Necessary and Sufficient Conditions for Oscillations of Functional Differential Equations

2018 ◽  
Vol 99 (3) ◽  
pp. 12-23
Author(s):  
G. M. Moremedi ◽  
◽  
Ioannis Stavroulakis ◽  
Z. K. Zhunussova ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Functional Differential ◽  
Necessary And Sufficient

scholarly journals Asymptotically power solutions of higher order neutral functional differential equations

2004 ◽  
Vol 35 (4) ◽  
pp. 383-389
Author(s):  
Zhi-Qiang Zhu ◽  
Sui Sun Cheng
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Necessary And Sufficient Conditions ◽  
Neutral Functional Differential Equations ◽  
Polynomial Solutions ◽  
Functional Differential ◽  
Necessary And Sufficient

Necessary and sufficient conditions are derived for the existence of asymptotically polynomial solutions of a class of neutral functional differential equations.

scholarly journals Spectral methods for nonlinear functionals and functional differential equations

2021 ◽  
Vol 8 (2) ◽  
Author(s):  
Daniele Venturi ◽  
Alec Dektor
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Approximate Solutions ◽  
Stability And Convergence ◽  
High Dimensional ◽  
Functional Derivatives ◽  
Nonlinear Functionals ◽  
Functional Differential ◽  
Necessary And Sufficient

AbstractWe present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpose of this analysis is twofold: First, we prove that continuous nonlinear functionals, functional derivatives, and FDEs can be approximated uniformly on any compact subset of a real Banach space admitting a basis by high-dimensional multivariate functions and high-dimensional partial differential equations (PDEs), respectively. Second, we show that the convergence rate of such functional approximations can be exponential, depending on the regularity of the functional (in particular its Fréchet differentiability), and its domain. We also provide necessary and sufficient conditions for consistency, stability and convergence of cylindrical approximations to linear FDEs. These results open the possibility to utilize numerical techniques for high-dimensional systems such as deep neural networks and numerical tensor methods to approximate nonlinear functionals in terms of high-dimensional functions, and compute approximate solutions to FDEs by solving high-dimensional PDEs. Numerical examples are presented and discussed for prototype nonlinear functionals and for an initial value problem involving a linear FDE.

The closure of the generalised eigenspace of a class of infinitesimal generators

1991 ◽  
Vol 117 (1-2) ◽  
pp. 171-192 ◽  
Author(s):  
S. M. Verduyn Lunel
Keyword(s):  
Differential Equations ◽  
Infinitesimal Generator ◽  
Entire Functions ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Solution Map ◽  
Strongly Continuous Semigroups ◽  
Infinitesimal Generators ◽  
Functional Differential ◽  
Necessary And Sufficient

SynopsisIn this paper we study the fine geometric structure of a class of strongly continuous semigroups that satisfy the following property: the resolvent of the infinitesimal generator can be represented as the quotient of entire functions of finite exponential type. This class includes the solution map for functional differential equations and certain partial differential equations. In particular, we present necessary and sufficient conditions for one-to-oneness of the solution map and for completeness of the system of generalised eigenfunctions of the generator.

scholarly journals On variational formulations for functional differential equations

2007 ◽  
Vol 5 (1) ◽  
pp. 89-101 ◽  
Author(s):  
I. A. Kolesnikova ◽  
A. M. Popov ◽  
V. M. Savchin
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Variational Principles ◽  
Boundary Value ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Partial Functional Differential Equations ◽  
Functional Differential ◽  
Necessary And Sufficient ◽  
Variational Formulations

Necessary and sufficient conditions for the existence of integral variational principles for boundary value problems for given ordinary and partial functional differential equations are obtained. Examples are given illustrating the results.

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