arXiv subjects
Publications and source records attributed to Ewa Schmeidel.
In this paper an emergence of leader-following model based on graph theory on the arbitrary time scales is investigated. It means that the step size is not necessarily constant but it is a function of time. We propose and prove conditions ensuring a leader-following consensus for any time scales using Grönwall inequality. The presented results are illustrated by examples.
Using direct variational method we consider the existence of non-spurious solutions to the following Dirichlet problem $\ddot{x}\left( t\right) =f\left( t,x\left( t\right) \right) $, $x\left( 0\right) =x\left( 1\right) =0 $ where $f:\left[ 0,1\right] \times \mathbb{R} \rightarrow \mathbb{R}$ is a jointly continuous function convex in $x$ which does not need to satisfy any further growth conditions.
In this work we investigate the existence of solutions, their uniqueness and finally dependence on parameters for solutions of second order neutral nonlinear difference equations. The main tool which we apply is Darbo fixed point theorem.
A class of fourth--order neutral type difference equations with quasidifferences and deviating arguments is considered. Our approach is based on studying the considered equation as a system of a four--dimensional difference system. The sufficient conditions under which the considered equation has no quickly oscillatory solutions are given.
We provide sufficient conditions for a mapping acting between two Banach spaces to be a diffeomorphism.
\noindent Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form \begin{equation*} Δ\left(r_{n}\left(Δ\left(x_{n}+p_{n}x_{n-k}\right) \right) ^γ\right) +q_{n}x_{n}^α+a_{n}f(x_{n})=0. \end{equation*}% where $x:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$, $a,p,q:{\mathbb{N}}%_{0}\rightarrow {\mathbb{R}}$, $r:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}% \setminus \{0\}$, $f\colon {\mathbb{R}}\rightarrow {\mathbb{R}}$ is a continuous function, and $k$ is a given positive integer, $γ\leq 1$ is ratio of odd positive integers, $α$ is a nonnegative constant. %$\sum a_{n}\left(t\right)$ converges uniformly on ${\mathbb{R}}$. %Here $\bN_0\colon =\left\{0,1,2, \dots \right\}$ and $\bN_k \colon = \left\{k, k+1, -k+2, \dots \right\}$ where $k$ is a given positive integer. Sufficient conditions for the existence of a bounded solution are obtained. Also a special type of stability and asymptotic stability are studied. Some earlier results are generalized. We note that the solution which we obtain does not directly correspond to a fixed point of a certain continuous operator since it is partially iterated. The method which we develop allows for considering through techniques connected with the measure of noncompactness also difference equations with memory. {\small \textbf{Keywords} Difference equation, measures of noncompactness, Darbo's fixed point theorem, boundedness, stability} {\small \textbf{AMS Subject classification} 39A10, 39A22, 39A30}