Abstract

The purpose of this article is to give the details of finding the transcendental entire solutions with finite order for the systems of nonlinear partial differential-difference equations where are polynomials in ; are positive integers, and . We obtain that there exist some pairs of the transcendental entire solutions of finite order for the above system, which is a very powerful supplement to the previous theorems given by Xu and Cao and Xu and Yang.

1. Introduction

In 1970, Yang [1] proved that the functional equations have no nonconstant entire solutions, if are positive integers satisfying . After this result, with the aid of the Nevanlinna theory and the difference analogues of the Nevanlinna theory (see [2–6]), there were rapid developments on complex differential and difference equations in one and several complex variables. Some classical results and topics in different fields are considered in difference versions, for example, difference Riccati equations, difference Painlevé equations, and difference Fermat equations (see [7–14]). Recently, Cao and Xu [15–17] investigated the existence of the entire and meromorphic solutions for some Fermat-type partial differential-difference equations by utilizing the Nevanlinna theory and difference Nevanlinna theory of several complex variables [18, 19] and obtained the following theorems which is an extension of the previous results given by Liu and his collaborators (see [20–24]).

Theorem 1 (see ([16], Theorem 1.1)). Let . Then, the Fermat-type partial differential-difference equationdoes not have any transcendental entire solution with finite order, where and are two distinct positive integers.

Theorem 2 (see ([15], Theorem 3.2)). Let . Suppose that is a nontrivial meromorphic solution of the Fermat type partial difference equationsorwhere , and is a nonzero meromorphic function on with respect to the solution , that is . If , then

Remark 3. Let and , then the above equations becomewhich can be called as the partial difference equations of Fermat type.

In 2020, the first author and his coauthors discussed the transcendental entire solutions with finite order for the systems of partial differential difference equations and gave the conditions on the existence of the finite-order transcendental entire solutions for the following systems, which are some extension and improvements of the previous results given by Xu and Cao and Gao [16, 25].

Theorem 4 (see ([26], Theorem 1.2)). Let , and be positive integers. If the following system of Fermat-type partial differential-difference equationssatisfies one of the conditions(i);(ii) for , .Then, system (6) does not have any pair of transcendental entire solution with finite order.

Remark 5. Here, is called as a pair of finite-order transcendental entire solutions for systemif are transcendental entire functions and .

Remark 6. The condition implies . Thus, a question rises naturally: what will happen on the existence of transcendental entire solutions with finite order when in system (6)?

In fact, we give the following example to explain that system (6) has a pair of transcendental entire solutions with finite order when and , that is,

Example 1. LetThen, is a pair of transcendental entire solutions of system (8) with and .

Corresponding to system (6), we further consider the following system of the partial differential difference equation

where are two nonzero polynomials in and obtained.

Theorem 7. Let , be positive integers satisfies one of the conditions(i);(ii) for , .Then, system (10) does not have any pair of transcendental entire solutions with finite order.

The following example shows that the conditions for and are precise and the existence of finite-order transcendental entire solutions for the system (10) when , and , that is,

Example 2. LetThen, is a pair of transcendental entire solutions of system (11) with and .

Remark 8. In Sections 3 and 4, we give the details proceeding for obtaining a class of finite-order transcendental entire solutions for systems (8) and (11).

Next, we continue to discuss the existence of the finite-order transcendental entire solutions for several systems including both the difference operator and the partial differential such as

where are constants in . It is easy to find the finite-order transcendental entire solutions for systems (13) and (14). For , system (5) has a pair of finite-order transcendental entire solutions of the formsand for , system (14) has a pair of finite-order transcendental entire solutions of the formswhere satisfy and . Furthermore, we can give the finite-order transcendental entire solutions for systems (13) and (14) when and easily.

Example 3. The functionis a pair of transcendental entire solutions with for system (13) when , , and .

Example 4. The functionis a pair of transcendental entire solutions with for system (14) when , and .

Corresponding to systems (13) and (14), we can also obtain the solutions of the following systems

where are constants in . In fact, for , then systems (19) has a pair of solutions with the formswhere are two period functions with period , and for and , then system (20) has a pair of solutions with the formswhere are two period functions with period , and , are constants satisfying

2. Proof of Theorem 7

The following lemmas will be used in this paper.

Lemma 9 ([27, 28]). Let be a nonconstant meromorphic function on and let be a multi-index with length . Assume that for some . Then,holds for all outside a set of finite logarithmic measure , where

Lemma 10 ([18, 19]). Let be a nonconstant meromorphic function with finite order on such that , and let . Then, for ,holds for all outside a set of finite logarithmic measure .

Lemma 11 (see [29]). Let be a nonconstant meromorphic function on . Take a positive integer and take polynomials of and its partial derivatives:where are finite sets of distinct elements and are meromorphic functions on with . Assume that satisfies the equationsuch that , and are differential polynomials, that is, their coefficients satisfy . If , thenholds for all possibly outside of a set with finite logarithmic measure.

Proof. Let be a pair of transcendental entire functions with finite-order satisfying system (10). Here, we will discuss two following cases.

Case 1. . In view of Lemma 10, the following conclusions thatholds for all outside of a possible exceptional set of finite logarithmic measure . Thus, we can deduce from (29) thatfor all . By using Lemma 9 and Lemma 11, it follows from (30) thatfor all . Similarly, we haveIn view of (31) and (32), it yieldsIn view of , this is impossible since are transcendental entire functions.