数学论文_英文翻译

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毕业设计(论文)附录

（翻译）

课 题 名 称 一些周期性的二阶线性微分方程解的方法

学 生 姓 名 万 益

目 录

1．毕业设计（论文）附录（翻译）英文

2．毕业设计（论文）附录（翻译）中文

2014年 5月25日

Some Properties of Solutions of Periodic Second Order

Linear Differential Equations

1. Introduction and main results

In 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,

(f)and (f)to denote respectively the order 16]. In addition, we will use the notation (f)，

of growth, the lower order of growth and the exponent of convergence of the zeros of a meromorphic function f， e(f)（[see 8]），the e-type order of f(z), is defined to be

e(f) limlogT(r,f) r r

Similarly, e(f)，the e-type exponent of convergence of the zeros of meromorphic function f, is defined to be

log N(r,1/f) e(f) lim r r

We say thatf(z)has regular order of growth if a meromorphic functionf(z)satisfies

(f) limlogT(r,f) r logr

We consider the second order linear differential equation

f Af 0

Where A(z) B(e z)is a periodic entire function with period 2 i/ . 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 [2{11, 13, 17{19]. WhenA(z)is rational in e，Bank and Laine [6] proved the following theorem

Theorem A LetA(z) B(e z)be a periodic entire function with period 2 i/ and rational in e z z.IfB( )has poles of odd order at both and 0, then for every solutionf(z)( 0)of (1.1), (f)

Bank [5] generalized this result: The above conclusion still holds if we just suppose that both and 0are poles ofB( ), and at least one is of odd order. In addition, the stronger conclusion

log N(r,1/f) o(r) (1.2)

holds. WhenA(z)is transcendental ine, Gao [10] proved the following theorem

Theorem B Let B( ) g(1/ ) jb j 1j,whereg(t)is a transcendental entire function p z

zwith (g) 1, p is an odd positive integer andbp 0，Let A(z) B(e).Then any

non-trivia solution fof (1.1) must have (f) . In fact, the stronger conclusion (1.2) holds.

An example was given in [10] showing that Theorem B does not hold when (g)is any positive integer. If the order (g) 1 , but is not a positive integer, what can we say? Chiang and Gao [8] obtained the following theorems

zTheorem C Let A(z) B(e),whereB( ) g1(1/ ) g2( ),g1andg2are entire

functionsg2transcendental and (g2)not equal to a positive integer or infinity, andg1arbitrary. (i) (g2) 1. (a) If f is a non-trivial solution of (1.1) with e(f) (g2);

thenf(z)andf(z 2 i)are linearly dependent. (b) Iff1andf2are any two linearly independent solutions of (1.1), then e(f) (g2). Suppose

(g2) 1 (a) If f is a non-trivial solution of (1.1)

with e(f) 1,f(z)andf(z 2 i)are linearly dependent. Iff1andf2are any two linearly independent solutions of (1.1),then e(f1f2) 1.

Theorem D Letg( )be a transcendental entire function and its order be not a positive integer or (ii) Suppose infinity. LetA(z) B(e z); whereB( ) g(1/ ) pjb and p is an odd positive jj 1

integer. Then (f) or 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 when (g)is infinity.

The main purpose of this paper is to improve above results in the case whenB( )is transcendental. Specially, we find a condition under which Theorem D still holds in the case when (g)is a positive integer or infinity. We will prove the following results in Section 3.

Theorem 1 Let A(z) B(e),whereB( ) g1(1/ ) g2( ),g1andg2are entire functions withg2transcendental and z (g2)not equal to a positive integer or infinity, andg1arbitrary. If Some properties of solutions of periodic second order linear differential equationsf(z) and f(z 2 i)are two linearly independent solutions of (1.1), then

e(f)

Or

e(f) 1 (g2) 1 2

We remark that the conclusion of Theorem 1 remains valid if we assume (g1)

is not equal to a positive integer or infinity, andg2arbitrary and still assumeB( ) g1(1/ ) g2( ),In the case wheng1is transcendental with its lower order not equal to an integer or infinity andg2is arbitrary, we need only to consider B*( ) B(1/ ) g1( ) g2(1/ )in0 , 1/ .

Corollary 1 LetA(z) B(e),whereB( ) g1(1/ ) g2( ),g1andg2are

entire functions with g2 transcendental and

(a)

(b) z (g2)no more than 1/2, and g1 arbitrary. If f is a non-trivial solution of (1.1) with e(f) ,thenf(z) and f(z 2 i)are linearly dependent. Iff1andf2are any two linearly independent solutions of (1.1),

then e(f1f2) .

Theorem 2 Letg( )be a transcendental entire function and its lower order be no more than 1/2.

zLetA(z) B(e),whereB( ) g(1/ ) pb j 1jjand p is an odd positive integer,

then (f) 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

B( ) g( ) b j j

j 1p

We note that Theorem 2 generalizes Theorem D when (g)is a positive integer or infinity but (g) 1/2. Combining Theorem D with Theorem 2, we have

zCorollary 2 Letg( )be a transcendental entire function. LetA(z) B(e) where

B( ) g(1/ ) j 1bj jand p is an odd positive integer. Suppose that either (i) or (ii) below holds:

(i) (g) is not a positive integer or infinity;

(ii) (g) 1/2;

then (f) for each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.

2. Lemmas for the proofs of Theorems

Lemma 1 ([7]) Suppose thatk 2and thatA0,.....Ak 2are entire functions of period2 i,and that f is a non-trivial solution of p

y(k) Aj(z)y(j)(z) 0

i 0k 2

Suppose further that f satisfieslog N(r,1/f) o(r); that A0 is non-constant and rational

zine，and that ifk 3，thenA1,.....Ak 2are constants. Then there exists an integer q with1 q k such thatf(z) and f(z q2 i)are linearly dependent. The same conclusion

z holds ifA0is transcendental ine，and f satisfieslogN(r,1/f) o(r)，and if k 3，then

through a setL1r

k 2. haveT(r,Aj) o(T(r,Aj))forj 1,.....as

zof infinite measure, 1we and be Lemma 2 ([10]) LetA(z) B(e z)be a periodic entire function with period 2 i transcendental ine, B( )is transcendental and analytic on0 .IfB( )has a pole of

odd order at or 0(including those which can be changed into this case by varying the period ofA(z) andEq. (1.1) has a solutionf(z) 0which satisfies log N(r,1/f) o(r), thenf(z) and f(z )are linearly independent.

3. Proofs of main results

The proof of main results are based on [8] and [15].

Proof of Theorem 1 Let us assume e(f) .Sincef(z) and f(z 2 i)are linearly independent, Lemma 1 implies that f(z)and f(z 4 i)must be linearly dependent. LetE(z) f(z)f(z 2 i),ThenE(z)satisfies the differential equation

E (z)2E (z)c2

, (2.1) 4A(z) () 2 E(z)E(z)E(z)2

Where c 0is the Wronskian off1andf2(see [12, p. 5] or [1, p. 354]), andE(z 2 i) c1E(z)or some non-zero constantc1.Clearly, E /E

and E /Eare both periodic functions with period2 i,whileA(z)is periodic by definition.

2Hence (2.1) shows thatE(z)is also periodic with period 2 i.Thus we can find an analytic

function ( )in0

yields ,so thatE(z)2 (ez)Substituting this expression into (2.1) c2 3 4B( ) 2()2 2 (2.2) 4

Since bothB( )and ( )are analytic inC* :1 ,the Valiron theory [21, p. 15] gives their representations as

n B( ) R( )b( )， ( ) n1R1( ) ( )， （2.3）

n1are some integers, R( )andR1( )are functions that are analytic and non-vanishing wheren，

on C* { }，b( )and ( ) are entire functions. Following the same arguments as used in [8], we have

T( , ) N( ,1/ ) T( ,b) S( , )， （2.4）

whereS( , ) o(T( , )).Furthermore, the following properties hold [8]

e(f) e(E) e(E2) max{ eR(E2), eL(E2)},

eR(E2) 1( ) ( ),

Where eR(E2)(resp, eL(E2)) is defined to be

log NR(r,1/E2)log NR(r,1/E2)lim(resp, lim), r r rr

Some properties of solutions of periodic second order linear differential equations

)(resp. NL(r,1/E2)denotes a counting function that only counts the zeros

2of E(z)in the right-half plane (resp. in the left-half plane), 1( )is the exponent of convergence of the zeros of inC*, which is defined to be

log N( ,1/ ) 1( ) lim log

Recall the condition e(f) ,we obtain ( ) . whereNR(r,1/E

Now substituting (2.3) into (2.2) yields 2

nR 3nR c2

4 R( )b( ) n1 (1 1 ) 2(1 1 )2 R1 4 R1 R1( ) ( )

R1 n1R1n1 R1 R1 2n1(n1 1) ( 2 2 2 ) （2.5） 2 R1 R1 R1 n

Proof of Corollary 1 We can easily deduce Corollary 1 (a) from Theorem 1 .

Proof of Corollary 1 (b). Supposef1andf2are linearly independent and e(f1f2) ,then e(f1) ,and

Corollary 1 (a) that

Letfj(z)and e(f2) .We deduce from the conclusion of fj(z 2 i)are linearly dependent, j = 1; 2. E(z) f1(z)f2(z).Then we can find a non-zero constant c2such thatE(z 2 i) c2E(z).Repeating the same arguments as used in Theorem 1 by using the fact

2that E(z)is also periodic, we obtain

e(E) 1 (g2) 1 2,a contradiction since (g2) 1/2.Hence e(f1f2) .

Proof of Theorem 2 Suppose there exists a non-trivial solution f of (1.1) that satisfies log N(r,1/f) o(r). We deduce e(f) 0, so f(z)andf(z 2 i) are linearly dependent by Corollary 1 (a). However, Lemma 2 implies that f(z)andf(z 2 i)are linearly

independent. This is a contradiction. Hence logN(r,1/f) o(r)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.

References

[1] 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]. Results Math., 1996, 29(1-2): 42{55.

[3] BAESCH A, STEINMETZ N. Exceptional solutions of nth order periodic linear differential