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.

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>

A Liapunov functional for a scalar differential difference equation

1978 ◽  
Vol 79 (3-4) ◽  
pp. 307-316 ◽  
Author(s):  
E. F. Infante ◽  
J. A. Walker
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quadratic Functional ◽  
Liapunov Functional ◽  
Differential Difference Equation ◽  
The Difference ◽  
Rates Of Decay ◽  
Necessary And Sufficient

SynopsisGiven the scalar, retarded differential difference equation x'(t)=ax(t) +bx(t−τ), a quadratic functional in explicit form is obtained that yields necessary and sufficient conditions for the asymptotic stability of this equation. This functional a Liapunov functional, is obtained through the study of the Liapunov functions associated with a difference equation approximation of the difference differential equation. The functional then obtained not only yields necessary and sufficient conditions for asymptotic stability, but provides estimates for rates of decay of the solutions as well as conditions, for asymptotic stability independent of the magnitude of the delay τ.

Inverse problem about two-spectra for finite Jacobi matrices with zero diagonal

2016 ◽  
Vol 24 (6) ◽  
Author(s):  
Adil Huseynov
Keyword(s):  
Inverse Problem ◽  
Finite Order ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Jacobi Matrices ◽  
The Matrix ◽  
Explicit Procedure ◽  
Necessary And Sufficient

AbstractThe necessary and sufficient conditions for solvability of the inverse problem about two-spectra for finite order real Jacobi matrices with zero-diagonal elements are established. An explicit procedure of reconstruction of the matrix from the two-spectra is given.

Existence of moments in a stationary stochastic difference equation

1990 ◽  
Vol 22 (1) ◽  
pp. 129-146 ◽  
Author(s):  
Hans Arnfinn Karlsen
Keyword(s):  
Difference Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stochastic Difference Equation ◽  
Dependent Variables ◽  
Gaussian Case ◽  
Necessary And Sufficient

The stationary stochastic difference equation Xt = YtXt–1 + Wt is analyzed with emphasis on conditions ensuring that ||Xt||p <∞. Some general results are obtained and then applied to different classes of input processes {(Yt, Wt)}. Especially both necessary and sufficient conditions are given in the Gaussian case. We also obtain results concerning moments of products of dependent variables.

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