二阶线性微分方程英文翻译

1、Some Properties of Solutions of Periodic Second Order Linear Differential Equations1. Introduction and main resultsIn 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 funct

2、ions 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 o

3、f the zeros of meromorphic function , is defined to beWe say thathas regular order of growth if a meromorphic functionsatisfiesWe consider the second order linear differential equationWhere is a periodic entire function with period . The complex oscillation theory of (1.1) was first investigated by

4、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 proved the following theoremTheorem A Letbe a periodic entire function with period and rational in .Ifhas poles of odd order at both

5、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 theoremTheor

6、em 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

7、is not a positive integer, what can we say? Chiang and Gao 8 obtained the following theoremsTheorem C Let ,where,andare entire functionstranscendental andnot equal to a positive integer or infinity, andarbitrary.(i) Suppose . (a) If f is a non-trivial solution of (1.1) with; thenandare linearly depe

8、ndent. (b) Ifandare any two linearly independent solutions of (1.1), then .(ii) Suppose (a) If f is a non-trivial solution of (1.1) with,andare linearly dependent. Ifandare any two linearly independent solutions of (1.1),then.Theorem D Letbe a transcendental entire function and its order be not a po

9、sitive integer or infinity. Let; whereand p is an odd positive integer. Thenor each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.Examples were also given in 8 showing that Theorem D is no longer valid whenis infinity.The main purpose of this paper is to improve above

10、 results in the case whenis transcendental. Specially, we find a condition under which Theorem D still holds in the case when is a positive integer or infinity. We will prove the following results in Section 3.Theorem 1 Let ,where,andare entire functions withtranscendental andnot equal to a positive

11、 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 or infinity, a

12、ndarbitrary 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-trivial solutio

13、n 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 fa

14、ct, 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 and p is an o

15、dd 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 functions of peri

16、od,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 if ，then asth

17、rough 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

18、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, (2.1)Where is

19、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)

20、 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 fo

21、llowing properties hold 8,Where (resp, ) is defined to be(resp, ),Some properties of solutions of periodic second order linear differential equationswhere(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 conver

22、gence of the zeros of in, which is defined to beRecall 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 conclu

23、sion 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-trivia

24、l 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 aut

25、hors would like to thank the referees for helpful suggestions to improve this paper.References1 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. R

26、esults Math., 1996, 29(1-2): 4255.3 BAESCH A, STEINMETZ N. Exceptional solutions of nth order periodic linear differential equations J.Complex Variables Theory Appl., 1997, 34(1-2): 717.4 BANK S B. On the explicit determination of certain solutions of periodic differential equations J. Complex Varia

27、bles Theory Appl., 1993, 23(1-2): 101121.5 BANK S B. Three results in the value-distribution theory of solutions of linear differential equations J.Kodai Math. J., 1986, 9(2): 225240.6 BANK S B, LAINE I. Representations of solutions of periodic second order linear differential equations J. J. Reine

28、Angew. Math., 1983, 344: 121.7 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): 6585.8 CHIANG Y M, GAO Shi'an. On a problem in complex oscillation theory of periodic sec

29、ond order lineardifferential equations and some related perturbation results J. Ann. Acad. Sci. Fenn. Math., 2002, 27(2):273290.一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

30、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211，13，1719。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）

31、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C 设，其中，函数和函数是整函数先验和不等于一个正整数或无穷大，并函数任意。（一） 假设（a）如果函数f是一个非平凡解在（1.1），那么和是线性相关。（b）如果函数和函数在（1.1）是两个线性无关函数，则存在这样一个条件。（二） 假设（a）如果函数f有一个非平凡解在（1.1）且，和是线性相关的。 如果函数和函数在（1.1）在（1.1）是两个线性无关函数，则存在这样一个条件。定理 D 让是一个超越整函数和它的秩序是正整数或无穷大。设，和p是一个

32、奇正整数。然后或F得到每一个非平凡解在（1.1）。事实上，在（1.2）中已经有证明的结论。例子表明在高8定理D不再成立，当是无穷的。本文的主要目的是改善上述结果的情况下，当是超越。特别地，我们找到的条件下定理D仍然成立的情况下，当是一个正整数或无穷大。我们将证明在第3节的结果如下：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