復(fù)域差分和差分方程的研究

陳宗煊, 黃志波

(華南師范大學(xué)數(shù)學(xué)科學(xué)學(xué)院,廣東廣州 510631)

復(fù)域差分和差分方程的研究

陳宗煊*, 黃志波

(華南師范大學(xué)數(shù)學(xué)科學(xué)學(xué)院,廣東廣州 510631)

介紹了近十年來復(fù)域差分、q-差分、差分方程及q-差分方程的主要研究成果,其中包括亞純函數(shù)對(duì)數(shù)導(dǎo)數(shù)引理的差分模擬;Clunie引理和Mohon′ko引理的差分模擬; 慢增長(zhǎng)亞純函數(shù)的差分、均差分的零點(diǎn)和不動(dòng)點(diǎn)的性質(zhì); 差分多項(xiàng)式的值分布性質(zhì);差分Riccati方程與差分Painlevé方程亞純解的性質(zhì);復(fù)域q-差分及q-差分方程的解析性質(zhì).

復(fù)域差分; 差分方程; 復(fù)域q-差分;q-差分方程; 亞純函數(shù)值分布

1925年,NEVANLINNA[1]發(fā)表了關(guān)于亞純函數(shù)理論的論文,后來發(fā)展為亞純函數(shù)Nevanlinna理論.隨后, 亞純函數(shù)Nevanlinna理論被運(yùn)用到線性和非線性微分方程亞純解的值分布、唯一性和存在性等問題的討論, 獲得豐富的成果[2-8].

然而,利用亞純函數(shù)Nevanlinna理論,對(duì)差分方程的研究可以追溯到二十世紀(jì)初期[9-10].由于缺乏有力的研究工具,復(fù)域差分方程理論的發(fā)展極其緩慢.雖然在20世紀(jì)七、八十年代,BANK和KAUFMAN[11], SHIMOMURA[12]和YANAGIHARA[13]獲得一些關(guān)于差分方程亞純解存在性的初始結(jié)果. 特別地,BANK和KAUFMAN[11]證明了, 對(duì)任意關(guān)于z的有理函數(shù)R(z),差分方程

f(z+1)-f(z)=R(z)

(1)

總存在滿足條件T(r,f)=O(r)的亞純解f(z). YANAGIHARA[13]證明了, 對(duì)任意關(guān)于f(z)的有理函數(shù)R(f(z)),差分方程

f(z+1)=R(f(z))

(2)

有非平凡的亞純解.

2000年, ABLOWITZ等[14]利用亞純函數(shù)Nevanlinna 理論研究二階非線性差分方程的可積性問題, 標(biāo)志著亞純函數(shù)Nevanlinna 理論作用到復(fù)域差分方程研究的真正實(shí)現(xiàn). 隨后,經(jīng)過許多復(fù)分析和數(shù)學(xué)物理領(lǐng)域的專家和學(xué)者近十幾年的努力, 亞純函數(shù)Nevanlinna 理論的差分模擬的研究取得重大突破,從而為復(fù)域差分方程的研究提供了有力的理論工具. 由此,復(fù)域差分、q-差分、復(fù)域差分方程及q-差分方程的研究,以及與之對(duì)應(yīng)的唯一性理論的差分模擬的研究,逐漸成為熱門研究課題.

近十年來,亞純函數(shù)Nevanlinna理論為復(fù)域差分、q-差分、復(fù)域差分方程及q-差分方程的研究和發(fā)展提供了一個(gè)有力的工具,主要體現(xiàn)在亞純函數(shù)的對(duì)數(shù)導(dǎo)數(shù)引理的差分模擬和Clunie-Mohon′ko引理的差分模擬. 下面將介紹其差分形式和q-差分形式.

有限級(jí)亞純函數(shù)的對(duì)數(shù)導(dǎo)數(shù)引理的差分模擬, 是由CHIANG和FENG[15]、HALBURD和KORHONEN[16-17]分別建立的. 這里, 我們給出它的最終形式.

對(duì)所有的r成立,至多除去一個(gè)對(duì)數(shù)測(cè)度為有限的集合.

對(duì)于有限級(jí)亞純函數(shù)f(z)的差分算子, HALBURD和KORHONEN[17]得到如下的對(duì)數(shù)導(dǎo)數(shù)引理.

對(duì)所有的r成立, 至多除去一個(gè)對(duì)數(shù)測(cè)度為有限的集合.

如果f(z)為無窮級(jí)的亞純函數(shù), HALBURD等[18]和KORHONEN[19]獲得對(duì)數(shù)導(dǎo)數(shù)引理的差分模擬的更一般的形式.

對(duì)所有的r成立,至多除去一個(gè)集合E滿足

如果f(z)的超級(jí)σ2(f)<1和ε>0,那么

對(duì)所有的r成立, 至多除去一個(gè)對(duì)數(shù)測(cè)度為有限的集合.

定理4[19]設(shè)f(z)為非常數(shù)亞純函數(shù),ω(z)=czn+pn-1zn-1+…+p0和φ(z)=czn+qn-1zn-1+…+q0是2個(gè)非常數(shù)多項(xiàng)式. 如果

那么

對(duì)所有的r成立,至多除去一個(gè)對(duì)數(shù)測(cè)度為有限的集合.

Clunie引理和Mohon′ko引理在微分多項(xiàng)式的值分布和微分方程亞純解的增長(zhǎng)級(jí)估計(jì)方面具有重要的作用.Clunie 引理的差分模擬最初形式可以參考文獻(xiàn)[16]的定理3.1和文獻(xiàn)[20]的定理2.3,對(duì)于更進(jìn)一步的結(jié)果, 可以參考文獻(xiàn)[21]-[23]. 下面給出其最初結(jié)論.

定理5 設(shè)f(z)是增長(zhǎng)級(jí)為σ(<∞)的亞純函數(shù),且滿足方程

U(z,f)P(z,f)=Q(z,f),

其中U(z,f),P(z,f)和Q(z,f)是差分多項(xiàng)式滿足degfQ(z,f)≤degfU(z,f)=n. 更進(jìn)一步地,U(z,f)僅含一項(xiàng)次數(shù)最大的項(xiàng). 那么,對(duì)任意ε>0,

m(P(z,f))=O(rσ-1+ε)+S(r,f)

對(duì)所有的r成立, 至多除去一個(gè)對(duì)數(shù)測(cè)度為有限的集合.

下面給出Mohon′ko引理的差分模擬.

定理6[16,20]設(shè)f(z)是有限級(jí)亞純函數(shù),且滿足方程P(z,f)=0,其中P(z,f)是差分多項(xiàng)式.對(duì)給定的小函數(shù)a(z),如果P(z,a(z))?0, 那么

下面將給出對(duì)數(shù)導(dǎo)數(shù)引理、Clunie引理和Mohon′ko引理的q-差分模擬.

在一對(duì)數(shù)密度為1的集合上成立.

定理8[24]設(shè)f(z)是非常數(shù)零級(jí)亞純函數(shù)且滿足fn(z)P(z,f)=Q(z,f),其中P(z,f)和Q(z,f)是關(guān)于f(z)的q-差分多項(xiàng)式且degfQ(z,f)≤n,那么

m(r,P(z,f))=o(T(r,f))

在一對(duì)數(shù)密度為1的集合上成立.

定理9[24]設(shè)f(z)是非常數(shù)零級(jí)亞純函數(shù)且滿足P(z,f)=0,其中P(z,f)是關(guān)于f(z)的q-差分多項(xiàng)式.對(duì)給定的小函數(shù)a(z),如果P(z,a(z))?0, 那么

在一對(duì)數(shù)密度為1的集合上成立.

對(duì)于亞純函數(shù)f(z),其位移f(z+c)和q-位移f(qz)的特征函數(shù)等的估計(jì)[15,25]、Nevanlinna第二基本定理[17,24]、Wiman-Valiron方法的差分模擬[26]等,這里不再闡述. 基于上述理論的建立,復(fù)域差分、q-差分、差分方程及q-差分方程和唯一性理論的差分模擬取得豐富的研究成果.

1 復(fù)域差分多項(xiàng)式

鑒于對(duì)數(shù)導(dǎo)數(shù)引理和Clunie-Mohon′ko引理等差分模擬的建立,對(duì)復(fù)域差分多項(xiàng)式的解析性質(zhì)的研究,主要體現(xiàn)在差分多項(xiàng)式的值分布和唯一性兩方面.

1.1復(fù)域差分多項(xiàng)式的值分布

具有無窮多個(gè)零點(diǎn).

同時(shí)提出了如下猜想：

猜想1[27]假設(shè)f(z)是超越整函數(shù)且滿足σ(f)<1,那么G(z)有無窮多個(gè)零點(diǎn).

若所考慮的函數(shù)f(z)為亞純函數(shù),BERGWEILER和LANGLEY還證明了:

隨后，LANGLEY[28]、CHEN和SHON[29]推廣了上述結(jié)果,并給出很多新穎的證明方法.特別地,CHEN和SHON[29]首次研究了具有慢增長(zhǎng)性級(jí)的亞純函數(shù)的差分和均差分的不動(dòng)點(diǎn)問題, 從而部分地證明了上述猜想.

具有無窮多個(gè)零點(diǎn)和無窮多個(gè)不動(dòng)點(diǎn).

(i)至多有有限個(gè)極點(diǎn)zj,zk滿足zj-zk=c;

那么

具有無窮多個(gè)零點(diǎn)和無窮多個(gè)不動(dòng)點(diǎn).

對(duì)于差分多項(xiàng)式的值分布的研究, 要?dú)w結(jié)于對(duì)HAYMAN[30]所探究的微分多項(xiàng)式值分布的差分模擬[31-33]. 隨后, 亞純函數(shù)的差分多項(xiàng)式的值分布獲得一系列有趣的結(jié)果[23,33-42]. 下面介紹HAYMAN關(guān)于微分多項(xiàng)式的幾個(gè)經(jīng)典結(jié)果的差分模擬.

例1[32]設(shè)f(z)=ez+1,則

H(z)=f(z)f(z+iπ)-1=(1+ez)(1-ez)-1=-e2z.

這個(gè)例子說明,定理14和定理15中的條件n≥2不能省略. 因此,CHEN等進(jìn)一步獲得如下結(jié)果.

1.2復(fù)域差分多項(xiàng)式的唯一性

設(shè)f(z)和g(z)為2個(gè)非常數(shù)的亞純函數(shù),a為任意復(fù)數(shù). 如果f(z)-a和g(z)-a具有相同的零點(diǎn)(計(jì)算重?cái)?shù)), 則稱f(z)和g(z)具有CM分擔(dān)a. 如果f(z)-a和g(z)-a具有相同的零點(diǎn)(不計(jì)算重?cái)?shù)),則稱f(z)和g(z)具有IM分擔(dān)a. 亞純函數(shù)唯一性理論的典型結(jié)果是Nevanlinna五值定理和四值定理[8]. 亞純函數(shù)唯一性理論的已有文獻(xiàn)指出,對(duì)于任意2個(gè)亞純函數(shù), 如果其分擔(dān)值的個(gè)數(shù)減少, 需要增加額外的假設(shè).基于這一事實(shí),HEITTOKANGAS等[43-44]首先研究了亞純函數(shù)f(z)和它的位移算子f(z+c)的唯一性,后來進(jìn)一步提升已有的結(jié)果, 得到

近幾年來, 很多學(xué)者在這一方面做出很多有意義的結(jié)果[42,45-54].

關(guān)于亞純函數(shù)唯一性理論的差分模擬, Brück猜想[55]的差分模擬具有重要的地位. 下面介紹幾個(gè)經(jīng)典的結(jié)果,更多的結(jié)果可參考文獻(xiàn)[33]、[44]、[48]-[49]、[56].

其中A為一非零常數(shù).

2 復(fù)域差分方程

復(fù)域微分方程的復(fù)振蕩理論在過去30多年發(fā)展迅速, 獲得一系列的成果[2,5-6].這些成果揭示了微分方程亞純解的增長(zhǎng)級(jí)和其系數(shù)增長(zhǎng)級(jí)之間的關(guān)系,刻畫了微分方程亞純解的零點(diǎn)、 極點(diǎn)和不動(dòng)點(diǎn)等性質(zhì). 近十年來,隨著亞純函數(shù)Nevanlinna理論的差分模擬的建立,對(duì)函數(shù)差分方程亞純解性質(zhì)的研究, 也進(jìn)入一個(gè)繁榮時(shí)期.

2.1復(fù)域差分方程

ABLOWITZ等[14]考慮復(fù)域離散方程作為復(fù)域時(shí)滯方程,運(yùn)用亞純函數(shù)Nevanlinna理論對(duì)其研究,為復(fù)域差分方程的研究提供了有力的工具.隨后,HALBURD和KORHONEN[57]利用奇異測(cè)試的方法證明了:如果二階差分方程

f(z+1)+f(z-1)=R(z,f)

(3)

存在有限級(jí)亞純解f(z),那么f(z)或者滿足差分Riccati方程

或者可經(jīng)過一線性變換,將方程(3)轉(zhuǎn)化為一些典型的線性差分方程和差分Painlevé方程.

2.1.1 復(fù)域差分方程亞純解的存在性 對(duì)復(fù)域差分方程亞純解存在性的研究,主要體現(xiàn)在有理解的存在性及其表示[58-61]、漸近解的存在性[57，61-62]和亞純解的表示[15,63-66]. 下面列出一些典型結(jié)果.

定理21[58]設(shè)a,b,c是常數(shù)且a,b不全為零.

(i)如果a≠0,那么差分Painlevé I方程

(4)

沒有有理解;

(ii)如果a=0和b≠0,那么差分Painlevé I方程(4)有一個(gè)非零常數(shù)解f(z)=A滿足

2A2-cA-b=0,

其他有理解具有形式f(z)=P(z)/Q(z),其中P(z)和Q(z)是互素的多項(xiàng)式且其次數(shù)滿足degP

定理22[59]設(shè)δ=±1,A(z)=m(z)/n(z)是不可約的有理函數(shù),其中m(z)和n(z)是多項(xiàng)式,其次數(shù)分別為degm(z)=m和degn(z)=n.

(i)假設(shè)m≥n且m-n是偶數(shù)或零.如果差分Riccati方程

(5)

有一個(gè)不可約的有理解f(z)=P(z)/Q(z),其中P(z)和Q(z)是多項(xiàng)式且其系數(shù)分別為degP(z)=p和degQ(z)=q, 那么p-q=(m-n)/2;

(ii)假設(shè)m≤n且m-n=k(≥2)是一正整數(shù).如果差分Riccati方程(5)有一個(gè)不可約的有理解f(z)=P(z)/Q(z),那么q-p=k-1或q-p=1;

(iii)假設(shè)m>n且m-n是奇數(shù),或m

SHIMOMURA[61]考慮差分Painlevé方程

(6)

(7)

(8)

(9)

定理23[61]如果α≠0,那么差分方程(6)～(8)沒有有理解．

對(duì)于一階非線性差分方程

f(z+1)=R(f(z)),

(10)

其中R(f(z))是關(guān)于f(z)的常系數(shù)有理函數(shù),HALBURD和KORHONEN[62]獲得方程(10)非平凡亞純解滿足一定的漸近性質(zhì).

(f(z)-γ)λ-z→α, Rez→-∞.

對(duì)于差分方程亞純解的表示形式,ISHIZAKI[63]獲得差分Riccati方程亞純解的表示性質(zhì).

定理26[63]設(shè)A(z)為一亞純函數(shù).如果差分Riccati方程

(11)

存在3個(gè)不同的亞純解f1(z)、f2(z)和f3(z), 那么方程(11)的任一亞純解f(z)可以表示為

f(z)=

(12)

其中Q(z)是周期為1的亞純函數(shù). 反之,對(duì)于任意周期為1的亞純函數(shù), 我們定義函數(shù)f(z)具有式(12)的形式,那么f(z)是方程(11)的亞純解.

2.1.2 復(fù)域差分方程亞純解的值分布 對(duì)于差分方程(包括:一般的差分方程、差分Painlevé方程和差分Riccati方程)亞純解的值分布,主要體現(xiàn)在差分方程亞純解的增長(zhǎng)級(jí)、零點(diǎn)收斂指數(shù)、極點(diǎn)收斂指數(shù)、不動(dòng)點(diǎn)收斂指數(shù)和Borel例外值的存在性, 詳見文獻(xiàn)[11]、[15]、[20]、[57]-[60]、[62]、[64]-[71]等, 這里不再贅述.

2.2復(fù)域函數(shù)方程

復(fù)域q-差分方程(函數(shù)方程)的研究可追溯到RITT[72]研究的自治Schr?der方程f(qz)=R(f(z)),其中R(f(z))是關(guān)于f(z)的常系數(shù)有理函數(shù),和VALIRON[73]研究的非自治Schr?der方程f(qz)=R(z,f(z)),其中R(z,f(z))是關(guān)于z、f(z)的有理函數(shù),它們是與復(fù)動(dòng)力系統(tǒng)緊密聯(lián)系的方程[73-76].

2.2.1 復(fù)域q-差分方程亞純解的存在性 具有有理系數(shù)的線性q-差分方程(函數(shù)方程), 其亞純解不一定存在. 近十年來,BEIGWEILER、GUNDERSEN和HEITTOKANGAS等[77-79]在一定的系數(shù)約束下,研究了q-差分方程亞純解的存在性.下面給出一些經(jīng)典結(jié)果.

定理27[78]設(shè)f(z)是q-差分方程

f(qz)=A(z)+γf(z)+δf(z)2,

(13)

(i)在方程(13)中,如果

那么方程(13)有2個(gè)不同的亞純解;

(ii)在方程(13)中, 如果(1-γ)2=4,那么方程(13)只有一個(gè)亞純解.

定理28[79]設(shè)a0(z),a1(z),…,an(z)是復(fù)常數(shù)滿足

和Q(z)=g1(z)是一整函數(shù).那么q-差分方程

(14)

只有唯一的整函數(shù)解.

定理29[77]考慮q-差分方程

f(q2z)+a(z)f(qz)+b(z)f(z)=0,

(15)

(i)如果不存在整數(shù)n滿足q2n+a0qn+b0=0, 那么方程(15)沒有任何超越亞純解;

(ii)如果b0≠0且存在整數(shù)n滿足q2n+a0qn+b0=0,那么方程(15)存在一個(gè)超越亞純解;

(iii)如果b0=0,那么方程(15)沒有任何超越亞純解.

對(duì)于q-差分方程亞純解的表示形式,HUANG[80]獲得q-差分Riccati方程亞純解的表示性質(zhì):

(16)

存在3個(gè)不同的亞純解g1(z)、g2(z)和g3(z),那么方程(16)的任一亞純解f(z)可以表示為

g(z)=

(17)

其中φ(z)是亞純函數(shù)且滿足φ(qz)=φ(z).反之,對(duì)于任意亞純函數(shù)φ(z)且滿足φ(qz)=φ(z),我們定義函數(shù)f(z)具有式(17)的形式,那么f(z)是方程(16)的亞純解.

2.2.2 復(fù)域q-差分方程亞純解的值分布 隨著亞純函數(shù)q-差分模擬的建立,q-差分方程亞純解的值分布性質(zhì)也得到廣泛探討, 詳見文獻(xiàn)[47]、[77]-[86]等, 這里不再贅述.

3 結(jié)束語

亞純函數(shù)Nevanlinna理論差分模擬的建立,為復(fù)域(q-)差分多項(xiàng)式和復(fù)域(q-)差分方程的發(fā)展開辟了新天地.但由于對(duì)復(fù)域差分方程的研究,沒有普適的研究方法,針對(duì)不同形式的差分方程,我們需要尋求其特有的解法,特別是差分方程亞純解的存在性方面,還有大量的工作需要去探討.

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Keywords： complex difference; difference equations; complexq-difference;q-difference equations; value distribution of meromorphic functions

StudyonComplexDifferencesandDifferenceEquations

CHEN Zongxuan*, HUANG Zhibo

(School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China)

The researches on the complex difference, complexq-difference, difference equations andq-difference equations in recent decades are mainly introduced. These results include the difference analogue of the logarithmic derivative; the difference analogue of Clunie lemma; the difference counterpart of Mohon′ko lemma; the properties on the zeros, fixed-points on complex differences and divided difference of meromorphic functions with small order; the properties on the value distribution of difference polynomials, the properties on the meromorphic solutions of difference Riccati and Painlevé equations; the results onq-differences and meromorphic solutions ofq-difference equations.

2013-07-04

國(guó)家自然科學(xué)基金項(xiàng)目(11171119)

*通訊作者:陳宗煊，教授，Email:chzx@vip.sina.com.

1000-5463(2013)06-0026-08

O174.5

A

10.6054/j.jscnun.2013.09.004

【中文責(zé)編：莊曉瓊 英文責(zé)編：肖菁】