Unicity of Meromorphic Solutions of the Pielou Logistic Equation

Li, Sheng; Chen, Baoqin

doi:https://doi.org/10.1155/2020/4253967

Discrete Dynamics in Nature and Society

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 4253967 | https://doi.org/10.1155/2020/4253967

Unicity of Meromorphic Solutions of the Pielou Logistic Equation

Sheng Li¹and Baoqin Chen¹

Academic Editor: Chris Goodrich

Received13 Jan 2020

Accepted17 Feb 2020

Published22 Mar 2020

Abstract

This paper mainly considers the unicity of meromorphic solutions of the Pielou logistic equation , where , and are nonzero polynomials. It shows that the finite order transcendental meromorphic solution of the Pielou logistic equation is mainly determined by its poles and 1-value points. Examples are given for the sharpness of our result.

1. Introduction

For a meromorphic function , we use standard notations of the Nevanlinna theory, such as , , and (see, e.g., [1–3]). Let denote any quantity that satisfies as possibly outside of an exceptional set of finite logarithmic measure. And we define the order of growth of by

Also we know that the unicity of solutions of a given equation is always one of its most essential properties. This paper is to discuss the unicity of meromorphic solutions of the Pielou logistic equationwhere , and are nonzero polynomials. Equation (2) is an important equation generalized from the famous Verhulst-Pearl equation, which is the most popular continuous model of growth of a population:

By denoting , we can get from (2) thatwhich is a linear difference equation.

On the growth, zeros, and poles of meromorphic solutions of (2) and (4), Chen proved numbers of significant results in [4]. Then, Cui and Chen [5, 6] began to consider the unicity of meromorphic solutions concerning their zeros, 1-value points, and poles and proved.

Theorem 1. (see [5]). Let be a finite order transcendental meromorphic solution of the equationwhere are nonzero polynomials such that If a meromorphic function shares CM with , then either or .

Theorem 2. (see [6]). Let be a finite order transcendental meromorphic solution of the equationwhere are nonzero polynomials such that If a meromorphic function shares CM with , then one of the following cases holds:(i)(ii)(iii)There exists a polynomial and a constant satisfying such thatwhere are constants.

Here and in the following, and are said to share the value a CM (IM), provided that and have the same zeros counting multiplicities (ignoring multiplicities). And and are said to share the value CM (IM), provided that and have the same poles with the same multiplicities (ignoring multiplicities).

Cui and Chen’s work is a natural product of generalization work (see, e.g., [1, 3, 7–11]) on the famous Nevanlinna’s 5 IM (4 CM) Theorem (see, e.g., [3, 12]) during the past, about 90 years, especially of the hot research studies on the complex differences and complex difference equations (see, e.g., [1, 4, 8–10, 13–15]) recently. They have given examples to show that all cases of Theorem A and Theorem B can happen, and the numbers of shared values cannot be reduced. Li and Chen [16] turned to consider the following question: What can we say about the unicity of finite order transcendental meromorphic solutions of the equationwhere are rational functions? And we proved some interesting results and also provided some examples for sharpness of them. Two of those results read are as follows.

Theorem 3. (see [16]). Let and be two finite order transcendental meromorphic solutions of equation (8), where . Suppose that and share CM. Then, either orwhere are constants such that , and the coefficients of (8) satisfy

Theorem 4. (see [16]). Let and be two finite order transcendental meromorphic solutions of equation (8), where

If and share CM, then .

Remark 1. Notice that and share 0 CM if and only if and share CM; and share CM if and only if and share 0 CM; and and share 1 CM if and only if and share 1 CM. As a result, for the unicity of finite order transcendental meromorphic solutions equation (2), we only need to consider the case that two CM shared values are Indeed, we prove the following Theorem 5, whose proof is different from that in [5, 6, 16].

Theorem 5. Let and be two finite order transcendental meromorphic solutions of equation (2). If and share CM and one of the following cases holds:(i)(ii) and has infinitely many poles of multiplicity (iii), is not an integer, and has at most finitely many simple poles, then

We give some examples for the sharpness of Theorem 5 as follows.

Example 1. (1) and satisfy the equation Here, and share CM such that they have infinitely many poles and and . This example shows that Theorem 5 may not hold for the case .(2) and satisfy the equation Here, and share CM such that they have infinitely many simple poles and and This example shows that Theorem 5 may not hold for the case if most (except finitely many) poles of are simple or is an integer.

Remark 2. It is interesting to ask a question: whether the shared condition “CM” is replaced by “IM” in Theorem 5. We have tried hard but failed to find some negative examples for this question. We conjecture that the conclusions in Theorem 5 still hold when the shared condition “CM” is replaced by “IM.”

2. Proof of Theorem 5

To prove Theorem 5, we need the following lemma of Clunie (see, e.g., [1, 2]).

Lemma 1. (see [1, 2]). Let be a transcendental meromorphic solution of the equationwhere and are polynomials in f and its derivatives with meromorphic coefficients, say , such that for all . If the total degree of as a polynomial in f and its derivatives is then

Proof of Theorem 5. Since and are finite order transcendental solutions of equation (2) and share CM, without loss of generality, assume that , and we getwhere is a polynomial such that , and are rational functions.
If , then our conclusion holds.
If , then , and from (17), we haveHere and in the following, we use the notationsfor any given meromorphic function for convenience.
Submitting (18) into (16), we havewhereFrom (15) and (20), we obtainor equally,Next, we discuss three cases.

Case 1. Then, , , andThus, (23) is of the formWe claim that . Otherwise, , and then and (25) yields that .
If there is a point such that , then . We can easily deduce from (2), (16), and (18) thatwhich gives a contradiction to the fact that .
If 1 is a Picard exceptional value of , then 1 is also a Picard exceptional value of . What is more, from (2) and (16), we see that is a Picard exceptional value of and . Since , has no other Picard exceptional value. Choose a point such that thenThis indicates that is not a Picard exceptional value of , a contradiction.
Now, we have proved that From (25), we getSince is transcendental, we see that . Setwhere are constants such that
Substituting (28) into (15), we getwhich gives whereSince is a rational function, there exist some and such that for all , , and we haveNotice thatas From (31) and (32), we can deduce thata contradiction to the fact that

Case 2. and has infinitely many poles of multiplicity From (23), we havewhere

Subcase 1. is a constant. Then, are rational functions and hence have at most finitely many poles. Choose a pole of with multiplicity , denoted by , such that Then, is a pole of with multiplicity and a pole of with multiplicity However, from (35), we see that it is impossible.

Subcase 2. is a nonconstant polynomial such that Then, fromwe see that has at most n zeros of multiplicity Then, are meromorphic functions which have at most finitely many poles of multiplicity . Choose a pole of with multiplicity denoted by such that is not a pole of of multiplicity Then, is a pole of with multiplicity and a pole of with multiplicity at most However, from (35) and we find that it is also impossible.

Case 3. is not an integer, and has at most finitely many simple poles. Then, since From Case 2, we can suppose that has at most finitely many poles of multiplicity and use (35) directly. Now, has at most finitely many poles.
On the one hand, we haveOn the other hand, since , it is easy to find thatApplying Lemma 1 to (35), we getwhich contradicts to (38). Our proof of Theorem 5 is thus completed.

3. Conclusion

Our result shows that the finite order transcendental meromorphic solution of equation (2) is mainly determined by its poles and 1-value points.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors have drafted the manuscript, read, and approved the final manuscript.

Acknowledgments

This work was supported by the Natural Science Foundation of Guangdong Province (2018A030307062).

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Copyright

Copyright © 2020 Sheng Li and Baoqin Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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