Solutions for systems of complex Fermat type partial differential-difference equations with two complex variables

<abstract>By making use of the Nevanlinna theory and difference Nevanlinna theory of several complex variables, we investigate some properties of the transcendental entire solutions for several systems of partial differential difference equations of Fermat type, and obtain some results about the existence and the forms of transcendental entire solutions of the above systems, which improve and generalize the previous results given by Cao, Gao, Liu [<xref ref-type="bibr" rid="b5">5</xref>,<xref ref-type="bibr" rid="b24">24</xref>,<xref ref-type="bibr" rid="b39">39</xref>]. Some examples are given show that there exist some significant differences in the forms of transcendental entire solutions with finite order of the systems of equations with between several complex variables and a single complex variable.</abstract>

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Entire solutions for several second-order partial differential-difference equations of Fermat type with two complex variables

Advances in Difference Equations ◽

10.1186/s13662-020-03201-y ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Hong Yan Xu ◽

Da Wei Meng ◽

Sanyang Liu ◽

Hua Wang

Keyword(s):

Finite Order ◽

Difference Equations ◽

Nevanlinna Theory ◽

Meromorphic Functions ◽

Complex Variables ◽

Second Order ◽

Several Complex Variables ◽

Entire Solutions ◽

Partial Differential

AbstractThis paper is concerned with description of the existence and the forms of entire solutions of several second-order partial differential-difference equations with more general forms of Fermat type. By utilizing the Nevanlinna theory of meromorphic functions in several complex variables we obtain some results on the forms of entire solutions for these equations, which are some extensions and generalizations of the previous theorems given by Xu and Cao (Mediterr. J. Math. 15:1–14, 2018; Mediterr. J. Math. 17:1–4, 2020) and Liu et al. (J. Math. Anal. Appl. 359:384–393, 2009; Electron. J. Differ. Equ. 2013:59–110, 2013; Arch. Math. 99:147–155, 2012). Moreover, by some examples we show the existence of transcendental entire solutions with finite order of such equations.

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Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in C2

Axioms ◽

10.3390/axioms10020126 ◽

2021 ◽

Vol 10 (2) ◽

pp. 126

Author(s):

Hong Li ◽

Hongyan Xu

Keyword(s):

Positive Integer ◽

Finite Order ◽

Difference Equations ◽

Functional Equations ◽

Growth Order ◽

Entire Solutions ◽

Partial Differential ◽

Integer Order ◽

Significant Difference ◽

Quadratic Trinomial

This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z), and the partial differential difference equations f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z12=eg(z),f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z2+∂f(z)∂z1+∂f(z)∂z22=eg(z). We establish some theorems about the forms of the finite order transcendental entire solutions of these functional equations. We also list a series of examples to explain the existence of the finite order transcendental entire solutions of such equations. Meantime, some examples show that there exists a very significant difference with the previous literature on the growth order of the finite order transcendental entire solutions. Our results show that some functional equations can admit the transcendental entire solutions with any positive integer order. These results make a few improvements of the previous theorems given by Xu and Cao, Liu and Yang.

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Results on the solutions of several second order mixed type partial differential difference equations

AIMS Mathematics ◽

10.3934/math.2022110 ◽

2021 ◽

Vol 7 (2) ◽

pp. 1907-1924

Author(s):

Wenju Tang ◽

◽

Keyu Zhang ◽

Hongyan Xu ◽

◽

...

Keyword(s):

Difference Equation ◽

Finite Order ◽

Difference Equations ◽

Mixed Type ◽

Existence Of Solutions ◽

Second Order ◽

Entire Solutions ◽

Partial Differential ◽

Differential Difference Equation ◽

Positive Integers

<abstract>This article is concerned with the existence of entire solutions for the following complex second order partial differential-difference equation <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left(\frac{\partial^2 f(z_1, z_2)}{\partial z_1^2}+\frac{\partial^2 f(z_1, z_2)}{\partial z_2^2}\right)^{l}+f(z_1+c_1, z_2+c_2)^{k} = 1, $\end{document} </tex-math></disp-formula> where $ c_1, c_2 $ are constants in $ \mathbb{C} $ and $ k, l $ are positive integers. In addition, we also investigate the forms of finite order transcendental entire solutions for several complex second order partial differential-difference equations of Fermat type, and obtain some theorems about the existence and the forms of solutions for the above equations. Meantime, we give some examples to explain the existence of solutions for some theorems in some cases. Our results are some generalizations of the previous theorems given by Qi [<xref ref-type="bibr" rid="b23">23</xref>], Xu and Cao [<xref ref-type="bibr" rid="b35">35</xref>], Liu, Cao and Cao [<xref ref-type="bibr" rid="b17">17</xref>].</abstract>

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THE TUMURA–CLUNIE THEOREM IN SEVERAL COMPLEX VARIABLES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972714000446 ◽

2014 ◽

Vol 90 (3) ◽

pp. 444-456 ◽

Cited By ~ 4

Author(s):

PEI-CHU HU ◽

CHUNG-CHUN YANG

Keyword(s):

Meromorphic Function ◽

Complex Variable ◽

Nevanlinna Theory ◽

Meromorphic Functions ◽

Complex Variables ◽

Several Complex Variables ◽

Several Variables

AbstractIt is a well-known result that if a nonconstant meromorphic function $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f$ on $\mathbb{C}$ and its $l$th derivative $f^{(l)}$ have no zeros for some $l\geq 2$, then $f$ is of the form $f(z)=\exp (Az+B)$ or $f(z)=(Az+B)^{-n}$ for some constants $A$, $B$. We extend this result to meromorphic functions of several variables, by first extending the classic Tumura–Clunie theorem for meromorphic functions of one complex variable to that of meromorphic functions of several complex variables using Nevanlinna theory.

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