Jaromír Baštinec

Exponential stability of linear discrete systems with constant coefficients and single delay

Abstract In the paper the exponential stability and exponential estimation of the norm of solutions to a linear system of difference equations with single delay x ( k + 1 ) = A x ( k ) + B x ( k − m ) , k = 0 , 1 , … is studied, where A , B are square constant matrices and m ∈ N . New sufficient conditions for exponential stability are derived using the method of Lyapunov functions. Illustrative examples are given as well.

Existence of positive solutions of discrete linear equations with a single delay

In the paper, we use a classical comparison result to prove the existence of positive solutions to a particular and very frequently investigated class of linear difference equations with a positive coefficient and a single delay. The relevant result is given in the form of an inequality (with a suitable auxiliary comparison function) for the equation coefficient. A parallel to known results is included as well as a discussion of future directions.

On the Critical Case in Oscillation for Differential Equations with a Single Delay and with Several Delays

New nonoscillation and oscillation criteria are derived for scalar delay differential equations 𝑥(𝑡)

Subdominant positive solutions of the discrete equation Δu(k

A delayed discrete equation Δ u ( k + n ) = − p ( k ) u ( k ) with positive coefficient p is considered. Sufficient conditions with respect to p are formulated in order to guarantee the existence of positive solutions if k → ∞ . As a tool of the proof of corresponding result, the method described in the authors previous papers is used. Except for the fact of the existence of positive solutions, their upper estimation is given. The analysis shows that every positive solution of the indicated family of positive solutions tends to zero (if k → ∞ ) with the speednot smaller than the speed characterized by the function k · ( n / ( n + 1 ) ) k . A comparison with the known results is given and some open questions are discussed.

Estimates of Exponential Stability for Solutions of Stochastic Control Systems with Delay

A nonlinear stochastic differential-difference control system with delay of neutral type is considered. Sufficient conditions for the exponential stability are derived by using Lyapunov-Krasovskii functionals of quadratic form with exponential factors. Upper bound estimates for the exponential rate of decay are derived.

A Final Result on the Oscillation of Solutions of the Linear Discrete Delayed Equation Δ()=−()(−) with a Positive Coefficient

A linear ( 𝑘 + 1 ) th-order discrete delayed equation Δ 𝑥 ( 𝑛 ) = − 𝑝 ( 𝑛 ) 𝑥 ( 𝑛 − 𝑘 ) where 𝑝 ( 𝑛 ) a positive sequence is considered for 𝑛 → ∞ . This equation is known to have a positive solution if the sequence 𝑝 ( 𝑛 ) satisfies an inequality. Our aim is to show that, in the case of the opposite inequality for 𝑝 ( 𝑛 ) , all solutions of the equation considered are oscillating for 𝑛 → ∞ .

Exponential Stability and Estimation of Solutions of Linear Differential Systems of Neutral Type with Constant Coefficients

This paper investigates the exponential-type stability of linear neutral delay differential systems with constant coefficients using Lyapunov-Krasovskii type functionals, more general than those reported in the literature. Delay-dependent conditions sufficient for the stability are formulated in terms of positivity of auxiliary matrices. The approach developed is used to characterize the decay of solutions (by inequalities for the norm of an arbitrary solution and its derivative) in the case of stability, as well as in a general case. Illustrative examples are shown and comparisons with known results are given.

On a delay population model with quadratic nonlinearity

AbstractA nonlinear delay differential equation with quadratic nonlinearity,n x˙(t)=r(t)[∑k=1mαkx(hk(t))−βx2(t)],t≥0,n is considered, where αk and β are positive constants, hk:[0,∞)→R are continuous functions such that t−τ≤hk(t)≤t, τ=const, τ>0, for any t>0 the inequality hk(t)<t holds for at least one k, and r:[0,∞)→(0,∞) is a continuous function satisfying the inequality r(t)≥r0=const for an r0>0. It is proved that the positive equilibrium is globally asymptotically stable without any further limitations on the parameters of this equation.

Stability and exponential stability of linear discrete systems with constant coefficients and single delay

This paper investigates the exponential stability and exponential estimate of the norms of solutions to a linear system of difference equations with single delay x ( k + 1 ) = A x ( k ) + B x ( k - m ) , k = 0 , 1 , ? where A, B are square constant matrices and m ? N . Sufficient conditions for exponential stability are derived using the method of Lyapunov functions and its efficiency is demonstrated by examples.

Oscillation of Solutions of a Linear Second-Order Discrete-Delayed Equation

A linear second-order discrete-delayed equation with a positive coefficient is considered for . This equation is known to have a positive solution if fulfils an inequality. The goal of the paper is to show that, in the case of the opposite inequality for , all solutions of the equation considered are oscillating for .