Urszula Ledzewicz Drábek Pavel Avner Friedman Marek Galewski Maria do Rosário de Pinho Bogdan Przeradzki Ewa Schmeidel
Asymptotically zero solution of a class of higher nonlinear neutral difference equations with quasidifferences
Ewa Schmeidel Robert Jankowski
A class of higher order nonlinear neutral difference equations with quasidifferences is studied. Sufficient conditions under which considered equation has a solution which converges to zero are presented.
keywords: Darbo's fixed point theorem. measures of noncompactness asymptotic behavior Higher order difference equation
Urszula Ledzewicz Marek Galewski Andrzej Nowakowski Andrzej Swierniak Agnieszka Kalamajska Ewa Schmeidel
Most mathematicians who in their professional career deal with differential equations, PDEs, dynamical systems, stochastic equations and a variety of their applications, particularly to biomedicine, have come across the research contributions of Avner Friedman to these fields. However, not many of them know that his family background is actually Polish. His father was born in the small town of Włodawa on the border with Belarus and lived in another Polish town, Łomza, before he emigrated to Israel in the early 1920's (when it was still the British Mandate, Palestine). His mother came from the even smaller Polish town Knyszyn near Białystok and left for Israel a few years earlier. In May 2013, Avner finally had the opportunity to visit his father's hometown for the first time accompanied by two Polish friends, co-editors of this volume. His visit in Poland became an occasion to interact with Polish mathematicians. Poland has a long tradition of research in various fields related to differential equations and more recently there is a growing interest in biomedical applications. Avner visited two research centers, the Schauder Center in Torun and the Department of Mathematics of the Technical University of Lodz where he gave a plenary talk at a one-day conference on Dynamical Systems and Applications which was held on this occasion. In spite of its short length, the conference attracted mathematicians from the most prominent research centers in Poland including the University of Warsaw, the Polish Academy of Sciences and others, and even some mathematicians from other countries in Europe. Avner had a chance to get familiar with the main results in dynamical systems and applications presented by the participants and give his input in the scientific discussions. This volume contains some of the papers related to this meeting and to the overall research interactions it generated. The papers were written by mathematicians, mostly Polish, who wanted to pay tribute to Avner Friedman on the occasion of his visit to Poland.

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Monotonic solutions of a higher-order neutral difference system
Robert Jankowski Barbara Łupińska Magdalena Nockowska-Rosiak Ewa Schmeidel

A class of a higher-order nonlinear difference system with delayed arguments where the first equation of the system is of a neutral type is considered. A classification of non-oscillatory solutions is given and results on their boundedness or unboundedness are derived. The obtained results are illustrated by examples.

keywords: Neutral difference equation nonlinear system non-oscillatory
On the existence of weighted asymptotically constant solutions of Volterra difference equations of nonconvolution type
Ewa Schmeidel Karol Gajda Tomasz Gronek
A Volterra difference equation of the form $$x(n+2)=a(n)+b(n)x(n+1)+c(n)x(n)+\sum\limits^{n+1}_{i=1}K(n,i)x(i)$$ where $a, b, c, x \colon\mathbb{N} \to\mathbb{R}$ and $K \colon \mathbb{N}\times\mathbb{N}\to \mathbb{R}$ is studied. For every admissible constant $C \in \mathbb{R}$, sufficient conditions for the existence of a solution $x \colon\mathbb{N} \to\mathbb{R}$ of the above equation such that \[ x(n)\sim C \, n \, \beta(n), \] where $\beta(n)= \frac{1}{2^n}\prod\limits_{j=1}^{n-1}b(j)$, are presented. As a corollary of the main result, sufficient conditions for the existence of an eventually positive, oscillatory, and quickly oscillatory solution $x$ of this equation are obtained. Finally, a conditions under which considered equation possesses an asymptotically periodic solution are given.
keywords: periodic solutions. nonoscillatory linear equation Volterra difference equation oscillatory
Existence of uncountably many asymptotically constant solutions to discrete nonlinear three-dimensional system with $p$-Laplacian
Magdalena Nockowska-Rosiak Piotr Hachuła Ewa Schmeidel

This work is devoted to the study of the existence of uncountably many asymptotically constant solutions to discrete nonlinear three-dimensional system with p-Laplacian.

keywords: p-Laplacian three-dimensional system of difference equations
Periodic solutions of a $2$-dimensional system of neutral difference equations
Małgorzata Migda Ewa Schmeidel Małgorzata Zdanowicz
The 2-dimensional system of neutral type nonlinear difference equations with delays in the following form
$\left\{ \begin{align}&Δ≤(x_1(n)-p_1(n)\,x_1(n-τ_1))=a_1(n)\,f_1(x_1(n-σ_1),x_2(n-σ_2))\\&Δ≤(x_2(n)-p_2(n)\,x_2(n-τ_2))=a_2(n)\,f_2(x_1(n-σ_3),x_2(n-σ_4)),\end{align} \right.$
is considered. In this paper we use Schauder's fixed point theorem to study the existence of periodic solutions of the above system.
keywords: System of difference equation delays 2-dimensional neutral type periodic solutions existence

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