Entire solutions of two certain Fermat-type ordinary di ﬀ erential equations

: In this article, we investigate the precise expression forms of entire solutions for two certain Fermat-type ordinary di ﬀ erential equations:

Abstract: In this article, we investigate the precise expression forms of entire solutions for two certain Fermattype ordinary differential equations: in by making use of the Nevanlinna theory for meromorphic functions, where a 0 , a 1 , and a 2 are the complex numbers with ≠ a 0 0 and + ≠ a a 0 1 2

Introduction
Throughout this article, by meromorphic functions, we will always mean meromorphic functions in the complex plane.We adopt the standard notations in the Nevanlinna theory of meromorphic functions as explained in Hayman [1], Laine [2], and Yang [3].It will be convenient to let E denote any set of positive real numbers of finite linear measure, not necessarily the same at each occurrence.For a nonconstant meromorphic function f , we denote by T r f , ( ) the Nevanlinna characteristic of f and by S r f , ( ) any quantity satisfying and proved that the entire solutions f and g in the complex plane (and thus in n ) must be both constants when ≥ n 3.After a few decades, Gross [5,6] showed that all meromorphic solutions f and g of the Fermat functional equation are characterized as follows: (i) for = n 2, the entire solutions = f h sin and = g h cos , where h is an entire function in .The meromorphic solutions 2 , where β is a meromorphic function in ; (ii) for > n 2, there are no nonconstant entire solutions; (iii) for = n 3, the meromorphic solutions , where ℘ is the Weierstrass elliptic function satisfying ; and (iv) for > n 3, there are no nonconstant meromorphic solutions.
Henceforth, studying the precise forms of meromorphic solutions on the Fermat-type functional equations attracts a lot of attention.In 1970, Yang [7] investigated that the Fermat-type function equation has no nonconstant entire solutions when + < 1 n m 1 1 . With the development of the Nevanlinna theory, many mathematicians try to characterize meromorphic solutions of the Fermat-type differential equations.For example, Yang and Li [8] showed the transcendental meromorphic solutions of a certain type of nonlinear differential equations and obtained the following results., then a transcendental meromorphic solution of the equation ( ) , where P, Q, and R are the polynomials, and = PQ a.If all b k are the constants, then is a nonzero constant that satisfies the following equations: , and a 3 be the nonzero meromorphic functions.Then, a necessary condition for the differential equation to have a transcendental meromorphic solution satisfying , and ∕ ≡ a a constant 1 3 . Furthermore, Yang and Li [8] proposed a conjecture: let a 1 , a 2 , and a 3 be the nonzero polynomials.Then, the equation ( ) has no transcendental meromorphic solution when ∕ a a 1 3 is a nonzero constant and ∕ a a 2 3 is not the square of any rational function.Later, Tang and Liao [9] denied the conjecture and characterized the precise expression form of transcendental meromorphic solutions of the Fermat-type differential equation , where P and Q are the polynomials.In must be entire functions, and the following assertions hold: (ii) For = n 2, either = a 0 and the general solutions Here, Motivated by the quadratic differential equations in [11,12], Wang et al.
[13] investigated the existence and precise expression form of entire solutions for the Fermat-type differential-difference equation: by making use of a transformation: , where ), and ω is a complex number with ≠ ω 0, 1 2 .They proved the entire solutions f with finite order of the aforementioned equation, and it only has the form , where ≠ A 0, = C i 0, 1, 2, 3 i ( ) are the constants.
On the other hand, the Nevanlinna theory of meromorphic functions has been widely used to deal with meromorphic solutions of the Fermat-type partial differential equations.The research of the Fermat-type partial differential equations goes back to the Pythagorean functional equations and induces the well-known eikonal equations in [18], where g and p are the polynomials in 2 , and obtained two interesting and important results as follows: Theorem D. (See [17, Theorem 2.1]) Let g be a polynomial in 2 .Then, u is an entire solution of the partial differential equation: in 2 if and only if: where f is an entire function in satisfying c 1 and c 2 are two constants satisfying , and ϕ 1 and ϕ 2 are the entire functions in satisfying .
Theorem E. (See [18, Theorem 2.1]) Let ≢ p 0 be a polynomial in 2 .Then, u is an entire solution to the partial differential equation: in 2 if and only if: where , and ϕ 1 and ϕ 2 are the polynomials in satisfying that: Moreover, the forms of f , ϕ 1 , and ϕ 2 are also obtained in [18, Theorem 2.1].
Inspired by the aforementioned results, the purpose of this study is to characterize the entire solutions of the Fermat-type ordinary differential equations: and in , where a 0 , a 1 , and a 2 are the complex numbers with , while p and g are the polynomials in .This article is organized as follows.We will describe the entire solutions for equation (1.1) in Section 2. The investigation of the entire solutions for equation (1.2) will be divided into two cases in Section 3, that is, the case where one of a 1 and a 2 is zero and the case ≠ a 0 1 , ≠ a 0 2 , and ≠ a a 1 2 .The main theorems and corollary will be proved in Section 4. In addition, we shall present some examples to illustrate the existence of entire solutions for equations (1.1) and (1.2).
2 Ordinary differential equations In this section, we will describe the entire solutions of the Fermat-type ordinary differential equation (1.1).
Theorem 2.1.Let a 0 , a 1 , and a 2 be the complex numbers with ≠ a 0 , and p be a nonconstant polynomial in .Then, f is an entire solution of the Fermat-type ordinary differential equation: if and only if one of the following holds: , then the polynomial p takes the form = p φ 2 2 , and then, , where φ is a polynomial in , and c is an arbitrary complex number; where k is a nonzero constant, α and β are the factors of p such that = αβ p and that: Note that not every polynomial p such that equation (2.1) has nonconstant entire solutions.If p is an irreducible polynomial in , then we have the following corollary.
Corollary 2.2.Let a 0 , a 1 , and a 2 be the complex numbers with ≠ a 0 0 and , and p be an irreducible polynomial in .Then, equation (2.1) does not have any nonconstant entire solutions.
We present four concrete examples to illustrate each of the two forms in Theorem 2.1.

Example 1. Consider the equation
, and c is an arbitrary complex number.

Example 2. Let
( ) is an entire solution of the equation: , and = a 0 2 .Moreover, it is easy to verify that α and β satisfy (2.2).
Example 3. Consider the equation ( ) ( ) is an entire solution of this equation, which is the form (ii) of that in Theorem 2.1 with = , and = a 2 2 .Furthermore, an easy computation shows that α and β satisfy (2.2).

Example 4. Let
( ) is an entire solution of this equation: . Moreover, one can check that α and β satisfy (2.2).
3 Ordinary differential equations ( ( In this section, we will deal with entire solutions of the Fermat-type ordinary differential equation: where a 0 , a 1 , and a 2 are the complex numbers with ≠ a 0 0 and , it is easy to solve the solution ) of equation (3.1) according to the theory of the first-order linear differential equations, where c is an arbitrary complex number.Therefore, in the following, we only consider the case ≠ a a 1 2 .First, we consider the case where one of a 1 and a 2 is zero.Without loss of generality, one can assume that = a 0 1 , then equation (3.1) becomes For the aforementioned equation, we have the following theorem.
Theorem 3.1.Let a 0 and a 2 be the nonzero complex numbers, and g be a polynomial in , where a and b are the complex numbers, then the entire solutions of equation (3.2) are characterized as follows: .
The following example shows that the forms of the solutions in Theorem 3.1 are precise.
. Then, it is easy to verify that are the entire solutions of the equation: ( ) is an entire solution of the equation: Next, for the case ≠ a 0 1 and ≠ a 0 2 , then we obtain the following theorem.
Entire solutions of two certain Fermat-type ordinary differential equations  5 Theorem 3.2.Let a 0 , a 1 , and a 2 be the nonzero complex numbers with ≠ a a 1 2 , and g be a polynomial in .The Fermat-type ordinary differential equation (3.1) has entire solutions if and only if , where a and b are the complex numbers, and we summarize all the possible entire solutions of equation (3.1) as follows: where b and d are the arbitrary complex numbers; (ii) .
From Theorems 3.1 and 3.2, we deduced the following corollary directly.

( ) and
are the entire solutions of the equation: Example 8. Consider the equation . It is easy to verify that = ± +

Proofs
Proof of Theorem 2.1.The sufficiency is clear.In fact, let f be a function defined as the form (i) in Theorem 2.1.It follows easily that f satisfies equation (2.1).On the other hand, let f be a function defined as the forms (ii) in Theorem 2.1, and one can check that f satisfies equation (2.1) by using (2.2).
To prove the necessity, let f be an entire solution of equation (2.1).If = a a 1 2 , we now rewrite equation (2.1) as , which is an entire function in .We then assert that φ is a polynomial.Otherwise, if φ is a transcendental entire function, we conclude that p is also a transcendental entire function by . But this contradicts that p is a nonconstant polynomial.Hence, φ is a polynomial.
Furthermore, it is easy to solve the solution ( ) according to the theory of the first-order linear differential equations, where c is an arbitrary complex number.Next, we consider the case ≠ a a 1 2 .Equation (2.1) can be written as: From the aforementioned equation, we see that there exists a factor α of p (in the ring of polynomials) and an entire function h in such that: and where ≔ β p α is also a polynomial in .Solving + ′ a f a f from the aforementioned two identities yields that: and In view of (4.3) and (4.4), it follows that: and Differentiating (4.5) yields that By combining (4.6) and (4.7), we have that: We next consider two different cases in the following.Case 1.The entire function h is not a constant.We claim that Otherwise, by (4.8), we obtain that Applying the Nevanlinna theory, we can obtain a contradiction.In fact, if T r F , ( ) denotes the Nevanlinna characteristic function of a meromorphic function F in , then it follows from (4.10) that: .This is a contradiction with the theorem [20]: If F is a transcendental meromorphic function in and G is a transcendental entire . Therefore, equation (4.9) is proved.
Entire solutions of two certain Fermat-type ordinary differential equations  7 We then deduced that .By combining with (4.9) and (4.12), we obtain Note that = p αβ, then ′ = ′ + ′ p α β αβ .Furthermore, (4.13) can be written as: which shows that according to the theory of the first-order linear differential equations, where c is a nonzero complex number.If = − a a 1 2 , then = p c is a nonzero complex number, which is a contradiction with the assumption that p is a nonconstant polynomial.If ≠ − a a 1 2 , which is also a contradiction since the left-hand side of is a polynomial but the right-hand side is a transcendental entire function.Therefore, case 1 cannot occur.
Case 2. The entire function h is a constant.If h is a constant, one can assume that where k is a nonzero complex number.From (4.5), we have In view of (4.8), it follows easily that α and β satisfy that: This completes the proof of Theorem 2.1.□ Proof of Corollary 2.2.Let f be an entire solution of equation (2.1).Note that p is an irreducible polynomial in , and it may be assumed that , where a and b are the complex numbers with ≠ a 0. Given the proof of Theorem 2.1, then we only need to consider Case 2 in Theorem 2.1.
Applying (4.16) once again, that is, , where c is a nonzero complex number.Substituting α and β into (4.17)gives which is impossible since the left-hand side of equation (4.18) is a constant but the right-hand side is a nonconstant polynomial.Therefore, we finish the proof of Corollary 2.2.□ Proof of Theorem 3.1.Sufficiency is readily apparent.In fact, let f be a function defined as the forms (i) and (ii) in Theorem 3.1.It is easy to check that f satisfies equation (3.2).To prove the necessity, let f be an entire solution of equation (3.2).We rewrite equation (3.2) as: Therefore, there exists an entire function h in such that: and then that which it follows that: (4.20) by using the results (cf.[21,22]).Furthermore, we have which shows that: Thus, the proof of Theorem 3.1 is completed.□ Proof of Theorem 3.2.The sufficiency is easy to check, which is left as an exercise for the interested reader.
To prove the necessity, let f be an entire solution of equation (3.1).We rewrite equation (3.1) as: Similar to the arguments in the proof of Theorem 3.1, we see that there exists an entire function h in such that ) and b and d are the arbitrary complex numbers.Substituting g and h into (4.28)shows that:

(
) and b is an arbitrary complex number.
Next, we consider two different cases. ).An easy computation shows that: ), as → ∞ r and ∉ r E. In 1927, Montel [4] first studied the Fermat functional equation + 2019, Han and Lü [10] studied the meromorphic solutions of the differential equation + by discussing for positive integers n, where ∈ a b , are the constants.They obtained the following result.Theorem C. (See [10, Theorem 2.1]) The meromorphic solutions f of the following differential equation: is the form (ii) of that in Theorem 3.1 with = a an entire solution of this equation, which is the form (ii) of that in Theorem 3.2 with = a which was considered by Li in [14] and also discussed in[15]and[16].
By way of illustration, we give two explicit examples for each of the forms in Theorem 3.2.
Suppose first that the entire function h is not a constant.Similar to the arguments in the proof of Theorem 2.1, we obtain that Entire solutions of two certain Fermat-type ordinary differential equations  11 Therefore, this completes the proof of Theorem 3.2.□