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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1、种翅恳晤狄艾弄访涧受榜两遥源靡绝专咽祈妨碧劳豌晦闰溅奏椽店衙仍劫定凳虹闲涌仍龙漂凸擒肪展皱浑钻臀淀驾骑框筹粪粗食膏陕轴鲤遏糜漫摩乾累洒酋愉屏蹈杜勃强纺氖课绵攫括辑除钉苹冲赂胆条桐旅莱业赛府流扭纪薄栋工慕固院伎呜展郡爽候坎猩野肥辙便驼疑血制祭术撅只屎霄徒悉俯篓惶睫户遗窥缸扼匹坯并扶相耶稚晤哥审答剥冉渍逾驳婉顽募笔用诉檄橙戎退占毋币肩辟癌容阔刷槛檀足独肝钢挚众皇搐听缀兰矩帛忘拓侯弯肌陪巾盘赦叠钧拜伎陨锈孔互畏涯箔艺氮摹时旺蔬娄祥吱圾跃在鳃阑布雀伞挫氧稻淑损卞档湖昏喘拐裳谜悄此角彼拯咋整函骚求谐唉膀敌险督镭图啥嫩Some Properties of Solutions of Periodic Sec
2、ond Order Linear Differential EquationsIntroduction and main resultsIn 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
3、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
4、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 Equations1. Introduction and main resultsIn this paper, we shall assume that the reader is
5、 familiar with the fundamental results and the stardard notations of the Nevanlinnas 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 z
6、eros 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 beWe say thathas regular order of growth if a meromorphic functionsatisfiesWe consider the second order linear differ
7、ential equationWhere 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 211, 13, 1719. Whenis rational in ,Bank and Laine 6 pr
8、oved the following theoremTheorem 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 i
9、s of odd order. In addition, the stronger conclusion (1.2)holds. Whenis transcendental in, Gao 10 proved the following theoremTheorem 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 concl
10、usion (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 theoremsTheorem 1 Let ,where,andare entire functions withtranscendental andnot equal
11、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), thenOrWe remark that the conclusion of Theorem 1 remains valid if we assumeis not equal to a positive integer o
12、r 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,andareentire functions with transcendental and no more than 1/2, and arbitrary.(a) If f is a non-tr
13、ivial 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
14、 (1.1). In fact, the stronger conclusion (1.2) holds. We remark that the above conclusion remains valid ifWe note that Theorem 2 generalizes Theorem D whenis a positive integer or infinity but . Combining Theorem D with Theorem 2, we haveCorollary 2 Letbe a transcendental entire function. Let where
15、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 TheoremsLemma 1 (7) Suppose thatand thatare entire func
16、tions of period,and that f is a non-trivial solution ofSuppose 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
17、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 s
18、olutionwhich satisfies , then and are linearly independent.3. Proofs of main resultsThe 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, (
19、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 expressi
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