Issue 
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 3, June 2022



Page(s)  195  200  
DOI  https://doi.org/10.1051/wujns/2022273195  
Published online  24 August 2022 
Mathematics
CLC number: O 174.5
On Entire Solutions of Two Certain Types of NonLinear DifferentialDifference Equations
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou
215009 , Jiangsu, China
^{†} To whom correspondence should be addressed. Email: alexehuang@sina.com
Received:
20
January
2022
In this paper, we mainly investigate entire solutions of the following two nonlinear differentialdifference equations and , where is an integer, are nonzero constants, is a nonvanishing polynomial and is a nonconstant polynomial. Under some additional hypotheses, we analyze the existence and expressions of transcendental entire solutions of the above equations.
Key words: entire solutions / nonlinear differentialdifference equations / order
Biography: LI Jingjing, female, Master candidate, research direction: complex analysis. Email: 1309171701@qq.com
Foundation item: Supported by the National Natural Science Foundation of China (11971344)
© Wuhan University 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction and Main Results
In this paper, Nevanlinna value distribution theory is a useful tool and its standard notations as well as wellknown theorems can be found in Refs.[1,2]. Let be a meromorphic function in the complex plane . The proximity function, counting function of poles and Nevanlinna characteristic function with regard to are denoted by , and , respectively. We also use to denote any small quantity of satisfying , as , possibly outside of a set of finite logarithmic measure. The order of growth of and the exponent of convergence of zeros of are defined by
A large number of studies concerning the solvability and existence for entire and meromorphic solutions of nonlinear complex differential equations, difference equations and differentialdifference equations can be found in Refs.[312].
The equation has exactly three nonconstant entire solutions , according to Yang and Li^{ [11]}. It is worth noting that , that is, is a linear combinations of and . As a result, it is meaningful to investigate the existence of the solutions of the differential equation
where are nonzero constants, and denotes a differential in with degree .
For Eq.(1), Li^{ [6]} obtained the following theorem.
Theorem 1^{ [6]} Let be an integer, be a differential polynomial in of degree at most , and are three nonzero constants. If is an entire solution of equation (1), then , where and are constants and .
Later, Li^{ [13]} proved by weakening the requirement on the degree of in Theorem A.
Theorem 2^{ [13]} Let be an integer, be a differential polynomial in of degree at most , and are three nonzero constants. If is a meromorphic solution of equation (1) and , then there exist two nonzero constants and a small function of such that .
Then it is natural to ask: is there a similar result if are polynomials? The question was answered by the following theorem.
Theorem 3^{ [14]} Let be an integer, be a differential polynomial in of degree . Let and be two nonzero polynomials, be two nonzero constants with rational. Then the differential equation
has no transcendental entire solutions.
The exponential polynomial plays a key role in the study of nonlinear complex differential equations and many interesting properties have been mentioned, for example, in 2012, Wen, Heittokangas and Laine^{ [15]} investigated and classified the finite order entire solutions of the following equation
in terms of growth and zero distribution, where is an integer, , are polynomials.
Recall that an exponential polynomial of the form
where are polynomials in such that is called an exponential polynomial of degree .
In the following, we define two classes of transcendental entire functions:
is a nonconstant polynomial
is a nonconstant polynomial and
Theorem 4^{ [15]} Let be an integer, let , and let be polynomials such that is not a constant and . Then we identify the finite order entire solutions of equation (3) as follows:
(i) Every solution satisfies and is of mean type;
(ii) Every solution satisfies if and only if ;
(iii) A solution belongs to if and only if . In particular, this is the case if ;
(iv) If a solution belongs to and if is any other finite order entire solution to (3), then , where ;
(v) If is an exponential polynomial solution of the form (4), then . Moreover, if , then .
Following that, Liu ^{[16] }and Chen et al ^{[5]} investigated the cases when in Eq.(3) is replaced by or , respectively, and obtained a special form of solutions of the equations.
According to the foregoing results, all equations have only one main term on the left side.
Chen et al ^{[17]} proposed the following question based on whether a similar result can be obtained when there are two main terms.
In 2021, Chen, Hu and Wang^{ [17]} considered the following differentialdifference equation
and obtained the following result.
Theorem 5^{ [17]} If is a transcendental entire solution with finite order of Eq.(5), then the following conclusions hold:
(i) If for and for , then every solution satisfies.
(ii) If and is a solution of (5), which belongs to , then
or
where .
Inspired by the above results, we consider the following types of equations:
and
Theorem 6 Let and be integers, and be nonzero constants such that . Suppose that and . Then Eq.(6) has no transcendental entire solutions.
Theorem 7 Let and be integers, be a nonvanishing polynomial and be a nonconstant polynomial. Suppose that and be nonzero constants such that . If Eq.(7) admits a finite order transcendental entire function with , then the following conclusions hold:
(i) If for , then every solution satisfies .
(ii) If a solution belongs to , then
or
where .
1 Preliminary Lemmas
In order to prove our results, we introduce the following Lemmas.
Lamma 1 ^{[18] } Let be a nonconstant meromorphic function and let . Then if the growth order of is finite, we have
And if the growth order of is infinite, we have
possibly outside a set of finite linear measure.
Lamma 2 ^{[19]} Let be a transcendental meromorphic function of finite order. Then
where denotes any quantity as , possible outside of a set of finite logarithmic measure.
In the study of complex differentialdifference equations, the Clunie lemma^{[10]} plays an important part. Let's recall the Clunie lemma's precise definition.
Lamma 3 (Clunie's lemma) ^{[10]} Let be a transcendental meromorphic solution of finite order of differentialdifference equation
where are polynomials in and its derivatives and its shifts with small meromorphic coefficients. If the total degree of as a polynomial in and its derivatives and its shifts is , then
for all out of a possible exceptional set of finite logarithmic measure.
Lamma 4 ^{[18]} Let be meromorphic functions, and let be entire functions satisfying
(i) ;
(ii) when , then is not a constant;
(iii) when , then
where is of finite linear measure or logarithmic measure.
Then .
2 Proof of Theorem 6
Let be a transcendental entire solution of Eq.(6).
Set ,. Then we write Eq.(6) as
Differentiating (8) yields
Eliminating from (8) and (9), we get
Differentiating (10) yields
Eliminating from (10) and (11), we get
Note that
and
Substituting into (12), we have
where
Note that and is a differentialdifference polynomial in and the total degree is at most 1. By Lemma 3, we obtain
If , then
which yields a contradiction.
Thus, . From (13), we get . Then has the form
where and are constants.
By integration, we have
where and are constants.
Similarly, by and (12) we can also obtain , then we have
where and are constants. From (17) and (18), we obtain
Dividing both sides of (19) by , we get
Since , and , by (20) and Lemma 4, we obtain , that is . Similarly, dividing both sides of (19) by , we can get . So , this is a contradiction.
Therefore, Eq.(7) has no transcendental entire solutions.
3 Proof of Theorem 7
Suppose that is a transcendental entire solution of finite order of Eq.(7). In what follows, we firstly prove that .
Case A Suppose that . By Lemma 1, Lemma 2 and (7), we have
then .
Note that , therefore . Denote . Rewriting (7) in the following form:
Differentiating (22) yields
where
Eliminating from (22) and (23), we obtain
Subcase A1 If , by (24) and Lemma 4, we have
which contradicts with .
Subcase A2 If , by (24), we can get
From (25) and Lemma 4, we have
Dividing with on both sides of (26), we obtain
Since , which yields a Riccati differential equation:
where .
By a simple calculation, we can get
then
where are constants.
If , then , this is a contradiction.
If , then , which contradicts with .
Case B Suppose that . Denote ,. Equation (7) can be written as:
Differentiating (29) yields
where .
Eliminating from (29) and (30), we get
where .
Note that and is a differentialdifference polynomial in and the total degree is at most 1. By Lemma 3, we obtain
and .
If , since is a transcendental entire function, then
which is a contradiction.
If , from (31) yields , then
By integration, we see that there exists a constant such that
Substituting (32) into (7) yields
Since is a transcendental entire function with , according to Hadamard decomposition theorem, can be written in the form
where is the canonical product formed by zeros of and is a nonconstant polynomial which satisfies
Substituting (34) into (33), we have
Combining (35) with (36), we can see that the left order of (36) is greater than 1, but the right order is 1, which is a contradiction.
Therefore, . From (7), Lemma 1 and Lemma 2, we obtain
Note that , then , that is .
The conclusion (i) is proved. Next we will prove the conclusion (ii).
If belongs to , and noting that , we define and , where , and .
Substituting and into (7) yields
Dividing both sides of (38) by , we get
We distinguish four cases below.
Case 1 Suppose that and . By Lemma 4 and (39), we obtain , which is a contradiction.
Case 2 Suppose that and .
If , by (39) and Lemma 4, we have , this is a contradiction.
If , then , substituting these into (39) yields
by Lemma 4 and (20), we have
then reduces to a nonzero constant, and .
Case 3 Suppose that and .
If , by (39) and Lemma 4, we have
this is a contradiction.
If , , substituting these into (39) yields
by Lemma 4 and (41), we have
then reduces to a nonzero constant, and .
Case 4 Suppose that and .
If and are pairwise distinct from each other. By Lemma 4 and (39), we have
this is a contradiction.
If only two of and coincide, without loss of generality, suppose that , then (39) can be written as:
From Lemma 4 and (42) yield , this is a contradiction.
If , then we write (39) as:
By Lemma 4, we have , which implies a contradiction. The proof is complete.
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