數(shù)學專業(yè)英語論文含中文版

1、1. 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 functions 12, 14, 16. In addition, we will use the notation，and to denote respectively t

2、he 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 beWe say thathas regular order

3、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 Bank and Laine 6. Studies concerning (1.1) have een carried on and various oscillat

4、ion 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 and , then for every solutionof (1.1), Bank 5 generalized this result: The above co

5、nclusion 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 theoremTheorem B Let ,whereis a transcendental entire function with, is an odd positive integer

6、 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

7、 theoremsTheorem 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), thenOrWe remark that the

8、conclusion of Theorem 1 remains valid if we assumeis 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,andareentire

9、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

10、/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 ifWe note that Theorem 2 generalizes Theorem D whenis a positive integer or infinity but . Combining Theorem D

11、 with Theorem 2, we haveCorollary 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) hol

12、ds.2. Lemmas for the proofs of TheoremsLemma 1 (7) Suppose thatand thatare entire functions 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 lin

13、early 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

14、 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 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 impli

15、es 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

16、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 en

17、tire 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 equationswhere(resp. denotes a counting function that only cou

18、nts 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 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 .Pr

19、oof 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 peri

20、odic, 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 e

21、ach 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.References1 ARSCOTT F M. Periodic Di®erential Equations M. The Macmillan Co., New York, 1964.2 BAESCH A. On the e

22、xplicit determination of certain solutions of periodic differential equations of higher order J. Results 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 t

23、he explicit determination of certain solutions of periodic differential equations J. Complex Variables 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

24、. Representations of solutions of periodic second order linear differential equations J. J. Reine 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

25、(1): 6585.8 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):273290.一些周期性的二階線性微分方程解的方法1 簡介和主要成果在本文中，我們假設讀者熟悉的函數(shù)的數(shù)值分布理論12，14，16的基本成果和數(shù)學符號。此外，我們

26、將使用的符號，and ，表示的順序分別增長，低增長的一個純函數(shù)的零點收斂指數(shù)，（8），E型的f(z)，被定義為同樣，E型的亞純函數(shù)的零點收斂指數(shù)，被定義為我們說，如果一個亞純函數(shù)滿足增長的正常秩序我們考慮的二階線性微分方程在是一個整函數(shù)在。在（1.1）的反復波動理論的第一次探討中由銀行和萊恩6。已經進行了研究在（1.1）中，并已取得各種波動定理在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設這函數(shù)是一個周期性函數(shù)，周期為在整個函數(shù)存在。如果有奇數(shù)階極點在和，然后對于任何一個結果答案在(1.1)中廣義這樣的結果：上述結論仍然認為，如果我們只是假設，既和的極點，并且至少

27、有一個是奇數(shù)階。此外，較強的結論 （1.2）認為。當是超越在，高10證明了如下定理定理B設，其中是一個超越整函數(shù)與，是奇正整并且，設，那么任何微分解在（1.1）的函數(shù)必須有。事實上，在（1.2）已經有證明的結論。是在10一個例子表明當定理B不成立時，是任意正整數(shù)。如果在另一方面，但如果沒有一個正整數(shù)，我們可以說些什么呢？蔣和高8得到以下定理定理1設，其中，和先驗和不等于一個正整數(shù)或無窮，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線性無關的解在（1.1），然后或者我們的說法，定理1的結論仍然有效，如果我們假設函數(shù)不等于一個正整數(shù)或無窮大，任意和承擔的情況下，當其低階不等于一個整

28、數(shù)或無窮超然是任意的，我們只需要考慮在，。推論1設，其中，函數(shù)和函數(shù)是整個先驗和不超過1 / 2，并且任意的。（一） 如果函數(shù)f是一個非平凡解在（1.1）中，那么和是線性相關。（二） 如果和是兩個線性無關解在（1.1）中，那么。定理2設是一個超越整函數(shù)及其低階不超過1 / 2。設，其中和p是一個奇正整數(shù)，則為每個非平凡解F到在（1.1）中。事實上，在（1.2）中證明正確的結論。  我們注意到，上述結論仍然有效的假設我們注意到，我們得出定理2推廣定理D，當是一個正整數(shù)或無窮，但結合定理2定理的研究。推論2設是一個超越整函數(shù)。設，其中和p是一個奇正整數(shù)。假設要么（一）或（二）中

29、認為：（一）不是正整數(shù)或無窮;（二）然后為每一個非平凡解在（1.1）中函數(shù)f對于。事實上，在（1.2）中已經有證明的結論。2 引理為定理的證明引理1（7），和的假設是整個周期，并且函數(shù)f是有一個非平凡解進一步假設函數(shù)f滿足;，是在非恒定和理性的，而且，如果，且是常數(shù)。則存在一個整數(shù)q與 ，和是線性相關。相同的結論認為，如果是超越，和f滿足，如果，然后通過一個無限措施的集合為，且引理2（10）設是一個周期為在（包括那些可以改變這種情況下極奇數(shù)階設是定期與整函數(shù)周期在的先驗。在（1.1）中由不同的時期，有一個滿足，那么和是線性無關的解。3主要結果的證明主要結果的證明的基礎上8和15。定理1的證明讓

30、我們假設。正弦和是線性無關的，引理1意味著和必須是線性相關的。設，則滿足微分方程, (2.1)其中是和(見12, p. 5 or 1, p. 354)，且或某些非零的常數(shù)。顯然，和是兩個周期，而是定義函數(shù)。在（2.1），也定期與周期。因此，我們可以找到一個解析函數(shù)在，使代入（2.1）得這種表達 (2.2)由于和在，理論21，p.15給出了他們的結論， （2.3）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， （2.4）其中，此外，下列結論由8得,其中是定義為(resp,),定期二階線性微分方程解的一些性質其中，(resp. 表示一個計數(shù)功能，只計算在右半平面的零

31、點（在左半平面），是在的零點收斂指數(shù)，它的定義為由條件，我們得到?，F(xiàn)在（2.3）代入（2.2）中 （2.5）推論1的證明我們可以很容易地推導出定理1的推論1（一）推論1的證明（B）。假設和與線性無關，那么，我們證明推論1的結論（一），與線性相關，J =1;2。假設，然后我們可以找到的一個非零的常數(shù)，重復同樣的論點定理1中使用的事實，也是能找到，我們得到與自矛盾，因此。定理2的證明假設存在一個非平凡解的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意味著和是線性無關的。這是一對矛盾。因此，認為都有非平凡解的F在（1.1）中，這就完成了定理2的證明。CONTROLLABI

32、LITY OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYAbstract In this article, we give sucient conditions for controllability of some partial neutral functional dierential equations with innite delay. We suppose that the linear part is not necessarily densely dened but satises the re

33、solvent estimates of the Hille-Yosida theorem. The results are obtained using the integrated semigroups theory. An application is given to illustrate our abstract result.Key words Controllability; integrated semigroup; integral solution; innity delay1 IntroductionIn this article, we establish a resu

34、lt about controllability to the following class of partial neutral functional dierential equations with innite delay: (1)where the state variabletakes values in a Banach spaceand the control is given in ,the Banach space of admissible control functions with U a Banach space. C is a bounded linear op

35、erator from U into E, A : D(A) E E is a linear operator on E, B is the phase space of functions mapping (, 0 into E, which will be specied later, D is a bounded linear operator from B into E dened byis a bounded linear operator from B into E and for each x : (, T E, T > 0, and t 0, T , xt represe

36、nts, as usual, the mapping from (, 0 into E dened byF is an E-valued nonlinear continuous mapping on.The problem of controllability of linear and nonlinear systems represented by ODE in nit dimensional space was extensively studied. Many authors extended the controllability concept to innite dimensi

37、onal systems in Banach space with unbounded operators. Up to now, there are a lot of works on this topic, see, for example, 4, 7, 10, 21. There are many systems that can be written as abstract neutral evolution equations with innite delay to study 23. In recent years, the theory of neutral functiona

38、l dierential equations with innite delay in innite dimension was developed and it is still a eld of research (see, for instance, 2, 9, 14, 15 and the references therein). Meanwhile, the controllability problem of such systems was also discussed by many mathematicians, see, for example, 5, 8. The obj

39、ective of this article is to discuss the controllability for Eq. (1), where the linear part is supposed to be non-densely dened but satises the resolvent estimates of the Hille-Yosida theorem. We shall assume conditions that assure global existence and give the sucient conditions for controllability

40、 of some partial neutral functional dierential equations with innite delay. The results are obtained using the integrated semigroups theory and Banach xed point theorem. Besides, we make use of the notion of integral solution and we do not use the analytic semigroups theory.Treating equations with i

41、nnite delay such as Eq. (1), we need to introduce the phase space B. To avoid repetitions and understand the interesting properties of the phase space, suppose that is a (semi)normed abstract linear space of functions mapping (, 0 into E, and satises the following fundamental axioms that were rst in

42、troduced in 13 and widely discussed in 16.(A) There exist a positive constant H and functions K(.), M(.):,with K continuous and M locally bounded, such that, for any and ,if x : (, + a E, and is continuous on , +a, then, for every t in , +a, the following conditions hold:(i) ,(ii) ,which is equivale

43、nt to or every(iii) (A) For the function in (A), t xt is a B-valued continuous function for t in , + a.(B) The space B is complete. Throughout this article, we also assume that the operator A satises the Hille-Yosida condition :(H1) There exist and ，such that and （2）Let A0 be the part of operator A

44、in dened byIt is well known that and the operator generates a strongly continuous semigroup on .Recall that 19 for all and ,one has and .We also recall that coincides on with the derivative of the locally Lipschitz integrated semigroup generated by A on E, which is, according to 3, 17, 18, a family

45、of bounded linear operators on E, that satises(i) S(0) = 0,(ii) for any y E, t S(t)y is strongly continuous with values in E,(iii) for all t, s 0, and for any > 0 there exists a constant l() > 0, such that or all t, s 0, .The C0-semigroup is exponentially bounded, that is, there exist two cons

46、tants and ,such that for all t 0. Notice that the controllability of a class of non-densely dened functional dierential equations was studied in 12 in the nite delay case.2 Main Results We start with introducing the following denition.Denition 1 Let T > 0 and B. We consider the following denition

47、.We say that a function x := x(., ) : (, T ) E, 0 < T +, is an integral solution of Eq. (1) if(i) x is continuous on 0, T ) ,(ii) for t 0, T ) ,(iii) for t 0, T ) ,(iv) for all t (, 0.We deduce from 1 and 22 that integral solutions of Eq. (1) are given for B, such that by the following system (3)

48、Where.To obtain global existence and uniqueness, we supposed as in 1 that(H2).(H3)is continuous and there exists > 0, such thatfor 1, 2 B and t 0. (4)Using Theorem 7 in 1, we obtain the following result.Theorem 1Assume that (H1), (H2), and (H3) hold. Let B such that D D(A). Then, there exists a u

49、nique integral solution x(., ) of Eq. (1), dened on (,+) .Denition 2Under the above conditions, Eq. (1) is said to be controllable on theinterval J = 0, , > 0, if for every initial function B with D D(A) and for anye1 D(A), there exists a control u L2(J,U), such that the solution x(.) of Eq. (1)

50、satises.Theorem 2Suppose that(H1), (H2), and (H3) hold. Let x(.) be the integral solution ofEq. (1) on (, ) , > 0, and assume that (see 20) the linear operator W from U into D(A)dened by， （5）nduces an invertible operatoron ，such that there exist positive constantsand satisfyingand ，then, Eq. (1)

51、is controllable on J providedthat， （6）Where.ProofFollowing 1, when the integral solution x(.) of Eq. (1) exists on (, ) , > 0, it is given for all t 0, byOr Then, an arbitrary integral solution x(.) of Eq. (1) on (, ) , > 0, satises x() = e1 if andonly ifThis implies that, by use of (5), it su

52、ces to take, for all t J,in order to have x() = e1. Hence, we must take the control as above, and consequently, theproof is reduced to the existence of the integral solution given for all t 0, byWithout loss of generality, suppose that 0. Using similar arguments as in 1, we can seehat, for every,and

53、 t 0, ,As K is continuous and,we can choose > 0 small enough, such that.Then, P is a strict contraction in,and the xed point of P gives the unique integralolution x(., ) on (, that veries x() = e1.Remark 1Suppose that all linear operators W from U into D(A) dened by0 a < b T, T > 0, induce

54、invertible operatorson,such that thereexist positive constants N1 and N2 satisfying and ,taking,N large enough and following 1. A similar argument as the above proof can be used inductivelyin,to see that Eq. (1) is controllable on 0, T for all T > 0.AcknowledgementsThe authors would like to thank

55、 Prof. Khalil Ezzinbi and Prof.Pierre Magal for the fruitful discussions.References1 Adimy M, Bouzahir H, Ezzinbi K. Existence and stability for some partial neutral functional dierentialequations with innite delay. J Math Anal Appl, 2004, 294: 4384612 Adimy M, Ezzinbi K. A class of linear partial n

56、eutral functional dierential equations with nondensedomain. J Dif Eq, 1998, 147: 2853323 Arendt W. Resolvent positive operators and integrated semigroups. Proc London Math Soc, 1987, 54(3):3213494 Atmania R, Mazouzi S. Controllability of semilinear integrodierential equations with nonlocal conditions.Electronic J of Di Eq, 2005, 2005: 195 Balachandran K, Anandhi E R. Controllability of neutral integrodierential innite delay systems in Banach spaces. Taiwane