二階線性微分方程英文翻譯

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):273290.一些周期性的二階線性微分方程解的方法1 簡(jiǎn)介和主要成果在本文中，我們假設(shè)讀者熟悉的函數(shù)的數(shù)值分布理論12，14，16的基本成果和數(shù)學(xué)符號(hào)。此外，我們將使用的符號(hào)，and ，表示的順序分別增長(zhǎng)，低增長(zhǎng)的一個(gè)純函數(shù)的零點(diǎn)收斂指數(shù)，（8），E型的f(z)，被定義為同樣，E型的亞純函數(shù)的零點(diǎn)收斂指數(shù)，被定義為我們說(shuō)，如果一個(gè)亞純函數(shù)滿足增長(zhǎng)的正常秩序

30、我們考慮的二階線性微分方程在是一個(gè)整函數(shù)在。在（1.1）的反復(fù)波動(dòng)理論的第一次探討中由銀行和萊恩6。已經(jīng)進(jìn)行了研究在（1.1）中，并已取得各種波動(dòng)定理在211，13，1719。在函數(shù)中正確的，銀行和萊恩6證明了如下定理定理A 設(shè)這函數(shù)是一個(gè)周期性函數(shù)，周期為在整個(gè)函數(shù)存在。如果有奇數(shù)階極點(diǎn)在和，然后對(duì)于任何一個(gè)結(jié)果答案在(1.1)中廣義這樣的結(jié)果：上述結(jié)論仍然認(rèn)為，如果我們只是假設(shè)，既和的極點(diǎn)，并且至少有一個(gè)是奇數(shù)階。此外，較強(qiáng)的結(jié)論 （1.2）認(rèn)為。當(dāng)是超越在，高10證明了如下定理定理B設(shè)，其中是一個(gè)超越整函數(shù)與，是奇正整并且，設(shè)，那么任何微分解在（1.1）的函數(shù)必須有。事實(shí)上，在（1.2）

31、已經(jīng)有證明的結(jié)論。是在10 一個(gè)例子表明當(dāng)定理B不成立時(shí)，是任意正整數(shù)。如果在另一方面，但如果沒(méi)有一個(gè)正整數(shù)，我們可以說(shuō)些什么呢？蔣和高8得到以下定理定理C 設(shè)，其中，函數(shù)和函數(shù)是整函數(shù)先驗(yàn)和不等于一個(gè)正整數(shù)或無(wú)窮大，并函數(shù)任意。（一） 假設(shè)（a）如果函數(shù)f是一個(gè)非平凡解在（1.1），那么和是線性相關(guān)。（b）如果函數(shù)和函數(shù)在（1.1）是兩個(gè)線性無(wú)關(guān)函數(shù)，則存在這樣一個(gè)條件。（二） 假設(shè)（a）如果函數(shù)f有一個(gè)非平凡解在（1.1）且，和是線性相關(guān)的。 如果函數(shù)和函數(shù)在（1.1）在（1.1）是兩個(gè)線性無(wú)關(guān)函數(shù)，則存在這樣一個(gè)條件。定理 D 讓是一個(gè)超越整函數(shù)和它的秩序是正整數(shù)或無(wú)窮大。設(shè)，和p是一個(gè)

32、奇正整數(shù)。然后或F得到每一個(gè)非平凡解在（1.1）。事實(shí)上，在（1.2）中已經(jīng)有證明的結(jié)論。例子表明在高8定理D不再成立，當(dāng)是無(wú)窮的。本文的主要目的是改善上述結(jié)果的情況下，當(dāng)是超越。特別地，我們找到的條件下定理D仍然成立的情況下，當(dāng)是一個(gè)正整數(shù)或無(wú)窮大。我們將證明在第3節(jié)的結(jié)果如下：定理1設(shè)，其中，和先驗(yàn)和不等于一個(gè)正整數(shù)或無(wú)窮，任意整函數(shù)。如果定期二階線性微分方程和的解不是一些屬性是兩個(gè)線性無(wú)關(guān)的解在（1.1），然后或者我們的說(shuō)法，定理1的結(jié)論仍然有效，如果我們假設(shè)函數(shù)不等于一個(gè)正整數(shù)或無(wú)窮大，任意和承擔(dān)的情況下，當(dāng)其低階不等于一個(gè)整數(shù)或無(wú)窮超然是任意的，我們只需要考慮在，。推論1設(shè)，其中，函數(shù)和函數(shù)是整個(gè)先驗(yàn)和不超過(guò)1 / 2，并且任意的。（一） 如果函數(shù)f是一個(gè)非平凡解在（1.1）中，那么和是線性相關(guān)。（二） 如果和是兩個(gè)線性無(wú)關(guān)解在（1.1）中，那么。定理2