scholarly journals Integro-differential equations of Volterra type

1970 ◽  
Vol 3 (1) ◽  
pp. 9-22 ◽  
Author(s):  
M. Rama Mohana Rao ◽  
Chris P. Tsokos
Keyword(s):  
Differential System ◽  
Basic Result ◽  
Sufficient Conditions ◽  
Continuous Operator ◽  
Continuous Functions ◽  
Uniform Asymptotic Stability ◽  
Space Of Continuous Functions ◽  
Volterra Type ◽  
The Stability ◽  
Image Position

The aim of this paper is concerned with studying the stability properties of an integro-differential system by reducing it into a scalar integro-differential equation. A theorem is stated about the existence of a maximal solution of such systems and a basic result on integro-differential inequalities. Utilizing these results we obtain sufficient conditions for uniform asymptotic stability of the trivial solution of the integro-differential system of the form where , with , , C(J) denotes the space of continuous functions, A a continuous operator such that A maps C(J) into C(J). The fruitfulness of the results of the paper are illustrated with two applications.

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 On uniform asymptotic stability for nonlinear integro-differential equations of Volterra type

2019 ◽  
pp. 161-166
Author(s):  
Natalia Sedova
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Auxiliary Function ◽  
Uniform Asymptotic Stability ◽  
Volterra Type ◽  
Razumikhin Technique ◽  
The Stability ◽  
The Right

The specifics of the application of Razumikhin technique to the stability analysis of Volterra type integrodifferential equations are considered. The equation can be nonlinear and nonautonomous. We propose new sufficient conditions for uniform asymptotic stability of the zero solution using the phase space of a special construction and constraints on the right side of the equation. In the presented constraints we can analyze stability, relying not only on the behavior of the auxiliary function along the solutions, but also on the properties of the so called limiting equations.

scholarly journals The Borel structure of iterates of continuous functions

1989 ◽  
Vol 32 (3) ◽  
pp. 483-494 ◽  
Author(s):  
Paul D. Humke ◽  
M. Laczkovich
Keyword(s):  
Banach Space ◽  
Continuous Functions ◽  
Baire Space ◽  
Space Of Continuous Functions ◽  
Borel Structure ◽  
Image Position ◽  
Set Of Points ◽  
Positive Integers

Let C[0,1] be the Banach space of continuous functions defined on [0,1] and let C be the set of functions f∈C[0,1] mapping [0,1] into itself. If f∈C, fk will denote the kth iterate of f and we put Ck = {fk:f∈C;}. The set of increasing (≡ nondecreasing) and decreasing (≡ nonincreasing) functions in C will be denoted by ℐ and D, respectively. If a function f is defined on an interval I, we let C(f) denote the set of points at which f is locally constant, i.e.We let N denote the set of positive integers and NN denote the Baire space of sequences of positive integers.

Global attraction for systems with almost C1 vector fields

2005 ◽  
Vol 135 (4) ◽  
pp. 815-832
Author(s):  
Zhanyuan Hou
Keyword(s):  
Linear Systems ◽  
Global Attractor ◽  
Differential System ◽  
Vector Fields ◽  
Single Point ◽  
Sufficient Conditions ◽  
Global Attraction ◽  
Almost Everywhere ◽  
Image Position ◽  
Special Case

Sufficient conditions are given for an autonomous differential system to have a single point global attractor (repeller) with f continuously differentiable almost everywhere. These results incorporate those of Hartman and Olech as a special case even when the condition f ∈ C1(D, ℝN) is fully met. Moreover, these criteria are simplified for a class of region-wise linear systems in ℝN.

scholarly journals On the Uniform Stability of Impulsive Lotka-Volterra Type Systems with Supremums

2018 ◽  
Vol 4 (2) ◽  
Author(s):  
Ivanka Stamova ◽  
Gani Stamov
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Type Systems ◽  
Uniform Stability ◽  
Volterra Model ◽  
Uniform Asymptotic Stability ◽  
Volterra Type

In this paper we propose an impulsive n- species Lotka-Volterra model with supremums. By using Lyapunov method we give sufficient conditions for uniform stability and uniform asymptotic stability of the positive states.

The Use of Partial Stability in the Analysis of Interconnected Systems

2020 ◽  
Vol 143 (4) ◽  
Author(s):  
Bo Wang ◽  
Hashem Ashrafiuon ◽  
Sergey G. Nersesov
Keyword(s):  
Sufficient Conditions ◽  
Control Design ◽  
Interconnected Systems ◽  
System State ◽  
Uniform Asymptotic Stability ◽  
Partial Stability ◽  
Control Framework ◽  
Control Objective ◽  
The Stability ◽  
Robot Platform

Abstract In this paper, we develop sufficient conditions for uniform asymptotic stability of interconnected dynamical systems that are not in cascade form. We show that the stability analysis of a two-subsystem interconnection can be reduced to ensuring the stability of the first nonisolated subsystem with respect to its own state vector (partial stability) and the stability of the isolated second subsystem. In addition, based on the above results, we provide a control design framework for nonlinear systems where the control objective reduces to stabilization of only a part of the system state while guaranteeing the stability for the entire state of the system. We validate the efficacy of the proposed control framework via simulations and experiments using the wheeled mobile robot platform.

scholarly journals Asymptotic Stability of the Solutions of Neutral Linear Fractional System with Nonlinear Perturbation

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 390
Author(s):  
Andrey Zahariev ◽  
Hristo Kiskinov
Keyword(s):  
Asymptotic Stability ◽  
Differential System ◽  
Global Asymptotic Stability ◽  
Linear Part ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Neutral System ◽  
Perturbed System ◽  
The Stability ◽  
Several Delays

In this article existence and uniqueness of the solutions of the initial problem for neutral nonlinear differential system with incommensurate order fractional derivatives in Caputo sense and with piecewise continuous initial function is proved. A formula for integral presentation of the general solution of a linear autonomous neutral system with several delays is established and used for the study of the stability properties of a neutral autonomous nonlinear perturbed linear fractional differential system. Natural sufficient conditions are found to ensure that from global asymptotic stability of the zero solution of the linear part of a nonlinearly perturbed system it follows global asymptotic stability of the zero solution of the whole nonlinearly perturbed system.

Alternating Chebyshev Approximation with A Non-Continuous Weight Function

1976 ◽  
Vol 19 (2) ◽  
pp. 155-157 ◽  
Author(s):  
Charles B. Dunham
Keyword(s):  
Weight Function ◽  
Best Approximation ◽  
Closed Interval ◽  
Chebyshev Approximation ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Negative Function ◽  
Chebyshev Problem ◽  
Image Position ◽  
Continuous Weight

Let [α, β] be a closed interval and C[α, β] be the space of continuous functions on [α, β], For g a function on [α, β] defineLet s be a non-negative function on [α, β]. Let F be an approximating function with parameter space P such that F(A, .)∊ C[α, β] for all A∊P. The Chebyshev problem with weight s is given f ∊ C[α, β], to find a parameter A* ∊ P to minimize e(A) = ||s * (f - F(A, .))|| over A∊P. Such a parameter A* is called best and F(A*,.) is called a best approximation to f.

Existence and properties of semi-bounded global solutions to the functional differential equation with Volterra-type operators on the real line

2017 ◽  
Vol 147 (6) ◽  
pp. 1119-1168
Author(s):  
Maitere Aguerrea ◽  
Robert Hakl
Keyword(s):  
Real Line ◽  
Asymptotic Properties ◽  
Global Solutions ◽  
Continuous Operator ◽  
Volterra Type ◽  
Existence Of A Solution ◽  
Carathéodory Conditions ◽  
Functional Differential ◽  
Image Position ◽  
Continuous Operators

Consider the equationwhere are linear positive continuous operators and f : Cloc(ℝ;ℝ) → Lloc(ℝ;ℝ) is a continuous operator satisfying the local Carathéodory conditions. Efficient conditions guaranteeing the existence of a global solution, which is bounded and non-negative in the neighbourhood of –∞, to the equation considered are established provided that ℓ0, ℓ1 and f are Volterra-type operators. The existence of a solution that is positive on the whole real line is discussed as well. Furthermore, the asymptotic properties of such solutions are studied in the neighbourhood of –∞. The results are applied to certain models appearing in the natural sciences.

scholarly journals Stability Analysis for Impulsive Stochastic Reaction-Diffusion Differential System and Its Application to Neural Networks

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yanke Du ◽  
Yanlu Li ◽  
Rui Xu
Keyword(s):  
Neural Networks ◽  
Differential System ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Impulsive Effects ◽  
Stability Criteria ◽  
Mixed Time Delays ◽  
The Stability ◽  
Stochastic Reaction

This paper is concerned with the stability of impulsive stochastic reaction-diffusion differential systems with mixed time delays. First, an equivalent relation between the solution of a stochastic reaction-diffusion differential system with time delays and impulsive effects and that of corresponding system without impulses is established. Then, some stability criteria for the stochastic reaction-diffusion differential system with time delays and impulsive effects are derived. Finally, the stability criteria are applied to impulsive stochastic reaction-diffusion Cohen-Grossberg neural networks with mixed time delays, and sufficient conditions are obtained for the exponentialp-stability of the zero solution to the neural networks. An example is given to illustrate the effectiveness of our theoretical results. The systems we studied in this paper are more general, and some existing results are improved and extended.

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