλ)=o(T(r,ω)) as r tends to infinity outside of an exceptional set E of finite logarithmic measure.

At the same year, Zheng and Chen [21] consider the value distribution of meromorphic solutions of zero order of a kind of q-difference equations and obtained the following result which is an extension of Theorem 1.

Theorem 2 (see [<xref ref-type="bibr" rid="B28">21</xref>, Theorem 1]).

Suppose that f is a nonconstant meromorphic solution of zero order of a q-difference equation of the form
(3)∑λ∈Icλ(z)f(qz)iλ,1f(q2z)iλ,2⋯f(qnz)iλ,n=P(z,f(z))Q(z,f(z))=(ak(z)(f(z))k+ak+1(z)(f(z))k+1+⋯+as(z)(f(z))sak(z)(f(z))k+ak+1(z)(f(z))k+1)×(b0(z)+b1(z)f(z)+⋯+bt(z)(f(z))t)-1,
where I={(iλ1,iλ2,…,iλn)} is a finite index set and iλ1+iλ2+⋯+iλn=σ>0 for all λ∈I and q(≠0,1)∈ℂ. Moreover, suppose that 0≤k≤s,ak(z)as(z)bt(z)≢0, the P(z,f) and Q(z,f) have no common factors, and that all meromorphic coefficients in (3) are of growth of o(T(r,f)) on a set of logarithmic density 1. If
(4)max{t,s-σ}>min{σ,k},
then
(5)N(r,f)≠o(T(r,f))
on any set of logarithmic density 1.

Remark 3.

The logarithmic density of a set F is defined by
(6)limsupr→∞1logr∫[1,r]∩F1tdt.

Recently, Gao [22–24] and others [25, 26] also investigated the growth and existence of meromorphic solutions of some systems of complex difference equations; one system of complex difference equation is based on (1) and obtained some interesting results.

Inspired by the idea of [21–24, 27], we will investigate the properties of meromorphic solutions of systems of a class of complex q-shift difference equations of the form
(7)Ω1(z,w1,w2)=R1(z,w1),Ω2(z,w1,w2)=R2(z,w2),
where q(≠0,1),cj(j=1,…,n)∈ℂ,I,J are two finite sets of multi-indexes (i1,…,in), (j1,…,jn), and Ω1(z,w1,w2),Ω2(z,w1,w2) are two homogeneous difference polynomials to be defined as
(8)Ω1(z,w1,w2)=Ω1(z,w1(qz+c1),w2(qz+c1),…,w1(qnz+cn),w2(qnz+cn))=∑(i)a(i)(z)∏k=12(wk(qz+c1))ik1⋯(wk(qnz+cn))ikn,Ω2(z,w1,w2)=Ω2(z,w1(qz+c1),w2(qz+c1),…,w1(qnz+cn),w2(qnz+cn))=∑(j)b(j)(z)∏k=12(wk(qz+c1))jk1⋯(wk(qnz+cn))jkn.
The coefficients {a(i)},{b(j)} are small with respect to w1, w2 in the sense that T(r,a(i))=o(T(r,wl)), T(r,b(j))=o(T(r,wl)), l=1,2, as r tends to infinity outside of an exceptional set E of finite logarithmic measure. The weights of Ω1(z,w1,w2), Ω2(z,w1,w2) are defined by
(9)σ11=max(i){∑l=1ni1l},σ12=max(i){∑l=1ni2l},σ21=max(j){∑l=1nj1l},σ22=max(j){∑l=1nj2l},R1(z,w1)=P1(z,w1)Q1(z,w1)=(ck11(z)(w1(z))k1+ck1+11(z)(w1(z))k1+1+⋯+cs11(z)(w1(z))s1(w1(z))k1+1)×((z)(w1(z))t1d01(z)+d11(z)w1(z)+⋯,+dt11(z)(w1(z))t1)-1,R2(z,w2)=P2(z,w2)Q2(z,w2)=(ck22(z)(w2(z))k2+ck2+12(z)(w2(z))k2+1+⋯+cs2(z)(w2(z))s2(w1(z))k1+1)×((w2(z))t2d02(z)+d12(z)w2(z)+⋯+dt22(z)(w2(z))t2)-1.
The coefficients {ckii(z)}, {dtii(z)} are meromorphic functions and small functions,
(10)S(r)=∑T(r,a(i))+∑T(r,b(j))+∑T(r,ckii)+∑T(r,dtii).

Now, we will show our main results as follows.

Theorem 4.

Let (w1,w2) be meromorphic solution of systems (7) satisfying ρ=ρ(w1,w2)=0. Moreover, suppose that 0≤ki≤si,ck1i(z)cs1i(z)dtii(z)≢0,i=1,2, the Pi(z,wi) and Qi(z,wi) are polynomials in wi(z) with meromorphic coefficients having no common factors, and that all meromorphic coefficients in (7) are of growth of o(T(r,f)) for all r on a set of logarithmic density 1 or outside of an exceptional set of logarithmic density 0. If
(11)max{t1,s1-σ11}>min{σ11,k1}+σ11+σ12,max{t2,s2-σ22}>min{σ22,k2}+σ22+σ21,
then N(r,w1)=o(T(r,w1)) and N(r,w2)=o(T(r,w2)) cannot hold both at the same time, for all r possibly outside of an exceptional set of logarithmic density 0, where the order of meromorphic solution (w1,w2) of systems (7) is defined by
(12)ρ=ρ(w1,w2)=max{ρ(w1),ρ(w2)},ρ(wi)=limsupr→∞logT(r,wi)logr,i=1,2.

Theorem 5.

Let (w1,w2) be meromorphic solution of systems (7) satisfying ρ=ρ(w1, w2)=0. Moreover, suppose that 0≤ki≤si,cki(z)csi(z)dti(z)≢0,i=1,2, the Pi(z,wi) and Qi(z,wi) are polynomials in wi(z) with meromorphic coefficients having no common factors, and that all meromorphic coefficients in (7) are of growth of o(T(r,f)) for all r on a set of logarithmic density 1 or outside of an exceptional set of logarithmic density 0, and
(13)A=2σ11-max{s1,t1+σ11}+min{σ11,k1},B=2σ22-max{s2,t2+σ22}+min{σ22,k2}.
If
(14)A<0,B<0,AB>9σ21σ12,
then m(r,wk)=o(T(r,wk)), k=1,2 hold for r that runs to infinity possibly outside of an exceptional set of logarithmic density 0.

2. Some LemmasLemma 6 (Valiron-Mohon’ko) ([<xref ref-type="bibr" rid="B16">28</xref>]).

Let f(z) be a meromorphic function. Then for all irreducible rational functions in f,
(15)R(z,f(z))=∑i=0mai(z)f(z)i∑j=0nbj(z)f(z)j,
with meromorphic coefficients ai(z), bj(z), the characteristic function of R(z,f(z)) satisfies that
(16)T(r,R(z,f(z)))=dT(r,f)+O(Ψ(r)),
where d=max{m,n} and Ψ(r)=maxi,j{T(r,ai),T(r,bj)}.

Lemma 7 (see [<xref ref-type="bibr" rid="B18">27</xref>]).

Let f(z) be a nonconstant zero-order meromorphic function and q∈ℂ∖{0}. Then
(17)m(r,f(qz+η)f(z))=o(T(r,f))=S(r,f),
on a set of logarithmic density 1 or outside of an exceptional set of logarithmic density 0.

Lemma 8 (see [<xref ref-type="bibr" rid="B22">29</xref>]).

Let f(z) be a transcendental meromorphic function of zero order, and let q, η be two nonzero complex constants. Then
(18)T(r,f(qz+η))=T(r,f(z))+S(r,f),N(r,f(qz+η))≤N(r,f)+S(r,f),
on a set of logarithmic density 1 or outside of a possibly exceptional set of logarithmic density 0.

3. The Proof of Theorem <xref ref-type="statement" rid="thm1.3">4</xref>

From the definitions of Ωi(z,w1,w2), by Lemma 7, it follows that
(19)m(r,Ω1(z,w1,w2)w1σ11)≤σ12m(r,w2)+o(T(r,w1)),r∉E1′,(20)m(r,Ω2(z,w1,w2)w2σ22)≤σ21m(r,w1)+o(T(r,w2)),r∉E2′,
where E1′, E2′ are two sets of logarithmic density 0. By Lemma 6, we have
(21)T(r,Ω1(z,w1,w2)w1σ11)=T(r,P1(z,w1)Q1(z,w1)w1σ11)=(max{t1+σ11,s1}-min{σ11,k1})×T(r,w1)+o(T(r,w1)),r∉E3′,(22)T(r,Ω2(z,w1,w2)w2σ22)=T(r,P2(z,w2)Q2(z,w2)w2σ22)=(max{t2+σ22,s2}-min{σ22,k2})×T(r,w2)+o(T(r,w2)),r∉E4′,
where E3′, E4′ are two sets of logarithmic density 0. Thus, from the assumptions of Theorem 4, combining (19) and (21), (20) and (22), respectively, we have
(23)N(r,Ω1(z,w1,w2)w1σ11)≥(1+σ12+σ11)T(r,w1)-σ12m(r,w2)+o(T(r,w1)),r∉E1=E1′∪E3′,N(r,Ω2(z,w1,w2)w2σ22)≥(1+σ21+σ22)T(r,w1)-σ21m(r,w1)+o(T(r,w2)),r∉E2=E2′∪E4′.

Since ρ=ρ(w1,w2)=0, from Lemma 8, we have
(24)N(r,Ω1(z,w1,w2)w1σ11)≤N(r,Ω1(z,w1,w2))+σ11N(r,1w1)≤σ11N(r,w1)+σ12N(r,w2)+σ11N(r,1w1)+o(T(r,w1))+o(T(r,w2)),r∉E5′,N(r,Ω2(z,w1,w2)w2σ22)≤N(r,Ω2(z,w1,w2))+σ22N(r,1w2)≤σ22N(r,w2)+σ21N(r,w1)+σ22N(r,1w2)+o(T(r,w1))+o(T(r,w2)),r∉E6′,
where E5′, E6′ are the sets of logarithmic density 0.

From (23) and (24), it follows that
(25)(1+σ12+σ11)T(r,w1)≤σ11N(r,w1)+σ12N(r,w2)+σ11N(r,1w1)+σ12m(r,w2)+o(T(r,w1))+o(T(r,w2))≤σ11N(r,w1)+σ12T(r,w2)+σ11T(r,w1)+o(T(r,w1))+o(T(r,w2)),r∉E3=E1∪E5′,(1+σ21+σ22)T(r,w1)≤σ22N(r,w2)+σ21N(r,w1)+σ22N(r,1w2)+σ21m(r,w1)+o(T(r,w1))+o(T(r,w2))≤σ22N(r,w2)+σ21T(r,w1)+σ22T(r,w2)+o(T(r,w1))+o(T(r,w2)),r∉E4=E2∪E6′.

Suppose now on the contrary to the assertion of Theorem 4 that N(r,w1)=o(T(r,w1)) and N(r,w2)=o(T(r,w2)), from (25); it follows that
(26)(1+σ12)T(r,w1)≤σ12T(r,w2)+o(T(r,w1))+o(T(r,w2)),(1+σ21)T(r,w2)≤σ21T(r,w1)+o(T(r,w1))+o(T(r,w2)),
that is,
(27)(1+σ12+o(1))T(r,w1)≤(σ12+o(1))T(r,w2),(1+σ21+o(1))T(r,w2)≤(σ21+o(1))T(r,w1).
From (27), we can get that
(28)(1+σ12)(1+σ21)≤σ12σ21.
From the previous inequality, we can get a contradiction.

Therefore, this completes the proof of Theorem 4.

4. The Proof of Theorem <xref ref-type="statement" rid="thm1.4">5</xref>

Since ρ=ρ(w1,w2)=0, from the assumptions concerning the coefficients of systems (7), by Lemma 7, and from the definitions of logarithmic measure and logarithmic density, we have
(29)N(r,Ω1(z,w1,w2)w1σ11)≤σ11[N(r,w1)+N(r,1w1)]+σ12[N(r,w2)+N(r,1w2)]+σ12N(r,w2)+o(T(r,w1))+o(T(r,w2)),r∉E5,
where E5 is a set of logarithmic density 0.

From (29), we have
(30)N(r,Ω1(z,w1,w2)w1σ11)≤σ11[N(r,w1)+N(r,1w1)]+σ12[2N(r,w2)+N(r,1w2)]+o(T(r,w1))+o(T(r,w2))≤σ11[2T(r,w1)-m(r,w1)]+σ12[3T(r,w2)-2m(r,w2)]+o(T(r,w1))+o(T(r,w2)),r∉E5.

From (19) and (29), we have
(31)N(r,Ω1(z,w1,w2)w1σ11)+σ12m(r,w2)≥N(r,Ω1(z,w1,w2)w1σ11)+m(r,Ω1(z,w1,w2)w1σ11)=T(r,P1(z,w1)Q1(z,w1)w1σ11)=(max{t1+σ11,s1}-min{σ11,k1})×T(r,w1)+o(T(r,w1)),r∉F1=E1′∪E5.
From the previous inequality and (30), we have for r∉F1(32)(max{t1+σ11,s1}-min{σ11,k1})T(r,w1)-σ12m(r,w2)≤σ11[2T(r,w1)-m(r,w1)]+σ12[3T(r,w2)-2m(r,w2)]+o(T(r,w1))+o(T(r,w2)).

By using the same argument as in the previously mentioned, there exists a set F2 of logarithmic density 0, for r∉F2, and we have
(33)(max{t2+σ22,s2}-min{σ22,k2})T(r,w2)-σ21m(r,w1)≤σ22[2T(r,w2)-m(r,w2)]+σ21[3T(r,w1)-2m(r,w1)]+o(T(r,w1))+o(T(r,w2)).
From (32) and (33), we have
(34)σ11m(r,w1)≤[2σ11-(max{t1+σ11,s1}-min{σ11,k1})+o(1)]T(r,w1)+(3σ12+o(1))T(r,w2),r∉F1,[(max{t2+σ22,s2}-min{σ22,k2})-2σ22+o(1)]T(r,w2)≤(3σ21+o(1))T(r,w1)-σ21m(r,w1),r∉F2.
From (34), we have
(35)σ11m(r,w1)≤[2σ11-(max{t1+σ11,s1}-min{σ11,k1})+o(1)σ11-(max{t1+σ11,s1}-min{σ11,k1})]T(r,w1)+((3σ12+o(1))×[(3σ21+o(1))T(r,w1)-σ21m(r,w1)](3σ12+o(1)))×((max{t2+σ22,s2}-min{σ22,k2})-2σ22)-1,r∉F=F1∪F2,
that is,
(36)(σ11-3σ12σ21B)m(r,w1)≤[A-9σ12σ21+o(1)B]T(r,w1),r∉F=F1∪F2,
where A=2σ11-max{s1,t1+σ11}+min{σ11,k1} and B=2σ22-max{s2,t2+σ22}+min{σ22,k2}. From (14) and (36), we have
(37)m(r,w1)=o(T(r,w1))
for all r outside of F, a set of logarithmic density 0.

Similarly, we can obtain
(38)m(r,w2)=o(T(r,w2)
for all r possibly outside of F′, a set of logarithmic density 0.

Thus, this completes the proof of Theorem 5.

Acknowledgments

This work was supported by the NSFC (Grant no. 61202313) and the Natural Science Foundation of Jiangxi Province in China (2010GQS0119, 20122BAB211036, 20122BAB201016, and 20122BAB201044).

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