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 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 Entire Solutions of the Second-Order Fermat-Type Differential-Difference Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Guoqiang Dang ◽  
Jinhua Cai
Keyword(s):  
Difference Equation ◽  
Finite Order ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
System Of Equations ◽  
Entire Solutions ◽  
Differential Difference Equation ◽  
Necessary And Sufficient

In this paper, the entire solutions of finite order of the Fermat-type differential-difference equation f″z2+△ckfz2=1 and the system of equations f1″z2+△ckf2z2=1 and f2″z2+△ckf1z2=1 have been studied. We give the necessary and sufficient conditions of existence of the entire solutions of finite order.

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 Oscillation theorems for second order difference equations woth negative neutral term

2015 ◽  
Vol 46 (4) ◽  
pp. 441-451 ◽  
Author(s):  
Ethiraju Thandapani ◽  
Devarajulu Seghar ◽  
Sandra Pinelas
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Second Order ◽  
Neutral Difference Equation ◽  
Order Difference ◽  
Oscillation Criteria ◽  
Neutral Term ◽  
Oscillation Theorems ◽  
Positive Integers

In this paper we obtain some new oscillation criteria for the neutral difference equation \begin{equation*} \Delta \Big(a_n (\Delta (x_n-p_n x_{n-k}))\Big)+q_n f(x_{n-l})=0 \end{equation*} where $0\leq p_n\leq p0$ and $l$ and $k$ are positive integers. Examples are presented to illustrate the main results. The results obtained in this paper improve and complement to the existing results.

scholarly journals Kamenev-Type Oscillation Criteria for Generalized Second Order Sublinear -Difference Equations

2018 ◽  
Vol 7 (4.10) ◽  
pp. 1046
Author(s):  
A. Benevatho Jaison ◽  
Sk. Khadar Babu ◽  
V. Chandrasekar
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Second Order ◽  
Nonlinear Difference Equation ◽  
Riccati Transformation ◽  
Transformation Techniques ◽  
Oscillation Criteria ◽  
Positive Integers ◽  
Type Oscillation

By means of Riccati transformation techniques, authors establish some new oscillation criteria for generalized second order nonlinear -difference equation when  and  are quotient of odd positive integers.   

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.

Improved oscillation criteria for second order nonlinear delay difference equations with non-positive neutral term

2016 ◽  
Vol 56 (1) ◽  
pp. 155-165 ◽  
Author(s):  
E. Thandapani ◽  
S. Selvarangam ◽  
R. Rama ◽  
M. Madhan
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Second Order ◽  
Delay Difference Equation ◽  
Oscillation Criteria ◽  
Neutral Term ◽  
Positive Integers ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

Abstract In this paper, we present some oscillation criteria for second order nonlinear delay difference equation with non-positive neutral term of the form $$\Delta (a_n (\Delta z_n )^\alpha ) + q_n f(x_{n - \sigma } ) = 0,\;\;\;n \ge n_0 > 0,$$ where zn = xn − pnxn−τ, and α is a ratio of odd positive integers. Examples are provided to illustrate the results. The results obtained in this paper improve and complement to some of the existing results.

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

2021 ◽  
Vol 6 (11) ◽  
pp. 11796-11814
Author(s):  
Hong Li ◽  
◽  
Keyu Zhang ◽  
Hongyan Xu ◽  
◽  
...  
Keyword(s):  
Finite Order ◽  
Difference Equations ◽  
Complex Variable ◽  
Nevanlinna Theory ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Systems Of Equations ◽  
Entire Solutions ◽  
Partial Differential ◽  
Single Complex

<abstract><p>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 <sup>[<xref ref-type="bibr" rid="b5">5</xref>,<xref ref-type="bibr" rid="b24">24</xref>,<xref ref-type="bibr" rid="b39">39</xref>]</sup>. 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.</p></abstract>

scholarly journals Oscillatory behavior of second order unstable type neutral difference equations

2005 ◽  
Vol 36 (1) ◽  
pp. 57-68
Author(s):  
E. Thandapani ◽  
S. Pandian ◽  
R. K. Balasubramanian
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Difference Equations ◽  
Neutral Type ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Neutral Difference Equations ◽  
Unbounded Solutions ◽  
Unstable Type ◽  
Positive Integers

This paper deals with the oscillatory behavior of all bounded/ unbounded solutions of second order neutral type difference equation of the form$$ \Delta (a_n(\Delta_c y_n+py_{n-k}))^\alpha)-g_nf(y_{\sigma(n)})=0, $$where $ p $ is real, $ \alpha $ is a ratio of odd positive integers, $ k $ is a positive integer and $ \{\sigma(n)\} $ is a sequence of integers. Examples are provided to illustrate the results.

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