Notes on the Existence of Entire Solutions for Several Partial Differential-Difference Equations
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Notes on the Existence of Entire Solutions for Several Partial Differential-Difference Equations Hong Yan Xu1 · Hua Wang2 Received: 11 March 2020 / Accepted: 19 August 2020 © Iranian Mathematical Society 2020
Abstract This paper is mainly devoted to discuss the existence of the finite order transcendental entire solutions for the partial differential-difference equations
∂ f (z 1 , z 2 ) ∂ f (z 1 , z 2 ) n + + f (z 1 + c1 , z 2 + c2 )m = 1, ∂z 1 ∂z 2 ∂ f (z 1 , z 2 ) n + f (z 1 + c1 , z 2 + c2 )m = 1. ∂z 1
It is confirmed that the existence of the finite order transcendental entire solutions for the above equations in the case n = 2 and m = 1. This is a very powerful supplement to the previous theorems given by Xu and Cao [L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), pages, 1-14.]. Keywords Existence · Entire solution · Partial differential-difference Mathematics Subject Classification 35M30 · 32W50 · 39A45
Communicated by Jialin Hong.
B
Hong Yan Xu [email protected] Hua Wang [email protected]
1
School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, Jiangxi, People’s Republic of China
2
Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen 333403, Jiangxi, People’s Republic of China
123
Bulletin of the Iranian Mathematical Society
1 Introduction In the past several decades, considerable attention had been paid to the existence of entire solutions for Fermat-type equation x n + y m = 1. Especially, A. Wiles and R. Taylor [19,20] pointed out this equation does not admit nontrivial solution in rational numbers for m = n ≥ 3, and this equation does exist nontrivial rational solution for m = n = 2. For the functional equation f n + g m = 1, Yang [27] in 1970 proved that there are no nonconstant entire solutions if m, n are positive integers satisfying 1 1 m + n < 1. Around 2010, Liu and his collaborators investigated the existence of solutions for a series of complex difference equations and complex differential–difference equations by using the difference Nevanlinna theory for meromorphic functions (see [4,6,7]) and obtained a lot of interest original results (see [12–14]). In order to be consistent with the following text, here we only list one of results given by Liu: Theorem 1.1 (see [13, p. 148]). For the equation f (z)n + f (z + c)m = 1, where m, n are positive integers. (i) If n > m > 1 or n = m > 2, then there is no nonconstant entire solution. (ii) If n > m, then there is no transcendental entire solution with finite order. Theorem 1.2 (see [13, Theorem 1.2]). The equation f (z)n + f (z + c)m = 1 has no transcendental entire solutions with finite order, provided that m = n, where n, m are positive integers. Theorem 1.3 (see [13, Theorem 1.4]). The equation f (z)n + [ f (z + c) − f (z)]m = 1 has no transcendental entire solutions with finite order, provided that m = n > 1, where n, m are positive integers. Recently, Cao and Xu
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