Exponential Stability of Linear Discrete Systems with Variable Delays via Lyapunov Second Method

The paper investigates the exponential stability of a linear system of difference equations with variable delays xk+1=Axk+∑i=1sBikxk-mik, k=0,1,… , where s∈N, A is a constant square matrix, Bik are square matrices, mik∈N∪0, and mik≤m for an m∈N. New criteria for exponential stability are derived using the method of Lyapunov functions and formulated in terms of the norms of matrices of linear terms and matrices solving an auxiliary Lyapunov equation. An exponential-type estimate of the norm of solutions is given as well. The efficiency of the derived criteria is numerically demonstrated by examples and their relations to the well-known results are discussed

One case of appearance of positive solutions of delayed discrete equations

summary:When mathematical models describing various processes are analysed, the fact of existence of a positive solution is often among the basic features. In this paper, a general delayed discrete equation $$\Delta u(k+n)=f(k,u(k),u(k+1),\dots ,u(k+ n))$$ is considered. Sufficient conditions concerning $f$ are formulated in order to guarantee the existence of a positive solution for $k\rightarrow \infty$. An upper estimate for it is given as well. The appearance of the positive solution takes its origin in the nature of the equation considered since the results hold only for delayed equations (i.e. for $n>0$) and not for the case of an ordinary equation (with $n=0$)

Long-time behaviour of solutions of delayed-type linear differential equations

This paper investigates the asymptotic behaviour of the solutions of the retarded-type linear differential functional equations with bounded delays x˙(t) = −L(t, xt) when t → ∞. The main results concern the existence of two significant positive and asymptotically different solutions x = ϕ (t), x = ϕ ∗∗(t) such that limt→∞ ϕ ∗∗(t)/ϕ (t) = 0. These solutions make it possible to describe the family of all solutions by means of an asymptotic formula. The investigation basis is formed by an auxiliary linear differential functional equation of retarded type y˙(t) = L (t, yt) such that L (t, yt) ≡ 0 for an arbitrary constant initial function yt . A commented survey of the previous results is given with illustrative examples

Long-time behaviour of solutions of delayed-type linear differential equations

This paper investigates the asymptotic behaviour of the solutions of the retarded-type linear differential functional equations with bounded delays $$\dot x(t)=-L(t,x_t)$$ when $t\to\infty$. The main results concern the existence of two significant positive and asymptotically different solutions $x=\varphi^*(t)$, $x=\varphi^{**}(t)$ such that $\lim_{t\to\infty}\varphi^{**}(t)/\varphi^{*}(t)=0$. These solutions make it possible to describe the family of all solutions by means of an asymptotic formula. The investigation basis is formed by an auxiliary linear differential functional equation of retarded type $\dot y(t)=L^{\ast}(t,y_t)$ such that $L^{\ast}(t,y_t)\equiv 0$ for an arbitrary constant initial function $y_t$. A commented survey of the previous results is given with illustrative examples

Exponential solutions of equation ẏ(t)=β(t)[y(t−δ)−y(t−τ)]

AbstractAsymptotic behaviour of solutions of first-order differential equation with two deviating arguments of the form ẏ(t)=β(t)y(t−δ)−y(t−τ) is discussed for t→∞. A criterion for representing solutions in exponential form is proved. As consequences, inequalities for such solutions are given. Connections with known results are discussed and a sufficient condition for existence of unbounded solutions, generalizing previous ones, is derived. An illustrative example is considered, too

A retract principle on discrete time scales

In this paper we discuss asymptotic behavior of solutions of a class of scalar discrete equations on discrete real time scales. A powerful tool for the investigation of various qualitative problems in the theory of ordinary differential equations as well as delayed differential equations is the retraction method. The development of this method is discussed in the case of the equation mentioned above. Conditions for the existence of a solution with its graph remaining in a predefined set are formulated. Examples are given to illustrate the results obtained

Initial data generating bounded solutions of linear discrete equations

A lot of papers are devoted to the investigation of the problem of prescribed behavior of solutions of discrete equations and in numerous results sufficient conditions for existence of at least one solution of discrete equations having prescribed asymptotic behavior are indicated. Not so much attention has been paid to the problem of determining corresponding initial data generating such solutions. We fill this gap for the case of linear equations in this paper. The initial data mentioned are constructed with use of two convergent monotone sequences. An illustrative example is considered, too