Subnormal Solutions of Second-Order Nonhomogeneous Linear Differential Equations with Periodic Coefficients
© Zhi-Bo Huang et al. 2009
Received: 8 February 2009
Accepted: 24 May 2009
Published: 29 June 2009
The study of the properties of the solutions of a linear differential equation with periodic coefficients is one of the difficult aspects in the complex oscillation theory of differential equations. However, it is also one of the important aspects since it relates to many special functions. Some important researches were done by different authors; see, for instance, [4–9].
Wittich has given the general forms of all subnormal solutions of (1.1) that are shown in the following theorem.
Theorem 1 A (see ).
G. Gundersen and E. M. Steinbart refined Theorem A and obtained the exact forms of subnormal solutions of (1.1) as follows.
Theorem 1 B (see ).
where , and are polynomials in such that are not both constants. They found the exact forms of all subnormal solutions of (1.5), that is, what is mentioned in [6, Theorem ?2.2, Theorem ?2.3 and Theorem ?2.4].
In this paper, we obtain the forms of subnormal solutions of nonhomogeneous linear differential equation (1.6) when . We have the following theorem.
2. Lemmas for the Proof
In order to prove Theorem 1.1, we need some lemmas.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
As an application of Lemma 2.4, one has the following lemma.
Since is entire and is linearly dependent with , can be written as (see [11, page 382]), where is a constant and is analytic on . Then we have the representation from Lemma 2.4.
So and are polynomials in and , respectively, and by (2.13), but and have the exact representations that depend on the relations of , and . If , then (2.14) is of the form (2.11), and (2.12) gives (2.10). If , then we repeat the above process finite times until we obtain (2.10) and (2.11). This completes the proof of Lemma 2.6.
3. Proof of Theorem
In this section, we will prove Theorem 1.1.
- (i)Suppose that is a subnormal solution of (1.6) with and . If is a polynomial solution of (1.6), then must be a constant, which is of the form (1.8). Thus we suppose that is transcendental. It follows from Lemma 2.2(i) that there exists a set that has linear measure zero such that if , then there is a constant such that for all satisfying and , we have
Now, we will discuss the following two cases.
Now, we give some examples to show that Theorem 1.1 is correct.
This is an example of Theorem 1.1(i).
The authors are very grateful to the referee for his (her) many valuable comments and suggestions which greatly improved the presentation of this paper. The project was supposed by the National Natural Science Foundation of China (no. 10871076), and also partly supposed by the School of Mathematical Sciences Foundation of SCNU, China.
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