Advances in Pure Mathematics
Vol.09 No.07(2019), Article ID:93887,8 pages
10.4236/apm.2019.97030
Unicity of Meromorphic Solutions of Some Nonlinear Difference Equations
Baoqin Chen
Faculty of Mathematics and Computer Science, Guangdong Ocean University, Zhangjiang, China
Copyright © 2019 by author(s) and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: June 18, 2019; Accepted: July 22, 2019; Published: July 25, 2019
ABSTRACT
This paper is to study the unicity of transcendental meromorphic solutions to some nonlinear difference equations. Let be a nonzero rational function. Consider the uniqueness of transcendental meromorphic solutions to some nonlinear difference equations of the form . For two finite order transcendental meromorphic solutions of the equation above, it shows that they are almost equal to each other except for a nonconstant factor, if they have the same zeros and poles counting multiplicities, when . Two relative results are proved, and examples to show sharpness of our results are provided.
Keywords:
Unicity, Meromorphic Solution, Difference Equation
1. Introduction
It is well known that a given nonconstant monic polynomial is determined by its zeros. But it is not true for transcendental entire or meromorphic functions. Take and for example, they are essentially different even have the same zeros, 1-value points and poles. This indicates that it is complex and interesting to determine a transcendental meromorphic function uniquely. Nevanlinna then proves his famous Nevanlinna’s 5 CM (4 IM) Theorem (see e.g. [1] [2] ):
Theorem A: Let w(z) and u(z) be two nonconstant meromorphic functions. If w(z) and u(z) share 5 values IM (4 values CM, respectively) in the extended complex plane, then , where T is a Möbius transformation, respectively).
Here and in the following, for two nonconstant meromorphic functions w(z) and u(z), and a complex constant a, we say w(z) and u(z) share a IM (CM), if w(z)-a and u(z)-a have the same zeros ignoring multiplicities (counting multiplicities); and we say w(z) and u(z) share ∞ IM(CM), if they have the same poles ignoring multiplicities (counting multiplicities).
Our aim is to study the unicity of meromorphic solutions to the nonlinear difference equation of the form
, (1.1)
where R(z) is a nonzero rational function and The Equation (1.1) comes from the family of Painlevé III equations which are given by Ronkainen in [3] when he classifies the difference equation
where R(z, w) is irreducible and rational in w and meromorphic in z. This is a natural idea which comes from the topic on the growth, value distribution and unicity on the meromorphic solutions to difference equations (see e.g. [4] [5] [6] [7] [8] ). The first result is as follows.
Theorem 1.1. Let w(z) and u(z) be two finite order transcendental meromorphic solutions to the Equation (1.1), where . If w(z) and u(z) share 0, ∞ CM, then , where is a constant such that .
The following examples show that all cases in Theorem 1.1. can happen, and the “CM” cannot be relaxed to “IM”.
Example 1. In the following examples, and share 0, ∞ CM, while and share 0, ∞ IM :
1) and satisfy the difference equation
here such that .
2) and satisfy the difference equation
here such that .
3) and satisfy the difference equation
here such that .
4) and satisfy the difference equation
here such that .
Theorem 1.2. Let w(z) and u(z) be two finite order transcendental meromorphic solutions to the Equation (1.1), where . If w(z) and u(z) share 0, ∞ CM, then
(1.2)
where are constants such that . What is more, if has a zero of multiplicity such that .
The following example shows that all conclusions in Theorem 1.2 can happen, and the “CM” cannot be relaxed to “IM”.
Example 2. Let and , , . Then and share 0, ∞ CM, while and share 0, ∞ IM (j = 1, 2, 3), and they solve the equation
Theorem 1.3. Let w(z) and u(z) be two finite order transcendental meromorphic solutions to the Equation (1.1), where . If w(z) and u(z) share 1, ∞ CM, then
(1.3)
where are constants such that:
1) , when ; 2) , when ; (3) , when
, where are some integers. What is more, if one of the following additional condition holds:
a) has a zero of multiplicity such that ;
b) there exist two constants such that and .
Remark 1. We have tried hard but failed to provide some similar results as Theorem 1.3 for the cases so far.
2. Proof of Theorem 1.1
Since w(z) and u(z) are finite order transcendental meromorphic functions and share 0, ∞ CM, we see that
where is a polynomial such that it is of degree
Next, we discuss case by case.
Case 1: m = −2. From (1.1) and (2.1) we get
which gives
Thus, we have
(2.2)
Since
from (2.2), it is easy to find that p = 0. Therefore, there exists some constant , such that and
That is, for , we have and .
Case 2: m = −1. Now, we obtain from (1.1) and (2.1) that
With this equation and similar reasoning as in Case 1, we can deduce that holds for some such that .
Case 3: m = 0. From (1.1) and (2.1), we have
Similarly, we can prove that holds for some such that .
Case 4: m = 1. Now (1.1) is of the form
(2.3)
Thus,
It follows from these two equations above and (2.1) that
with which we can show that holds for some such that . However, if , we find that
(2.4)
Combining (2.3) and (2.4), we get , which is impossible. Thus, .
3. Proof of Theorem 1.2
Notice that (2.1) still holds for this case. We can get from (1.1) and (2.1) that
Thus, we have
(3.1)
If , then our conclusion holds for . If , set
(3.2)
where are constants.
From (3.2), we see that
(3.3)
where q(z) is a polynomial such that when , or when .
Suppose that , we obtain from (3.1) and (3.3) that
which is impossible. Thus, , then from (3.1) and (3.3), we get immediately. To sum up, we prove that (1.2) holds.
Next, we use and prove our additional conclusion. From (1.2), we see that .
Differentiating both sides of (1.2), we can deduce that
and
By our assumption, (1.2}), (3.4) and the fact that , we have
Therefore, similarly, it follows from (3.5) that
As a result, we obtain
that is, . Hence, .
4. Proof of Theorem 1.3
Here, we need the lemma below, where the case that R(z) is a nonzero constant has been proved by Zhang and Yang [7] and the case that R(z) is a nonconstant rational function by Lan and Chen [8] .
Lemma 4.1. [7] [8] Let w(z) be a finite order transcendental meromorphic solution to
the Equation (1.1), where and a be a constant. Then
Proof of Theorem 1.3. Since and are finite order transcendental meromorphic functions and share 1, ∞ CM, we see that
(4.1)
where is a polynomial such that
(4.2)
where are constants and .
Case 1: m = 0. From (1.1) and (4.1), we obtain
(4.3)
and
(4.4)
where is a rational function. Combining (4.1}), (4.3) and (4.4), we have
(4.5)
Now, if , then and it follows from (4.5) that
(4.6)
Notice that . From (4.6), we can find that
This is a contradiction to the conclusion of Lemma 4.1. Thus, . From (4.2) there exists some integer such that
which yields obviously that . Therefore, we see that
and hence for some constant .
Case 2: m = −1. Now (1.1) is of the form
which gives
With this equation and a similar arguing as in Case 1, we can prove that for some integer and some constant .
Case 3: m = 1. Now (1.1) is of the form
which gives
And hence we have
It follows this equation that for some integer and some constant , and (1.3) holds.
Now, if has a zero of multiplicity such that , then from (4.1), we see that .
Rewrite (4.1) as the form
Differentiating both sides of the equation above, we have
Since
is a zero of with multiplicity
such that
, from the fact that
and (4.7), we find that
Thus, , and hence
. This implies that
.
Finally, we discuss the Case 2). Since and
, then from (4.1), we can deduce that
. Therefore, there exists an integer
such that
If, from the equation above, considering each form of
for
, we can find that
must be a nonzero rational number. This contradicts our assumption that
. Thus
, and hence
. This gives
again.
5. Conclusion
It is shown that the finite order transcendental meromorphic solution of the Equation (1.1) is mainly determined by its zeros (or 1-value points) and poles. Examples are provided to show sharpness of our results.
Acknowledgements
The author is very appreciated for the editors and reviewers for their constructive suggestions and comments for the readability of this paper.
Funding
This work was supported by the Natural Science Foundation of Guangdong Province (2018A030307062).
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
Cite this paper
Chen, B.Q. (2019) Unicity of Meromorphic Solutions of Some Nonlinear Difference Equations. Advances in Pure Mathematics, 9, 611-618. https://doi.org/10.4236/apm.2019.97030
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