Ewa Schmeidel

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.

On a class of fourth-order nonlinear difference equations

We consider a class of fourth-order nonlinear difference equations. The classification of nonoscillatory solutions is given. Next, we divide the set of solutions of these equations into two types: F+- and F−-solutions. Relations between these types of solutions and their nonoscillatory behavior are obtained. Necessary and sufficient conditions are obtained for the difference equation to admit the existence of nonoscillatory solutions with special asymptotic properties.

Existence of asymptotically periodic solutions of system of Volterra difference equations

A Volterra system of difference equations of the form where , and , is studied. Sufficient conditions for the existence of asymptotically periodic solutions of this system are presented. In addition, we present sufficient conditions for the nonexistence of an asymptotically periodic solution satisfying some auxiliary conditions.

Weighted Asymptotically Periodic Solutions of Linear Volterra Difference Equations

A linear Volterra difference equation of the form 𝑥 ( 𝑛 + 1 ) = 𝑎 ( 𝑛 ) + 𝑏 ( 𝑛 ) 𝑥 ( 𝑛 ) + ∑ 𝑛 𝑖 = 0 𝐾 ( 𝑛 , 𝑖 ) 𝑥 ( 𝑖 ) , where 𝑥 ∶ ℕ 0 → ℝ , 𝑎 ∶ ℕ 0 → ℝ , 𝐾 ∶ ℕ 0 × ℕ 0 → ℝ and 𝑏 ∶ ℕ 0 → ℝ ⧵ { 0 } is 𝜔 -periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on ∏ 𝜔 − 1 𝑗 = 0 𝑏 ( 𝑗 ) is assumed. The results generalize some of the recent results.

An application of measures of noncompactness in the investigation of boundedness of solutions of second-order neutral difference equations

AbstractThe purpose of this paper is to investigate a nonlinear second-order neutral difference equation of the form Δ(rnΔ(xn+pnxn−k))+anf(xn)=0, where x:N0→R, a:N0→R, p,r:N0→R∖{0}, f:R→R is a continuous function, and k is a given positive integer. Sufficient conditions for the existence of a bounded solution of this equation are obtained. Also, stability and asymptotic stability of this equation are studied. Additionally, the Emden-Fowler difference equation is considered as a special case of the above equation. The obtained results are illustrated by examples.MSC:39A10, 39A22, 39A30.

Existence of bounded solution of Volterra difference equations via Darbo's fixed-point theorem

A linear Volterra difference equation of non-convolution type of the form where , , and , is considered. Sufficient conditions for the existence of a bounded solution of this equation are given. Using this result, an asymptotic equivalence of some solution and given sequence, depending on terms of sequence b, is presented. Finally, the existence of a solution of the considered Volterra equation estimated by a given sequence is obtained.

On the Existence of Optimal Controls for the Fractional Continuous-Time Cucker–Smale Model

In this work the Cucker–Smale fractional optimal control problem is proposed and studied. We show that considered problem has an optimal solution and we derive necessary conditions for this solution.

The emergence on isolated time scales

In the paper the Cucker-Smale model on isolated time scales is studied. This dynamical system models a consensus of emergence in a population of autonomous agents. The results establishing conditions under which such consensus occurs are presented.

On the asymptotic behavior of solutions of linear difference equations

In the paper sufficient conditions for the difference equation