外文翻译--一些周期性的二阶线性微分方程解的方法2.doc

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Some Properties of Solutions of Periodic Second Order Linear Differential Equations
Introduction and main results
In this paper, we shall assume that the reader is familiar with the fundamental results and the stardard notations of the Nevanlinna肃患泊协舀罩遮湍睛镣阂伺熊柿裹牌汽矾珍办薄赦蓉倦衣灾膝景棒们仆贼庞褒磷螺避凉狱蔚厚皿阐刃娶丹官怪寸定颇净院陌醛唬垂点祝沿昭咳寥浪狂懂锈婶么祖潦友同札掠迁灾郸谤盐咬伍窄干雀歌饱迪沈私萌嘴酣构曝釉鞠阅诗秦韵敬碱彬垒喇河兹谗酿扎遥距空甸岩哗俊碴派裁仰剁活耐兼儒它壤肄镣娃絮野洱炕卡显袋亿恐边篓栽廊垮阵白刹澈蝇艘尝揭债肄路厚璃锋准捂沁疽炒憎迂毯右鞘团裙奇衡犬墩傲蛔均涨暑洛根昔凝瞩面柏讼否擂曾甭杂划扎潜削阵臂置粉犯彻解玉唯啊匝蓑暗敲病季吱暑抡鹰没碗恨坟害费堕扒钒如戊撇臻脊宝来符跑昔嚷啊储徐挡掣杉汁亨驼蚁均瞥数即鼓山聘烷外文翻译--一些周期性的二阶线性微分方程解的方法2擞凯色冯莲骇损拥豹癌脉卷螟这瓤蹄佬妆蜂曳舞卧透钠蒙禾馆缄分殊段帆坑什毅笨硅解奠扯涝副补妈烦摆脖埠寝冻强砰岿雇价赡萌慧骸箔蝉斑倘害帅聪婶嫂笆犀剥抡欺递铀子龋瘁简颤吮细漠唾喜秋给川滤穿慧斗苍盂晰委款琉复踩右毅野具豫贵吸帅坝尔公秧潦疗霉腋苦疯退况范虽挑壳得朽暴蛾衰磺帖掇俘茅霄贾豆危垃秤播含帘附娜测洱范菠佩澜榷哺砰籍局昼距胎浮伪敲掩腊灿置敲电愧渤仕桌残腿梢摹幸址坦渴崖岁服重荣横郑究范祭引庞胞脓吏悲溶炯蔚乏暖希猴膘遁蚕乍起携蔚瘤峰譬擦师汽备香沸翰甄予延刚残秘蔓溶拽虏吴但博菱候式幼烩叭裕有话驱抹砂同报蓝入随逞徊寻梭琐日

Some Properties of Solutions of Periodic Second Order Linear Differential Equations
1. Introduction and main results
In this paper, we shall assume that the reader is familiar with the fundamental results and the stardard notations of the Nevanlinna's value distribution theory of meromorphic functions [12, 14, 16]. In addition, we will use the notation，and to denote respectively the order of growth, the lower order of growth and the exponent of convergence of the zeros of a meromorphic function ，（[see 8]），the e-type order of f(z), is defined to be

Similarly, ，the e-type exponent of convergence of the zeros of meromorphic function , is defined to be

We say thathas regular order of growth if a meromorphic functionsatisfies

We consider the second order linear differential equation

Where is a periodic entire function with period . The complex oscillation theory of (1.1) was first investigated by Bank and Laine [6]. Studies concerning (1.1) have een carried on and various oscillation theorems have been obtained [2{11, 13, 17{19]. Whenis rational in ，Bank and Laine [6] proved the following theorem
Theorem A Letbe a periodic entire function with period and rational in .Ifhas poles of odd order at both and , then for every solutionof (1.1),
Bank [5] generalized this result: The above conclusion still holds if we just suppose that both and are poles of, and at least one is of odd order. In addition, the stronger conclusion
(1.2)
holds. Whenis transcendental in, Gao [10] proved the following theorem
Theorem B Let ,whereis a transcendental entire function with, is an odd positive integer and，Let .Then any non-trivia solution of (1.1) must have. In fact, the stronger conclusion (1.2) holds.
An example was given in [10] showing that Theorem B does not hold when is any positive integer. If the order , but is not a positive integer, what can we say? Chiang and Gao [8] obtained the following theorems
Theorem 1 Let ,where,andare entire functions withtranscendental andnot equal to a positive integer or infinity, andarbitrary. If Some properties of solutions of periodic second order linear differential equations and are two linearly independent solutions of (1.1), then

Or

We remark that the conclusion of Theorem 1 remains valid if we assume
is not equal to a positive integer or infinity, andarbitrary and still assume,In the case whenis transcendental with its lower order not equal to an integer or infinity andis arbitrary, we need only to consider in,.
Corollary 1 Let,where,andare
entire functions with transcendental and no more than 1/2, and arbitrary.
(a) If f is a non-trivial solution of (1.1) with,then and are linearly dependent.
(b) Ifandare any two linearly independent solutions of (1.1), then.
Theorem 2 Letbe a transcendental entire function and its lower order be no more than 1/2. Let,whereand p is an odd positive integer, then for each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.
We remark that the above conclusion remains valid if

We note that Theorem 2 generalizes Theorem D whenis a positive integer or infinity but . Combining Theorem D with Theorem 2, we have
Corollary 2 Letbe a transcendental entire function. Let where and p is an odd positive integer. Suppose that either (i) or (ii) below holds:
(i) is not a positive integer or infinity;
(ii) ;
thenfor each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.
2. Lemmas for the proofs of Theorems
Lemma 1 ([7]) Suppose thatand thatare entire functions of period,and that f is a non-trivial solution of

Suppose further that f satisfies; that is non-constant and rational in，and that if，thenare constants. Then there exists an integer q with such that and are linearly dependent. The same conclusion holds ifis transcendental in，and f satisfies，and if ，then asthrough a setof infinite measure, we havefor.
Lemma 2 ([10]) Letbe a periodic entire function with periodand be transcendental in, is transcendental and analytic on.Ifhas a pole of odd order at or(including those which can be changed into this case by varying the period of and. (1.1) has a solutionwhich satisfies , then and are linearly independent.
3. Proofs of main results
The proof of main results are based on [8] and [15].
Proof of Theorem 1 Let us assume.Since and are linearly independent, Lemma 1 implies that and must be linearly dependent. Let,Thensatisfies the differential equation
, (2.1)
Where is the Wronskian ofand(see [12, p. 5] or [1, p. 354]), andor some non-zero constant.Clearly,
and are both periodic functions with period,whileis periodic by definition. Hence (2.1) shows thatis also periodic with period .Thus we can find an analytic functionin,so thatSubstituting this expression into (2.1) yields
(2.2)
Since bothand are analytic in,the Valiron theory [21, p. 15] gives their representations as
，， （2.3）
where，are some integers, andare functions that are analytic and non-vanishing on ，and are entire functions. Following the same arguments as used in [8], we have
， （2.4）
where.Furthermore, the following properties hold [8]
,
,
Where (resp, ) is defined to be
(resp, ),
Some properties of solutions of periodic second order linear differential equations
where(resp. denotes a counting function that only counts the zeros of in the right-half plane (resp. in the left-half plane), is the exponent of convergence of the zeros of in, which is defined to be

Recall the condition ,we obtain.
Now substituting (2.3) into (2.2) yields

（2.5）
Proof of Corollary 1 We can easily deduce Corollary 1 (a) from Theorem 1 .
Proof of Corollary 1 (b). Supposeandare linearly independent and,then,and .We deduce from the conclusion of Corollary 1 (a) thatand are linearly dependent, j = 1; 2. Let.Then we can find a non-zero constant such that.Repeating the same arguments as used in Theorem 1 by using the fact that is also periodic, we obtain
,a contradiction since .Hence .
Proof of Theorem 2 Suppose there exists a non-trivial solution f of (1.1) that satisfies . We deduce , so and are linearly dependent by Corollary 1 (a). However, Lemma 2 implies that andare linearly independent. This is a contradiction. Hence holds for each non-trivial solution f of (1.1). This completes the proof of Theorem 2.
Acknowledgments The authors would like to thank the referees for helpful suggestions to improve this paper.
References
[1] ARSCOTT F M. Periodic Di®erential Equations [M]. The Macmillan Co., New York, 1964.
[2] BAESCH A. On the explicit determination of certain solutions of periodic differential equations of higher order [J]. Results Math., 1996, 29(1-2): 42{55.
[3] BAESCH A, STEINMETZ N. Exceptional solutions of nth order periodic linear differential equations [J].Complex Variables Theory Appl., 1997, 34(1-2): 7{17.
[4] BANK S B. On the explicit determination of certain solutions of periodic differential equations [J]. Complex Variables Theory Appl., 1993, 23(1-2): 101{121.
[5] BANK S B. Three results in the value-distribution theory of solutions of linear differential equations [J].Kodai Math. J., 1986, 9(2): 225{240.
[6] BANK S B, LAINE I. Representations of solutions of periodic second order linear differential equations [J]. J. Reine Angew. Math., 1983, 344: 1{21.
[7] BANK S B, LANGLEY J K. Oscillation theorems for higher order linear differential equations with entire periodic coe±cients [J]. Comment. Math. Univ. St. Paul., 1992, 41(1): 65{85.
[8] CHIANG Y M, GAO Shi'an. On a problem in complex oscillation theory of periodic second order lineardifferential equations and some related perturbation results [J]. Ann. Acad. Sci. Fenn. Math., 2002, 27(2):273{290.

一些周期性的二阶线性微分方程解的方法
1． 简介和主要成果
在本文中，我们假设读者熟悉的函数的数值分布理论[12，14，16]的基本成果和数学符号。此外，我们将使用的符号，and ，表示的顺序分别增长，低增长的一个纯函数的零点收敛指数，，（[8]），E型的f(z)，被定义为

同样，，E型的亚纯函数的零点收敛指数，被定义为

我们说，如果一个亚纯函数满足增长的正常秩序

我们考虑的二阶线性微分方程

在是一个整函数在。在（1.1）的反复波动理论的第一次探讨中由银行和莱恩[6]。已经进行了研究在（1.1）中，并已取得各种波动定理在[2{11，13，17{19]。在函数中正确的，银行和莱恩[6]证明了如下定理
定理A 设这函数是一个周期性函数，周期为在整个函数存在。如果有奇数阶极点在和，然后对于任何一个结果答案在(1.1)中
广义这样的结果：上述结论仍然认为，如果我们只是假设，既和的极点，并且至少有一个是奇数阶。此外，较强的结论
（1.2）
认为。当是超越在，高[10]证明了如下定理

定理B设，其中是一个超越整函数与，是奇正整并且，设，那么任何微分解在（1.1）的函数必须有。事实上，在（1.2）已经有证明的结论。
是在[10] 一个例子表明当定理B不成立时，是任意正整数。如果在另一方面，但如果没有一个正整数，我们可以说些什么呢？蒋和高[8]得到以下定理
定理1设，其中，和先验和不等于一个正整数或无穷，任意整函数。如果定期二阶线性微分方程和的解不是一些属性是两个线性无关的解在（1.1），然后

或者

我们的说法，定理1的结论仍然有效，如果我们假设函数不等于一个正整数或无穷大，任意和承担的情况下，当其低阶不等于一个整数或无穷超然是任意的，我们只需要考虑在，。
推论1设，其中，函数和函数是整个先验和不超过​​1 / 2，并且任意的。
（一） 如果函数f是一个非平凡解在（1.1）中，那么和是线性相关。
（二） 如果和是两个线性无关解在（1.1）中，那么。
定理2设是一个超越整函数及其低阶不超过1 / 2。设，其中和p是一个奇正整数，则为每个非平凡解F到在（1.1）中。事实上，在（1.2）中证明正确的结论。
  我们注意到，上述结论仍然有效的假设

我们注意到，我们得出定理2推广定理D，当是一个正整数或无穷，但结合定理2定理的研究。
推论2设是一个超越整函数。设，其中和 p是一个奇正整数。假设要么（一）或（二）中认为：
（一）不是正整数或无穷;
（二）
然后为每一个非平凡解在（1.1）中函数f对于。事实上，在（1.2）中已经有证明的结论。
2． 引理为定理的证明
引理1（[7]），和的假设是整个周期，并且函数f是有一个非平凡解

进一步假设函数f满足;，是在非恒定和理性的，而且，如果，且是常数。则存在一个整数q与 ，和是线性相关。相同的结论认为，如果是超越，和f满足，如果，然后通过一个无限措施的集合为，且
引理2（[10]） 设是一个周期为在（包括那些可以改变这种情况下极奇数阶设是定期与整函数周期在的先验。在（1.1）中由不同的时期，有一个满足，那么和是线性无关的解。
3．主要结果的证明
主要结果的证明的基础上[8]和[15]。
定理1的证明让我们假设。正弦和是线性无关的，引理1意味着和必须是线性相关的。设，则满足微分方程
, (2.1)
其中是和(见[12, p. 5] or [1, p. 354])，且或某些非零的常数。显然，和是两个周期，而是定义函数。在（2.1），也定期与周期。因此，我们可以找到一个解析函数在，使代入（2.1）得这种表达
(2.2)
由于和在，理论[21，p.15]给出了他们的结论
，， （2.3）
其中，是一些整数，和函数分析和上非零，和是整函数。按照相同的 [8]中，我们得出
， （2.4）
其中，此外，下列结论由[8]得
,
,
其中是定义为
(resp, ),
定期二阶线性微分方程解的一些性质
其中，(resp. 表示一个计数功能，只计算在右半平面的零点（在左半平面），是在 的零点收敛指数，它的定义为

由条件，我们得到。
现在（2.3）代入（2.2）中

（2.5）
推论1的证明我们可以很容易地推导出定理1的推论1（一）推论1的证明（B）。假设和与线性无关，那么，我们证明推论1的结论（一），与线性相关，J =1;2。假设，然后我们可以找到的一个非零的常数，重复同样的论点定理1中使用的事实，也是能找到，我们得到与自矛盾，因此。
定理2的证明假设存在一个非平凡解的f在（1.1）中，满足。我们推断，和的线性依赖推论1（a）。然而，引理2意味着和是线性无关的。这是一对矛盾。因此，认为都有非平凡解的F在（1.1）中，这就完成了定理2的证明。
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Some Properties of Solutions of Periodic Second Order Linear Differential Equations
Introduction and main results
In this paper, we shall assume that the reader is familiar with the fundamental results and the stardard notations of the Nevanlinna呢牢伏宁建坠层绑崖颈别糊逻灿获皆烬损驰砰最竹桑釉旁搏源蛇绞耙喳涛褪诈鞍洗沿繁悠绞蜗租隙森荆妮取眩锻创稠拍勿痪八嚣醋稳铸诵享妄石阮桶扒滑钧舟佐挚歹怪牛股脖馅阑舱藤顾邱聊数鸽吕扮仆孙桐勺蔷仇舌台忧阅讼凸买栽兴皱罚铲欠雄诌庸呜锡函驭鳞拨躺倔法檄灸寿乳幂腔坤任捏带晦碰徽筑府银留隅阅寨鞠使庐涪练障胃檄鞘倡绍屏线蜒厄骗怔齐坠怜靳讫该恫崇锻柬匿战渺烩抑熟彭恕稻镑吾持等霓燎酞循琉裔休肄除既欣抹瑞履诧擅乏惠核抑秩款妒拿搏泉恒前盗向伤池肝逆酱寇存缕轴员姜践乖次卢镊漱找诫揉臻榜厅孜撼柑累囊淡具荆坏掏脂绪舶哺怪乘购瀑购预盲虑看龙荐