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## Über dieses Buch

The book presents the proceedings of the 23rd International Conference on Difference Equations and Applications, ICDEA 2017, held at the West University of Timișoara, Romania, under the auspices of the International Society of Difference Equations (ISDE), July 24 - 28, 2017. It includes new and significant contributions in the field of difference equations, discrete dynamical systems and their applications in various sciences. Disseminating recent studies and related results and promoting advances, the book appeals to PhD students, researchers, educators and practitioners in the field.

## Inhaltsverzeichnis

### Carrying Simplices for Competitive Maps

Abstract
The carrying simplex is a finite-dimensional, attracting Lipschitz invariant manifold that is commonly found in both continuous and discrete-time competition models from Ecology. It can be studied using the graph transform and cone conditions often applied to study attractors in continuous-time finite and infinite-dimensional models from applied mathematics, including chemical reaction networks and reaction diffusion equations. Here we show that the carrying simplex can also be studied from the point of view of the graph transform and cone conditions. However, unlike many of the models mentioned above, we do not use—at least directly—a gap condition that is often used to establish existence of a globally and exponentially attracting manifold. Instead we use contraction of phase volume to ‘suck’ hypersurfaces together uniformly, and ultimately onto the carrying simplex. We give a proof of the existence of the carrying simplex for a class of competitive maps, viewed here as also normally monotone maps. The result is not new, but is carried out in the framework of the graph transform to indicate how the carrying simplex relates to other well-known classes of invariant manifolds. We also discuss the relation between hypersurfaces with positive normals, unordered hypersurfaces and also the type of maps that preserve these types of hypersurfaces. Finally we review several examples from models in Ecology where the carrying simplex is known to exist.
Stephen Baigent

### Comparison of Attractors of Asymptotically Equivalent Difference Equations

Abstract
A nonautonomous difference equation is aymptotically autonomous if its right-hand side becomes more and more like that of an autonomous difference equation as time increases. It can then be shown that the component sets of a pullback attractor of the nonautonomous converge to the attractor of the autonomous system. Various conditions ensuring this both in the Hausdorff semi-metric and the full Hausdorff metric are given. Asymptotic equivalence of nonautonomous difference equations is also considered in the backwards sense. Both single-valued and set-valued difference equations are investigated. The results are applied to a simplified example of a discrete time neural field lattice model with both single-valued and set-valued interaction terms.
Hongyong Cui, Peter E. Kloeden

### Global Dynamics of Discrete Dynamical Systems and Difference Equations

Abstract
We present a survey of new approaches to the investigation of the global dynamics of discrete dynamical systems or autonomous difference equations. To achieve our objectives, we have utilized singularity theory of Whitney, the notion of critical curves of Mira and Gardini, and the notion of the carrying simplex of Hirsch. Using a geometric approach, we extend the notion of monotonicity of Smith from planar systems to higher dimensional systems. The global dynamics of a special class of systems generated by triangular maps will be, thoroughly studied. Biological and economics models will be introduced to illustrate the effectiveness and applicability of our methods. Finally, we present some open problems and conjectures to stimulate more research in this area of paramount importance to the field of dynamical systems/difference equations.
Saber Elaydi

### Bifurcations in Smooth and Piecewise Smooth Noninvertible Maps

Abstract
In the present chapter we recall some basic concepts and results associated with the local and global bifurcations of attractors and their basins in smooth and nonsmooth noninvertible maps, continuous and discontinuous. Such maps appear to be important both from theoretical and applied points of view. Using numerous examples we show that noninvertibility and nonsmoothness, as well as discontinuity of the considered maps lead to peculiar bifurcation phenomena which cannot be observed in smooth invertible maps.
Laura Gardini, Iryna Sushko

### Difference Equations Everywhere: Some Motivating Examples

Abstract
This work collects several situations where discrete dynamical systems or difference equations appear. Most of them are different from the examples used in textbooks and from the usual mathematical models appearing in Biology or Economy. The examples are presented in detail, including some appropriate references. Although most of them are known, the fact of collecting all together aims to be a source of motivation for studying DDS and difference equations and to facilitate teaching these subjects.
Armengol Gasull

### On Cubic Difference Equations with Variable Coefficients and Fading Stochastic Perturbations

Abstract
We consider the stochastically perturbed cubic difference equation with variable coefficients
$$x_{n+1}=x_n(1-h_nx_n^2)+\rho _{n+1}\xi _{n+1}, \quad n\in \mathbb N,\quad x_0\in \mathbb R.$$
Here $$(\xi _n)_{n\in \mathbb N}$$ is a sequence of independent random variables, and $$(\rho _n)_{n\in \mathbb N}$$ and $$(h_n)_{n\in \mathbb N}$$ are sequences of nonnegative real numbers. We can stop the sequence $$(h_n)_{n\in \mathbb N}$$ after some random time $$\mathscr {N}$$ so it becomes a constant sequence, where the common value is an $$\mathscr {F}_\mathscr {N}$$-measurable random variable. We derive conditions on the sequences $$(h_n)_{n\in \mathbb N}$$, $$(\rho _n)_{n\in \mathbb N}$$ and $$(\xi _n)_{n\in \mathbb N}$$, which guarantee that $$\lim _{n\rightarrow \infty } x_n$$ exists almost surely (a.s.), and that the limit is equal to zero a.s. for any initial value $$x_0\in \mathbb R$$.
Ricardo Baccas, Cónall Kelly, Alexandra Rodkina

### On Uniform Exponential Splitting of Variational Nonautonomous Difference Equations in Banach Spaces

Abstract
In this paper we study a concept of uniform exponential splitting, as a generalization of uniform exponential dichotomy for a discrete cocycle C over a discrete semiflow S. Discrete characterizations of this concept are obtained from the point of view of the projectors families (invariant and strongly invariant) and some illustrative examples are given.
Larisa Elena Biriş, Traian Ceauşu, Claudia Luminiţa Mihiţ

### The Linear Constrained Control Problem for Discrete-Time Systems: Regulation on the Boundaries

Abstract
The chapter deals with the problem of regulation of linear systems around an equilibrium lying on the boundary of a polyhedral domain where linear constraints on the control and/or the state vectors are satisfied. In the first part of the chapter, the fundamental limitations for constrained control with active constraints at equilibrium are exposed. Next, based on the invariance properties of polyhedral and semi-ellipsoidal sets, design methods for guaranteeing convergence to the equilibrium while respecting linear control constraints are proposed. To this end, Lyapunov-like polyhedral functions, LMI methods and eigenstructure assignment techniques are applied.
George Bitsoris, Sorin Olaru, Marina Vassilaki

### Discrete Version of an Optimal Partitioning Problem

Abstract
Many recent works deal with problems concerning optimal partitions related to spectral quantities of domains in Euclidean spaces or on manifolds. Due to the complexity of these problems, few explicit solutions are known. Therefore, numerical algorithms have been developed in order to find approximations of optimal partitions. Such algorithms are based on discretizations of the domain and lead to finite dimensional difference equations. In the following, the coupling of the gradient descent method with a projection algorithm leads to a non-linear difference equation. Various properties of the discrete problem are discussed and numerical results illustrating the behaviour of the discretization scheme are shown.
Beniamin Bogosel

### On (h, k)-Dichotomy of Linear Discrete-Time Systems in Banach Spaces

Abstract
The paper considers a general concept of dichotomy with different growth rates for linear discrete-time systems in Banach spaces. Characterizations in terms of Lyapunov type sequences of norms are given. The approach is illustrated by various examples.
Violeta Crai, Mirela Aldescu

### Existence and Stability Properties of Almost Periodic Solutions in Discrete Almost Periodic Systems

Abstract
In order to obtain the existence of almost periodic solutions of both linear and nonlinear almost periodic discrete systems: $$x(n+1) = A(n)x(n)$$ and $$x_{i}(n+1)=\sum _{j=1}^{m}a_{ij}(n)g_{j}(x_{j}(n)) \quad \text{ for } \quad 1 \le i \le m$$, respectively, we shall consider between certain stability properties, which are referred to as uniformly asymptotically stable, and the diagonal dominance matrix condition.
Yoshihiro Hamaya

### A Hilbert Space Approach to Difference Equations

Abstract
We consider general difference equations $$u_{n+1} = F(u)_n$$ for $$n \in \mathbb {Z}$$ on exponentially weighted $$\ell _2$$ spaces of two-sided Hilbert space-valued sequences u and discuss initial value problems. As an application of the Hilbert space approach, we characterize exponential stability of linear equations and prove a stable manifold theorem for causal nonlinear difference equations.
Konrad Kitzing, Rainer Picard, Stefan Siegmund, Sascha Trostorff, Marcus Waurick

### Global Behavior of Some Nonautonomous Delay Difference Equations

Abstract
Our aim in this paper is to investigate the permanence, the extreme stability, and the periodicity of positive solutions of the nonautonomous delay difference equation of the form
$$x_{n+1}=a_{n}x_{n}f(x_{n-k}),\quad \quad n=0,1,...$$
where $$\left\{ a_{n}\right\}$$ is positive and periodic with period p,  and f satisfies some additional assumptions. The results are applied to some classical periodically forced population models with delay such as Pielou logistic model, Ricker’s model, and Hassel’s model.
Vlajko L. Kocic

### The Convergence of a Sequence of Iterated Polygons: A Discrete Combinatorial Analysis

Abstract
This paper deals with the convergence of a sequence of iterated polygons. Starting with a polygon in an euclidean space, another polygon may be constructed by connecting the mid-points of the edges of the initial polygon. If we iterate this construction we get a sequence of polygons, for which we prove, by using elementary computations, that is convergent. Moreover, if we consider other “weighted points” instead of mid-points we obtain that the corresponding sequence of iterated polygons converges to the centroid of the initial polygon. Moreover, we prove the same result even in a more general framework, for curved geodesic polygons, when the vertices of the polygon belong to a space with curved geometry.
Maria Mălin, Ionel Rovenţa, Mihai Tudor

### On Splitting with Different Growth Rates for Linear Discrete-Time Systems in Banach Spaces

Abstract
The aim of this work is to study a property of trisplitting with different growth rates for linear discrete-time systems in Banach spaces. Some characterizations for this concept are given and as consequences, we obtain some results for exponential trisplitting and for the trichotomy with different growth rates, particularly for exponential trichotomy.
Mihail Megan, Claudia Luminiţa Mihiţ, Raluca Lolea

### Lyapunov Functionals and Stability in Finite Delays Difference Equations

Abstract
In this research we prove general theorems regarding the stability of the zero solution of a functional difference equation with finite delay. In the analysis we assume the existence of a Lyapunov functional that satisfies certain conditions. Results on finite delay difference equations using Lyapunov functions or functionals are scarce. We apply our results to finite delay difference equations and to Volterra difference equations with finite delays.
Youssef N. Raffoul
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