scholarly journals On the stability of differential-difference equations

1976 ◽  
Vol 19 (3) ◽  
pp. 358-370 ◽  
Author(s):  
R. D. Braddock ◽  
P. Van Den Driessche
Keyword(s):  
Time Delay ◽  
Linear System ◽  
Difference Equations ◽  
Local Stability ◽  
Original Model ◽  
Time Lags ◽  
Local Properties ◽  
Linear Differential ◽  
The Stability ◽  
Stability Results

AbstractThe local properties of non-linear differential-difference equations are investigated by considering the location of the roots of the eigen-equation derived from the lineraised approximation of the original model. A general linear system incorporating one time delay is considered and local stability results are obtained for cases in which the coefficient matrices satisfy certain assumptions. The results have applications to recent Biological and Economic models incorporating time lags.

On the Necessary and Sufficient Conditions for the Stability of Linear Generalized Ordinary Differential, Linear Impulsive and Linear Difference Systems

2009 ◽  
Vol 16 (4) ◽  
pp. 597-616
Author(s):  
Shota Akhalaia ◽  
Malkhaz Ashordia ◽  
Nestan Kekelia
Keyword(s):  
Linear System ◽  
Difference Equations ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Difference Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Generalized Ordinary Differential Equations ◽  
Lyapunov Sense

Abstract Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

scholarly journals ON THE STABILITY OF THE DIFFERENTIAL PROCESS GENERATED BY COMPLEX INTERPOLATION

2020 ◽  
pp. 1-32
Author(s):  
Jesús M. F. Castillo ◽  
Willian H. G. Corrêa ◽  
Valentin Ferenczi ◽  
Manuel González
Keyword(s):  
Unit Circle ◽  
Banach Spaces ◽  
Local Stability ◽  
Interpolation Method ◽  
Complex Interpolation ◽  
Analytic Family ◽  
Köthe Spaces ◽  
The Stability ◽  
Stability Results ◽  
Differential Process

We study the stability of the differential process of Rochberg and Weiss associated with an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces, we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit circle while there is no (bounded) stability for families of four or more Köthe spaces. In the context of arbitrary pairs of Banach spaces, we present some local stability results and some global isometric stability results.

Stability results for an elastic–viscoelastic wave equation with localized Kelvin–Voigt damping and with an internal or boundary time delay

2020 ◽  
pp. 1-57
Author(s):  
Mouhammad Ghader ◽  
Rayan Nasser ◽  
Ali Wehbe
Keyword(s):  
Wave Equation ◽  
Time Delay ◽  
Frequency Domain ◽  
Viscoelastic Damping ◽  
One Dimensional ◽  
Viscoelastic Wave ◽  
Smooth Coefficients ◽  
The Stability ◽  
Stability Results ◽  
Dimensional Wave

We investigate the stability of a one-dimensional wave equation with non smooth localized internal viscoelastic damping of Kelvin–Voigt type and with boundary or localized internal delay feedback. The main novelty in this paper is that the Kelvin–Voigt and the delay damping are both localized via non smooth coefficients. Under sufficient assumptions, in the case that the Kelvin–Voigt damping is localized faraway from the tip and the wave is subjected to a boundary delay feedback, we prove that the energy of the system decays polynomially of type t − 4 . However, an exponential decay of the energy of the system is established provided that the Kelvin–Voigt damping is localized near a part of the boundary and a time delay damping acts on the second boundary. While, when the Kelvin–Voigt and the internal delay damping are both localized via non smooth coefficients near the boundary, under sufficient assumptions, using frequency domain arguments combined with piecewise multiplier techniques, we prove that the energy of the system decays polynomially of type t − 4 . Otherwise, if the above assumptions are not true, we establish instability results.

scholarly journals Endogenous Reactivity in a Dynamic Model of Consumer’s Choice

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Ahmad K. Naimzada ◽  
Fabio Tramontana
Keyword(s):  
Dynamic Model ◽  
Optimal Choice ◽  
Original Model ◽  
Rational Behavior ◽  
Learning Mechanisms ◽  
Learning Mechanism ◽  
The Stability ◽  
Consumer Model ◽  
Asymptotical Convergence ◽  
Stability Results

We move from a boundedly rational consumer model (Naimzada and Tramontana, 2008, 2010) characterized by a gradient-like decisional process in which, under particular parameters conditions, the asymptotical convergence to the optimal choice does not happen but it does under a least squared learning mechanism. In the present paper, we prove that even a less sophisticated learning mechanism leads to convergence to the rational choice and also prove that convergence is ensured when both learning mechanisms are available. The stability results that we obtain give more strength to the rational behavior assumption of the original model; in fact, the less demanding is the learning mechanism ensuring convergence to the rational behavior, the higher is the probability that even quite naive consumers will learn the composition of their optimum consumption bundles.

scholarly journals Existence and Stability Results on Hadamard Type Fractional Time-Delay Semilinear Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1242
Author(s):  
Nazim Mahmudov ◽  
Areen Al-Khateeb
Keyword(s):  
Time Delay ◽  
Differential Equations ◽  
Explicit Form ◽  
Nonlinear Equations ◽  
Matrix Function ◽  
Linear Differential Equations ◽  
Existence And Stability ◽  
Linear Differential ◽  
Stability Results

A delayed perturbation of the Mittag-Leffler type matrix function with logarithm is proposed. This combines the classic Mittag–Leffler type matrix function with a logarithm and delayed Mittag–Leffler type matrix function. With the help of this introduced delayed perturbation of the Mittag–Leffler type matrix function with a logarithm, we provide an explicit form for solutions to non-homogeneous Hadamard-type fractional time-delay linear differential equations. We also examine the existence, uniqueness, and Ulam–Hyers stability of Hadamard-type fractional time-delay nonlinear equations.

scholarly journals Conditions for Factorization of Linear Differential-Difference Equations

2013 ◽  
Vol 54 (1) ◽  
pp. 93-99
Author(s):  
Klara R. Janglajew ◽  
Kim G. Valeev
Keyword(s):  
Linear System ◽  
Small Parameter ◽  
Difference Equations ◽  
Integral Manifold ◽  
Sufficient Conditions ◽  
System Of Differential Equations ◽  
Deviating Argument ◽  
Constant Coefficients ◽  
Complex Deviating Argument ◽  
Linear Differential

Abstract The paper deals with a linear system of differential equations of the form with constant coefficients, a small parameter and complex deviating argument. Sufficient conditions for factorizing of this system are presented. These conditions are obtained by construction of an integral manifold of solutions to the considered system.

On the asymptotic stability of linear system of fractional-order difference equations

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Raghib Abu-Saris ◽  
Qasem Al-Mdallal
Keyword(s):  
Asymptotic Stability ◽  
Linear System ◽  
Fractional Order ◽  
Difference Equations ◽  
Equilibrium Solution ◽  
Initial Condition ◽  
System Of Difference Equations ◽  
Order Difference ◽  
Order Linear ◽  
The Stability

AbstractIn this paper we investigate the stability of the equilibrium solution of the νth order linear system of difference equations $(\Delta _{a + \nu - 1}^\nu y)(t) = \Lambda y(t + \nu - 1);t \in \mathbb{N}_a ,a \in \mathbb{R},and\Lambda \in \mathbb{R}^{p \times p} ,$ subject to the initial condition $y(a + \nu - 1) = y - 1,$, where 0 < ν < 1 and y−1 ∈ ℝp.

scholarly journals On the Global Asymptotic Stability of Solutions of Some Difference Equations with Intrinsic Initial Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Vasile Berinde
Keyword(s):  
Global Stability ◽  
Difference Equations ◽  
Software Package ◽  
Numerical Experiments ◽  
Initial Conditions ◽  
Remarkable Feature ◽  
Order Difference ◽  
First Order ◽  
The Stability ◽  
Stability Results

Our aim in this paper is to study the asymptotic global stability of the positive solutions for a class of first-order nonlinear difference equations with a remarkable feature: the initial conditions are intrinsic and not explicitly given. Global stability results are obtained in a particular case and then for a general class of first-order difference equations. We also provide the results of some numerical experiments obtained by the mini software package FIXPOINT to illustrate asymptotic global stability as well as the rate of convergence. To the best of our knowledge, our approach is the first one in the literature on the stability of difference equations without explicit initial conditions and might generate an interesting new direction of further studies.

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