scholarly journals Exponential Polynomials and Nonlinear Differential-Difference Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Junfeng Xu ◽  
Jianxun Rong
Keyword(s):  
Difference Equation ◽  
Finite Order ◽  
Difference Equations ◽  
Polynomial Solution ◽  
Nonlinear Difference Equation ◽  
Exponential Polynomial ◽  
Entire Solutions ◽  
Exponential Polynomials

In this paper, we study finite-order entire solutions of nonlinear differential-difference equations and solve a conjecture proposed by Chen, Gao, and Zhang when the solution is an exponential polynomial. We also find that any exponential polynomial solution of a nonlinear difference equation should have special forms.

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

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><p>This article is concerned with the existence of entire solutions for the following complex second order partial differential-difference equation</p> <p><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></p> <p>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 <sup>[<xref ref-type="bibr" rid="b23">23</xref>]</sup>, Xu and Cao <sup>[<xref ref-type="bibr" rid="b35">35</xref>]</sup>, Liu, Cao and Cao <sup>[<xref ref-type="bibr" rid="b17">17</xref>]</sup>.</p></abstract>

scholarly journals Meromorphic solutions of certain nonlinear difference equations

2020 ◽  
Vol 18 (1) ◽  
pp. 1292-1301
Author(s):  
Huifang Liu ◽  
Zhiqiang Mao ◽  
Dan Zheng
Keyword(s):  
Difference Equation ◽  
Finite Order ◽  
Difference Equations ◽  
Nonlinear Difference Equation ◽  
Meromorphic Solutions ◽  
Nonlinear Difference Equations ◽  
Zero Distribution ◽  
Forward Difference

Abstract This paper focuses on finite-order meromorphic solutions of nonlinear difference equation {f}^{n}(z)+q(z){e}^{Q(z)}{\text{&#x0394;}}_{c}f(z)=p(z) , where p,q,Q are polynomials, n\ge 2 is an integer, and {\text{&#x0394;}}_{c}f is the forward difference of f. A relationship between the growth and zero distribution of these solutions is obtained. Using this relationship, we obtain the form of these solutions of the aforementioned equation. Some examples are given to illustrate our results.

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

Axioms ◽  
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.

scholarly journals Qualitative Inspection Fourth order Non Linear Difference Equations

2019 ◽  
Vol 8 (11) ◽  
pp. 3467-3469
Keyword(s):  
Difference Equation ◽  
Fourth Order ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Nonlinear Difference Equation ◽  
Linear Difference Equations ◽  
All Solutions ◽  
Non Linear

In this paper, the authors obtained some new sufficient conditions for the oscillation of all solutions of the fourth order nonlinear difference equation of the form ( ) ( 1 ) 0 3  anxn  pnxn  qn f xn  n = 0,1,2, … ., where an, pn, qn positive sequences. The established results extend, unify and improve some of the results reported in the literature. Examples are provided to illustrate the main result.

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

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.

scholarly journals Homoclinic solutions of 2nth-order difference equations containing both advance and retardation

2016 ◽  
Vol 14 (1) ◽  
pp. 520-530 ◽  
Author(s):  
Yuhua Long ◽  
Yuanbiao Zhang ◽  
Haiping Shi
Keyword(s):  
Difference Equation ◽  
Critical Point ◽  
Difference Equations ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Nonlinear Difference Equation ◽  
Mountain Pass Lemma ◽  
Order Difference ◽  
Existence And Multiplicity ◽  
New Criteria

AbstractBy using the critical point method, some new criteria are obtained for the existence and multiplicity of homoclinic solutions to a 2nth-order nonlinear difference equation. The proof is based on the Mountain Pass Lemma in combination with periodic approximations. Our results extend and improve some known ones.

On Properties of Entire Solutions of Difference Equations and Difference Polynomials

2015 ◽  
Vol 65 (3) ◽  
Author(s):  
Xiaoguang Qi ◽  
Yinhong Cao ◽  
Yong Liu
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Linear Difference Equation ◽  
Entire Solutions ◽  
Non Linear ◽  
Difference Polynomials

AbstractIn this paper, we consider the existence of entire solutions of certain type of non-linear difference equation of the form: f(z)

Oscillation Theorems of Nonlinear Difference Equations of Second Order

2003 ◽  
Vol 10 (2) ◽  
pp. 343-352
Author(s):  
S. H. Saker
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Second Order ◽  
Nonlinear Difference Equation ◽  
Riccati Transformation ◽  
Nonlinear Difference Equations ◽  
Transformation Techniques ◽  
Oscillation Criteria ◽  
Special Cases ◽  
Oscillation Theorems

Abstract Using the Riccati transformation techniques, we establish some new oscillation criteria for the second-order nonlinear difference equation Some comparison between our theorems and the previously known results in special cases are indicated. Some examples are given to illustrate the relevance of our results.

scholarly journals Dual exponential polynomials and a problem of Ozawa

2021 ◽  
pp. 1-19
Author(s):  
Janne Heittokangas ◽  
Katsuya Ishizaki ◽  
Kazuya Tohge ◽  
Zhi-Tao Wen
Keyword(s):  
Differential Equations ◽  
Polynomial Solution ◽  
Linear Differential Equations ◽  
The Other ◽  
Exponential Polynomial ◽  
Exponential Polynomials ◽  
Open Problems ◽  
Duality Property ◽  
Complex Linear ◽  
Linear Differential

Complex linear differential equations with entire coefficients are studied in the situation where one of the coefficients is an exponential polynomial and dominates the growth of all the other coefficients. If such an equation has an exponential polynomial solution $f$ , then the order of $f$ and of the dominant coefficient are equal, and the two functions possess a certain duality property. The results presented in this paper improve earlier results by some of the present authors, and the paper adjoins with two open problems.

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