二阶线性微分方程英文翻译(2)

j 1

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

Corollary 2 Letg( )be a transcendental entire function. LetA(z) B(ez) 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

zholds ifA0is transcendental ine，and f satisfieslog N(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

z

zof infinite measure, 1we and be Lemma 2 ([10]) LetA(z) B(e)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 logN(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 2E(z)E(z)E(z)

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

n1R1 32n1R1 2c2

4 R( )b( ) n1 ( ) ( ) 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 that E(z)2is 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

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