Josef Diblík

Controllability of Linear Discrete Systems with Constant Coefficients and Pure Delay

The purpose of this contribution is to develop a controllability method for linear discrete systems with constant coefficients and with pure delay. To do this, a representation of solutions with the aid of a discrete matrix delayed exponential is used. Such an approach leads to new conditions of controllability. Except for a criterion of relative controllability, a control function is constructed as well.

On a Third-Order System of Difference Equations with Variable Coefficients

We show that the system of three difference equations , , and , , where all elements of the sequences , , , , , and initial values , , , , are real numbers, can be solved. Explicit formulae for solutions of the system are derived, and some consequences on asymptotic behavior of solutions for the case when coefficients are periodic with period three are deduced.

On Some Solvable Difference Equations and Systems of Difference Equations

Here, we give explicit formulae for solutions of some systems of difference equations, which extend some very particular recent results in the literature and give natural explanations for them, which were omitted in the previous literature.

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.

REPRESENTATION OF SOLUTIONS OF LINEAR DISCRETE SYSTEMS WITH CONSTANT COEFFICIENTS AND PURE DELAY

The purpose of this contribution is to develop a method for construction of solutions of linear discrete systems with constant coefficients and with pure delay. Solutions are expressed with the aid of a special function called the discrete matrix delayed exponential having between every two adjoining knots the form of a polynomial. These polynomials have increasing degrees in the right direction. Such approach results in a possibility to express initial Cauchy problem in the closed form.

On the New Control Functions for Linear Discrete Delay Systems

New control functions are derived for linear difference equations with delay, which can be used to construct a nontrivial solution of a boundary value problem with zero boundary conditions. Later, a special case of invariant linear subspace is considered and corresponding control functions are constructed. Results for weakly nonlinear problems are also discussed.

On a solvable system of rational difference equations

We show that the following system of difference equationswhere , , , and sequences , , and are real, can be solved in closed form. For the case when the sequences , , and are constant and , we apply obtained formulas in the investigation of the asymptotic behaviour of well-defined solutions of the system. We also find domain of undefinable solutions of the system. Our results considerably extend and improve some recent results in the literature.

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.

Positive and oscillating solutions of differential equations with delay in critical case

Abstract This article is devoted to the problem of existence of positive solution for common classes of nonlinear retarded functional differential equations. Some criteria of its existence are proved, as well as some comparison results. The obtained results are applied to the linear case. Moreover, the existence of positive and oscillating solutions of a differential equation with delay x (t)=−a(t)x(t−τ) in the critical case is considered. Some comparisons with known results are given.

On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence

Abstract Schauder’s fixed point technique is applied to asymptotical analysis of solutions of a linear Volterra difference equation x ( n + 1 ) = a ( n ) + b ( n ) x ( n ) + ∑ i = 0 n K ( n , i ) x ( i ) where n ∈ N 0 , x : N 0 → R , a : N 0 → R , K : N 0 × N 0 → R , and b : N 0 → R ⧹ { 0 } is ω -periodic. In the paper, sufficient conditions are derived for the validity of a property of solutions that, for every admissible constant c ∈ R , there exists a solution x = x ( n ) such that x ( n ) ∼ c + ∑ i = 0 n - 1 a ( i ) β ( i + 1 ) β ( n ) , where β ( n ) = ∏ j = 0 n - 1 b ( j ) , for n → ∞ and inequalities for solutions are derived. Relevant comparisons and illustrative examples are given as well.