On the Meromorphic Solutions of Fermat-Type Differential Equations

Abstract

In this paper, we investigate the meromorphic solutions of the Fermat-type differential equations ( ) ( ) ( ) e 0 n m Az B f z f z c c + ′ + + = ≠ over the complex plane  for positive integers , m n , and , , A B c are constants. Our results improve and extend some earlier results given by Liu et al. Moreover, some examples are presented to show the preciseness of our results.

Solutions of Fermat-Type Partial Differential–Difference Equations in $$\pmb {{\mathbb {C}}^n}$$

“…Recently, the difference analogues of Nevanlinna theory, specially the development of difference analogous lemma of the logarithmic derivative has been established by Halburd and Korhonen [10,11], and Chiang and Feng [4], independently. Since then many researchers have started to study the existence of entire and meromorphic solutions of complex difference as well as complex differentialdifference equations, and obtained a number of important and interesting results in the literature (see [23,25,26,34]).…”

Transcendental entire solutions of quadratic functional equations

“…x + 2αu x u y + u 2 y = 1. Building upon this, Liu and Yang [30] further investigated the existence and form of solutions for quadratic trinomial functional equations. It is proved in [30] that if α = ±1, 0, then the equation f (z) 2 + 2αf (z)f ′ (z) + f ′ 2 (z) = 1 has no transcendental meromorphic solutions.…”

“…Building upon this, Liu and Yang [30] further investigated the existence and form of solutions for quadratic trinomial functional equations. It is proved in [30] that if α = ±1, 0, then the equation f (z) 2 + 2αf (z)f ′ (z) + f ′ 2 (z) = 1 has no transcendental meromorphic solutions. However, the equation f (z) 2 + 2αf (z)f (z + c) + f (z + c) 2 = 1 must have transcendental entire solutions of order equal to one.…”

Entire solutions of several quadratic binomial and trinomial partial differential-difference equations in $$ {\mathbb {C}}^2 $$

“…must take the form f (z) = 12 sin(2z +bi), where c = (2p+1)π, p being an integer, and b is a constant. Although there are many important and interesting results on the existence of entire and meromorphic solutions of complex difference as well as differentialdifference equations (see [25][26][27][28][29][30][31]39]), only a few number of results are there on the existence and forms of entire and meromorphic solutions of quadratic trinomial functional equation. To the best of our knowledge, Saleeby [37] first investigated the existence and forms of entire and meromorphic solutions of quadratic trinomial functional equation…”

“…By utilizing Theorem A, in 2016, Liu-Yang [31] obtained more precise properties of entire and meromorphic solutions of…”