scholarly journals On the Non-Exponential Convergence of Asymptotically Stable Solutions of Linear Scalar Volterra Integro-Differential Equations

2002 ◽  
Vol 14 (2) ◽  
pp. 109-118 ◽  
Author(s):  
John A.D. Appleby ◽  
David W. Reynolds
Keyword(s):  
Differential Equations ◽  
Exponential Convergence ◽  
Asymptotically Stable ◽  
Stable Solutions ◽  
Linear Scalar

scholarly journals Modulo Periodic Poisson Stable Solutions of Quasilinear Differential Equations

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1535
Author(s):  
Marat Akhmet ◽  
Madina Tleubergenova ◽  
Akylbek Zhamanshin
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Existence And Uniqueness ◽  
Asymptotically Stable ◽  
Stable Solutions ◽  
Theoretical Results

In this paper, modulo periodic Poisson stable functions have been newly introduced. Quasilinear differential equations with modulo periodic Poisson stable coefficients are under investigation. The existence and uniqueness of asymptotically stable modulo periodic Poisson stable solutions have been proved. Numerical simulations, which illustrate the theoretical results are provided.

scholarly journals On asymptotically stable sets

1970 ◽  
Vol 17 (2) ◽  
pp. 181-186 ◽  
Author(s):  
D. Desbrow
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Metric Space ◽  
Physical System ◽  
Asymptotically Stable ◽  
Stable Sets ◽  
Compact Closure ◽  
First Order ◽  
Closed Sets ◽  
The Future

In this paper we study closed sets having a neighbourhood with compact closure which are positively asymptotically stable under a flow on a metric space X. For an understanding of this and the rest of the introduction it is sufficient for the reader to have in mind as an example of a flow a system of first order, autonomous ordinary differential equations describing mathematically a time-independent physical system; in short a dynamical system. In a flow a set M is positively stable if the trajectories through all points sufficiently close to M remain in the future in a given neighbourhood of M. The set M is positively asymptotically stable if it is positively stable and, in addition, trajectories through all points of some neighbourhood of M approach M in the future.

scholarly journals Stability Analysis of an Alcoholism Model with Public Health Education and NSFD Scheme

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Zhaofeng An ◽  
Suxia Zhang ◽  
Jinhu Xu
Keyword(s):  
Public Health ◽  
Health Education ◽  
Differential Equations ◽  
Discrete Model ◽  
Public Health Education ◽  
Continuous Model ◽  
Basic Reproductive Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nsfd Scheme

In this paper, an alcoholism model of SEAR type with different susceptibilities due to public health education is investigated, with the form of continuous differential equations as well as discrete differential equations by applying the Mickens nonstandard finite difference (NSFD) scheme to the continuous equations. Threshold dynamics of the continuous model are performed by constructing Lyapunov functions. The analysis of a discrete model indicates that the alcohol-free equilibrium is globally asymptotically stable if the basic reproductive number R0<1, and conversely, the alcohol-present equilibrium is globally asymptotically stable if R0>1, revealing the consistency and efficiency of the discrete model to preserve the dynamical properties of the corresponding continuous model. In addition, stability preserving and the impact of the parameters related with public health education are conducted by numerical simulations.

scholarly journals Stability for a Class of Differential Equations with Nonconstant Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Jin Liang ◽  
Tzon-Tzer Lu ◽  
Yashan Xu
Keyword(s):  
Differential Equations ◽  
Positive Constant ◽  
Zero Solution ◽  
Continuous Functions ◽  
Delay Equation ◽  
Asymptotically Stable ◽  
Uniformly Stable

Stability is investigated for the following differential equations with nonconstant delayx't=qtFxt-ptfxt-τt,wherep:[0,+∞)→[0,+∞),q:[0,+∞)→R,τ:[0,+∞)→[0,r], andFandf:R→Rwithxfx>0   for   x≠0   and   x≤a(ais a positive constant) are continuous functions. A criterion is given for the zero solution of this delay equation being uniformly stable and asymptotically stable.

scholarly journals Integro-differential equations for the self-organisation of liver zones by competitive exclusion of cell-types

1987 ◽  
Vol 29 (2) ◽  
pp. 156-194 ◽  
Author(s):  
L. Bass ◽  
A. J. Bracken ◽  
K. Holmåker ◽  
B. R. F. Jefferies
Keyword(s):  
Differential Equations ◽  
Enzymatic Activity ◽  
Cell Types ◽  
Stationary Solutions ◽  
Stable Set ◽  
Model Parameters ◽  
Asymptotically Stable ◽  
Hepatic Sinusoid ◽  
Jump Discontinuities ◽  
Cell Densities

AbstractA model is developed for the seif-organisation of zones of enzymatic activity along a liver capillary (hepatic sinusoid) lined with cells of two types, which contain different enzymes and compete for sites on the wall of the sinusoid. An effectively non-local interaction between the cells arises from local consumption of oxygen from blood flowing throug1 the sinusoid, which gives rise to gradients of oxygen concentration in turn influencing rates of division and of death of the two cell-types. The process is modelled by a pair of coupled non-linear integro-differential equations for the cell-densities as functions of time and position along the sinusoid. Existence of a unique, bounded, non-negative solution of the equations is proved, for prescribed initial values. The equations admit infinitely many stationary solutions, but it is shown that all except one are unstable, for any given set of the model parameters. The remaining solution is shown to be asymptotically stable against a large class of perturbations. For certain ranges of the model parameters, the asymptotically stable stationaxy solution has a zonal structure, with cells of one type located entirely upstream of cells of the other type, and with jump discontinuities in the cell densities at a certain distance along the sinusoid. Such sinusoidal zones can account for zones of enzymatic activity observed in the intact liver. Exceptional cases are found for singular choices of model parameters, such that stationary cell-densities cannot be asymptotically stable individually, but together form an asymptotically stable set. Certain mathematical questions are left open, notably the behaviour of large deviations from stationary solutions, and the global stability of such solutions. Possible generalisations of the model are described.

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