Two problems are discussed in this paper. In the first problem, we consider one homogeneous and one non-homogeneous differential equations and study when the solutions of these differential equations can have (nearly) the same zeros. In the second problem, we consider two linear second-order differential equations and investigate when the solutions of these differential equations can take the value 0 and a non-zero value at (nearly) the same points.
We apply the Nevanlinna theory and properties of entire solutions of linear differential equations.
In the first problem, the results determine all pairs of such equations having solutions with the same zeros or nearly the same zeros. Regarding the second problem, the results also show all pairs of such equations having solutions taking the value 0 and a non-zero value at (nearly) the same points.
Keywords:Nevanlinna theory; differential equations
There has been much research [1-8] on zeros of solutions of linear differential equations with entire coefficients. The principal paper  that was published in 1982 by Bank and Laine has stimulated many studies on this kind of problems. The reader is referred to [10-12] for background on some applications of the Nevanlinna theory. We use the standard notation of the Nevanlinna theory from .
In 1955, Wittich  proved the following theorem.
(ii) Iffhas finite order, thenAis a polynomial.
(iii) Ifais a non-zero complex number, thenftakes the valueainfinitely often, and in fact, outside a set of finite measure,
The following facts follow from the asymptotic representation for solutions of the equation
LetPbe a polynomial of degreen, and letwbe a non-trivial solution of the equation (1). Then, whas order of growth equal to. Moreover, ifwis a solution of (1) which has infinitely many zeros, then we have
Our previous paper  studied homogeneous linear differential equations having solutions with nearly the same zeros and proved several results, including the following.
Recently, the same problem, but with non-homogeneous first-order differential equations, has been studied in , including the following result.
Then the following conclusions hold.
In this paper, our first result (Theorem 2.1) looks at the same problem but with one homogeneous and one non-homogeneous differential equations. In particular, we consider the first equation to be homogeneous of the second-order with a polynomial coefficient and the second equation to be non-homogeneous of the first-order with entire coefficients.
A further result (Theorem 2.2) studies the case where the solutions of two second-order homogeneous differential equations can take the value 0 and a non-zero value at (nearly) the same points.
2 Our results
Our first result is the following theorem.
ThenAis a polynomial and there exists a polynomialQsuch that
Example 2.1 Take Q to be a polynomial. Let
Note that v has the same zeros as w. Now, we have
We now state our second result.
Theorem 2.2Suppose thatis a polynomial of degreen, andAis an entire function, and suppose thatwsolves (1) andvsolves (3), and. Letandwhave, with finitely many exceptions, the same zeros and the same multiplicities. Then one of the following holds.
3 Proof of Theorem 2.1
Proof We have
We also have, using (1), (5), (8), (9) and (10),
We divide (11) by L, and by using (12) and (13), we get
The next step is to estimate the growth of M.
We know that from . Therefore,
Write (5) as
Then, there exists a constant c such that
Also, using (6), we can write
Therefore, we get
We use Lemma 2.3 in [, p.38] with to get
This completes the proof of Claim 3.1.
Using Claim 3.1 and (14), we get
Also, by Theorem 1.2, we get
Therefore, we must have
We now divide (17) by B to get
But then, we can write
Then we also can write
Substitute (18) in (16), we obtain
So, we get
Substituting (21) in (19), we obtain
Therefore, A is a polynomial.
Since B has no zeros, from (20) we can write
where Q is a polynomial.
Since w and H solve the same equation and are linearly independent (because w has zeros but H does not), we can write
Now, we have
So, we can write
Therefore, we have
Hence, using (22), we obtain
Now, from (23) and (24), we notice that w and v have the same zeros.
Also, differentiating (24), using (22), we have
Comparing this with (5), we get
This completes the proof of Theorem 2.1. □
4 A lemma needed to prove Theorem 2.2
In order to prove Theorem 2.2, we must state and prove the following lemma. We include a proof for completeness.
Now, we have two cases to consider.
5 Proof of Theorem 2.2
We first note that, outside a set of finite measure, by Theorem 1.1,
Assume henceforth that w has infinitely many zeros. Then (26) implies that , and so A is a polynomial of degree at most n by the Wiman-Valiron theory . Also, since has infinitely many zeros.
Now, two cases have to be considered.
where δ is constant.
Now, by using Lemma 4.1, we get
We must now try six cases:
There are four more cases:
Case (II). Suppose that P is non-constant. We will show that this leads to a contradiction. Let
where L is a rational function and Q is an entire function.
Also, from (31), we have
Now, we have two cases to consider.
is a rational function, which is a contradiction since w has infinitely many zeros.
Then we can write (33) as
are rational functions and Q is a polynomial.
Now, we have two cases to consider:
which is a contradiction since w has infinitely many zeros.
Now, (1) and (35) give
because if not, w has finitely many zeros, a contradiction. Also,
From (36), we get
Since w and G solve the same equation but w has infinitely many zeros and G has finitely many zeros, w and G are linearly independent, and we can write
where c is a non-zero constant. So,
By integrating, we get
Also, using (31), (38) and (39),
We differentiate (40) to get
is a rational function.
So, from (3) and (41), we get
From (44), we have
From (42), we get
We integrate to get
where a is a constant. But this contradicts the fact that H and K are rational functions and G is a transcendental function. This completes the proof of Claim 5.1.
Once we have Claim 5.1, (41) gives
By (45), v has finitely many zeros, so we can write
Now, we can write this as
Hence, P is constant by , which contradicts our assumption in Case (II) that P is non-constant. □
The author declares that he has no competing interests.
The author would like to thank his supervisor Prof. Jim Langley for his support and guidance. Also, he would like to thank King Abdulaziz University for financial support for his PhD study.
Rossi, J: Second order differential equations with transcendental coefficients . Proc. Am. Math. Soc.. 97, 61–66 (1986). Publisher Full Text
doi:10.1186/1029-242X-2011-134BioMed Central Full Text
Hayman, WK: The local growth of power series: a survey of the Wiman-Valiron method . Can. Math. Bull.. 17(3), 317–358 (1974). Publisher Full Text