scholarly journals Uniform ultimate boundedness and periodicity in functional-differential equations

1990 ◽  
Vol 42 (1) ◽  
pp. 93-100 ◽  
Author(s):  
Theodore Allen Burton ◽  
Bo Zhang
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Ultimate Boundedness ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential

Periodic solutions to functional differential equations

1985 ◽  
Vol 101 (3-4) ◽  
pp. 253-271 ◽  
Author(s):  
O. A. Arino ◽  
T. A. Burton ◽  
J. R. Haddock
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Ultimate Boundedness ◽  
Space Of Continuous Functions ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential ◽  
Image Position

SynopsisWe consider a system of functional differential equationswhere G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

scholarly journals Stability, boundedness and existence of unique periodic solutions to a class of fourth order functional differential equations

2021 ◽  
Vol 40 (2) ◽  
pp. 271-303
Author(s):  
Adeleke Timothy Ademola
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Ultimate Boundedness ◽  
Uniform Asymptotic Stability ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential ◽  
Theoretical Results

In this paper a novel class of fourth order functional differential equations is discussed. By reducing the fourth order functional differential equation to system of first order, a suitable complete Lyapunov functional is constructed and employed to obtain sufficient conditions that guarantee existence of a unique periodic solution, asymptotic and uniform asymptotic stability of the zero solutions, uniform boundedness and uniform ultimate boundedness of solutions. The obtained results are new and include many prominent results in literature. Finally, two examples are given to show the feasibility and reliability of the theoretical results.

scholarly journals Ultimate Boundedness of the Systems Governed by Stochastic Differential Equations

1972 ◽  
Vol 47 ◽  
pp. 111-144 ◽  
Author(s):  
Yoshio Miyahara
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stochastic Differential Equations ◽  
Functional Differential Equations ◽  
Ultimate Boundedness ◽  
Functional Differential ◽  
The Stability

The stability of the systems given by ordinary differential equations or functional-differential equations has been studied by many mathematicians. The most powerful tool in this field seems to be the Liapunov’s second method (see, for example [6]).

Compact boundedness in periodic functional differential equations with infinite delay

1990 ◽  
Vol 114 (3-4) ◽  
pp. 291-297
Author(s):  
Junji Kato
Keyword(s):  
Phase Space ◽  
Differential Equations ◽  
Infinite Delay ◽  
Functional Differential Equations ◽  
Periodic Systems ◽  
Compact Set ◽  
Ultimate Boundedness ◽  
Local Compactness ◽  
Non Local ◽  
Functional Differential

SynopsisIt is the aim of this article to consider some problems arising from the non local-compactness of the phase space for functional differential equations. The compact boundedness, that is, the boundedness depending on each compact set involving the initial values, is proved to be implied from the ultimate boundedness for periodic systems of functional differential equations on Cγ: = {φ ∊ C((–∞,0]) Note that it is known that the compactness cannot be dropped in the above. An example is also given to show that the asymptotic stability is not necessarily uniform even for periodic functional differential equations on Co.

Comparison Theorems of Liapunov-Razumikhin Type for NFDEs With Infinite Delay

1995 ◽  
Vol 47 (3) ◽  
pp. 500-526 ◽  
Author(s):  
John R. Haddock ◽  
Shigui Ruan ◽  
Jianhong Wu ◽  
Huaxing Xia
Keyword(s):  
Phase Space ◽  
Infinite Delay ◽  
Functional Differential Equations ◽  
Comparison Theorems ◽  
Asymptotic Behavior Of Solutions ◽  
Boundedness Of Solutions ◽  
Ultimate Boundedness ◽  
Neutral Functional Differential Equations ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential

AbstractSome comparison theorems of Liapunov-Razumikhin type are provided for uniform (asymptotic) stability and uniform (ultimate) boundedness of solutions to neutral functional differential equations with infinite delay with respect to a given phase space pair. Examples are given to illustrate how the comparison theorems and stability and boundedness of solutions depend on the choice(s) of phase space(s) and are related to asymptotic behavior of solutions to some difference and integral equations.

Stability of Implicit Difference Equations Generated by Parabolic Functional Differential Problems

2007 ◽  
Vol 7 (1) ◽  
pp. 68-82
Author(s):  
K. Kropielnicka
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Initial Boundary ◽  
Numerical Examples ◽  
Convergence Results ◽  
Nonlinear Parabolic ◽  
Implicit Difference Methods ◽  
Neumann Type ◽  
Difference Methods ◽  
Functional Differential

AbstractA general class of implicit difference methods for nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type is constructed. Convergence results are proved by means of consistency and stability arguments. It is assumed that given functions satisfy nonlinear estimates of Perron type with respect to functional variables. Differential equations with deviated variables and differential integral problems can be obtained from a general model by specializing given operators. The results are illustrated by numerical examples.

Sign in / Sign up

Export Citation Format

Share Document