The Existence of Meromorphic Solutions to Non-Linear Delay Differential Equations ()
1. Introduction
In actual life, a completely linear system does not exist due to part of the system itself have varying degrees of non-linear properties and the influence of external conditions. At the same time, through the industrial production process and in natural social sciences, there are many practical systems like the well-known network control transmission system, water and power system, communication system, urban traffic management system, etc., which are related to the state of a certain moment in the past, and the characteristics of the system are called delay. We can see that it is very necessary to study nonlinear time-delay system, among which, nonlinear delay differential equation is a vital tool for studying nonlinear time-delay system, we study nonlinear delay differential equation to characterize a part of the corresponding nonlinear time-delay system, so as to obtain the characteristics of nonlinear time-delay system and solve some practical problems.
The differential Painlevé equations over the complex domain and the Painlevé type equations are a special and significant class of nonlinear delay differential equations with important applications in physics. In 2000, Ablowitz et al. [1] applied Nevanlinna theory in difference equations of complex domains, studied the following equations:
and obtained some results on the degree of the right side of the equations.
Subsequently, some well-known theories and approaches which are widely used in the study of differential and difference equations have emerged such as the difference version of the logarithmic derivative lemma (Halburd, Korhonen [2] and Chiang, Feng [3] ) and so on ( [4] [5] [6] ).
In the year of 2007, Halburd and Korhonen [7] discovered a discrete version of the Painlevé III and obtained the following theorem:
Theorem 1.1. Let
be an admissible finite-order meromorphic solution of the equation
where the coefficients are meromorphic functions,
and
. If the order of the poles of
is bounded, then either
satisfies a difference Riccati equation
where
, (
is a set of small functions of
), or equation can be transformed by a bilinear change in
to one of the equations
where
and
are arbitrary finite-order periodic functions such that
and
have period 2 and
has period 1.
In 2017, Halburd and Korhonen [8] applied Nevanlinna theory ( [9] [10] ) to consider the existence of the meromorphic solutions of complex differential-difference equations with the hyper-order less than 1 and obtain the following theorem.
Theorem 1.2. Let
be a non-rational meromorphic solution of
where
is rational,
is a polynomial in
having rational coefficients in z, and
is a polynomial in
with roots that are non-zero rational functions of z and not roots of
. If the hyper-order of
is less than one, then
.
Inspiring by the above work, Liu and Song [11] contemplated the non-linear difference equations
.
We can’t help but considering how the result would be different when
in the above equation turns to a more general
. That’s the main purpose of this paper. Here are the main contents and conclusions of this article:
Theorem 1.3. Suppose that k is a positive integer and that
is a rational function. Let
be a transcendental meromorphic solution of
(1.1)
where
and
are two coprime polynomials in
having rational coefficients in z. If
, then
and one of the following holds for
is not a root of
.
(i) when
or
, we have
(ii) when
, we have
In particular, when
is a root of
, its multiplicity is at most k.
In fact, a meromorphic function which satisfies
is called minimal-hypertype where
is Nevanlinna characteristics of
. Now we list some examples below to demonstrate that our results are accurate.
Example 1.1. The meromorphic function
with
solves
We can obtain that
since
.
Example 1.2. The meromorphic function
solves
and
, where
is an arbitrary rational function. Obviously, we have
and
This paper is organized as follows. In Section 2, we outline the lemma we need to use. The main results discussed on different situations are summarized in Section 3.
2. Auxiliary Lemmas
We present some lemmas which play important role in the following. The first one is the difference version of the logarithmic derivative lemma for meromorphic functions with minimal hyper-type due to Zheng and Korhonen [12] .
Lemma 2.1. [ [12] , Theorem 1.2] Let
be a meromorphic function. If
,
then
holds for a constant c as
, where E is a subset of
with the zero upper density, that is
Lemma 2.2. [ [12] , Lemma 2.1] Let
be a nondecreasing positive function in
and logarithmic convex with
. Assume that
Set
Then given a real number
, one has
,
where
is a subset of
with the zero upper density.
Lemma 2.3. [ [13] , Lemma 19] Let
be a non-rational meromorphic solution of
,
where
is a differential-difference polynomial in
with rational coefficients, and let
be rational functions satisfying
for all
.
If there exists
and
such that
then
.
If the right side of (1.1) is a polynomial in
, we can obtain the following fact.
Lemma 2.4. Let
be a non-rational meromorphic solution of the equation
(2.1)
where k is a positive integer, where
is a rational function and
is a polynomial of
on z. If
, then
.
The proof of Lemma 2.4. Assume that
. Firstly, we consider that
have finitely many poles and zeros, it’s obvious that there exist a rational function
and an entire function
such that
On the basis of Equation (2.1), we will get
(2.2)
Using the Lemma 2.1, one knows
At this point and the fact that
is transcendental, it follows from (2.2) that
This is a contradiction since
for
.
In the following, we consider that either
has infinitely many zeros or
has infinitely many poles (or both). Since the coefficients of (2.1) are rational, we can always choose a zero or a pole
of
in such a way that there is no cancellation with the coefficients. At this time, we continue to discuss the following two different situations:
Case 1.
is a pole of
with multiplicity
, then
has a pole with multiplicity
at
.
Subcase 1.1.
is a pole of
with multiplicity t (
) and
is a pole of
with multiplicity
. By shifting (2.1) forward and backward, we have
Analyzing the poles on both sides of the above equations and we will know
is a pole of
with multiplicity
,
is a pole of
with multiplicity
. Continue iterating over the equation, one knows
is a pole of
with multiplicity
,
is a pole of
with multiplicity
,
is a pole of
with multiplicity
,
is a pole of
with multiplicity
, and so on. Hence,
for all
. It follows that
This contradicts to
, so the assumption is not valid.
Subcase 1.2.
is a pole of
with multiplicity
,
is a zero of
with multiplicity
. By shifting (2.1) up, one can deduce that
is a pole of
with multiplicity
,
is a pole of
with multiplicity
,
is a pole of
with multiplicity
, and so on. Thus,
for all
, and so
This contradicts to
.
Subcase 1.3.
is a pole of
with multiplicity
,
is a zero of
with
, as the proof process of Subcases 1.2, we can push out the contradicts so that the hypothesis
is not valid.
Case 2.
has a zero in
with multiplicity q, then
has k-th poles at
. This implies that at least one of
and
is a pole of
. In the same discussion as in case 1, we can also derive that
This contradicts to
. The Lemma 2.4 is proved.
Finally, we consider the case in which
of (1.1) has a non-zero repeated root as a polynomial in
.
Lemma 2.5. Suppose that
are two positive integers. Let
be a non-rational meromorphic solution of
(2.3)
where
,
are rational functions, and
is a polynomial in
such that
. If
, then
On basis of Lemma 2.5, one can deduce
provided that
. Next we give the details of the proof of Lemma 2.5 below.
The proof of Lemma 2.5. We can transform (2.3) into
, where
is a differential difference polynomial. Notice that
, so the first condition of Lemma 2.3 is satisfied.
Suppose that
is a zero of
with multiplicity
, and that neither
nor any of the coefficients in (2.3) have a zero or pole at
. Furthermore, if the coefficient functions in (2.3) don’t have a zero or a pole at
and
, then
is called a generic zeros. Since the coefficients of (2.3) are rational, for the case of non-generic zeros, we can know that for integrated counting functions, it will bring an error term
at most. Hence, we only need to consider the generic zeros in the following.
Case 1. Assume that
Let
be a generic zero of
of order
, it follows from (2.3) that
is a pole of
with multiplicity
. This implies that at least one of
and
is a pole of
.
○ Assume
is a pole of
with multiplicity
and
is a pole of
with multiplicity
. By shifting (2.3) forward and backward once time, one can deduce that
has a pole of order k at
and a pole of order k at
. By continuing the iteration, it follows that
has either a finite value or a pole at
(or
). Consequently,
since
. Due to the Lemma 2.3 we have
, it contradicts the assumption.
○ Assume one of
and
is a pole of
with multiplicity
.
Iterating (2.3) as before we will know that
is a pole of
with multiplicity k or
is a pole of
with multiplicity k, so
since
and
. By Lemma 2.3, we also have
. This is a contradiction, so
Case 2. Assume that
We also let
be a generic zero of
of order
, it’s easy to see that
is a pole of
with multiplicity
. It indicates that at least one of
and
is a pole of
. By this time, we also expand into several subcases:
○ Assume
is a pole of
with multiplicity
and
is a pole of
with multiplicity
. Shifting (2.3) forward and backward and we can deduce that
has a pole of order
at
and a pole of order
at
. Thus, we have
since
. Combining with the Lemma 2.3 and we will get
which is a contradiction.
○ Assume one of
and
is a pole of
with multiplicity
. By iterating (2.3) as before, we have that
is a pole of
with multiplicity
or
is a pole of
with multiplicity
, so
since
and
. Following from the Lemma 2.3 and we will obtain
. This is a contradiction, so
In conclusion, we have proved the Lemma 2.5.
3. The Proof of Theorem 1.3
For the equation (1.1), we proceed to prove that
. Taking the Nevanlinna characteristic function of both sides of (1.1), we have
since the coefficients of
and
are rational functions. Furthermore, in view of Lemma 2.2 and the lemma on the logarithmic derivative,
(3.1)
We can see that
has a pole in
if and only if
has a pole or zero in
, so
(3.2)
Together with (3.1) and (3.2), we have
and thus
.
Next, let us complete the proof of the Theorem 1.3. If
, it follows from Lemma 2.4 that
. When
,
we consider the following two cases.
Case 1.
is not a root of
.
Subcase 1.1. Assume that
, without loss of generality, we set
, where
is a rational function. Thus (1.1) can be rewritten as
(3.3)
It is easy to see that
is not a solution of (3.3), so the first condition of Lemma 2.3 is satisfied. Suppose that
and that
is a generic zero of
with multiplicity
. It follows that
has a pole at
of order at least
.
If
is a pole of
with multiplicity
and
is a pole of
with multiplicity
. By Shifting (3.3) up, we have
(3.4)
and thus
has a pole of order
at
. Similarly, from (3.3) one can obtain that
(3.5)
and that
has a pole of order
at
. Therefore,
Combining with the Lemma 2.3 and we will get
, which contradicts to the fact in Theorem 1.3.
If
or
is a pole of
with multiplicity
, it follows from (3.4) and (3.5) that
is a pole of
with multiplicity
or
is a pole of
with multiplicity
, then we can also obtain
which is impossible, so we have
thus, in view of this fact and Lemma 2.4, the first result (i ) of Theorem 1.3 is proved.
Subcase 1.2. Assume that
. If
of (1.1) has at least a non-zero repeated root as a polynomial in
, it follows from Lemma 2.5 that
Set
. Now we consider the case of all non-zero roots of
are simple, say
, then (1.1) can be written as
(3.6)
where
is a polynomial in z. It is obviously that
are not solutions of the above equation, so it satisfies the first condition of Lemma 2.3. The aim is to prove the inequality
Let
be generic zero of
with multiplicity
. If
, considering the zeros of
of (3.6), for example
, we know that
has a pole of order at least
at
.
Assume
is a pole of
with multiplicity
and
is a pole of
with multiplicity
. Shifting (3.6) forward and backward and we can obtain that
has a pole of order k at
and a pole of order k at
. Thus, we have
(3.7)
Assume
or
is a pole of
with multiplicity
. By iterating (3.6) as before, we have that
is a pole of
with multiplicity k or
is a pole of
with multiplicity k, we also have
(3.8)
Similarly, we can also obtain (3.7) and (3.8) for any
. So, we get
through the Lemma 2.3, which is a contradiction. Consequently, we obtain
.
Next, we turn to the proof of another side of the inequality, that is
Assume that
and let
be a generic zero of
with multiplicity
. From (3.6), we know that
has a pole of order at least
in
. Here we only consider the case of
is a pole of
with multiplicity at least
. By shifting (3.6) up,
It follows that
is a pole of
with multiplicity at least
, and thus
since
. Lemma 2.3 indicates that
, which contradicts the assumption. Therefore, the result
is proved.
Case 2.
is a root of
. We shall prove that
is a zero of
with the multiplicity at most k. Suppose that
,
and
is a polynomial in
with degree at most
. Then (1.1) can be rewritten as
(3.9)
Let
be a generic zero of
with multiplicity
. Then
is a pole of
with multiplicity at lease
since
. Without loss of generality, we only consider the case when
has a pole of order
at
. By shifting (3.9) up, one has
(3.10)
If
, it follows from the above equation that
is a pole of
with multiplicity
, and thus
could be finite. This means that
where
. Notice that
is not a solution of (3.9). Using Lemma 2.3, we can obtain
, which is a contradiction.
If
for
, it follows from (3.10) that
is a pole of
with multiplicity
. Hence,
,
where
. On the basis of Lemma 2.3 we can also get a contradiction. This completes the proof of Theorem 1.3.
Acknowledgements
The author also wants to express thanks to the anonymous referees for their suggestions and comments that improved the quality of the paper.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 12171050, 12071047) and the Fundamental Research Funds for the Central Universities (Grant No. 500421126).