Open Access

Subnormal Solutions of Second-Order Nonhomogeneous Linear Differential Equations with Periodic Coefficients

Journal of Inequalities and Applications20092009:416273

DOI: 10.1155/2009/416273

Received: 8 February 2009

Accepted: 24 May 2009

Published: 29 June 2009

Abstract

We obtain the representations of the subnormal solutions of nonhomogeneous linear differential equation , where and are polynomials in such that and are not all constants, . We partly resolve the question raised by G. G. Gundersen and E. M. Steinbart in 1994.

1. Introduction

We use the standard notations from Nevanlinna theory in this paper (see [13]).

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, [49].

Now, we firstly consider the second-order homogeneous linear differential equations
(1.1)

where and are polynomials in and are not both constants. It is well known that every solution of (1.1) is an entire function.

Let be an entire function. We define
(1.2)

to be the -type order of .

If is a solution of (1.1) and if satisfies , then we say that is a subnormal solution of (1.1). For convenience, we also say that is a subnormal solution of (1.1).
  1. H.

    Wittich has given the general forms of all subnormal solutions of (1.1) that are shown in the following theorem.

     

Theorem 1 A (see [9]).

If is a subnormal solution of (1.1), where and are polynomials in and are not both constants, then must have the form
(1.3)
where is an integer and are constants with .
  1. G.

    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 [6]).

In addition to the statement of Theorem A, the following statements hold with regard to the subnormal solutions of (1.1).

(i)If and then any subnormal solution of (1.1) must have the form
(1.4)

where is an integer and are constants with

(ii)If and , then any subnormal solution of (1.1) must be a constant.

(iii)If , then the only subnormal solution of (1.1) is .

Whether the conclusions of Theorem A and Theorem B can be generalized or not, Gundersen and Steinbart considered the second-order nonhomogeneous linear differential equations
(1.5)

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 [6], they also have raised the following problem, that is, what about the forms of the subnormal solutions of the equation
(1.6)

where , and are polynomials in such that , , , and are not all constants?

In [7], we have obtained the exact forms of all subnormal solutions of homogeneous equation
(1.7)

where , and are polynomials in and are not all constants.

In this paper, we obtain the forms of subnormal solutions of nonhomogeneous linear differential equation (1.6) when . We have the following theorem.

Theorem 1.1.

Suppose that is a subnormal solution of (1.6), where , , and are polynomials in such that and are not all constants.

(i)If and , then must have the form
(1.8)

where is a constant, and are polynomials in .

(ii)If and , then must have the form
(1.9)

where is a constant, and are constants that may or may not be equal to zero, may be equal to zero or may be a polynomial in , , and are polynomials in with .

2. Lemmas for the Proof

In order to prove Theorem 1.1, we need some lemmas.

Lemma 2.1 (see [7]).

Suppose that is a subnormal solution of (1.7), where , , and are polynomials in and are not all constants.

(i)If and then any subnormal solution must be a constant.

(ii)If and then must have the form
(2.1)

where is a polynomial in with .

Lemma 2.2 (see [10]).

Let be a transcendental meromorphic function, let be a given real constant, and let . Then there exists a constant such that the following two statements hold (where ).

(i)There exists a set that has linear measure zero such that if , then there is a constant such that for all satisfying and one has
(2.2)

(ii)There exists a set that has finite logarithmic measure such that (2.2) holds for all satisfying .

Lemma 2.3 (see [6]).

Let with as and let Let

(2.3)
and set
(2.4)
Let be analytic on the set . Suppose that is unbounded on the set . Then there exists an infinite sequence of points with as such that
(2.5)

Lemma 2.4 (see [8]).

Consider the nth- order differential equation of the form
(2.6)
where are polynomials in and with . Suppose that is an entire and subnormal solution of (2.6) and that can be expressed as , where is a constant and is analytic on . Then has the form
(2.7)

where is a constant and and are polynomials in .

As an application of Lemma 2.4, one has the following lemma.

Lemma 2.5.

Suppose that is an entire subnormal solution of (2.6), where are polynomials in and with , and that and are linearly dependent. Then has the form
(2.8)

where is a constant and and are polynomials in .

Proof.

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.

Lemma 2.6.

Suppose that is a solution of (1.6), where , , , , , and are polynomials in such that , , and are not all constants. If
(2.9)
then there exists a polynomial such that
(2.10)
where is a solution of
(2.11)

where and are polynomials in with

Proof.

Let and set
(2.12)
where is the constant such that
(2.13)
It follows from (1.6) and (2.12) that
(2.14)
where
(2.15)

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.

Proof.
  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
    (3.1)
     

where is a constant and . It also follows from Lemma 2.2(ii) that there exists a set that has finite logarithmic measure such that (3.1) holds for all satisfying .

Now let be an infinite sequence satisfying such that for all and as Let be a small constant such that and . Set
(3.2)
and set
(3.3)

From above, we have that (3.1) holds on the set

We now assert that is bounded on the set . On the contrary, it follows from Lemma 2.3 that there exists a sequence of points with as such that
(3.4)
(3.5)
By (1.6), we have for all ,
(3.6)

It follows from (3.4)–(3.6) and that (3.6) yields as on the set This is a contradiction.

By the maximum modulus principle, is bounded in the angular domain
(3.7)
However, we know
(3.8)

where the integral of is defined on the simple contour , extending from a point to a point in the complex domain.

So we obtain
(3.9)

as in the angular domain .

Thus , from the Cauchy integral formula, we obtain
(3.10)
as in the angular domain . By (1.6), (3.8), and (3.9), we have for some constant
(3.11)

as in the angular domain . Together with (3.8) and (3.11), is bounded in the angular domain .

If , it follows from Lemma 2.5 that must have the form (1.8).

If , since is a subnormal solution of (1.6), so is . Thus,
(3.12)

will be a subnormal solution of (1.7). Since we suppose that , we will discuss the following two cases.

Case 1.

If we have, by Lemma 2.1(i),
(3.13)
where is a constant. Hence , that is,
(3.14)
From this, can be written as (see [11, page 382]), where is a constant and is analytic on . Thus, can be written as , where is a constant and is analytic on . It follows from Lemma 2.4 that
(3.15)

where is a constant, and are polynomials in . Thus, has the form of (1.8).

Case 2.

If we obtain from Lemma 2.1(ii) that
(3.16)

where is a polynomial in with .

However, we can assert that in (3.16). Otherwise, there exists such that
(3.17)
By (3.16), we have
(3.18)
Thus from (3.16) and (3.18), we have
(3.19)
By repeating this process finite times, we obtain that for any integer ,
(3.20)
We have, by (3.17) and (3.20),
(3.21)
as This is a contradiction to the fact that is bounded in the angular domain . This shows that is not possible when under the hypotheses. This completes the proof of part (i).
  1. (ii)
    We firstly suppose that . Since is a subnormal solution of (1.6), so is . Set
    (3.22)
     

Then is a subnormal solution of (1.7). Now if , this shows that and has the form of (1.9) by Lemma 2.5. Thus, we suppose that in the following.

Now, assume that .

If it follows from the proof of Case 1 of Theorem 1.1(i) that has the form of (1.9).

If we obtain from Lemma 2.1(ii) that
(3.23)

where is a polynomial in with .

Set
(3.24)

where is an integer and are constants with

Let and set
(3.25)

where

Now, we will discuss the following two cases.

Case A.

We consider in (3.24). Let be a constant defined by
(3.26)
and set
(3.27)
Since is a subnormal solution of (1.7), it follows from (3.27) that satisfies
(3.28)
We obtain from (3.23)–(3.28) that
(3.29)

where and are polynomials in with

Set
(3.30)

It follows from (1.6), (3.25), (3.29) and (3.30) that

Set
(3.31)
So , and satisfies
(3.32)
We have by (3.27) that is a subnormal solution of (1.7), is a subnormal solution of (3.29). Moreover, is also a subnormal solution of (3.32) by (3.30) and is a subnormal solution of (1.6). Thus, we deduce from Theorem 1.1(i) and (3.32) with that has the form
(3.33)
where is a constant, and are polynomials in . Hence (3.23), (3.24), (3.27), (3.30), and (3.33) yield
(3.34)

where and are constants, , and are polynomials in with . This is the form of (1.9).

Case B.

We consider in (3.24). Let be a constant defined by
(3.35)
where is a number such that is the first coefficient in (3.24) which is not equal to zero. Set
(3.36)
Similar to the proof of Case A of Theorem 1.1(ii), we have
(3.37)

where and are constants, and, are polynomials in with . Set . Then is a polynomial in by the hypotheses of in (3.36). This is the form of (1.9). We have proved Theorem 1.1(ii) when

Now we suppose that . By Lemma 2.6, there exists a polynomial in , satisfies (2.10) and (2.11).

Since and since we have proved Theorem 1.1 holds in the cases when holds, we can apply this result to (2.11).

If , it follows from Theorem 1.1(i) that
(3.38)
where is a constant, and are polynomials in . By (2.10) and (3.38), we obtain that
(3.39)

where is a constant, and are polynomials in . This is a form of (1.9).

If , it follows from the proof of Theorem 1.1(ii) when that
(3.40)

where and are polynomials in with , and are constants that may or may not be equal to zero. By (2.10) and (3.40), we obtain that has the form of (1.9). Theorem 1.1(ii) is completed.

Now, we give some examples to show that Theorem 1.1 is correct.

Example 3.1.

Let , then satisfies
(3.41)

This is an example of Theorem 1.1(i).

Example 3.2.

Let , then satisfies
(3.42)

This is an example of Theorem 1.1(ii) with and .

Example 3.3.

Let , then satisfies
(3.43)

This is an example of Theorem 1.1 (ii) with and .

Declarations

Acknowledgements

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.

Authors’ Affiliations

(1)
School of Mathematical Sciences, South China Normal University
(2)
Department of Applied Mathematics, South China Agricultural University

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Copyright

© Zhi-Bo Huang et al. 2009

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