Entire Solutions for Complex Systems of the Second-Order Partial Differential Difference Equations of Fermat Type

Abstract

is article is mainly concerned with the existence and the forms of entire solutions for several systems of the second-order partial differential difference equations of Fermat type (α(z 2 f 1 (z 1 , z 2)/zz 2 1) + β(z 2 f 1 (z 1 , z 2)/zz 2 2)) n 1 + f 2 (z 1 + c 1 , z 2 + c 2) m 1 � 1 (α(z 2 f 2 (z 1 , z 2)/zz 2 1) + β(z 2 f 2 (z 1 , z 2)/zz 2 2)) n 2 + f 1 (z 1 + c 1 , z 2 + c 2) m 2 � 1 and (z 2 f 1 (z 1 , z 2)/zz 2 1) 2 + f 2 (z 1 + c 1 , z 2 + c 2) 2 � 1 (z 2 f 2 (z 1 , z 2)/zz 2 1) 2 + f 1 (z 1 + c 1 , z 2 + c 2) 2 � 1. Our results about the existence and the forms of solutions for these systems generalize the previous theorems given by Xu and Cao, Gao, Liu, and Yang. In addition, we give some examples to explain the existence of solutions of this system in each case.

Entire solutions of system of Fermat-type difference and partial differential-difference equations in $ \mathbb{C}^2 $

“…Liu and Xu [32], further extended Theorem I by considering Fermat-type systems of second-order partial differential-difference equation, obtained the following result.…”

“…As far as our knowledge is concerned, although there are some important and remarkable results about the existence and forms of transcendental entire solutions of Fermat-type difference and partial differential-difference equations in several complex variables (see [15,45,48,49,51,57]), there are only few results about the solutions of the system of Fermat-type equations in the literature (see [5,32,50]). Some of these results are listed as follows.…”

“…Besides finding the solutions of Fermat-type difference equations, Fermat-type partial differential-difference equations are also studied by many researchers (see [25,32,45,49,57]). For example, Xu and Cao [45] have investigated the entire solutions of Fermat-type partial differential-difference equation…”

Results on Solutions for Several Systems of Partial Differential-Difference Equations with Two Complex Variables

“…and let x + y � (x 1 + y 1 , x 2 + y 2 ) for any x � (x 1 , x 2 ), y � (y 1 , y 2 ) ∈ C 2 . In this paper, based on the results of Xu and Cao [18] and Liu and Xu [15,17], we further obtain some more generalized results, considering mixing higher order partial diferential-diference equations (system). We obtain the following.…”

“…x 1 + I 2 x 2 � p, have been studied (see [15][16][17]), and a number of results concerning the existence and forms of solutions for partial diferential equations in several complex variables have been studied. In very recent years, with the rapid development of the diference Nevanlinna theory, Xu and Cao [18] discussed the existence of the entire solutions for the Fermat-type partial diferential-diference equation:…”