有限迭代級整系數(shù)線性微分方程解的性質(zhì)

張曉梅,孫道椿

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

有限迭代級整系數(shù)線性微分方程解的性質(zhì)

張曉梅,孫道椿

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

研究了有限迭代級整系數(shù)的復(fù)線性微分方程,應(yīng)用Nevalinna和Ahlfors的角域理論,得到了有關(guān)解的迭代級,零點(diǎn)迭代收斂指數(shù)以及角域中的零點(diǎn)分布的結(jié)果.

微分方程;迭代級;迭代收斂指數(shù);角域分布

1 引言及主要結(jié)果

本文考慮微分方程

其中A(z)為迭代級有窮的整函數(shù).自1982年以來,當(dāng)A(z)為多項(xiàng)式或σ(A)＜∞時,人們對方程(1.1)的解的零點(diǎn)分布做了許多研究.而對于一般的情況當(dāng)σp(A)＜∞(1＜p＜∞,p∈N) 時,其結(jié)果甚少.因此作者在本文主要討論當(dāng)系數(shù)A(z)為迭代級有窮的整函數(shù)時,方程(1.1)的解的零點(diǎn)分布.

我們使用Nevanlinna理論以及復(fù)方程理論中的標(biāo)準(zhǔn)記號和基本內(nèi)容.為了敘述結(jié)果,還需要以下定義和記號.

定義1[1]整函數(shù)f(z)的迭代級σp(f)定義為

2 主要引理

3 定理的證明

定理1的證明根據(jù)引理1,有

[1]K innunen L.Linear differential equations with solutions of finite iterated order[J].Southeast Asian Bu ll. M ath.,1998,22:385-405.

[2]Wu S J.On the location of zeros of solution of f''+A f=0 where A(z)is entire[J].Math.Scand,1994, 74:293-312.

[3]Goldberg A A,Ostrovskii IV.The Distribution of Value of Meromorphic Functions[M].Moscow:Izdat Nauk press,1970(in Russian).

[4]Tsu jiM.Potential Theory in Modern Function Theory[M].Tokyo:M aruzen p ress,1959.

[5]Haym an W K.M eorm orphic Functions[M].Oxford:C larendon Press,1964.

[6]Lain I.Nevanlinna Theory and Com p lex Differential Equations[M].New York:Walter de Gruyter,1993.

[7]梁建軍,劉名生.高階亞純系數(shù)非齊次線性微分方程的復(fù)震蕩[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2006,22(4):464-470.

The properties of solu tions of linear differen tial equations with entire coefficients of finite iterated order

ZHANG Xiao-mei,SUN Dao-chun

(School of Mathem atical Sciences,South China Norm al University,Guangzhou 510631,China)

The com p lex linear differential equations with entire coefficients of finite iterated order are investigated.By app lying the theory in an angular domain of Nevalinna and Ahlfors,some results concerning the iterated order of solutions,the iterated convergence exponent of the zeros of solutions and the zeros distribution of solutions in an angular domain are given.

differential equation,iterated order,iterated convergence exponent,angular distribution

O174.52

A

1008-5513(2009)02-0237-07

2007-09-25.

國家自然科學(xué)基金(10471048),廣東省自然科學(xué)基金(04010474).

張曉梅(1981-),博士,研究方向：函數(shù)論.

2000M SC:30D 35