Research Article | Open Access

Guoqiang Dang, Jinhua Cai, "Entire Solutions of the Second-Order Fermat-Type Differential-Difference Equation", *Journal of Mathematics*, vol. 2020, Article ID 4871812, 8 pages, 2020. https://doi.org/10.1155/2020/4871812

# Entire Solutions of the Second-Order Fermat-Type Differential-Difference Equation

**Academic Editor:**Basil K. Papadopoulos

#### Abstract

In this paper, the entire solutions of finite order of the Fermat-type differential-difference equation and the system of equations have been studied. We give the necessary and sufficient conditions of existence of the entire solutions of finite order.

#### 1. Introduction

In this work, we assume that the readers are familiar with general definitions and fundamental theories of Nevanlinna theory [1â€“3]. is a meromorphic function which means is meromorphic in the finite complex plane . If has no poles, we call is an entire function. We denote by , any function satisfying, , outside of a possible exceptional set of finite logarithmic measure. For a meromorphic function , we define its shift by and its difference operators by

The classic Fermat-type functional equationhas been intensively studied in recent years. If , equation (2) has no transcendental meromorphic solutions [4]. If , equation (2) has no transcendental entire solutions [4]. If , all entire solutions are the forms of and , where is any entire function [5].

Yang [6] studied the following generalized Fermat-type functional equation:where and are positive integers and obtained the following theorem.

Theorem 1. *If , then equation (3) has no nonconstant entire solutions and .*

For further research solutions of the Fermat-type functional equation, Yang and Li [7] considered the following special Fermat-type functional equation:and they obtained the following theorem.

Theorem 2. *The transcendental meromorphic solutions of (4) must satisfy , where and are nonzero constants.*

In the following, is a nonzero constant, unless otherwise specified.

Liu [8] investigated the following special Fermat-type functional equation:

He obtained that each transcendental entire solution of (5) with the finite order must satisfy , where , and . Liu et al. [9] proved that the nonconstant finite-order entire solutions of (5) must have order one. Then, Liu et al. [10] considered the entire solutions of the following difference equations:and obtained the following results.

Theorem 3. *Equation (6) has no nonconstant entire solution if or ; has no transcendental entire solution with finite order if ; and has the general solutions , where is any entire function periodic with period .*

Theorem 4. *Equation (7) has no transcendental entire solution with finite order, provided that , where and are positive integers.*

Liu et al. [10] obtained that satisfies the equation when ; satisfies the equation when , where ; and satisfies the equation when , where or and is an integer.

Liu et al. [10] also investigated the following special Fermat-type functional equation:and obtained the following theorem.

Theorem 5. *Equation (8) has no transcendental entire solutions with finite order, provided that , where and are positive integers.*

Liu et al. built exact solutions for equation (8); for , the equation has a transcendental entire solution , where and ; for , the equation admits a transcendental entire solution , where , and , where is odd; and for , the equation admits a transcendental entire solution , where is an integer, and is a constant.

In 2019, Liu et al. [11] researched the entire solutions with finite order of the Fermat-type differential-difference equationand the following system of differential-difference equations:

In 2019, Dang and Chen [12] obtained the meromorphic solutions of the following special Fermat-type functional equation:

In the following, we will consider the entire solutions with finite order of the Fermat-type differential-difference equationand obtain the following result.

Theorem 6. * be entire solutions of finite order of differential-difference equation (12) if and only if be the following forms:where , and are constants, , , and ; , where for , for , and for .*

Obviously, from Theorem 6, we immediately get the following example.

*Example 1. *The transcendental entire function solutions with finite order of the differential-difference equationmust satisfywhere , and are constants, , , and ; , where .

Then, we will study the following system of differential-difference equations:and obtain the next result.

Theorem 7. * be the transcendental entire solutions of finite order of the system of differential-difference equation (16) if and only if be the following forms:where , , and ;where , , and ; for ; , where for ; , where for ; and for , where , and are constants, .*

Obviously, from Theorem 7, we immediately get the following example.

*Example 2. *Let . Then, the transcendental entire function solutions with finite order of the system of differential-difference equationsmust be in the following forms:where , , and ;where , , and .

#### 2. Lemmas

Lemma 1. *(see [1]). Let be a meromorphic function, , being nonconstant, satisfying and . If andwhere and , then .*

Lemma 2. *(see [1]). Suppose that are meromorphic functions and are entire functions satisfying the following conditions:*(1)*(2)**The orders of are less than those of for **Then, .*

Lemma 3. *(Hadamardâ€™s factorization theorem; see [1]). Let be an entire function of finite order with zeros and -fold zero at the origin. Then,where is the canonical product of formed with nonnull of and is a polynomial of degree less than .*

#### 3. Proof of Theorem 6

Suppose that is an entire solution with finite order which satisfies (12). We rewrite (12) as follows:

It follows that and have no zeros. By Lemma 3, we havewhere is a polynomial.

From (25), we get

*Case 1. *Suppose that is a transcendental entire function. Then, it follows from (26) that is a nonconstant polynomial. Let ; then, . It follows from (26) thatIt follows from (27) thatThen,Combining (28) and (30), we getThis impliesTherefore,If , then for , we haveand by (33) and Lemma 2, we obtain , which is contradicting.

Hence, . Let , where ; then, , , , and . Then, by (33), we haveThis impliesThen,This impliesThen,This impliesThen,Obviously, , and . From (41) and Lemma 2, we obtain thatBy (42), it is easy to get and . It follows from (42) that . Thus, it follows from (26) that

*Case 2. *Suppose that is a polynomial. Then, it follows from (26) that is a constant, and .â€‰If , then . It follows from (12) that .â€‰If , then . It follows from (12) that .â€‰If , then . It follows from (12) that .Thus, Theorem 6 is proved.

#### 4. Proof of Theorem 7

Suppose that be an entire solution with finite order which satisfies equation (16). We rewrite (16) as follows:

It follows that

By Lemma 3, we havewhere and are polynomials.

From (46), we get

*Case 3. *Suppose that is a transcendental entire function with finite order. Then, it follows from (47)â€“(50) that and are two nonconstant polynomials. Let ; then, .

It follows from (47)â€“(50) thatThen,From (51) and (55), we getFrom (52) and (56), we getIt follows thatThen,If are nonconstant polynomials for any , then we obtain that are nonconstants for any . Assume ; then, does not vanish; then, we multiply on both sides of (61), and we haveThen, by Lemma 1, we have ; this contradicts with .

Hence, there exist such that or , where is a constant.

Suppose that . Obviously, . We assert that . Otherwise, we assume that ; then, for any and for any .

From (63), we haveThen, it follows from (64) and Lemma 2 that . This implies is a constant, and , which is contradicting; therefore, the assertion is proved.

Accordingly, and . Thus, we can assume that ; then,