外文翻譯--一些周期性的二階線性微分方程解的方法

1、<p><b>　　濱州學(xué)院</b></p><p><b>　　外 文 翻 譯</b></p><p>　　學(xué)號(hào)：2009010477 </p><p>　　姓名：曲洋 </p><p>　　Some Properties of Solutions of Perio

2、dic Second Order Linear Differential Equations</p><p>　　Introduction and main results</p><p>　　In this paper, we shall assume that the reader is familiar with the fundamental results and the sta

3、rdard 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

4、 the exponent of convergence of the zeros of a meromorphic function ，（[see 8]），the e-type order of f(z), is defined to be </p><p>　　Similarly, ，the e-type exponent of convergence of the zeros of meromorphic

5、function , is defined to be</p><p>　　We say thathas regular order of growth if a meromorphic functionsatisfies</p><p>　　We consider the second order linear differential equation</p><p

6、>　　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

7、have been obtained [2{11, 13, 17{19]. Whenis rational in ，Bank and Laine [6] proved the following theorem</p><p>　　Theorem A Letbe a periodic entire function with period and rational in .Ifhas poles of odd o

8、rder at both and , then for every solutionof (1.1), </p><p>　　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 orde

9、r. In addition, the stronger conclusion</p><p><b>　　(1.2)</b></p><p>　　holds. Whenis transcendental in, Gao [10] proved the following theorem</p><p>　　Theorem B Let ,whe

10、reis 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.</p><p>　　An example was given i

11、n [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</p><p>　　Theorem 1

12、 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 line

13、arly independent solutions of (1.1), then</p><p><b>　　Or</b></p><p>　　We remark that the conclusion of Theorem 1 remains valid if we assume</p><p>　　is not equal to a po

14、sitive 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,.</p><p>　　Co

15、rollary 1 Let,where,andare</p><p>　　entire functions with transcendental and no more than 1/2, and arbitrary.</p><p>　　If f is a non-trivial solution of (1.1) with,then and are linearly depend

16、ent.</p><p>　　Ifandare any two linearly independent solutions of (1.1), then.</p><p>　　Theorem 2 Letbe a transcendental entire function and its lower order be no more than 1/2. Let,whereand p is

17、 an odd positive integer, then for each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.</p><p>　　We remark that the above conclusion remains valid if</p><p>　　We n

18、ote that Theorem 2 generalizes Theorem D whenis a positive integer or infinity but . Combining Theorem D with Theorem 2, we have</p><p>　　Corollary 2 Letbe a transcendental entire function. Let where and p i

19、s an odd positive integer. Suppose that either (i) or (ii) below holds:</p><p>　　(i) is not a positive integer or infinity;</p><p><b>　　(ii) ;</b></p><p>　　thenfor each

20、 non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.</p><p>　　Lemmas for the proofs of Theorems</p><p>　　Lemma 1 ([7]) Suppose thatand thatare entire functions of peri

21、od,and that f is a non-trivial solution of</p><p>　　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 ar

22、e linearly dependent. The same conclusion holds ifis transcendental in，and f satisfies，and if ，then asthrough a setof infinite measure, we havefor.</p><p>　　Lemma 2 ([10]) Letbe a periodic entire function wi

23、th 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

24、, then and are linearly independent.</p><p>　　Proofs of main results</p><p>　　The proof of main results are based on [8] and [15].</p><p>　　Proof of Theorem 1 Let us assume.Since an

25、d are linearly independent, Lemma 1 implies that and must be linearly dependent. Let,Thensatisfies the differential equation</p><p>　　, (2.1)</p><p>　　Where is the Wronskian ofand(see

26、[12, p. 5] or [1, p. 354]), andor some non-zero constant.Clearly, </p><p>　　and are both periodic functions with period,whileis periodic by definition. Hence (2.1) shows thatis also periodic with period .Thu

27、s we can find an analytic functionin,so thatSubstituting this expression into (2.1) yields</p><p><b>　　(2.2)</b></p><p>　　Since bothand are analytic in,the Valiron theory [21, p. 15]

28、 gives their representations as</p><p>　　，， （2.3）</p><p>　　where，are some integers, andare functions that are analytic and non-vanishing on ，and are entire functions. Following the same a

29、rguments as used in [8], we have</p><p>　　， （2.4）</p><p>　　where.Furthermore, the following properties hold [8]</p><p><b>　　,</b></p><p><b>

30、　　,</b></p><p>　　Where (resp, ) is defined to be</p><p><b>　　(resp, ),</b></p><p>　　Some properties of solutions of periodic second order linear differential equat

31、ions</p><p>　　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 define

32、d to be</p><p>　　Recall the condition ,we obtain.</p><p>　　Now substituting (2.3) into (2.2) yields</p><p><b>　　（2.5）</b></p><p>　　Proof of Corollary 1 We c

33、an easily deduce Corollary 1 (a) from Theorem 1 .</p><p>　　Proof of Corollary 1 (b). Supposeandare linearly independent and,then,and .We deduce from the conclusion of Corollary 1 (a) thatand are linearly dep

34、endent, 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</p><p>　　,a contradiction since .Henc

35、e .</p><p>　　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 lin

36、early independent. This is a contradiction. Hence holds for each non-trivial solution f of (1.1). This completes the proof of Theorem 2.</p><p>　　Acknowledgments The authors would like to thank the referees

37、for helpful suggestions to improve this paper.</p><p>　　References</p><p>　　[1] ARSCOTT F M. Periodic Di®erential Equations [M]. The Macmillan Co., New York, 1964.</p><p>　　[2]

38、 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.</p><p>　　[3] BAESCH A, STEINMETZ N. Exceptional solu

39、tions of nth order periodic linear differential equations [J].Complex Variables Theory Appl., 1997, 34(1-2): 7{17.</p><p>　　[4] BANK S B. On the explicit determination of certain solutions of periodic differ

40、ential equations [J]. Complex Variables Theory Appl., 1993, 23(1-2): 101{121.</p><p>　　[5] BANK S B. Three results in the value-distribution theory of solutions of linear differential equations [J].Kodai Mat

41、h. J., 1986, 9(2): 225{240.</p><p>　　[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.</p><p&g

42、t;　　[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.</p><p>　　[8] C

43、HIANG 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.<

44、/p><p>　　一些周期性的二階線性微分方程解的方法</p><p><b>　　簡(jiǎn)介和主要成果</b></p><p>　　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<

45、/p><p>　　同樣，，E型的亞純函數(shù)的零點(diǎn)收斂指數(shù)，被定義為</p><p>　　我們說(shuō)，如果一個(gè)亞純函數(shù)滿足增長(zhǎng)的正常秩序</p><p>　　我們考慮的二階線性微分方程</p><p>　　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[2{11，13，1

46、7{19]。在函數(shù)中正確的，銀行和萊恩[6]證明了如下定理</p><p>　　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</p><p>　　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</p><p><b>　　（1.

47、2）</b></p><p>　　認(rèn)為。當(dāng)是超越在，高[10]證明了如下定理</p><p>　　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）已經(jīng)有證明的結(jié)論。</p><p>　　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)

48、些什么呢？蔣和高[8]得到以下定理</p><p>　　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</p><p><b>　　或者</b></p><p>　　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

49、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，。</p><p>　　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。</p><p>　　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)。</p><p>　　如果和是兩個(gè)線性無(wú)關(guān)解在（1.1）中，那么。</p><p>　　定理2設(shè)是一

50、個(gè)超越整函數(shù)及其低階不超過(guò)1 / 2。設(shè)，其中和p是一個(gè)奇正整數(shù)，則為每個(gè)非平凡解F到在（1.1）中。事實(shí)上，在（1.2）中證明正確的結(jié)論。  我們注意到，上述結(jié)論仍然有效的假設(shè)</p><p>　　我們注意到，我們得出定理2推廣定理D，當(dāng)是一個(gè)正整數(shù)或無(wú)窮，但結(jié)合定理2定理的研究。</p><p>　　推論2設(shè)是一個(gè)超越整函數(shù)。設(shè)，其中和 p是一個(gè)奇正整數(shù)。假設(shè)要么

51、（一）或（二）中認(rèn)為：</p><p>　?。ㄒ唬┎皇钦麛?shù)或無(wú)窮;</p><p><b>　?。ǘ?lt;/b></p><p>　　然后為每一個(gè)非平凡解在（1.1）中函數(shù)f對(duì)于。事實(shí)上，在（1.2）中已經(jīng)有證明的結(jié)論。</p><p><b>　　引理為定理的證明</b></p>&

52、lt;p>　　引理1（[7]），和的假設(shè)是整個(gè)周期，并且函數(shù)f是有一個(gè)非平凡解</p><p>　　進(jìn)一步假設(shè)函數(shù)f滿足;，是在非恒定和理性的，而且，如果，且是常數(shù)。則存在一個(gè)整數(shù)q與 ，和是線性相關(guān)。相同的結(jié)論認(rèn)為，如果是超越，和f滿足，如果，然后通過(guò)一個(gè)無(wú)限措施的集合為，且</p><p>　　引理2（[10]） 設(shè)是一個(gè)周期為在（包括那些可以改變這種情況下極奇數(shù)階設(shè)是定期與整函

53、數(shù)周期在的先驗(yàn)。在（1.1）中由不同的時(shí)期，有一個(gè)滿足，那么和是線性無(wú)關(guān)的解。</p><p><b>　　3．主要結(jié)果的證明</b></p><p>　　主要結(jié)果的證明的基礎(chǔ)上[8]和[15]。</p><p>　　定理1的證明讓我們假設(shè)。正弦和是線性無(wú)關(guān)的，引理1意味著和必須是線性相關(guān)的。設(shè)，則滿足微分方程</p><p

54、>　　, (2.1)</p><p>　　其中是和(見(jiàn)[12, p. 5] or [1, p. 354])，且或某些非零的常數(shù)。顯然，和是兩個(gè)周期，而是定義函數(shù)。在（2.1），也定期與周期。因此，我們可以找到一個(gè)解析函數(shù)在，使代入（2.1）得這種表達(dá)</p><p><b>　　(2.2)</b></p><p>　　

55、由于和在，理論[21，p.15]給出了他們的結(jié)論</p><p>　　，， （2.3）</p><p>　　其中，是一些整數(shù)，和函數(shù)分析和上非零，和是整函數(shù)。按照相同的 [8]中，我們得出</p><p>　　， （2.4）</p><p>　　其中，此外，下列結(jié)論由[8]得</p><p

56、><b>　　,</b></p><p><b>　　,</b></p><p><b>　　其中是定義為</b></p><p><b>　　(resp, ),</b></p><p>　　定期二階線性微分方程解的一些性質(zhì)</p>&

57、lt;p>　　其中，(resp. 表示一個(gè)計(jì)數(shù)功能，只計(jì)算在右半平面的零點(diǎn)（在左半平面），是在 的零點(diǎn)收斂指數(shù)，它的定義為</p><p><b>　　由條件，我們得到。</b></p><p>　　現(xiàn)在（2.3）代入（2.2）中</p><p><b>　　（2.5）</b></p><p>

58、;　　推論1的證明我們可以很容易地推導(dǎo)出定理1的推論1（一）推論1的證明（B）。假設(shè)和與線性無(wú)關(guān)，那么，我們證明推論1的結(jié)論（一），與線性相關(guān)，J =1;2。假設(shè)，然后我們可以找到的一個(gè)非零的常數(shù)，重復(fù)同樣的論點(diǎn)定理1中使用的事實(shí)，也是能找到，我們得到與自矛盾，因此。</p><p>　　定理2的證明假設(shè)存在一個(gè)非平凡解的f在（1.1）中，滿足。我們推斷，和的線性依賴推論1（a）。然而，引理2意味著和是線性無(wú)關(guān)的