外文翻譯二階線(xiàn)性微分方程-英文

1、中國(guó)石油大學(xué)勝利學(xué)院本科畢業(yè)設(shè)計(jì)（論文）外文翻譯英文文獻(xiàn)翻譯Some Properties of Solutions of Periodic Second Order Linear Differential Equations一些周期性的二階線(xiàn)性微分方程解的方法文獻(xiàn)來(lái)源： 百度文庫(kù) 作 者： 李文娟 譯 者： 張琪 指導(dǎo)教師： 武斌 11Some Properties of Solutions of Periodic Second Order Linear Differential Equations1. Introduction and main resultsIn this paper,

2、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 the order of growth, the lower order of growth a

3、nd the exponent of convergence of the zeros of a meromorphic function ，（see 8），the e-type order of f(z), is defined to be (1)Similarly, ，the e-type exponent of convergence of the zeros of meromorphic function , is defined to be (2)We say thathas regular order of growth if a meromorphic functionsatis

4、fies (3)We consider the second order linear differential equation (4)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 have been obtained

5、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 conclusion still holds if we just

6、suppose that both and are poles of, and at least one is of odd order. In addition, the stronger conclusion (5)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 solu

7、tion 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 theoremsTheorem C Let ,where,anda

8、re 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 dependent. (b) Ifandare any two linearly independent solutions of (1.1), then .(ii) Suppose (a) If f is a non-trivial sol

9、ution of (.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 positive integer or infinity. Let; whereand p is an odd positive integer. Thenor each non-trivial solution f to (1.1). I

10、n 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 results in the case whenis transcendental. Specially, we find a condition under which Theorem D still holds in the ca

11、se 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 integer or infinity, andarbitrary. If Some properties of solutions of periodic second order linear differential equat

12、ions and are two linearly independent solutions of (1.1), then (6)We remark that the 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 infin

13、ity 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 solution of (1.1) with,then and are linearly dependent.(b) Ifandare any two linearly independent solutions of (1.1), then.T

14、heorem 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 fact, the stronger conclusion (1.2) holds.We remark that the above conclusion remains valid if (7)We note that Theorem

15、 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 odd positive integer. Suppose that either (i) or (ii) below holds:(i) is not a positive integer or infinity;(ii) ;

16、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 period,and that f is a non-trivial solution of (8)Suppose further that f satisfies; that is non-constant and rational

17、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 asthrough a setof infinite measure, we havefor.Lemma 2 (10) Letbe a periodic entire function with periodand be tra

18、nscendental 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 , then and are linearly independent.3.Proofs of main resultsThe proof of main results are based on 8

19、 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, (9)Where is the Wronskian ofand(see 12, p. 5 or 1, p. 354), andor some non-zero constant.Clearly, and are both periodic funct

20、ions 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) yields (10)Since bothand are analytic in,the Valiron theory 21, p. 15 gives their representations as ， (11)where

21、，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 (12)where.Furthermore, the following properties hold 8, (13)Where (resp, ) is defined to be(resp, ), (14)Some properties of solutions of periodic

22、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 convergence of the zeros of in, which is defined to be (15)Recall the condition ,we obtain.Now substituting (2.3)

23、 into (2.2) yields (16)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 conclusion of Corollary 1 (a) thatand are linearly dependent, j = 1; 2. Let.Then we can find a non-zero const

24、ant 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-trivial solution f of (1.1) that satisfies . We deduce , so and are linearly dependent by Corollary 1 (a). Ho

25、wever, 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 authors would like to thank the referees for helpful suggestions to improve this paper.In this article, we

26、 establish a result about controllability to the following class of partial neutral functional euations with innite delay: (17)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 lin

27、ear operator 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 by (18)is a bounded linear operator from B into E and for each x : (, T E, T > 0, and t 0, T ,

28、 xt represents, as usual, the mapping from (, 0 into E dened by (19)F 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 t

29、o innite dimensional 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 n

30、eutral functional ations 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 o

31、bjective 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 conditions for controllability of so

32、me partial neutral functional 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 innite delay such

33、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 introduced in 13 an

34、d 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 equivalent to or every(ii

35、i) (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 (20)Let A0 be the part of operator A in dened by (21)

36、It is well known that and the operator generates a strongly continuous semigroup on .Recall that 19 for all and ,one has and (22)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 of bound

37、ed 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 constants an

38、d ,such that for all t 0. Notice that the controllability of a class of non-densely dened functional Differential tions was studied in 12 in the finite delay case.4.Main Results We start with introducing the following denition.Denition 1 Let T > 0 and B. We consider the following denition.We say

39、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 (23)Where .

40、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. (24)Using Theorem 7 in 1, we obtain the following result.Theorem 1 Assume that (H1), (H2), and (H3) hold. Let B such that D D(A). Then, there exists a uni

41、que integral solution x(., ) of Eq. (1), dened on (,+) .Denition 2 Under the above conditions, Eq. (1) is said to be controllable on the interval J = 0, , > 0, if for every initial function B with D D(A) and for any e1 D(A), there exists a control u L2(J,U), such that the solution x(.) of Eq. (1) satises .Theorem 2 Suppose that(H1), (H2), and (H3) hold. Let x(.) be the integral solution of Eq. (1) on (, ) , > 0, and assu