scholarly journals Accurate solution estimates for nonlinear nonautonomous vector difference equations

2004 ◽  
Vol 2004 (7) ◽  
pp. 603-611 ◽  
Author(s):  
Rigoberto Medina ◽  
M. I. Gil'
Keyword(s):  
Difference Equations ◽  
Discrete Dynamical System ◽  
Reaction Diffusion ◽  
Accurate Solution ◽  
Asymptotically Stable ◽  
Region Of Attraction ◽  
Constant Matrix ◽  
Freezing Method ◽  
Vector Difference ◽  
Solution Estimates

The paper deals with the vector discrete dynamical systemxk+1=Akxk+fk(xk). Thewell-known result by Perron states that this system is asymptotically stable ifAk≡A=constis stable andfk(x)≡f˜(x)=o(‖x‖). Perron's result gives no information about the size of the region of asymptotic stability and norms of solutions. In this paper, accurate estimates for the norms of solutions are derived. They give us stability conditions for (1.1) and bounds for the region of attraction of the stationary solution. Our approach is based on the “freezing” method for difference equations and on recent estimates for the powers of a constant matrix. We also discuss applications of our main result to partial reaction-diffusion difference equations.

scholarly journals Accurate solution estimates for vector difference equations

2003 ◽  
Vol 2003 (48) ◽  
pp. 3059-3066
Author(s):  
Rigoberto Medina
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Reaction Diffusion ◽  
Accurate Solution ◽  
Region Of Attraction ◽  
Constant Matrix ◽  
Partial Reaction ◽  
Volterra Difference Equation ◽  
Vector Difference ◽  
Solution Estimates

Accurate estimates for the norms of the solutions of a vector difference equation are derived. They give us stability conditions and bounds for the region of attraction of the stationary solution. Our approach is based on estimates for the powers of a constant matrix. We also discuss applications of our main results to partial reaction-diffusion difference equations and to a Volterra difference equation.

Dynamical analysis of a reaction–diffusion SEI epidemic model with nonlinear incidence rate

2021 ◽  
pp. 2150041
Author(s):  
Jianpeng Wang ◽  
Binxiang Dai
Keyword(s):  
Steady State ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Reaction Diffusion ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Nonlinear Incidence Rate ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
Positive Steady State

In this paper, a reaction–diffusion SEI epidemic model with nonlinear incidence rate is proposed. The well-posedness of solutions is studied, including the existence of positive and unique classical solution and the existence and the ultimate boundedness of global solutions. The basic reproduction numbers are given in both heterogeneous and homogeneous environments. For spatially heterogeneous environment, by the comparison principle of the diffusion system, the infection-free steady state is proved to be globally asymptotically stable if [Formula: see text] if [Formula: see text], the system will be persistent and admit at least one positive steady state. For spatially homogenous environment, by constructing a Lyapunov function, the infection-free steady state is proved to be globally asymptotically stable if [Formula: see text] and then the unique positive steady state is achieved and is proved to be globally asymptotically stable if [Formula: see text]. Finally, two examples are given via numerical simulations, and then some control strategies are also presented by the sensitive analysis.

scholarly journals Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 20 ◽  
Author(s):  
Michael Gil’
Keyword(s):  
Hilbert Space ◽  
Difference Equation ◽  
Difference Equations ◽  
Lyapunov Equation ◽  
Stability Test ◽  
Norm Estimates ◽  
Point Estimates ◽  
Adjoint Operators ◽  
Solution Estimates

The paper is devoted to the discrete Lyapunov equation X - A * X A = C , where A and C are given operators in a Hilbert space H and X should be found. We derive norm estimates for solutions of that equation in the case of unstable operator A, as well as refine the previously-published estimates for the equation with a stable operator. By the point estimates, we establish explicit conditions, under which a linear nonautonomous difference equation in H is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in H . Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators.

scholarly journals EXTREMAL EQUILIBRIA FOR DISSIPATIVE PARABOLIC EQUATIONS IN LOCALLY UNIFORM SPACES

2009 ◽  
Vol 19 (11) ◽  
pp. 1995-2037 ◽  
Author(s):  
JAN W. CHOLEWA ◽  
ANÍBAL RODRÍGUEZ-BERNAL
Keyword(s):  
Parabolic Equations ◽  
Uniform Space ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Stationary Solutions ◽  
Reaction Diffusion Equation ◽  
Asymptotically Stable ◽  
Uniform Spaces ◽  
Integrable Functions ◽  
Convergence Topology

We consider a reaction diffusion equation ut = Δu + f(x, u) in ℝN with initial data in the locally uniform space [Formula: see text], q ∈ [1, ∞), and with dissipative nonlinearities satisfying s f(x, s) ≤ C(x)s2 + D(x) |s|, where [Formula: see text] and [Formula: see text] for certain [Formula: see text]. We construct a global attractor [Formula: see text] and show that [Formula: see text] is actually contained in an ordered interval [φm, φM], where [Formula: see text] is a pair of stationary solutions, minimal and maximal respectively, that satisfy φm ≤ lim inft→∞ u(t; u0) ≤ lim supt→∞ u(t; u0) ≤ φM uniformly for u0 in bounded subsets of [Formula: see text]. A sufficient condition concerning the existence of minimal positive steady state, asymptotically stable from below, is given. Certain sufficient conditions are also discussed ensuring the solutions to be asymptotically small as |x| → ∞. In this case the solutions are shown to enter, asymptotically, Lebesgue spaces of integrable functions in ℝN, the attractor attracts in the uniform convergence topology in ℝN and is a bounded subset of W2,r(ℝN) for some r > N/2. Uniqueness and asymptotic stability of positive solutions are also discussed. Applications to some model problems, including some from mathematical biology are given.

Numerical Simulation of Noninteger Order System in Subdiffusive, Diffusive, and Superdiffusive Scenarios

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Kolade M. Owolabi ◽  
Abdon Atangana
Keyword(s):  
Reaction Diffusion ◽  
Order System ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Numerical Studies ◽  
Main Equation ◽  
Spatial Derivatives ◽  
Exponential Time Differencing Method ◽  
2D And 3D ◽  
Fractional Reaction

In this work, we investigate both the mathematical and numerical studies of the fractional reaction–diffusion system consisting of spatial interactions of three components’ species. Our main result is based on the analysis of the model for linear stability. Mathematical analysis of the main equation shows that the dynamical system is both locally and globally asymptotically stable. We further propose a theorem which guarantees the existence and permanence of the three species. We formulate a viable numerical methods in space and time. By adopting the Fourier spectral approach to discretize in space, the issue of stiffness associated with the fractional-order spatial derivatives in such system is removed. The resulting system of ordinary differential equations (ODEs) is advanced with the exponential time-differencing method of ADAMS-type. The complexity of the dynamics in the system which we discussed theoretically are numerically presented through some numerical simulations in 1D, 2D, and 3D to address the points and queries that may naturally arise.

scholarly journals Perron-Type Criterion for Linear Difference Equations with Distributed Delay

2007 ◽  
Vol 2007 ◽  
pp. 1-12
Author(s):  
Jehad O. Alzabut ◽  
Thabet Abdeljawad
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Trivial Solution ◽  
Distributed Delay ◽  
Linear Difference Equation ◽  
Asymptotically Stable ◽  
Linear Difference Equations ◽  
Type Criterion

It is shown that if a linear difference equation with distributed delay of the formΔx(n)=∑k=−d0Δkζ(n+1,k−1)x(n+k−1),n≥1, satisfies a Perron condition then its trivial solution is uniformly asymptotically stable.

scholarly journals Linear θ-Method and Compact θ-Method for Generalised Reaction-Diffusion Equation with Delay

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Fengyan Wu ◽  
Qiong Wang ◽  
Xiujun Cheng ◽  
Xiaoli Chen
Keyword(s):  
Diffusion Equation ◽  
Necessary Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Stability And Convergence ◽  
Asymptotically Stable ◽  
Sufficient And Necessary Conditions ◽  
Equation Solvability ◽  
Equation With Delay ◽  
Theoretical Results

This paper is concerned with the analysis of the linear θ-method and compact θ-method for solving delay reaction-diffusion equation. Solvability, consistence, stability, and convergence of the two methods are studied. When θ∈[0,1/2), sufficient and necessary conditions are given to show that the two methods are asymptotically stable. When θ∈[1/2,1], the two methods are proven to be unconditionally asymptotically stable. Finally, several examples are carried out to confirm the theoretical results.

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